\def\papertitle{Multiplicative bias correction for Inverse Gamma and Beta Prime kernel density estimators}
\def\paperauthorA{Lynda HARFOUCHE}
\def\paperauthorB{Yasmina ZIANE}
\def\paperauthorC{Nabil ZOUGAB}
\def\paperauthorD{Smail ADJABI}
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\title{\papertitle}
\affiliation{\paperauthorA, \paperauthorB, \paperauthorC, \paperauthorD}	% This command replaces \name{The DAFx Crew}
{Research unit LaMOS, Higher School of Computing and Digital Sciences\\ and Technologies (ESTIN), 06008, Amizour, Bejaia, Algeria.
06000\\ Bejaia, Algeria.\\
E-mail: lyndaharfouche@hotmail.com\\
Research unit LaMOS, Faculty of Exact Sciences, Bejaia University,
06000\\ Bejaia, Algeria.\\
%E-mail: yasmina-ziane@hotmail.com\\
Research unit LaMOS, Faculty of technology, Bejaia University, 06000\\
Bejaia, Algeria.\\
%E-mail: nabilzougab@yahoo.fr\\
Research unit LaMOS, Faculty of Exact Sciences, Bejaia University,
06000\\ Bejaia, Algeria.\\
%E-mail: adjabi@hotmail.com
}

\newtheorem{thm}{Theorem}
\newtheorem{remark}{Remark}
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\begin{abstract}
In this paper, we demonstrate that the multiplicative bias correction (MBC) approaches can be extended for both Inverse Gamma (IG) and Beta Prime (BP) kernel density estimators. First, some properties of the MBC-IG and MBC-BP kernel density estimators (bias, variance and mean integrated squared error) are shown. Second, the least square cross validation
 technique (LSCV) is adapted for the choice of bandwidth.
\end{abstract}

\section{Introduction}





Given a random sample of observations $\{X_{i}\}_{i=1}^{n}$ with univariate density $f$, whitch is supported on $\mathbb{T}=[0,\infty)$. A continuous
symmetric or asymmetric kernel estimator $\hat{f}_{h}(x)$ of $f(x)$ can be defined as follows:
\begin{equation*}\label{estker}
\hat{f}_{h}(x)=\frac{1}{n}\sum\limits_{i=1}^{n}K_{x,h}(X_{i}),
\end{equation*}
Where $K_{x,h}$ is the continuous symmetric or asymmetric kernel with the target $x$ and $h=h(n)>0$ is an arbitrary sequence of smoothing parameters (bandwidths).\\
Recently two classes of multiplicative bias correction ("MBC") techniques were proposed in order to estimate the univariate densities, with compact \cite{h10} and with bounded support \cite{hs14}, \cite{zaz2015} for generalized Birnbaum-Saunders kernel density estimators, \cite{HZAB2017} and \cite{HZA2018} in the context of probability mass function with discrete kernels. These two classes are initially investigated in \cite{ts80} and in \cite{jln95}, witch improve bias from $O(h^{2})$ as the bandwidth $h\rightarrow 0$ to $O(h^{4})$ for symmetric kernel functions. It is well known that the symmetric kernel estimator is inappropriate in the context of estimating unknown probability densities whitch are supported on $\mathbb{T}=[0,\infty)$ because it causes boundary bias without producing negative values of the estimate, and we accept that the use of asymmetric kernels functions has many advantages such as optimal rate of convergence in the mean integrated squared error sense for kernels of order two, witch were originally employed in \cite{chen1999} for nonparametric estimation of compact using Beta as kernel functions, and \cite{chen2000} with (gamma and modified gamma kernels). Two recent papers are respectively proposed by \cite{mhf} (Inverse Gamma (IG) kernel) and \cite{EM} (Beta Prime (BP) kernel) in the same context (In the context of unknown probability density estimation). They have shown that the optimal rate of convergence of the mean integrated squared error for this estimator is of order $O(n^{-4/(5)})$ where they assuming that the true density $f$ is twice continuous. \cite{h10}, \cite{hs14} and \cite{zaz2015} have demonstrated that the two classes of MBC approaches can be applied in the case of nonparametric density estimation with asymmetric kernel, where they assuming that $f$ is four times continuous with bounded derivatives, the both methods reach an optimal rate of convergence of mean integrated squared error of order $O(n^{-8/(9)})$.

