Package 'Ake'

Associated kernel estimations

Description

Continuous and discrete estimation of density dke.fun, probability mass function (p.m.f.) kpmfe.fun and regression reg.fun functions are performed using continuous and discrete associated kernels. The cross-validation technique hcvc.fun, hcvreg.fun and the Bayesian procedure hbay.fun are also implemented for bandwidth selection.

Details

The estimated density or p.m.f:

The associated kernel estimator $\widehat{f}_n$ of $f$ is defined as

$\widehat{f}_n(x) = \frac{1}{n}\sum_{i=1}^{n}{K_{x,h}(X_i)},$

where $K_{x,h}$ is one of the kernels kef defined below. In practice, we first calculate the global normalizing constant

${C}_n = \int_{x\in T}{\widehat{f}_n(x) \nu(dx)},$

where $T$ is the support of the density or p.m.f. function and $\nu$ is the Lebesgue or count measure on $T$. For both continuous and discrete associated kernels, this normalizing constant is not generally equal to 1 and it will be computed. The represented density or p.m.f. estimate is then $\tilde{f}_n=\widehat{f}_n/C_n$.

For discrete data, the integrated squared error (ISE) defined by

${ISE}_0 = \sum_{x\in N}{{\{\tilde{f}_n(x)} - f_0(x)\}^2}$

is the criteria used to measure the smoothness of the associated kernel estimator $\tilde{f}_n$ with the empirical p.m.f. $f_0$; see Kokonendji and Senga Kiessé (2011).

The estimated regressor:

Both in continuous and discrete cases, considering the relation between a response variable $y$ and an explanatory variable $x$ given by

$y=m(x)+\epsilon ,$

where $m$ is an unknown regession function on $T$ and $\epsilon$ the disturbance term with null mean and finite variance. Let $(x_1,y_1),\ldots,(x_n,y_n)$ be a sequence of independent and identically distributed (iid) random vectors on $T\times R$ with $m(x)=E(y|x)$. The well-known Nadaraya-Watson estimator using associated kernels is $\widehat{m}_n$ defined as

$\widehat{m}_n(x) = \sum_{i=1}^{n}{\omega_{x}(X_i)Y_i},$

where $\omega_{x}(X_i)=K_{x,h}(X_i)/\sum_{i=1}^{n}{K_{x,h}(X_i)}$ and $K_{x,h}$ is one of the associated kernels defined below.

Beside the criterion of kernel support, we retain the root mean squared error (RMSE) and also the practical coefficient of determination $R^2$ defined respectively by

$RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n}{\{y_i-\widehat{m}_n(x_i)\}^2}}$

and

$R^2=\frac{\sum_{i=1}^{n}{\{\widehat{m}_n(x_i)-\bar{y}\}^2}}{\sum_{i=1}^{n}(y_i-\bar{y})^2},$

where $\bar{y}=n^{-1}(y_1+\ldots+y_n)$; see Kokonendji et al. (2009).

Given a data sample, the package allows to compute the density or p.m.f. and regression functions using one of the seven associated kernels: extended beta, lognormal, gamma, reciprocal inverse Gaussian for continuous data, DiracDU for categorical data, and binomial and discrete triangular for count data. The bandwidth parameter is computed using the cross-validation technique. When the associated kernel function is binomial, the bandwidth parameter is also computed using the local Bayesian procedure. The associated kernel functions are defined below. The first four kernels are for continuous data and the last three kernels are for discrete case.

Extended beta kernel:

The extended beta kernel is defined on ${S}_{x,h,a,b}=[a,b]=T$ with $a<b<\infty$, $x \in T$ and $h>0$:

$BE_{x,h,a,b}(y) = \frac {(y-a)^{(x-a)/\{(b-a)h\}}(b-y)^{(b-x)/\{(b-a)h\}}} {(b-a)^{1+h^{-1}}B\left(1+(x-a)/(b-a)h,1+(b-x)/(b-a)h\right)}1_{S_{x,h,a,b}}(y),$

where $B(r,s)=\int_0^1 t^{r-1}(1-t)^{s-1}dt$ is the usual beta function with $r>0$, $s>0$ and $1_A$ denotes the indicator function of A. For $a=0$ and $b=1$, it corresponds to the beta kernel which is the probability density function of the beta distribution with shape parameters $1+x/h$ and $(1-x)/h$; see Libengué (2013).

