Class "BoundedDensity"

Description

This class deals with generic estimations of a bounded densities. The probability density function is approximated by providing a set of data points in a lower and upper bounded interval and their associated densities. Using this information, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function boundedDensity.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

Examples

# data points and its densities
a <- seq(0,1,0.01)
b <- dbeta(a,5,10)

# create the density model
model <- boundedDensity(x=a,densities=b)

# examples of usual functions
density(model,0.5)

distribution(model,0.2,discreteApproximation=FALSE)
distribution(model,0.2,discreteApproximation=TRUE)
 
# graphical representation
hist(b,freq=FALSE)
lines(model, col="red",lwd=2)

Class "BrVitale"

Description

This class deals with bias reduced version of Vitale (1975) Bernstein Polynomial approximation as described in Leblanc (2009). The polynomial estimator is computed using the provided data samples. Using this polynomial estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function brVitale.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

m:

the order of the polynomial approximation

M:

a numeric parameter for bias reduction. Usually this parameter is set to m/2 since it leads to optimal MISE (mean integrated squared error) properties

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getm

See "getm" for details

getM

See "getM" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Vitale, R. A. (1975). A Bernstein polynomial approach to density function estimation. tatistical Inference and Related Topics, 2, 87-99.

Leblanc, A. (2010). A bias-reduced approach to density estimation using Bernstein polynomials. Journal of Nonparametric Statistics, 22(4), 459-475.

Examples

# create the model 
model <- brVitale(dataPoints = tuna.r, m = 25, M = 25/2)


# examples of usual functions
density(model,0.5)

distribution(model,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(model, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(model,show=TRUE,includePoints=TRUE)

Class "Chen99Kernel"

Description

This class deals with Kernel estimators for bounded densities as described in Chen's 99 paper. The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function chen99Kernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

modified:

if TRUE, the modified version of the kernel estimator is used

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmodified

See "getmodified" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Chen, S. X. (1999). Beta kernel estimators for density functions. Computational Statistics & Data Analysis, 31, 131-145.

Examples

# create the model 
kernel.noModified <- chen99Kernel(dataPoints = tuna.r, b = 0.01, modified = FALSE)
kernel.Modified <- chen99Kernel(dataPoints = tuna.r, b = 0.01, modified = TRUE)

# examples of usual functions
density(kernel.noModified,0.5)
density(kernel.Modified,0.5)

distribution(kernel.noModified,1,discreteApproximation=FALSE)
distribution(kernel.noModified,1,discreteApproximation=TRUE)
 
distribution(kernel.Modified,1,discreteApproximation=FALSE)
distribution(kernel.Modified,1,discreteApproximation=TRUE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Chen99 Kernels Tuna Data")
lines(kernel.noModified,col="red",lwd=2)
lines(kernel.Modified,col="blue",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(list("KernelNoModified"=kernel.noModified,
                "KernelModified"=kernel.Modified),show=TRUE)

Class "HirukawaJLNKernel"

Description

This class deals with the JLN Kernel estimator as described in Hirukawa (2010). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function hirukawaJLNKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

modified:

if TRUE, the modified version of the kernel estimator is used

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmodified

See "getmodified" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Hirukawa, M. (2010). Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval. Computational Statistics & Data Analysis, 54(2), 473-495.

Examples

# create the model 
kernel.noModified <- hirukawaJLNKernel(dataPoints = tuna.r, b = 0.01, modified = FALSE)
kernel.Modified <- hirukawaJLNKernel(dataPoints = tuna.r, b = 0.01, modified = TRUE)

# examples of usual functions
density(kernel.noModified,0.5)
density(kernel.Modified,0.5)

distribution(kernel.noModified,1,discreteApproximation=FALSE)
distribution(kernel.noModified,1,discreteApproximation=TRUE)
 
distribution(kernel.Modified,1,discreteApproximation=FALSE)
distribution(kernel.Modified,1,discreteApproximation=TRUE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Chen99 Kernels Tuna Data")
lines(kernel.noModified, col="red",lwd=2)
lines(kernel.Modified,col="blue",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(list("noModified"=kernel.noModified, 
          "modified"=kernel.Modified), show=TRUE)

Class "HirukawaTSKernel"

Description

This class deals with the TS Kernel estimator as described in Hirukawa (2010). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function hirukawaTSKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

modified:

if TRUE, the modified version of the kernel estimator is used

c:

a numeric value between 0 and 1. This parameter is used in the TS approximation as a smoothing parameter

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmodified

See "getmodified" for details

getc

See "getc" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Hirukawa, M. (2010). Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval. Computational Statistics & Data Analysis, 54(2), 473-495.