The current work is also motivated by several points. First, The IG and BP kernel density estimators are often used in modeling the hydrological problems and modeling the frequency of certain behavioral acts because of its long tail. Second, they are free of boundary bias, have flexible shape, always nonnegative, and achieve the optimal rate of convergence for the MSE of order of $O(n^{-4/(5)})$ and MISE within the class of nonnegative kernel density estimator. The main aim of this paper is to extend the application of MBC approaches for Inverse Gamma (IG) kernel and Beta Prime (BP) kernel estimator as in \cite{h10}, \cite{hs14} and \cite{zaz2015}, in order to reach an optimal rate of convergence of mean integrated squared error of order $O(n^{-8/(9)})$. First, we provide the asymptotic properties of these estimators and show that the optimal rate of convergence of the mean integrated squared error is obtained.

This paper is organized as follows. Section 2 briefly recalls on IG and BP kernels for density estimation. In Section 3, we first introduce the MBC kernel density estimators based on IG and BP kernels. Second, we show some properties of the MBC-IG and MBC-BP kernels density estimators (bias, variance and mean integrated squared error). Third, we adapt the least square cross validation technique (LSCV) for the choice of bandwidth. Conclusion of our paper is given in section 4.

\section{A short review on IG and BP kernels}
This section is dedicated to present a brief recall on both IG and BP kernel density estimators.
\subsection{IG and BP kernel estimator}
Given a random sample $X_{1},\ldots ,X_{n}$, the IG and BP estimator of an unknown pdf $f$ with nonnegative support
are given by \cite{mhf} and \cite{EM} respectively by

\begin{equation}
% \nonumber to remove numbering (before each equation)
  \widehat{f}_{j}(x)=\frac{1}{n}\sum\limits_{i=1}^{n}K_{j(x,h)}(X_{i})\\
   %&=&\frac{1}{n}\sum_{i=1}^{n}cg\left(\frac{1}{h}\left(\frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right)\right)\frac{1}{2\sqrt{h}}\left(\frac{1}{\sqrt{x X_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right),~x>0,
   \label{eqfh}
\end{equation}

where $x>0$ is the target (point where the density is estimated), $h>0$ is a bandwidth (or smoothing parameter), $j=IG,BP$ and the explicit form of $K_{j(x,h)}(X_{i})$ is given in table \ref{tablkernels}.
\begin{table}[h!]
%\scriptsize
\caption{Univariate continuous kernels.}\label{tablkernels}
\begin{tabular}{lll}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Kernel$(j)$  &Explicit form\\
 \hline
 IG (\cite{mhf})&
$K_{IG(\alpha,\beta)}(y)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}(1/y)^{\alpha+1}\exp(-\beta/y),$\\
   & where $\alpha=\frac{h+x}{h}+1~~~~and~~~~ \beta=\frac{x(h+x)}{h}$.   \\
 BP (\cite{EM})
&$K_{BP(\lambda,\mu)}(~y)=\frac{y^{\lambda-1}(1+y)^{-\lambda-\mu}}{\beta(\lambda,\mu)},$\\
& where $\lambda=\frac{x^{2}}{h}+x+1~~~~and~~~~ \mu=\frac{x}{h}+\frac{1}{x+h}+1$.   \\
  \hline
\end{tabular}
\end{table}



The expressions of the bias and variance for $\widehat{f}_{j}(x)$ are derived by \cite{mhf} and \cite{EM}.
The asymptotic bias when $h\rightarrow0$ is given by