Gamma kernel:

The gamma kernel is defined on ${S}_{x,h}=[0, \infty)=T$ with $x \in T$ and $h>0$ by

$GA_{x,h}(y) = \frac {y^{x/h}} {\Gamma(1+x/h)h^{1+x/h}}exp\left(-\frac{y}{h} \right)1_{S_{x,h}}(y),$

where $\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}dt$ is the classical gamma function. The probability density function $GA_{x,h}$ is the gamma distribution with scale parameter $1+x/h$ and shape parameter $h$; see Chen (2000).

Lognormal kernel:

The lognormal kernel is defined on ${S}_{x,h}=[0,\infty)=T$ with $x \in T$ and $h>0$ by

$LN_{x,h}(y) = \frac {1} {yh\sqrt{2\pi}}exp\left\{-\frac{1}{2}\left(\frac{1}{h}log(\frac{y}{x})-h \right)^{2}\right\}1_{S_{x,h}}(y).$

It is the probability density function of the classical lognormal distribution with parameters $log(x)+h^{2}$ and $h$; see Libengué (2013).

Binomial kernel:

Let $x\in N:= \{0, 1, \ldots \}$ and ${S}_x = \{0, 1, \ldots, x + 1\}$. The Binomial kernel is defined on the support ${S}_x$ by

$B_{x,h}(y) = \frac {(x+1)!} {y!(x+1-y)!}\left(\frac{x+h}{x+1}\right)^y\left(\frac{1-h}{x+1}\right)^{(x+1-y)}1_{S_{x}}(y),$

where $h\in(0, 1]$. Note that $B_{x,h}$ is the p.m.f. of the binomial distribution with its number of trials $x+1$ and its success probability $(x+h)/(x+1)$; see Kokonendji and Senga Kiessé (2011).

Discrete triangular kernel:

For fixed arm $a\in N$, we define ${S}_{x,a} = \{x-a,\ldots, x, \ldots, x + a\}$. The discrete triangular kernel is defined on ${S}_{x,a}$ by

$DT_{x,h;a}(y) = \frac {(a+1)^h - |y-x|^h} {P(a,h)}1_{S_{x,a}}(y),$

where $x\in N$, $h>0$ and $P(a,h)=(2a+1)(a+1)^h - 2(1+2^h+ \cdots +a^h)$ is the normalizing constant. For $a=0$, the Discrete Triangular kernel $DT_{x,h;0}$ corresponds to the Dirac kernel on $x$; see Kokonendji et al. (2007), and also Kokonendji and Zocchi (2010) for an asymmetric version of discrete triangular.

DiracDU kernel:

For fixed number of categories $c\in \{2,3,...\}$, we define ${S}_{c} = \{0, 1, \ldots, c-1\}$. The DiracDU kernel is defined on ${S}_{c}$ by

$DU_{x,h;c}(y) = (1 - h)1_{\{x\}}(y)+\frac {h} {c-1}1_{S_{c}\setminus\{x\}}(y),$

where $x\in {S}_{c}$ and $h\in(0, 1]$. See Kokonendji and Senga Kiessé (2011), and also Aitchison and Aitken (1976) for multivariate case.

Note that the global normalizing constant is 1 for DiracDU.

The bandwidth selection:

Two functions are implemented to select the bandwidth: cross-validation and local Bayesian procedure. The cross-validation technique is used for all the associated kernels both in density and regression; see Kokonendji and Senga Kiessé (2011). The local Bayesian procedure is implemented to select the bandwidth in the estimation of p.m.f. when using binomial kernel; see Zougab et al. (2014).

In the coming versions of the package, adaptive Bayesian procedure will be included for bandwidth selection in density estimation when using gamma kernel. A global Bayesian procedure will also be implemented for bandwidth selection in regression when using binomial kernel.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

Maintainer: W. E. Wansouwé <[email protected]>

References

Aitchison, J. and Aitken, C.G.G. (1976). Multivariate binary discrimination by the kernel method, Biometrika 63, 413 - 420.

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

Chen, S. X. (2000). Probability density function estimation using gamma kernels, Annals of the Institute of Statistical Mathematics 52, 471 - 480.

Igarashi, G. and Kakizawa, Y. (2015). Bias correction for some asymmetric kernel estimators, Journal of Statistical Planning and Inference 159, 37 - 63.