Examples

# create the model 
kernel.noModified <- hirukawaTSKernel(dataPoints = tuna.r, b = 0.01, 
                      modified = FALSE, c = 0.5)
kernel.Modified <- hirukawaTSKernel(dataPoints = tuna.r, b = 0.01,
                      modified = TRUE, c = 0.5)

# examples of usual functions
density(kernel.noModified,0.5)
density(kernel.Modified,0.5)

distribution(kernel.noModified,1,discreteApproximation=FALSE)
distribution(kernel.noModified,1,discreteApproximation=TRUE)
 
distribution(kernel.Modified,1,discreteApproximation=FALSE)
distribution(kernel.Modified,1,discreteApproximation=TRUE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Chen99 Kernels Tuna Data")
lines(kernel.noModified,col="red",lwd=2)
lines(kernel.Modified,col="blue",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(list("noModified"=kernel.noModified, 
          "modified"=kernel.Modified), show=TRUE)

Class "JonesCorrectionMuller91BoundaryKernel"

Description

This class deals with nonnegative boundary correction of the muller91BoundaryKernel estimators for bounded densities. In this normalization, two kernel functions are needed. The first kernel funciton -K(u)- is the kernel function used in muller91BoundaryKernel (using left boundary, interior or right boundary kernel functions as needed). For the second kernel function, the popular choice L(u) = u * K(u) is taken. The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations. Note that the renormalization of this kernel estimator guarantees nonnegative values for the density function but the cumulative density function may takes values greater than 1.

Objects from the Class

Objects can be created by using the generator function jonesCorrectionMuller91BoundaryKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

mu:

a integer value indicating the degree of smoothness for the boundary kernel. mu can take the following values: 0 (uniform kernel), 1 (Epanechnikov kernel), 2 (biweight kernel) or 3 (triweight kernel)

normalizedKernel:

this slot is used to save a NormalizedBoundaryKernel object used in the normalization. It is only for internal use

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmu

See "getmu" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Jones, M. C. and Foster, P. J. (1996). A simple nonnegative boundary correction method for kernel density estimation. Statistica Sinica, 6, 1005-1013.

Muller, H. (1991). Smooth optimum kernel estimators near endpoints. Biometrika, 78(3), 521-530.

Examples

# create the model 
kernel <-jonesCorrectionMuller91BoundaryKernel(dataPoints = tuna.r, b = 0.01, mu = 2)


# examples of usual functions
density(kernel,0.5)

distribution(kernel,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(kernel, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(kernel, show=TRUE, includePoints=TRUE)

Class "JonesCorrectionMuller94BoundaryKernel"

Description

This class deals with nonnegative boundary correction of the muller94BoundaryKernel estimators for bounded densities. In this normalization, two kernel functions are needed. The first kernel funciton -K(u)- is the kernel function used in muller94BoundaryKernel (using left boundary, interior or right boundary kernel functions as needed). For the second kernel function, the popular choice L(u) = u * K(u) is taken. The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations. Note that the renormalization of this kernel estimator guarantees nonnegative values for the density function but the cumulative density function may takes values greater than 1.

Objects from the Class

Objects can be created by using the generator function jonesCorrectionMuller94BoundaryKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

mu:

a integer value indicating the degree of smoothness for the boundary kernel. mu can take the following values: 0 (uniform kernel), 1 (Epanechnikov kernel), 2 (biweight kernel) or 3 (triweight kernel)

normalizedKernel:

this slot is used to save a NormalizedBoundaryKernel object used in the normalization. It is only for internal use

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmu

See "getmu" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Jones, M. C. and Foster, P. J. (1996). A simple nonnegative boundary correction method for kernel density estimation. Statistica Sinica, 6, 1005-1013.

Muller, H. and Wang, J. (1994). Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics, 50(1), 61-76.

Examples

# data points to cache densities and distribution
cache <- seq(0,1,0.01)

# create the model 
kernel <-jonesCorrectionMuller94BoundaryKernel(dataPoints = tuna.r, b = 0.01, mu = 2, 
                                                dataPointsCache = cache)


# examples of usual functions
density(kernel,0.5)

distribution(kernel,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(kernel, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(kernel, show=TRUE, includePoints = TRUE)

Class "KakizawaB1"

Description

This class deals with B1 approximation to kernel density estimation as described in Kakizawa (2004). This is a Berstein polynomial approximation of the density function which uses BoundedDensity objects instead of a polynomial function. By contrast to the original Kakizawa's approach where only boundary kernels are used, here, any BoundedDensity object is allowed. Using this estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function kakizawaB1.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

gamma:

a numeric value between 0 and 1. This parameter is used in the B1 approximation using Bernstein polynomials

densityEstimator:

a BoundedDensity object used to estimate the density

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getm

See "getm" for details

getgamma

See "getgamma" for details

getdensityEstimator

See "getdensityEstimator" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Kakizawa, Y. (2004). Bernstein polynomial probability density estimation. Journal of Nonparametric Statistics, 16(5), 709-729.

Examples

# create the model 
# we use a MicroBetaChen99Kernel is used as estimator y KakizawaB1 approximation
est <- microBetaChen99Kernel(dataPoints = tuna.r, b = 0.01, modified = FALSE)
model <- kakizawaB1(dataPoints = tuna.r, m = 25, gamma = 0.25)


# examples of usual functions
density(model,0.5)

distribution(model,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(model, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(model, show=TRUE, includePoints=TRUE)

Class "KakizawaB2"

Description

This class deals with B2 approximation to kernel density estimation as described in Kakizawa (2004). This is a Berstein polynomial approximation of the density function which uses BoundedDensity objects instead of a polynomial function. By contrast to the original Kakizawa's approach where only boundary kernels are used, here, any BoundedDensity object is allowed. Using this estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function kakizawaB2.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

densityEstimator:

a BoundedDensity object used to estimate the density

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getm

See "getm" for details

getdensityEstimator

See "getdensityEstimator" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Kakizawa, Y. (2004). Bernstein polynomial probability density estimation. Journal of Nonparametric Statistics, 16(5), 709-729.