\begin{equation}\label{biasf}
    {\rm bias}(\widehat{f}_{j}(x))=q_{j}(x,f)h+o(h),
\end{equation}
where the explicit form of $q_{j}(x,f)$ is given in table \ref{qj}.
\begin{table}[h!]
%\scriptsize
\caption{Explicit form of $q_{j}(x,f)$.}\label{qj}
\begin{tabular}{lll}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Kernel$(j)$  &Explicit form\\
 \hline
 IG (\cite{mhf})&
$q_{IG}(x,f)=\frac{1}{2}f''(x)x,$\\
   %& where $\alpha=\frac{h+x}{h}+1~~~~and~~~~ \beta=\frac{x(h+x)}{h}$.   \\
 BP (\cite{EM})
&$q_{BP}(x,f)=f'(x)+(1+x)\frac{f''(x)}{2}.$\\
%& where $\lambda=\frac{x^{2}}{h}+x+1~~~~and~~~~ \mu=\frac{x}{h}\frac{1}{x+h}+1$.   \\
  \hline
\end{tabular}
\end{table}
Similarly, when $n\rightarrow\infty$ and $h\rightarrow 0$ the asymptotic variance is
\begin{equation}\label{varf}
    {\rm Var}(\widehat{f}_{j}(x))=p_{j}(x,h)f(x)+o\left(\frac{1}{nh^{1/2}}\right).
\end{equation}
where the explicit form of $p_{j}(x,h)$ is given in table \ref{pj}.\\
\begin{table}[h!]
%\scriptsize
\caption{Explicit form of $p_{j}(x,h)$.}\label{pj}
\begin{tabular}{lll}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Kernel$(j)$  &Explicit form\\
 \hline
 IG (\cite{mhf})&
$p_{IG}(x,h)=\frac{1}{ 2\sqrt{2\pi} }n^{-1}h^{-1/2}x^{-1/2},$\\
   %& where $\alpha=\frac{h+x}{h}+1~~~~and~~~~ \beta=\frac{x(h+x)}{h}$.   \\
 BP (\cite{EM})
&$p_{BP}(x,h)=\frac{1}{ 2\sqrt{\pi} }n^{-1}h^{-1/2}(1+x)^{-1/2}.$\\
%& where $\lambda=\frac{x^{2}}{h}+x+1~~~~and~~~~ \mu=\frac{x}{h}\frac{1}{x+h}+1$.   \\
  \hline
\end{tabular}
\end{table}
The mean integrated squared error (${\rm MISE}$) is also given in \cite{mhf} and \cite{EM} and  is expressed as

\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  {\rm MISE}(\widehat{f}_{j}) &=& \int\limits_{0}^{\infty}{\rm bias}^{2}(\widehat{f}_{j}(x))dx+ \int\limits_{0}^{\infty}{\rm Var}(\widehat{f}_{j}(x))dx \nonumber\\
   &=&h^{2}\int q_{j}^2(x,f)dx \nonumber\\
     &+&\int p_{j}(x,h)f(x)dx+o\left(h^{2}+\frac{1}{nh^{\frac{1}{2}}}\right)\label{misegbs}.
\end{eqnarray}
\textbf{Remark}\\
Under sufficient smoothness of the true density, they are shown that the order of magnitude in $MISE$ of kernel density estimator (\ref{eqfh}) is $o\left(h^{2}+\frac{1}{nh^{\frac{1}{2}}}\right)$. For these raison, two classes of MBC density estimators will be investigated in the next section. The two classes of MBC density estimators by construction nonnegative, and establish a faster convergence rate of $o\left(h^{4}+\frac{1}{nh^{\frac{1}{2}}}\right)$.


\section{MBC for IG and BP density estimators}
In this section, we apply two classes of MBC techniques for IG and BP kernel density estimator. The two classes are originally proposed by \cite{ts80} and \cite{jln95} for symmetric kernel density estimator. Note that these MBC techniques have also extended recently by \cite{h10} and \cite{hs14} for kernel density estimation on the unit interval using beta and modified beta kernels and for density estimation using asymmetric kernels (gamma, modified gamma, inverse gaussian, reciprocal inverse gaussian, log-Normal and Birnbaum-Saunders kernels), respectively.

\subsection{Estimators}
Based on the same idea of \cite{ts80} and \cite{h10}, the MBC kernel density estimator using the IG and BP kernels, which we simply denote TS-j kernel density estimators, can be adapted as follows:
\begin{equation}\label{MBC1}
    \widehat{f}_{TS-j}(x)=\left\{\widehat{f}_{j,h}(x)\right\}^{\frac{1}{1-a}}\left\{\widehat{f}_{j,h/a}(x)\right\}^{-\frac{a}{1-a}},
\end{equation}
where $\widehat{f}_{j,h}$ and $\widehat{f}_{j,h/a}$ denote the IG or BP kernel density estimators given by (\ref{eqfh}) with bandwidths $h$ and $h/a$, respectively, with $a\in(0,1)$ is a constant that does not depend on the
target $x$; see, e.g., \cite{h10}.