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Demétrio, C.G.B. (2009). Appropriate kernel regression on a count explanatory variable and applications, Advances and Applications in Statistics 12, 99 - 125.

Libengue, F.G. (2013). Méthode Non-Paramétrique par Noyaux Associés Mixtes et Applications, Ph.D. Thesis Manuscript (in French) to Université de Franche-Comté, Besançon, France and Université de Ouagadougou, Burkina Faso, June 2013, LMB no. 14334, Besançon.

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2014). Bayesian approach in nonparametric count regression with binomial kernel, Communications in Statistics - Simulation and Computation 43, 1052 - 1063.

---

Function for density estimation

Description

The (S3) generic function dkde.fun computes the density. Its default method does so with the given kernel and bandwidth $h$.

Usage

dke.fun(Vec, ...)
## Default S3 method:
dke.fun(Vec, h, type_data = c("discrete", "continuous"), 
ker = c("BE", "GA", "LN", "RIG"), x = NULL, a0 = 0, a1 = 1, ... )

Arguments

Vec	The data sample from which the estimate is to be computed.
h	The bandwidth or smoothing parameter.
type_data	The data sample type. Data can be continuous or discrete (categorical or count). Here, in this function , we deal with continuous data.
ker	A character string giving the smoothing kernel to be used which is the associated kernel: "BE" extended beta, "GA" gamma, "LN" lognormal and "RIG" reciprocal inverse Gaussian.
x	The points of the grid at which the density is to be estimated.
a0	The left bound of the support used for extended beta kernel. Default value is 0 for beta kernel.
a1	The right bound of the support used for extended beta kernel. Default value is 1 for beta kernel.
...	Further arguments.

Details

The associated kernel estimator $\widehat{f}_n$ of $f$ is defined in the above sections. We recall that in general, the sum of the estimated values on the support is not equal to 1. In practice, we compute the global normalizing constant $C_n$ before computing the estimated density $\tilde{f}_n$; see e.g. Libengué (2013).

Value

Returns a list containing:

data	The data - same as input Vec.
n	The sample size.
kernel	The asssociated kernel used to compute the density estimate.
h	The bandwidth used to compute the density estimate.
eval.points	The coordinates of the points where the density is estimated.
est.fn	The estimated density values.
C_n	The global normalizing constant.
hist	The histogram corresponding to the observations.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Libengué, F.G. (2013). Méthode Non-Paramétrique par Noyaux Associés Mixtes et Applications, Ph.D. Thesis Manuscript (in French) to Université de Franche-Comté, Besançon, France and Université de Ouagadougou, Burkina Faso, June 2013, LMB no. 14334, Besançon.

Examples

## A sample data with n=100.
V<-rgamma(100,1.5,2.6)
##The bandwidth can be the one obtained by cross validation.
h<-0.052
## We choose Gamma kernel.

est<-dke.fun(V,h,"continuous","GA")

---

Local Bayesian procedure for bandwidth selection

Description

The (S3) generic function hbay.fun computes the local Bayesian procedure for bandwidth selection.

Usage

hbay.fun(Vec, ...)
## Default S3 method:
hbay.fun(Vec, x = NULL, ...)

Arguments

Vec	The data sample from which the estimate is to be computed.
x	The points of the grid where the density is to be estimated.
...	Further arguments for (non-default) methods.

Details

hbay.fun implements the choice of the bandwidth $h$ using the local Bayesian approach of a kernel density estimator.

Value

Returns the bandwidth selected using the local Bayesian procedure.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

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

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2014). Bayesian approach in nonparametric count regression with binomial kernel, Communications in Statistics - Simulation and Computation 43, 1052 - 1063.

---

Cross-validation function for bandwidth selection for continuous data

Description

The (S3) generic function hcvc.fun computes the cross-validation bandwidth selector.

Usage

hcvc.fun(Vec,...)
## Default S3 method:
hcvc.fun(Vec, bw = NULL, type_data, ker, a0 = 0, a1 = 1, ...)

Arguments

Vec	The data sample from which the estimate is to be computed.
bw	The sequence of bandwidths where to compute the cross-validation. Default value is NULL.
type_data	The sample data type.
ker	The associated kernel.
a0	The left bound of the extended beta. Default value is 0.
a1	The right bound of the extended beta.Default value is 1.
...	Further arguments.