Examples

# create the model 
# we use a MicroBetaChen99Kernel is used as estimator y KakizawaB1 approximation
est <- microBetaChen99Kernel(dataPoints = tuna.r, b = 0.01, modified = FALSE)
model <- kakizawaB2(dataPoints = tuna.r, m = 25, estimator = est)


# examples of usual functions
density(model,0.5)

distribution(model,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(model, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(model, show=TRUE, includePoints=TRUE)

Class "KakizawaB3"

Description

This class deals with B3 approximation to kernel density estimation as described in Kakizawa (2004). This is a Berstein polynomial approximation of the density function which uses BoundedDensity objects instead of a polynomial function. By contrast to the original Kakizawa's approach where only boundary kernels are used, here, any BoundedDensity object is allowed. Using this estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function kakizawaB3.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

densityEstimator:

a BoundedDensity object used to estimate the density

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getm

See "getm" for details

getdensityEstimator

See "getdensityEstimator" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Kakizawa, Y. (2004). Bernstein polynomial probability density estimation. Journal of Nonparametric Statistics, 16(5), 709-729.

Examples

# create the model 
# we use a MicroBetaChen99Kernel is used as estimator y KakizawaB1 approximation
est <- microBetaChen99Kernel(dataPoints = tuna.r, b = 0.01, modified = FALSE)
model <- kakizawaB3(dataPoints = tuna.r, m = 25, estimator = est)


# examples of usual functions
density(model,0.5)

distribution(model,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(model, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(model, show=TRUE, includePoints=TRUE)

Class "MacroBetaChen99Kernel"

Description

This class deals with the density-wise normalization (macro beta) of the Chen's 99 Kernel estimator (as described in Gourierous and Monfort, 2006). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function macroBetaChen99Kernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

modified:

if TRUE, the modified version of the kernel estimator is used

normalizationConst:

this slot is used to save the density-wise normalization constant. It is only for internal use

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmodified

See "getmodified" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Chen, S. X. (1999). Beta kernel estimators for density functions. Computational Statistics & Data Analysis, 31, 131-145.

Gourieroux, C. and Monfort, A. (2006). (Non) consistency of the Beta Kernel Estimator for Recovery Rate Distribution. Working Paper 2006-31, Centre de Recherche en Economie et Statistique.

Examples

# create the model 
kernel.noModified <- macroBetaChen99Kernel(dataPoints = tuna.r, b = 0.01,
                        modified = FALSE)
kernel.Modified <- macroBetaChen99Kernel(dataPoints = tuna.r, b = 0.01,
                        modified = TRUE)

# examples of usual functions
density(kernel.noModified,0.5)
density(kernel.Modified,0.5)

distribution(kernel.noModified,1,discreteApproximation=FALSE)
distribution(kernel.noModified,1,discreteApproximation=TRUE)
 
distribution(kernel.Modified,1,discreteApproximation=FALSE)
distribution(kernel.Modified,1,discreteApproximation=TRUE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Chen99 Kernels Tuna Data")
lines(kernel.noModified, col="red",lwd=2)
lines(kernel.Modified,col="blue",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(list("noModified"=kernel.noModified, 
          "modified"=kernel.Modified), show=TRUE)

Class "MacroBetaHirukawaJLNKernel"

Description

This class deals with the density-wise normalization (macro beta) of the JLN Kernel estimator as described in Hirukawa (2010). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function macroBetaHirukawaJLNKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

modified:

if TRUE, the modified version of the kernel estimator is used

normalizationConst:

this slot is used to save the density-wise normalization constant. It is only for internal use

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmodified

See "getmodified" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Hirukawa, M. (2010). Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval. Computational Statistics & Data Analysis, 54(2), 473-495.

Examples

# create the model 
kernel.noModified <- macroBetaHirukawaJLNKernel(dataPoints = tuna.r, b = 0.01,
                        modified = FALSE)
kernel.Modified <- macroBetaHirukawaJLNKernel(dataPoints = tuna.r, b = 0.01,
                        modified = TRUE)

# examples of usual functions
density(kernel.noModified,0.5)
density(kernel.Modified,0.5)

distribution(kernel.noModified,1,discreteApproximation=FALSE)
distribution(kernel.noModified,1,discreteApproximation=TRUE)
 
distribution(kernel.Modified,1,discreteApproximation=FALSE)
distribution(kernel.Modified,1,discreteApproximation=TRUE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Chen99 Kernels Tuna Data")
lines(kernel.noModified, col="red",lwd=2)
lines(kernel.Modified,col="blue",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(list("noModified"=kernel.noModified, 
          "modified"=kernel.Modified), show=TRUE)

Class "MacroBetaHirukawaTSKernel"

Description

This class deals with the density-wise normalization (macro beta) of the TS Kernel estimator as described in Hirukawa (2010). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function macroBetaHirukawaTSKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

modified:

if TRUE, the modified version of the kernel estimator is used

c:

a numeric value between 0 and 1. This parameter is used in the TS approximation as a smoothing parameter

normalizationConst:

this slot is used to save the density-wise normalization constant. It is only for internal use

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmodified

See "getmodified" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Hirukawa, M. (2010). Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval. Computational Statistics & Data Analysis, 54(2), 473-495.