The second class of MBC techniques for symmetric kernel density estimators is attributed to \cite{jln95} (see also \cite{h10} and \cite{hs14} for asymmetric kernel density estimators). The analogue of their estimators
using IG and BP kernels, which we denote JLN-j kernel density estimators is given by
\begin{equation}\label{MBC2}
\widehat{f}_{JLN-j}(x)=\widehat{f}_{j}(x)\left\{\frac{1}{n}\sum\limits_{i=1}^{n}\frac{K_{j(x,h)}(X_{i})}{\widehat{f}_{j}(X_{i})}\right\},
\end{equation}
where $K_{j(x,h)}$ is the IG or BP kernel.

\subsection{Asymptotic properties}
The following theorems present the asymptotic bias and variance of the TS-j and JLN-j kernel estimators. We assume that
\begin{enumerate}
  \item [A1.] $f$ has four continuous and bounded derivatives.
  \item [A2.] The sequence of bandwidths $h=h(n)$ satisfies $\lim_{n\rightarrow\infty}h=0$.% and  $\lim_{n\rightarrow\infty}nh^{7/2}=\infty$.
\end{enumerate}
Note that these assumptions have been discussed in \cite{h10} and \cite{hs14}.
\begin{thm}\label{thm1}
Let $\widehat{f}_{TS-j}$ be the TS-j kernel estimator defined by (\ref{MBC1}). For a given $x>0$ and under assumptions A1 and A2, then:

(i) the  bias of the TS-j kernel estimators admit the following expansion
\begin{equation*}
    {\rm bias}(\widehat{f}_{TS-j})=\frac{1}{a}\left[\frac{1}{2}\left\{ \frac{\psi^{2}_{1,j}(x)}{f(x)}-\psi_{2,j}(x)\right\}\right]h^{2}+o(h^{2}),
\end{equation*}
where $\psi_{1,j}(x)$ and $\psi_{2,j}(x)$ are given in Table \ref{psi1} and \ref{psi2} respectively.\\
\begin{table}[h!]
%\scriptsize
\caption{Explicit form of $\psi_{1,j}(x)$.}\label{psi1}
\begin{tabular}{lll}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Kernel$(j)$  &Explicit form\\
 \hline
 IG &
$\frac{1}{2}xf''(x),$\\
   %& where $\alpha=\frac{h+x}{h}+1~~~~and~~~~ \beta=\frac{x(h+x)}{h}$.   \\
 BP
&$f'(x)+\frac{(x+1)}{2}f''(x)+x^{2}f'''(x)-\frac{x^{4}}{12}f''''(x).$\\
%& where $\lambda=\frac{x^{2}}{h}+x+1~~~~and~~~~ \mu=\frac{x}{h}\frac{1}{x+h}+1$.   \\
  \hline
\end{tabular}
\end{table}
\begin{table}[h!]
%\scriptsize
\caption{Explicit form of $\psi_{2,j}(x)$.}\label{psi2}
\begin{tabular}{lll}
  \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
Kernel$(j)$  &Explicit form\\
 \hline
 IG &
$\frac{1}{4}x^{2}f''''(x)-\frac{4}{3}xf'''(x),$\\
   %& where $\alpha=\frac{h+x}{h}+1~~~~and~~~~ \beta=\frac{x(h+x)}{h}$.   \\
 BP
&$-\frac{x^{3}}{8}f''''(x).$\\
%& where $\lambda=\frac{x^{2}}{h}+x+1~~~~and~~~~ \mu=\frac{x}{h}\frac{1}{x+h}+1$.   \\
  \hline
\end{tabular}
\end{table}
(ii) the  variance of the TS-j kernel estimators is given by
\begin{equation*}
{\rm Var}(\widehat{f}_{TS-j})=\gamma(a)p_{j}(x,h)f(x)+o\left(\frac{1}{nh^{1/2}}\right).
\end{equation*}
where $\gamma(a)=\frac{(1+a^{5/2})(1+a)^{1/2}-2\sqrt{2}a^{3/2}}{(1+a)^{1/2}(1-a)^{2}}$,
%$c=1/\int_{-\infty}^{\infty}g(y^{2})dy$ and $c_{g^{2}}=1/\int_{-\infty}^{\infty}g^{2}(y^{2})dy$.
\end{thm}


\begin{thm}\label{thm2}
Let $\widehat{f}_{JLN-j}$ be the JLN-j kernel estimator defined by (\ref{MBC2}). For a given $x>0$ and under assumptions A1 and A2, then:

(i) the  bias of the JLN-j kernel estimator is given by
\begin{equation*}
    {\rm bias}(\widehat{f}_{JLN-j})=-f(x)\psi_{1,j}(x,g)h^{2}+o(h^{2}),
\end{equation*}
where $g(x)=\psi_{1}(x,j)/f(x)$ and $\psi_{1,j}(x)$ is the same as given in Theorem 1.