Details

hcvc.fun implements the choice of the bandwidth $h$ using the cross-validation approach of a kernel density estimator.

Value

Returns a list containing:

hcv	value of bandwidth parameter.
CV	the values of cross-validation function.
seq_h	the sequence of bandwidths where the cross validation is computed.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

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

Chen, S. X. (2000). Gamma kernels estimators for density functions, Annals of the Institute of Statistical Mathematics 52, 471 - 480.

Libengué, F.G. (2013). Méthode Non-Paramétrique par Noyaux Associés Mixtes et Applications, Ph.D. Thesis Manuscript (in French) to Université de Franche-Comté, Besançon, France and Université de Ouagadougou, Burkina Faso, June 2013, LMB no. 14334, Besançon.

Igarashi, G. and Kakizawa, Y. (2015). Bias correction for some asymmetric kernel estimators, Journal of Statistical Planning and Inference 159, 37 - 63.

Examples

V=rgamma(100,1.5,2.6)
## Not run: 
hcvc.fun(V,NULL,"continuous","GA")

## End(Not run)

---

Cross-validation function for bandwidth selection in p.m.f. estimation

Description

The (S3) generic function hcvd.fun computes the cross-validation bandwidth selector in p.m.f. estimation.

Usage

hcvd.fun(Vec, ...)
## Default S3 method:
hcvd.fun(Vec, seq_bws = NULL, ker = c("bino", "triang", "dirDU"), a = 1, c = 2,...)

Arguments

Vec	The data sample from which the estimate is to be computed.
seq_bws	The sequence of bandwidths where to compute the cross-validation. Default value is NULL.
ker	The associated kernel
a	The arm of the discrete triangular kernel. Default value is 1.
c	The number of categories in DiracDU kernel.Default value is 2.
...	Further arguments.

Details

The hcvd.fun function implements the choice of the bandwidth $h$ using the cross-validation approach in p.m.f. estimate.

Value

Returns a list containing:

hcv	The optimal bandwidth parameter.
CV	The cross-validation function values.
seq_h	The sequence of bandwidths where the cross-validation is computed.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

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

Chen, S. X. (2000). Probability density function estimation using gamma kernels, Annals of the Institute of Statistical Mathematics 52, 471 - 480.

Libengué, F.G. (2013). Méthode Non-Paramétrique par Noyaux Associés Mixtes et Applications, Ph.D. Thesis Manuscript (in French) to Université de Franche-Comté, Besançon, France and Université de Ouagadougou, Burkina Faso, June 2013, LMB no. 14334, Besançon.

Igarashi, G. and Kakizawa, Y. (2015). Bias correction for some asymmetric kernel estimators, Journal of Statistical Planning and Inference 159, 37 - 63.

Examples

## Data can be simulated data or real data
## We use real data 
## and then compute the cross validation. 
Vec<-c(10,0,1,0,4,0,6,0,0,0,1,1,1,2,4,4,5,6,6,6,6,7,1,7,0,7,7,
7,8,0,8,12,8,8,9,9,0,9,9,10,10,10,10,0,10,10,11,12,12,10,12,12,
13,14,15,16,16,17,0,12)
## Not run: 
CV<-hcvd.fun(Vec,NULL,"bino")
CV$hcv

## End(Not run)
##The cross validation function can be also ploted.
## Not run: 
plot.fun(CV$seq_bws,CV$CV, type="l")

## End(Not run)

---

Cross-validation function for bandwidth selection in regresssion

Description

The (S3) generic function hcvreg.fun computes the bandwidth by cross-validation for the regression. Its default method does so. It allows to compute the optimal bandwidth using the cross-validation method. The associated kernels available are: "BE" extended beta, "GA" gamma, "LN" lognormal and "RIG" reciprocal inverse Gaussian, DiracDU, binomial and discrete triangular; see Kokonendji and Senga Kiessé (2011), and also Kokonendji et al. (2009).

Usage

hcvreg.fun(Vec, ...)
## Default S3 method:
hcvreg.fun(Vec, y, type_data = c("discrete", "continuous"), 
ker = c("bino", "triang", "dirDU", "BE", "GA", "LN", "RIG"),
 h = NULL, a0 = 0, a1 = 1, a = 1, c = 2, ...)