Examples

# create the model 
kernel.noModified <- macroBetaHirukawaTSKernel(dataPoints = tuna.r, b = 0.01,
                      modified = FALSE, c = 0.5)
kernel.Modified <- macroBetaHirukawaTSKernel(dataPoints = tuna.r, b = 0.01,
                      modified = TRUE, c = 0.5)

# examples of usual functions
density(kernel.noModified,0.5)
density(kernel.Modified,0.5)

distribution(kernel.noModified,1,discreteApproximation=FALSE)
distribution(kernel.noModified,1,discreteApproximation=TRUE)
 
distribution(kernel.Modified,1,discreteApproximation=FALSE)
distribution(kernel.Modified,1,discreteApproximation=TRUE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Chen99 Kernels Tuna Data")
lines(kernel.noModified,col="red",lwd=2)
lines(kernel.Modified,col="blue",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(list("noModified"=kernel.noModified, 
          "modified"=kernel.Modified), show=TRUE)

Class "MicroBetaChen99Kernel"

Description

This class deals with the kernel-wise normalization of the Chen's 99 Kernel estimator (as described in Gourierous and Monfort, 2006). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function microBetaChen99Kernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

modified:

if TRUE, the modified version of the kernel estimator is used

normalizationConstants:

this slot is used to save the kernel-wise normalization constants. It is only for internal use

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmodified

See "getmodified" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Chen, S. X. (1999). Beta kernel estimators for density functions. Computational Statistics & Data Analysis, 31, 131-145.

Gourieroux, C. and Monfort, A. (2006). (Non) consistency of the Beta Kernel Estimator for Recovery Rate Distribution. Working Paper 2006-31, Centre de Recherche en Economie et Statistique.

Examples

# create the model 
kernel.noModified <- microBetaChen99Kernel(dataPoints = tuna.r, b = 0.01,
                      modified = FALSE)
kernel.Modified <- microBetaChen99Kernel(dataPoints = tuna.r, b = 0.01,
                      modified = TRUE)

# examples of usual functions
density(kernel.noModified,0.5)
density(kernel.Modified,0.5)

distribution(kernel.noModified,1,discreteApproximation=FALSE)
distribution(kernel.noModified,1,discreteApproximation=TRUE)
 
distribution(kernel.Modified,1,discreteApproximation=FALSE)
distribution(kernel.Modified,1,discreteApproximation=TRUE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Chen99 Kernels Tuna Data")
lines(kernel.noModified, col="red",lwd=2)
lines(kernel.Modified,col="blue",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(list("noModified"=kernel.noModified, 
          "modified"=kernel.Modified), show=TRUE)

Class "Muller91BoundaryKernel"

Description

This class deals with Kernel estimators for bounded densities using boundary kernel described in Muller (1991). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations. Note that this kernel estimator is not normalized and therefore it is not a probability distribution (the cumulative density function may return values greater than 1).

Objects from the Class

Objects can be created by using the generator function muller91BoundaryKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

mu:

a integer value indicating the degree of smoothness for the boundary kernel. mu can take the following values: 0 (uniform kernel), 1 (Epanechnikov kernel), 2 (biweight kernel) or 3 (triweight kernel)

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmu

See "getmu" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Muller, H. (1991). Smooth optimum kernel estimators near endpoints. Biometrika, 78(3), 521-530.

Examples

# create the model 
kernel <- muller91BoundaryKernel(dataPoints = tuna.r, b = 0.01, mu = 2)


# examples of usual functions
density(kernel,0.5)

distribution(kernel,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(kernel, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(kernel, show=TRUE, includePoints=TRUE)

Class "Muller94BoundaryKernel"

Description

This class deals with Kernel estimators for bounded densities using boundary kernel described in Muller and Wang (1994). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations. Note that this kernel estimator is not normalized and therefore it is not a probability distribution (the cumulative density function may return values greater than 1).

Objects from the Class

Objects can be created by using the generator function muller94BoundaryKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

mu:

a integer value indicating the degree of smoothness for the boundary kernel. mu can take the following values: 0 (uniform kernel), 1 (Epanechnikov kernel), 2 (biweight kernel) or 3 (triweight kernel)

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmu

See "getmu" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Muller, H. and Wang, J. (1994). Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics, 50(1), 61-76.

Examples

# create the model 
kernel <- muller94BoundaryKernel(dataPoints = tuna.r, b = 0.01, mu = 2)


# examples of usual functions
density(kernel,0.5)

distribution(kernel,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(kernel, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(kernel, show=TRUE, includePoints=TRUE)

Class "NoBoundaryKernel"

Description

This class deals with Kernel estimators for bounded densities using boundary kernels where the same kernel function is used for all regions: left boundary, interior and right boundary. The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations. Note that this kernel estimator is not normalized and therefore it is not a probability distribution (the cumulative density function may return values greater than 1).