(ii) the  variance of the JLN-j kernel estimators has asymptotic form
\begin{equation*}
{\rm Var}(\widehat{f}_{JLN-j})=p_{j}(x,h)f(x)+o\left(\frac{1}{nh^{1/2}}\right).
\end{equation*}
%where $c=1/\int_{-\infty}^{\infty}g(y^{2})dy$ and $c_{g^{2}}=1/\int_{-\infty}^{\infty}g^{2}(y^{2})dy$.
\end{thm}

\subsection{Global propriety}
The criterion to use for the global propriety  is the mean integrated squared error (${\rm MISE}$) defined as
\begin{equation}\label{miseMBC}
{\rm MISE}(\widehat{f}_{MBC-j})=\int\limits_{0}^{\infty}{\rm bias}^{2}(\widehat{f}_{MBC-j}(x))dx+ \int\limits_{0}^{\infty}{\rm Var}(\widehat{f}_{MBC-j}(x))dx,
\end{equation}
where $\widehat{f}_{MBC-j}$ is the TS-j or the JLN-j kernel density estimators.

The mean integrated squared error (${\rm MISE}$) of the TS-j kernel estimators given in (\ref{MBC1}) is expressed as
\begin{equation}\label{miseMBC1}
% \nonumber to remove numbering (before each equation)
  {\rm MISE}(\widehat{f}_{TS-j})=\frac{h^{4}}{4a^{2}}\int\limits_{0}^{\infty}\left\{ \frac{\psi^{2}_{1,j}(x)}{f(x)}-\psi_{2,j}(x)\right\}^{2}dx+\gamma(a)\int\limits_{0}^{\infty} p_{j}(x,h)f(x)dx+o\left(\frac{1}{nh^{1/2}}+h^{4}\right).
\end{equation}

The optimal bandwidth minimizing the corresponding ${\rm MISE}$ (\ref{miseMBC1}) is such that
\begin{equation}\label{hTSIG}
h_{TS-IG}^{opt}=\left\{\frac{a^{2}\gamma(a)\int\limits_{0}^{\infty} \frac{f(x)}{x^{1/2}2\sqrt{2\pi}}dx}{8\int\limits_{0}^{\infty}\left\{ \frac{\psi^{2}_{1,IG}(x)}{f(x)}-\psi_{2,IG}(x)\right\}^{2}dx}\right\}^{2/9}n^{-2/9}. \end{equation}

\begin{equation}\label{hTSBP}
h_{TS-BP}^{opt}=\left\{\frac{a^{2}\gamma(a)\int\limits_{0}^{\infty} \frac{f(x)}{(1+x)^{1/2}2\sqrt{\pi}}dx}{8\int\limits_{0}^{\infty}\left\{ \frac{\psi^{2}_{1,BP}(x)}{f(x)}-\psi_{2,BP}(x)\right\}^{2}dx}\right\}^{2/9}n^{-2/9}. \end{equation}

Similarly, the  ${\rm MISE}$ of the JLN-j kernel estimators given in (\ref{MBC2}) is given by