Arguments

Vec	The explanatory variable.
y	The response variable.
type_data	The data sample type. Data can be continuous or discrete.
ker	A character string giving the smoothing kernel to be used which is the associated kernel: "BE" extended beta, "GA" gamma, "LN" lognormal and "RIG" reciprocal inverse Gaussian, "dirDU" DiracDU,"bino" binomial, "triang" discrete triangular.
h	The bandwidth or smoothing parameter.the smoothing bandwidth to be used, can also be a character string giving a rule to choose the bandwidth.
a0	The left bound of the support used for extended beta kernel. Default value is 0 for beta kernel.
a1	The right bound of the support used for extended beta kernel. Default value is 1 for beta kernel.
a	The arm of the discrete triangular kernel
c	The number of categories
...	Further arguments

Details

The selection of the bandwidth parameter is always crucial. If the bandwidth is small, we will obtain an undersmoothed estimator, with high variability. On the contrary, if the value is big, the resulting estimator will be very smooth and farther from the function that we are trying to estimate. The cross-validation function defined in the above sections is used to compute the optimal bandwidth for the associated kernels.

Value

Returns a list containing:

kernel	The associated kernel used to compute the optimal bandwidth.
hcv	The optimal bandwidth parameter obtained by cross-validation.
CV	The values of the cross-validation.
seq_bws	A sequence of bandwidths where the cross-validation is computed.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Demétrio, C.G.B. (2009). Appropriate kernel regression on a count explanatory variable and applications, Advances and Applications in Statistics 12, 99 - 125.

Examples

## Data can be simulated data or real data
## We use real data 
## and then compute the cross validation. 
data(milk)
x=milk$week
y=milk$yield
hcvreg.fun(x,y,"discrete",ker="triang",a=1)

---

Continuous and discrete associated kernel function

Description

This function computes the associated kernel function.

Usage

kef(x, t, h, type_data = c("discrete", "continuous"), 
ker = c("bino", "triang", "dirDU", "BE", "GA", "LN", "RIG"), 
a0 = 0, a1 = 1, a = 1, c = 2)

Arguments

x	The target.
t	A single value or the grid where the associated kernel function is computed.
h	The bandwidth or smoothing parameter.
type_data	The sample data type
ker	The associated kernel:"bino" Binomial, "triang" discrete triangular kernel, "BE" extended beta, "GA" gamma, "LN" lognormal and "RIG" reciprocal inverse Gaussian,"dirDU" DiracDU.
a0	The left bound of the support used for extended beta kernel. Default value is 0 for beta kernel.
a1	The right bound of the support used for extended beta kernel. Default value is 1 for beta kernel.
a	The arm in discrete triangular kernel. The default value is 1.
c	The number of categories in DiracDU kernel. The default value is 2.

Details

The associated kernel is one of the those which have been defined in the sections above : extended beta, gamma, lognormal, reciprocal inverse Gaussian, DiracDU, binomial and discrete triangular; see Kokonendji and Senga Kiessé (2011), and also Kokonendji et al. (2007).

Value

Returns the value of the associated kernel function at t according to the target and the bandwidth.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

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

Chen, S. X. (2000). Probability density function estimation using gamma kernels, Annals of the Institute of Statistical Mathematics 52, 471 - 480.

Igarashi, G. and Kakizawa, Y. (2015). Bias correction for some asymmetric kernel estimators, Journal of Statistical Planning and Inference 159, 37 - 63.

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Zocchi, S.S. (2007). Discrete triangular distributions and non-parametric estimation for probability mass function, Journal of Nonparametric Statistics 19, 241 - 254.

Libengué, F.G. (2013). Méthode Non-Paramétrique par Noyaux Associés Mixtes et Applications, Ph.D. Thesis Manuscript (in French) to Université de Franche-Comté, Besançon, France and Université de Ouagadougou, Burkina Faso, June 2013, LMB no. 14334, Besançon.

Examples

x<-5
h<-0.2
t<-0:10
kef(x,t,h,"discrete","bino")

---

The associated kernel function

Description

The (S3) generic function kern.fun computes the value of the associated kernel function. Its default method does so with a given kernel and bandwidth $h$.

Usage

kern.fun(x, ...)
## Default S3 method:
kern.fun(x, t, h, type_data = c("discrete", "continuous"),
 ker = c("bino", "triang", "dirDU", "BE", "GA", "LN", "RIG"), 
a0 = 0, a1 = 1, a = 1, c = 2, ...)