Objects from the Class

Objects can be created by using the generator function noBoundaryKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

mu:

a integer value indicating the degree of smoothness for the boundary kernel. mu can take the following values: 0 (uniform kernel), 1 (Epanechnikov kernel), 2 (biweight kernel) or 3 (triweight kernel)

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmu

See "getmu" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

Examples

# create the model 
kernel <- noBoundaryKernel(dataPoints = tuna.r, b = 0.01, mu = 2)


# examples of usual functions
density(kernel,0.5)

distribution(kernel,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(kernel, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(kernel, show=TRUE, includePoints=TRUE)

Class "NormalizedBoundaryKernel"

Description

This class deals with Kernel estimators for bounded densities using renormalized boundary kernel described in Kakizawa (2004). The kernel estimator is computed using the provided data samples. Using this kernel estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations. Note that, the renormalization of this kernel guarantees non-negative density values. However, despite its name, the normalized boundary kernel is not a probability distribution (the cumulative density function may return values greater than 1).

Objects from the Class

Objects can be created by using the generator function normalizedBoundaryKernel.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

b:

the bandwidth of the kernel estimator

mu:

a integer value indicating the degree of smoothness for the boundary kernel. mu can take the following values: 0 (uniform kernel), 1 (Epanechnikov kernel), 2 (biweight kernel) or 3 (triweight kernel)

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getb

See "getb" for details

getmu

See "getmu" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Kakizawa, Y. (2004). Bernstein polynomial probability density estimation. Journal of Nonparametric Statistics, 16(5), 709-729.

Examples

# create the model 
kernel <- normalizedBoundaryKernel(dataPoints = tuna.r, b = 0.01, mu = 2)


# examples of usual functions
density(kernel,0.5)

distribution(kernel,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(kernel,col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(kernel, show=TRUE, includePoints=TRUE)

Class "Vitale"

Description

This class deals with Vitale (1975) Bernstein Polynomial approximation as described in Leblanc (2009). The polynomial estimator is computed using the provided data samples. Using this polynomial estimator, the methods implemented in the class can be used to compute densities, values of the distribution function, quantiles, sample the distribution and obtain graphical representations.

Objects from the Class

Objects can be created by using the generator function vitale.

Slots

dataPointsCache:

a numeric vector containing points within the [lower.limit,upper.limit] interval

densityCache:

a numeric vector containing the density for each point in dataPointsCache

distributionCache:

a numeric vector used to cache the values of the distribution function. This slot is included to improve the performance of the methods when multiple calculations of the distribution function are used

dataPoints:

a numeric vector containing data samples within the [lower.limit,upper.limit] interval. These data samples are used to obtain the kernel estimator

m:

the order of the polynomial approximation

lower.limit:

a numeric value for the lower limit of the bounded interval for the data

upper.limit:

a numeric value for the upper limit of the bounded interval for the data

Methods

density

See "density" for details

distribution

See "distribution" for details

quantile

See "quantile" for details

rsample

See "rsample" for details

plot

See "plot" for details

getdataPointsCache

See "getdataPointsCache" for details

getdensityCache

See "getdensityCache" for details

getdistributionCache

See "getdistributionCache" for details

getdataPoints

See "getdataPoints" for details

getm

See "getm" for details

Author(s)

Guzman Santafe, Borja Calvo and Aritz Perez

References

Vitale, R. A. (1975). A Bernstein polynomial approach to density function estimation. Statistical Inference and Related Topics, 2, 87-99.

Leblanc, A. (2010). A bias-reduced approach to density estimation using Bernstein polynomials. Journal of Nonparametric Statistics, 22(4), 459-475.

Examples

# create the model 
model <- vitale(dataPoints = tuna.r, m = 25)


# examples of usual functions
density(model,0.5)

distribution(model,0.5,discreteApproximation=FALSE)
 
# graphical representation
hist(tuna.r,freq=FALSE,main="Tuna Data")
lines(model, col="red",lwd=2)

# graphical representation using ggplot2 
graph <- gplot(model, show=TRUE, includePoints=TRUE)

Generic bounded density constructor

Description

Function to access all the methods

Usage

bde(dataPoints,dataPointsCache=NULL,estimator,b=length(sample)^{-2/5}, 
    lower.limit=0, upper.limit=1,options=NULL)

Arguments

• betakernel:

  • "modified": a logical value indicating whether the modified kernel has to be used or not. False by default

  • "normalization": a string: "none", to use the original kernels, "densitywise" to use the macrobeta kernels and "kernelwise" to use the microbeta kernels. If not specified, no normalization is used

  • "mbc": a string indicating the multiplicative bias correction to be used: "none", no correction is used, "jnl" Hirukawa's JNL approach, "ts" Hirukawa's TS approach. If not specified, no correction is used

  • "c": a numeric value between 0 and 1 corresponding to the c parameter in the TS correction (it is only taken into consideration if TS correction is selected). Default value is set to 0.5

• vitale:

  • "biasreduced": a logical value. If true, Leblanc's bias reduced estimator is used; otherwise the original estimator is used. False by default

• boundarykernel:

  • "mu": numeric parameter to indicate the kind of kernel. Options are 0, for the rectangular function, 1 for Epanechnikov's kernel, 2 for the quadratic and 3 for the biquadratic. Default value is set at 1

  • "method": a string indicating the functions to be used: "Muller94" (default value), "Muller91", "Normalize" or "None"

  • "corrected": a logical value indicating whether Jones' non-negativity correction should be used. By default it is set to false

• kakizawa:

  • "method": a string indicating the function to be used "b1", "b2" or "b3" (default value).