\begin{equation}\label{miseMBC2}
% \nonumber to remove numbering (before each equation)
  {\rm MISE}(\widehat{f}_{JLN-j})=h^{4}\int\limits_{0}^{\infty}f^{2}(x)\psi^{2}_{1,j}(x,g)dx+\int\limits_{0}^{\infty}p_{j}(x,h) f(x)dx+o\left(\frac{1}{nh^{1/2}}+h^{4}\right).
\end{equation}
 By minimizing (\ref{miseMBC2}) in the bandwidth $h$, we obtain the optimal value
\begin{equation}\label{hJLNIG}
h_{JLN-IG}^{opt}=\left\{\frac{\int\limits_{0}^{\infty} \frac{f(x)}{x^{1/2}2\sqrt{2\pi}}dx}{8\int\limits_{0}^{\infty}f^{2}(x)\left[\frac{1}{2}xg''(x)\right]^{2}dx} \right\}^{2/9}n^{-2/9}.
\end{equation}
\begin{equation}\label{hJLNBP}
h_{JLN-BP}^{opt}=\left\{ \frac{\int\limits_{0}^{\infty} \frac{f(x)}{(1+x)^{1/2}2\sqrt{\pi}}dx}{8\int\limits_{0}^{\infty}f^{2}(x)\left[g'(x)+\frac{(x+1)}{2}g''(x)+x^{2}g'''(x)-\frac{x^{4}}{12}g''''(x)\right]^{2}dx} \right\}^{2/9}n^{-2/9}.
\end{equation}
Note that the bandwidths (\ref{hTSIG}), (\ref{hTSBP}), (\ref{hJLNIG}) and (\ref{hJLNBP}) can not be employed in practice. Then, next subsection presents a practical procedure to bandwidth selection.
\subsection{Choice of bandwidth for MBC-j kernel estimators}
The optimal bandwidths given by (\ref{hTSIG}), (\ref{hTSBP}), (\ref{hJLNIG}) and (\ref{hJLNBP}) depend on the unknown density $f$ and on
its derivatives $f'$, $f''$, $f'''$ and $f''''$, for these raison, they can not be exploited in practice. In this paper, we adapt the least square cross validation (LSCV) method.
 In the case of the LSCV technique, for a given  estimator $\widehat{f}_{MBC-j}$, which denote the TS-j or JLN-j kernel estimators, the optimal
bandwidth $h_{ucv}$ of $h$ is obtained by
 \begin{equation*}
h_{LSCV}=\arg \min\limits_{h}{\rm LSCV}(h),
\end{equation*}
where
\begin{eqnarray*}
{\rm LSCV}(h) &=&\int \widehat{f}^{2}_{MBC-j}(x)dx-\frac{2}{n}\sum\limits_{i=1}^{n}\widehat{f}^{(-i)}_{MBC-j}(X_{i}),
\end{eqnarray*}
where $\widehat{f}^{(-i)}_{MBC-j}(y)$ is the leave-one-out estimator computed as $\widehat{f}_{MBC-j}(y)$ by excluding the observation $X_{i}$. For the TS-j kernel estimators, the LSCV function is given

