Conake-package {Conake}

Continuous Associated Kernel Estimation

Description

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.

Details

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}.

Author(s)

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

Maintainer: W. E. Wansouwé <[e-mail]>

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.