  • "estimator": a Bounded Density estimator. See all accepted classes here with getSubclasses("BoundedDensity"). If no estimator is provided, a Muller94BoundaryKernel estimator with default parameters and the same dataPoints as those give for the Kakizawa estimator is used.

  • "gamma": in case that b1 function is used the gamma parameter is required. This parameter takes 0.5 as default value.

Synthetic dataset from a beta distribution

Description

This is a synthetic generated dataset sampling a beta distribution with parameters shape1 = 0.75 and shape2 = 0.65

Usage

beta_0.75_0.65

Format

A vector containing 10000 observations.

Synthetic dataset from a beta distribution

Description

This is a synthetic generated dataset sampling a beta distribution with parameters shape1 = 1 and shape2 = 10

Usage

beta_1_10

Format

A vector containing 10000 observations.

Synthetic dataset from a beta distribution

Description

This is a synthetic generated dataset sampling a beta distribution with parameters shape1 = 5 and shape2 = 10

Usage

beta_5_10

Format

A vector containing 10000 observations.

BoundedDensity generator method

Description

User friendly constructor method for BoundedDensity objects.

Usage

boundedDensity(x,densities,lower.limit=0,upper.limit=1)

Arguments

Details

See BoundedDensity class for more details.

BrVitale generator method

Description

User friendly constructor method for BrVitale objects.

Usage

brVitale(dataPoints, m=round(length(dataPoints)^(2/5)), M=NULL, dataPointsCache=NULL, 
          lower.limit = 0, upper.limit = 1)

Arguments

Details

See BrVitale class for more details.

chen99Kernel generator method

Description

User friendly constructor method for Chen99Kernel objects.

Usage

chen99Kernel(dataPoints, b=length(dataPoints)^(-2/5), dataPointsCache=NULL, 
              modified = FALSE, lower.limit = 0, upper.limit = 1)

Arguments

Details

See Chen99Kernel class for more details.

Probability Density Function (pdf)

Description

Density function for the given bounded density object.

Arguments

Methods

density(x,values)

Cumulative Density Function (cdf)

Description

Distribution function for the given bounded density object

Arguments

Details

If discreteApproximation is not specified it assumes the default value TRUE. When the distribution function is used with a BoundedDensity object, discreteApproximation value is and a discrete approximation is always obtained.

Methods

distribution(.Object,x,discreteApproximation=TRUE)

Eruption lengths of Old Faithful geyser

Description

The dataset comprises lengths (in minutes) of eruptions of Old Faithful geyser in Yellowstone National Park, USA. The data are within the interval [1.67,4.93].

Usage

eruption

Format

A vector containing 107 observations.

Source

The data were obtained from Silverman (1996) Table 2.2

References

Silverman, B. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall

Weisberg, S. (1980). Applied linear regression. John Wiley & Sons, Canada

Accesor method for M slot

Description

This method obtains the values stored in the M slot of a BrVitale object. This slot contains parameter for bias reduction.

Arguments

Methods

getM(.Object)

List of subclasses

Description

This method returns a list containing the name of the class given as parameter and all the subclasses. Virtual classes are excluded from the list.

Usage

getSubclasses(className)

Arguments

Examples

# show the names of the class BoundedDensity and all its subclasses
getSubclasses("BoundedDensity")

# show the names of the class Chen99Kernel and all its subclasses
getSubclasses("Chen99Kernel")

Accesor method for b slot

Description

This method obtains the values stored in the b slot of a bounded density object. This slot contains the bandwidth parameter for the kernel estimator.

Arguments

Methods

getb(.Object)

Accesor method for c slot

Description

This method obtains the values stored in the c slot of a HirukawaTSKernel object. This parameter is used in the kernel estimation as a smoothing parameter.

Arguments

Methods

getc(.Object)

Accesor method for dataPoints slot

Description

This method obtains the values stored in the DataPoints slot of a bounded density object. This slot contains the data sample used to estimate the density model.

Arguments

Methods

getdataPoints(.Object)

Accesor method for DataPointsCache slot

Description

This method obtains the values stored in the dataPointsCache slot of a bounded density object.

Arguments

Methods

getdataPointsCache(.Object)

Accesor method for densityCache slot

Description

This method obtains the values stored in the DensityCache slot of a bounded density object

Arguments

Methods

getdensityCache(.Object)

Accesor method for gamma slot

Description

This method obtains the class name of the object stored in the densityEstimator slot of a KakizawaB1, KakizawaB2 or KakizawaB3 object.

Arguments

Methods

getdensityEstimator(.Object)

Accesor method for distributionCache slot

Description

This method obtains the values stored in the DistributionCache slot of a bounded density object.