\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  {\rm LSCV}_{TS-j}(h) &=& \int\limits_{0}^{\infty}\left\{\widehat{f}_{j,h}(x)\right\}^{\frac{2}{1-a}}\left\{\widehat{f}_{j,h/a}(x)\right\}^{-\frac{2a}{1-a}}dx-\frac{2}{n(n-1)} \nonumber\\
  ~~~&\times&
\sum\limits_{i}\left[\left\{\sum\limits_{k\neq i}K_{j(X_{i},h)}(X_{k})\right\}^{\frac{1}{1-a}}\left\{\sum\limits_{k\neq i}K_{j(X_{i},h/a)}(X_{k})\right\}^{-\frac{a}{1-a}}\right].
\end{eqnarray}
In the case of JLN-j kernel estimators, the expression of LSCV is

\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
  {\rm LSCV}_{JLN-j}(h) &=& \frac{1}{n^{2}}\int\limits_{0}^{\infty}\widehat{f}_{j}(x)^{2}\left\{\sum\limits_{i=1}^{n}\frac{K_{j(x,h)}(X_{i})}{\widehat{f}_{j}(X_{i})}\right\}^{2}dx-\frac{2}{n(n-1)} \nonumber\\
  ~~~&\times&
\sum\limits_{i}\sum\limits_{k\neq i}K_{j(X_{i},h)}(X_{k})\frac{\widehat{f}_{j}(X_{i})}{\widehat{f}_{j}(X_{k})}.
\end{eqnarray}
%\section{Illustrations from simulated data}
%This section investigates the performances of the MBC approaches against the classical estimator considered in the previous section. We simulate data from pareto and weibull distributions. Some of these distributions have been also studied in several papers, see \cite{scai} for exemple.\\
%% (see Figure ??) described in A-E given below.
%%$(A)$ Beta prime density with parameters $(\alpha,\lambda)=(2,1)$ :
%%\begin{equation}\label{simData}
%%f(x)=\frac{x^{(\alpha-1)}}{(1+x)^{(\alpha+\lambda)}\beta(\alpha,\lambda)},\nonumber
%%\end{equation}
%$(A)$ Pareto density with parameters $(\lambda,\rho)=(1,2)$ :
%\begin{equation}\label{simData}
%f(x)=\frac{\rho \lambda^{\rho}}{(x+\lambda)^{\rho+1}},\nonumber
%\end{equation}
%$(B)$ Weibull density with parameters $(\alpha,\lambda)=(3,1)$:
%\begin{equation}\label{simData}
%f(x)= \frac{\alpha x^{(\alpha-1)}\exp(-x/\lambda)^{\alpha}}{\lambda^{\alpha}},\nonumber
%\end{equation}
%$(C)$ Gamma density with parameters $(\mu,\lambda)=(1,3)$:
%\begin{equation}\label{simData}
%f(x)=\frac{x^{(\mu-1)}\exp(-x/\lambda)}{\lambda^{\mu}\Gamma(\mu)}, \nonumber
%\end{equation}
%%$(D)$ Mixture of Gamma density with parameters $(\mu_{1},\lambda_{1})=(1,2.5)$ and $(\mu_{2},\lambda_{2})=(1,10)$:
%%\begin{equation}\label{simData}
%%f(x)=0.5\frac{x^{(\mu_{1}-1)}\exp(-x/\lambda_{1})}{\lambda_{1}^{\mu_{1}}\Gamma(\mu_{1})}+0.5\frac{x^{(\mu_{2}-1)}\exp(-x/\lambda_{2})}{\lambda_{2}^{\mu_{2}}\Gamma(\mu_{2})}, \nonumber
%%\end{equation}
%%$(E)$ Mixture Weibull density with parameters $(\mu_{1},\lambda_{1})=(1,1)$ and $(\mu_{2},\lambda_{2})=(1,3)$:
%%\begin{equation}\label{simData}
%%f(x)= 0.5\frac{\mu_{1} x^{(\mu_{1}-1)}\exp(-x/\lambda_{1})^{\mu_{1}}}{\lambda_{1}^{\mu_{1}}}+0.5\frac{\mu_{2} x^{(\mu_{2}-1)}\exp(-x/\lambda_{2})^{\mu_{2}}}{\lambda_{2}^{\mu_{2}}}.\nonumber
%%\end{equation}
%For each density, 500 replications of sample size $n=50, 200$ and $500$ are generated using R statistical software. The parameter $a$ is fixed as $a=0.5$ in the use of the TS-IG and TS-BP kernel estimator where these optimal value can be obtained in the sense of the mean integrated squared error (MISE); see, e.g., \cite{h10}. For the choice of the bandwidth, we use the LSCV method proposed in the previous section. We examine the performances of the estimators via integrated squared error (${\rm ISE}$) and integrated squared bias
% (${\rm ISB}$) given respectively by
% \begin{equation*}\label{ise}
%    {\rm ISE}:=\int \left[\widehat{f}(x)-f(x)\right]^{2} dx
%\end{equation*}
%and
%\begin{equation*}\label{isb}
%    {\rm ISB}:=\int \left[\mathbb{E}\{\widehat{f}(x)\}-f(x)\right]^{2} dx,
%\end{equation*}
%the Integrated Variance (${\rm IV}$) is also computed
%\begin{equation*}\label{isb}
%    {\rm IV}:=\int \left[\mathbb{V}ar\{\widehat{f}(x)\}\right] dx,
%\end{equation*}
%where $\widehat{f}$ is the MBC-IG or MBC-BP kernel estimators.
%\\