Arguments

x	The target
t	A single value or the grid where the discrete associated kernel function is computed.
h	The bandwidth or smoothing parameter.
type_data	The sample data type
ker	The associated kernel: "dirDU" DiracDU,"bino" Binomial, "triang" Discrete Triangular kernel, "BE" extended beta, "GA" gamma, "LN" lognormal and "RIG" reciprocal inverse Gaussian.
a0	The left bound of the support used for extended beta kernel. Default value is 0 for beta kernel.
a1	The right bound of the support used for extended beta kernel. Default value is 0 for beta kernel.
a	The arm in Discrete Triangular kernel. The default value is 1.
c	The number of categories in DiracDU kernel. The default value is 2.
...	Further arguments

Details

The associated kernel is one of the those which have been defined in the sections above : extended beta, gamma,lognormal, reciprocal inverse Gaussian, DiracDU, Binomial and Discrete Triangular; see Kokonendji and Senga Kiessé (2011), and also Kokonendji et al. (2007).

Value

Returns the value of the discrete associated kernel function at t according to the target and the bandwidth.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Zocchi, S.S. (2007). Discrete triangular distributions and non-parametric estimation for probability mass function, Journal of Nonparametric Statistics 19, 241 - 254.

Examples

x<-5
h<-0.2
t<-0:10
kern.fun(x,t,h,"discrete","bino")

---

Function for associated kernel estimation of p.m.f.

Description

The function estimates the p.m.f. in a single value or in a grid using discrete associated kernels. Three different associated kernels are available: DiracDU (for categorical data), binomial and discrete triangular (for count data).

Usage

kpmfe.fun(Vec,...)
## Default S3 method:
kpmfe.fun(Vec, h, type_data = c("discrete", "continuous"), 
        ker = c("bino", "triang", "dirDU"), x = NULL, a = 1, c = 2, ...)

Arguments

Vec	the data sample from which the estimate is to be computed.
h	The bandwidth or smoothing parameter. The smoothing bandwidth to be used, can also be a character string giving a rule to choose the bandwidth.
type_data	The data sample type. Data type is "discrete" (categorical or count).
ker	The associated kernel: "dirDU" DiracDU,"bino" binomial, "triang" discrete triangular.
x	The points of the grid at which the density is to be estimated.
a	The arm in discrete triangular kernel. The default value is 1.
c	The number of categories in DiracDU. The default value is 2.
...	Further arguments.

Details

The associated kernel estimator $\widehat{f}_n$ of $f$ is defined in the above sections. We recall that in general, the sum of the estimated values on the support is not equal to 1. In practice, we compute the global normalizing constant $C_n$ before computing the estimated p.m.f. $\tilde{f}_n$; see Kokonendji and Senga Kiessé (2011).

The bandwidth parameter in the function is obtained using the cross-validation technique for the three associated kernels. For binomial kernel, the local Bayesian approach is also implemented and is recommanded to select the bandwidth; see Zougab et al. (2012).

Value

Returns a list containing:

data	The number of observations.
n	The number of observations.
eval.points	The support of the estimated p.m.f.
h	The bandwidth
C_n	The global normalizing constant.
ISE_0	The integrated square error.
f_0	A vector of (x,f0(x)).
f_n	A vector of (x,fn(x)).
f0	The empirical p.m.f.
est.fn	The estimated p.m.f. containing estimated values after normalization.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Zocchi, S.S. (2007). Discrete triangular distributions and non-parametric estimation for probability mass function. Journal of Nonparametric Statistics 19, 241 - 254.

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2012). Binomial kernel and Bayes local bandwidth in discrete functions estimation. Journal of Nonparametric Statistics 24, 783 - 795.

Examples

## A sample data with n=60.
V<-c(10,0,1,0,4,0,6,0,0,0,1,1,1,2,4,4,5,6,6,6,6,7,1,7,0,7,7,
7,8,0,8,12,8,8,9,9,0,9,9,10,10,10,10,0,10,10,11,12,12,10,12,12,
13,14,15,16,16,17,0,12)


##The bandwidth can be the one obtained by cross validation.
h<-0.081
## We choose Binomial kernel.

est<-kpmfe.fun(Vec=V,h,"discrete","bino")
##To obtain the normalizing constant:
est

---

Average daily fat yields.