Arguments

Methods

getdistributionCache(.Object)

Accesor method for gamma slot

Description

This method obtains the values stored in the gamma slot of a KakizawaB1 object. This slot contains a parameter used in the B1 approximation using Bernstein polynomials.

Arguments

Methods

getgamma(.Object)

Accesor method for m slot

Description

This method obtains the values stored in the m slot of a BernsteinPolynomials object. This slot contains the order of the polynomial expansion.

Arguments

Methods

getm(.Object)

Accesor method for modified slot

Description

This method obtains the values stored in the modified slot of a Kernel density object. The value of this slot is TRUE if a modified version of the kernel estimator is used and FALSE otherwise.

Arguments

Methods

getgetmodified(.Object)

Accesor method for Mu slot

Description

This method obtains the values stored in the mu slot of a Boundary Kernel object. This slot contains the degree of smoothing for the boundary kernel estimator. mu can take the following values: 0 (uniform kernel), 1 (Epanechnikov kernel), 2 (bicuadratic kernel) or 3 (tricuadratic kernel).

Arguments

Methods

getmu(.Object)

Bounded Density Plotting based on ggplot2

Description

Function to plot bounded density probability density functions.

Arguments

Methods

gplot(.Object,show=FALSE,includePoints=FALSE,lwd=1,alpha=1)

References

Wickham, H. (2009). ggplot2: Elegant Graphics for Data Analysis. Springer.

HirukawaJLNKernel generator method

Description

User friendly constructor method for HirukawaJLNKernel objects.

Usage

hirukawaJLNKernel(dataPoints, b, dataPointsCache=NULL, modified = FALSE, 
                  lower.limit = 0, upper.limit = 1)

Arguments

Details

See HirukawaJLNKernel class for more details.

HirukawaTSKernel generator method

Description

User friendly constructor method for HirukawaTSKernel objects.

Usage

hirukawaTSKernel(dataPoints, c, b=length(dataPoints)^(-2/5), dataPointsCache=NULL, 
                  modified = FALSE, lower.limit = 0, upper.limit = 1)

Arguments

Details

See HirukawaTSKernel class for more details.

JonesCorrectionMuller91BoundaryKernel generator method

Description

User friendly constructor method for JonesCorrectionMuller91BoundaryKernel objects.

Usage

jonesCorrectionMuller91BoundaryKernel(dataPoints, mu=1, b=length(dataPoints)^(-2/5), 
                                      dataPointsCache=NULL, lower.limit = 0, 
                                      upper.limit = 1)

Arguments

Details

See JonesCorrectionMuller91BoundaryKernel class for more details.

JonesCorrectionMuller94BoundaryKernel generator method

Description

User friendly constructor method for JonesCorrectionMuller94BoundaryKernel objects.

Usage

jonesCorrectionMuller94BoundaryKernel(dataPoints, mu=1, b=length(dataPoints)^(-2/5), 
                                      dataPointsCache=NULL, lower.limit = 0, 
                                      upper.limit = 1)

Arguments

Details

See JonesCorrectionMuller94BoundaryKernel class for more details.

KakizawaB1 generator method

Description

User friendly constructor method for KakizawaB1 objects.

Usage

kakizawaB1(dataPoints,estimator=NULL,m=round(length(dataPoints)^(2/5)),gamma=0.5,
            dataPointsCache=NULL, lower.limit = 0, upper.limit = 1)

Arguments

Details

See KakizawaB1 class for more details.

KakizawaB2 generator method

Description

User friendly constructor method for KakizawaB2 objects.

Usage

kakizawaB2(dataPoints, estimator=NULL,m=round(length(dataPoints)^(2/5)),
            dataPointsCache=NULL, lower.limit = 0, upper.limit = 1)

Arguments

Details

See KakizawaB2 class for more details.

KakizawaB3 generator method

Description

User friendly constructor method for KakizawaB3 objects.

Usage

kakizawaB3(dataPoints, estimator=NULL,m=round(length(dataPoints)^(2/5)),
            dataPointsCache=NULL, lower.limit = 0, upper.limit = 1)

Arguments

Details

See KakizawaB3 class for more details.

Shiny launch application

Description

Runs the shiny service for the bde package.

Usage

launchApp(...)

Arguments

Add a Bounded Density pdf to a Plot

Description

Function to draw a bounded density probability density functions in the current plot.

Arguments

Methods

lines(x,...)

MacroBetaChen99Kernel generator method

Description

User friendly constructor method for MacroBetaChen99Kernel objects.

Usage

macroBetaChen99Kernel(dataPoints, b=length(dataPoints)^(-2/5), dataPointsCache=NULL, 
                      modified = FALSE, lower.limit = 0, upper.limit = 1)

Arguments

Details

See MacroBetaChen99Kernel class for more details.

MacroBetaHirukawaJLNKernel generator method

Description

User friendly constructor method for MacroBetaHirukawaJLNKernel objects.

Usage

macroBetaHirukawaJLNKernel(dataPoints, b=length(dataPoints)^(-2/5), dataPointsCache=NULL,
                            modified = FALSE, lower.limit = 0, upper.limit = 1)

Arguments

Details

See MacroBetaHirukawaJLNKernel class for more details.

MacroBetaHirukawaTSKernel generator method

Description

User friendly constructor method for MacroBetaHirukawaTSKernel objects.

