Conake-package {Conake} | R Documentation |

Continuous smoothing of probability density function defined on a compact `T=[a,b]`

or semi-infinite support `T=[0,\infty)`

is performed using four continuous associated kernels: extended beta, gamma, lognormal and reciprocal inverse Gaussian. The cross-validation technique is also implemented to select the smoothing parameter.

- The estimated density:
The 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 defined below. In practice, we first calculate the normalizing constant`{C}_n = \int_{x\in T}{\widehat{f}_n(x)dx},`

where T is the support of the density function. This normalizing constant is not generally equal to 1. The estimated density is then

`\tilde{f}_n=\widehat{f}_n/C_n`

.

Given a data sample, the Conake package allows to compute the density `dke`

using one of the four kernel functions: extended beta, gamma, lognormal and reciprocal inverse Gaussian. The bandwidth parameter is calculated using the cross-validation technique `cvbw`

.The kernel functions `kef`

are defined below.

- 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`

, the extended beta kernel 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`

:`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(.)`

is the classical gamma function. It is the probability density function of the gamma distribution with scale parameter`1+x/h`

and shape parameter`h`

; see Chen (2000) and also Libengué (2013).- Lognormal kernel :
The lognormal kernel is defined on

`{S}_{x,h}=[0,\infty)=T`

with`x \in T`

and`h>0`

:`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 densiy function of the classical lognormal distribution with mean

`log(x)+h^{2}`

and standard deviation`h`

; see Igarashi and Kakizawa (2015) and also Libengué (2013).- Reciprocal inverse Gaussian kernel:
The reciprocal inverse Gaussian kernel is defined on

`{S}_{x,h}=]0,\infty)=T`

with`x \in T`

and`h>0`

:`RIG_{x,h}(y) = \frac {1}{\sqrt{2\pi hy}} exp\left\{-\frac{\zeta(x,h)}{2h}\left(\frac{y}{\zeta(x,h)}-2+\frac{\zeta(x,h)}{y}\right)\right\}1_{S_{x,h}}(y),`

where

`\zeta(x,h)=(x^2+xh)^{1/2}`

. It is the probability densiy function of the classical reciprocal inverse Gaussian distribution with mean`1/\sqrt{x^2+xh}`

and standard deviation`1/h`

; see Igarashi and Kakizawa (2015) and also Libengué (2013).- The bandwidth selection:
The cross-validation technique

`cvbw`

is used for the bandwidth selection. The optimal parameter is the one which minimizes the cross-validation function defined by:`CV(h)=\int_{x\in T}{\{\widehat{f}_n(x)\}^{2}dx}-\frac{2}{n}\sum_{i=1}^{n}{\widehat{f}_{n,-i}(X_i)},`

where

`\widehat{f}_{n,-i}(X_i)=(n-1)^{-1}\sum_{j \ne i}^{n}K_{X_i,h}(X_j)`

is the density estimator computed without the observation`X_{i}`

.

W. E. Wansouwé, F.G. Libengué and C. C. Kokonendji

Maintainer: W. E. Wansouwé <ericwansouwe@gmail.com>

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.

[Package *Conake* version 1.0.1 Index]