Description

This data is the average daily fat yields (kg/day) of milk from a single cow for each of 35 weeks; see Kokonendji et al. (2009).

Usage

data(milk)

Format

A data frame with 35 observations on the following 2 variables.

week

Number of the week

yield

The yield quantity

Source

McCulloch, C.E. (2001). An Introduction to Generalized Linear Mixed Models, 46a Reuniao Anual da RBRAS - 9o SEAGRO, University of Sao Paulo - ESALQ, Piracicaba.

References

Kokonendji, C.C., Senga Kiessé, T. and Demétrio, C.G.B. (2009). Appropriate kernel regression on a count explanatory variable and applications, Advances and Applications in Statistics 12, 99 - 125.

Examples

data(milk)

---

Plot of density function

Description

The plot.dke.fun is to plot the associated kernel density estimation.

Usage

## S3 method for class 'dke.fun'
plot(x,main = NULL, sub = NULL, xlab = NULL, 
ylab = NULL, type = "l", las = 1, lwd = 1, col = "blue", lty = 1, ...)

Arguments

x	An object class dke.fun
main	The main parameter
sub	The sub title
xlab, ylab	The axis label
type	the type parameter
las	Numeric in 0,1,2,3; the style of axis labels.
lwd	The line width, a positive number, defaulting to 1.
col	A specification for the default plotting color.
lty	The line type.
...	Futher arguments

Value

Plot of associated kernel density function is sent to graphics window.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Zocchi, S.S. (2007). Discrete triangular distributions and non-parametric estimation for probability mass function. Journal of Nonparametric Statistics 19, 241 - 254.

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2012). Binomial kernel and Bayes local bandwidth in discrete functions estimation. Journal of Nonparametric Statistics 24, 783 - 795.

---

Plot of cross-validation function for bandwidth selection in density or p.m.f. estimation.

Description

The functions allows to plot the cross-validation both in discrete plot.hcvd.fun and continuous plot.hcvc.fun cases.

Usage

## S3 method for class 'hcvc.fun'
plot(x,  ...)
## S3 method for class 'hcvd.fun'
plot(x,  ...)

Arguments

x	an object
...	Further arguments

Details

Plot a graphic for cross-validation function

Value

returns a graphics

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Zocchi, S.S. (2007). Discrete triangular distributions and non-parametric estimation for probability mass function. Journal of Nonparametric Statistics 19, 241 - 254.

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2012). Binomial kernel and Bayes local bandwidth in discrete functions estimation. Journal of Nonparametric Statistics 24, 783 - 795.

---

Plot of associated kernel function

Description

The plot.kern.fun function loops through calls to the kern.fun function.

Usage

## S3 method for class 'kern.fun'
plot(x, ...)

Arguments

x	an object of class kern.fun (output from kern.fun).
...	Other graphics parameters

Value

Plot of associated the kernel function is sent to graphics window.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Demétrio, C.G.B. (2009). Appropriate kernel regression on a count explanatory variable and applications, Advances and Applications in Statistics 12, 99 - 125.

---

Plot of the function for associated kernel estimation of the p.m.f.

Description

The function plots the p.m.f. estimation in a single value or in a grid using discrete associated kernels. Three different associated kernels are available: DiracDU (for categorical data), binomial and discrete triangular (for count data).

Usage

## S3 method for class 'kpmfe.fun'
plot(x, ...)

Arguments

x	An object of class kpmfe.fun.
...	Further arguments

Details

Plot a graphic

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Zocchi, S.S. (2007). Discrete triangular distributions and non-parametric estimation for probability mass function. Journal of Nonparametric Statistics 19, 241 - 254.

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2012). Binomial kernel and Bayes local bandwidth in discrete functions estimation. Journal of Nonparametric Statistics 24, 783 - 795.

---

Plot for associated kernel regression

Description

Plot for associated kernel regression for univariate data. The plot.reg.fun function loops through calls to the reg.fun function.

Usage

## S3 method for class 'reg.fun'
plot(x,  ...)

Arguments

x	An object of class reg.fun
...	other graphics parameters

Details

The function allows to plot the regression

Value

Plot is sent to graphics window.

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Zocchi, S.S. (2007). Discrete triangular distributions and non-parametric estimation for probability mass function. Journal of Nonparametric Statistics 19, 241 - 254.