Usage

macroBetaHirukawaTSKernel(dataPoints, c, b=length(dataPoints)^(-2/5),
                          dataPointsCache=NULL, modified = FALSE, lower.limit = 0, 
                          upper.limit = 1)

Arguments

Details

See MacroBetaHirukawaTSKernel class for more details.

MicroBetaChen99Kernel generator method

Description

User friendly constructor method for MicroBetaChen99Kernel objects.

Usage

microBetaChen99Kernel(dataPoints, b=length(dataPoints)^(-2/5), dataPointsCache=NULL, 
                      modified = FALSE, lower.limit = 0, upper.limit = 1)

Arguments

Details

See MicroBetaChen99Kernel class for more details.

Mean Integrated Squared Error

Description

Computes the mean integrated squared error (MISE) for two given Bounded density objects.

Usage

mise(model1,model2,discreteApproximation = TRUE)

Arguments

Examples

# a general approximation to a Beta(1,10) distribution using BoundedDensity objects
cache <- seq(0,1,0.01)
dens  <- dbeta(cache,1,10)
bd    <- boundedDensity(x=cache,densities=dens)

# a BrVitale approximation to the Beta(1,10) distribution using a random data sample to 
# learn the model
dataSample <- rbeta(100,1,10)
kernel     <- hirukawaTSKernel(dataPoints=dataSample, b=0.1, c=0.3, 
                                dataPointsCache=cache, modified=FALSE)

# compute the mise
mise(bd,kernel,discreteApproximation=TRUE)
mise(bd,kernel,discreteApproximation=FALSE)

Muller91BoundaryKernel generator method

Description

User friendly constructor method for Muller91BoundaryKernel objects.

Usage

muller91BoundaryKernel(dataPoints,  mu=1, b=length(dataPoints)^(-2/5), 
                        dataPointsCache=NULL, lower.limit = 0, 
                        upper.limit = 1)

Arguments

Details

See Muller91BoundaryKernel class for more details.

Muller94BoundaryKernel generator method

Description

User friendly constructor method for Muller94BoundaryKernel objects.

Usage

muller94BoundaryKernel(dataPoints, mu=1, b=length(dataPoints)^(-2/5), 
                        dataPointsCache=NULL, lower.limit = 0, 
                        upper.limit = 1)

Arguments

Details

See Muller94BoundaryKernel class for more details.

NoBoundaryKernel generator method

Description

User friendly constructor method for NoBoundaryKernel objects.

Usage

noBoundaryKernel(dataPoints, mu=1, b=length(dataPoints)^(-2/5), 
                  dataPointsCache=NULL, lower.limit = 0, 
                  upper.limit = 1)

Arguments

Details

See NoBoundaryKernel class for more details.

NormalizedBoundaryKernel generator method

Description

User friendly constructor method for NormalizedBoundaryKernel objects.

Usage

normalizedBoundaryKernel(dataPoints, mu=1, b=length(dataPoints)^(-2/5), 
                          dataPointsCache=NULL, lower.limit = 0, 
                          upper.limit = 1)

Arguments

Details

See NormalizedBoundaryKernel class for more details.

Bounded Density Plotting

Description

Function to plot bounded density probability density functions.

Arguments

Methods

plot(x,main="Bounded density",type="l",xlab="X",ylab="Density",...)

Quantile

Description

Quantile function for the given bounded density object.

Arguments

Methods

quantile(x,p)

Random sample

Description

Random generator function for the given bounded density object.

Arguments

Methods

rsample(.Object,n)

Scaled data from suicide risk data

Description

The dataset comprises lengths (in days) of psychiatric treatment spells for patients used as controls in a study of suicide risks. The data have been scaled to the interval [0,1] by dividing each data sample by the maximum value.

Usage

suicide.r

Format

A vector containing 86 observations.

Source

The data were obtained from Silverman (1996) Table 2.1

References

Silverman, B. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall

Copas, J. B. and Fryer, M. J. (1980). Density estimation and suicide risks in psychiatric treatment. Journal of the Royal Statistical Society. Series A, 143(2), 167-176

Synthetic dataset from a truncated Gaussian distribution

Description

This is a synthetic generated dataset sampling a truncated Gaussian distribution on the interval [0,1] with mean=0 and sd=0.25

Usage

tgaussian

Format

A vector containing 10000 observations.

Scaled tuna data

Description

The tuna data come from an aerial line transect survey of Southern Bluefin Tuna in the Great Australian Bight and it is included in the boot package. The tuna.r data is a scaled version of the tuna data within the [0,1] interval. This new data set is obtained as follows:

library(boot)

tuna.r <- tuna$y/17

Usage

tuna.r

Format

A vector containing 64 observations.

Source

The data were obtained from

Chen, S.X. (1996). Empirical likelihood confidence intervals for nonparametric density estimation. Biometrica, 83, 329-341.

See Also

tuna

Vitale generator method

Description

User friendly constructor method for Vitale objects.

Usage

vitale(dataPoints, m=round(length(dataPoints)^(2/5)), dataPointsCache=NULL, 
        lower.limit = 0, upper.limit = 1)

Arguments

Details

See Vitale class for more details.