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2012). Binomial kernel and Bayes local bandwidth in discrete functions estimation. Journal of Nonparametric Statistics 24, 783 - 795.

---

Print for regression function

Description

The function allows to print the result of computation in regression as a data frame.

Usage

## S3 method for class 'reg.fun'
print(x, digits = NULL, ...)

Arguments

x	object of class reg.fun.
digits	The number of digits
...	Further arguments

Details

The associated kernel estimator $\widehat{m}_n$ of $m$ is defined in the above sections; see Kokonendji and Senga Kiessé (2011). The bandwidth parameter in the function is obtained using the cross-validation technique for the associated kernels.

Value

Returns a list containing:

data	The explanatory variable, printed as a data frame
y	The response variable, printed as a data frame
n	The size of the sample
kernel	The associated kernel
h	The smoothing parameter
eval.points	The grid where the regression is computed, printed as data frame
m_n	The estimated values, printed as data frame
Coef_det	The Coefficient of determination

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Demétrio, C.G.B. (2009). Appropriate kernel regression on a count explanatory variable and applications, Advances and Applications in Statistics 12, 99 - 125.

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2014). Bayesian approach in nonparametric count regression with Binomial Kernel, Communications in Statistics - Simulation and Computation 43, 1052 - 1063.

Examples

data(milk)
x=milk$week
y=milk$yield
##The bandwidth is the one obtained by cross validation.
h<-0.10
## We choose binomial kernel.
m_n<-reg.fun(x, y, "discrete",ker="bino", h)
print.reg.fun(m_n)

---

Function for associated kernel estimation of regression

Description

The function estimates the discrete and continuous regression in a single value or in a grid using associated kernels. Different associated kernels are available: extended beta, gamma, lognormal, reciprocal inverse Gaussian (for continuous data), DiracDU (for categorical data), binomial and also discrete triangular (for count data).

Usage

reg.fun(Vec, ...)
## Default S3 method:
reg.fun(Vec, y, type_data = c("discrete", "continuous"), 
ker = c("bino", "triang", "dirDU", "BE", "GA", "LN", "RIG"),
 h, x = NULL, a0 = 0, a1 = 1, a = 1, c = 2, ...)

Arguments

Vec	The explanatory variable.
y	The response variable.
type_data	The sample data type.
ker	The associated kernel: "dirDU" DiracDU,"bino" binomial, "triang" discrete triangular, etc.
h	The bandwidth or smoothing parameter.
x	The single value or the grid where the regression is computed.
a0	The left bound of the support used for extended beta kernel. Default value is 0 for beta kernel.
a1	The right bound of the support used for extended beta kernel. Default value is 0 for beta kernel.
a	The arm in Discrete Triangular kernel. The default value is 1.
c	The number of categories in DiracDU. The default value is 2.
...	Further arguments

Details

The associated kernel estimator $\widehat{m}_n$ of $m$ is defined in the above sections; see also Kokonendji and Senga Kiessé (2011). The bandwidth parameter in the function is obtained using the cross-validation technique for the seven associated kernels. For binomial kernel, the local Bayesian approach is also implemented; see Zougab et al. (2014).

Value

Returns a list containing:

data	The data sample, explanatory variable
y	The data sample, response variable
n	The size of the sample
kernel	The asociated kernel
h	The bandwidth
eval.points	The grid where the regression is computed
m_n	The estimated values
Coef_det	The coefficient of determination

Author(s)

W. E. Wansouwé, S. M. Somé and C. C. Kokonendji

References

Kokonendji, C.C. and Senga Kiessé, T. (2011). Discrete associated kernel method and extensions, Statistical Methodology 8, 497 - 516.

Kokonendji, C.C., Senga Kiessé, T. and Demétrio, C.G.B. (2009). Appropriate kernel regression on a count explanatory variable and applications, Advances and Applications in Statistics 12, 99 - 125.

Zougab, N., Adjabi, S. and Kokonendji, C.C. (2014). Bayesian approach in nonparametric count regression with binomial kernel, Communications in Statistics - Simulation and Computation 43, 1052 - 1063.

Examples

data(milk)
x=milk$week
y=milk$yield
##The bandwidth is the one obtained by cross validation.
h<-0.10
## We choose binomial kernel.
## Not run: 
m_n<-reg.fun(x, y, "discrete",ker="bino", h)

## End(Not run)

---