%Table \ref{tabise} presents the optimal average of ISE for three estimators the classical "C" and the two MBC estimators (TS and JLN) with respect to the five densities $(A)$, $(B)$ and $(C)$ according to different sample sizes $n \in \{50, 200, 500\}$. Each result is obtained from 500 replications.
%
%Table \ref{tabisb} shows the optimal average of ISB for estimations based on 500 replications.
%
%Table \ref{tabiv} reports the IV values for the classical and MBC estimators. For each model, the number of replications is again $500$ for sample sizes $n\in \{50, 200, 500\}$.
%\begin{table}[h!]
%\footnotesize
%\caption{Empirical ${\rm ISE}$ values based on 500 replications for the previous considered pmf's.}\label{tabise}
%\begin{tabular}{lllllll}
%    \hline
%    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
%    $f$& Kernel & Estimator & $n=50$ & $n=200$ & $n=500$ \\
%    \hline
%    \hline
%    $ (A)$  &  & $\widehat{f}_{C}$ &0.4415373  &0.3076841 & 0.1538420\\
%    & IG& $\widetilde{f}_{TS}$ &0.4281741 & \textbf{0.2928473} & \textbf{0.1464236}\\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.4249911}  &0.2948032  &0.1474016 \\
%    \\
%     &  & $\widehat{f}_{C}$ &0.2607013 &0.1303506  &0.0592502 \\
%    & BP& $\widetilde{f}_{TS}$ &\textbf{0.2521402}  &\textbf{0.1260701} &\textbf{0.0573046}\\
%    &  & $\widetilde{f}_{JLN}$ &0.2577394 &0.1288697 & 0.0585771\\
%    \\
%    $ (B)$&  & $\widehat{f}_{C}$ &0.0074509   & 4.105734e-03 & 3.016166e-03 \\
%    & IG& $\widetilde{f}_{TS}$ & \textbf{0.0068014}  & \textbf{3.759457e-03} &\textbf{2.896641e-03}  \\
%    &  & $\widetilde{f}_{JLN}$ & 0.0070209   & 3.998476e-03 &3.211156e-03  \\
%    \\
%     &  & $\widehat{f}_{C}$ & 0.0052592 & 0.0013570 &1.191194e-03 \\
%    & BP& $\widetilde{f}_{TS}$ & 0.0055457 & \textbf{0.0011732} &0.0010591 \\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.0033717}  &0.0011959 &\textbf{9.503874e-04} \\
%    \\
%    $ (C)$&  & $\widehat{f}_{C}$ & 0.1252512 &0.070118 &0.0545970\\
%    &IG & $\widetilde{f}_{TS}$  & \textbf{0.1021318} &\textbf{0.0546102} &\textbf{0.0443532}  \\
%    &  & $\widetilde{f}_{JLN}$ & 0.1241508 &0.0757947 &0.0641895 \\
%    \\
%     &  & $\widehat{f}_{C}$ &0.1171658 &0.0439182  &0.0222926 \\
%    & BP& $\widetilde{f}_{TS}$ &0.1059155  & 0.0539755 & 0.0275477\\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.0980294}  &\textbf{0.0340546} &\textbf{0.0213627} \\
%    %\\
%%    $(D)$  &  & $\widehat{f}_{C}$   &   & & \\
%%    &  IG &$\widetilde{f}_{TS}$ & &  & \\
%%    &  & $\widetilde{f}_{JLN}$ & &    & \\
%%    \\
%%     &  & $\widehat{f}_{C}$ & &  & \\
%%    & BP& $\widetilde{f}_{TS}$ &  &  & \\
%%    &  & $\widetilde{f}_{JLN}$ &  & & \\
%    \hline
%\end{tabular}
%\end{table}
%\begin{table}[h!]
%\footnotesize
%\caption{Empirical ${\rm ISB}$ values based on 500 replications for the previous considered pmf's.}\label{tabisb}
%\begin{tabular}{lllllll}
%    \hline
%    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
%    $f$& Kernel & Estimator & $n=50$  & $n=200$ & $n=500$ \\
%    \hline
%    \hline
%    $ (A)$  &  & $\widehat{f}_{C}$ & 0.4236996 &0.3037370 &0.1518685 \\
%    & IG& $\widetilde{f}_{TS}$ &\textbf{0.4027794} & \textbf{0.2870917}  & \textbf{0.1435458}\\
%    &  & $\widetilde{f}_{JLN}$ &0.4077250 & 0.2922072&0.1461036 \\
%    \\
%     &  & $\widehat{f}_{C}$ &0.2327249 & 0.1163624& 0.0528920\\
%    & BP& $\widetilde{f}_{TS}$ &\textbf{0.2086085}  & \textbf{0.1043042} &\textbf{0.0474110} \\
%    &  & $\widetilde{f}_{JLN}$ & 0.2365250 &0.1182625 &0.0537557 \\
%    \\
%    $ (B)$&  & $\widehat{f}_{C}$ & 0.0068435  &4.085574e-03  & 2.922041e-03\\
%    & IG& $\widetilde{f}_{TS}$ & \textbf{0.0059348}   &\textbf{3.736633e-03} &\textbf{2.836421e-03}  \\
%    &  & $\widetilde{f}_{JLN}$ &  0.0063723 &3.987261e-03 & 3.140386e-03 \\
%    \\
%     &  & $\widehat{f}_{C}$ &0.0045062 & 0.0012506 & 1.095889e-03\\
%    & BP& $\widetilde{f}_{TS}$ &0.0041756  &\textbf{0.0009716}  &0.0009252 \\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.0029694}  & 0.0010877&\textbf{8.738666e-04} \\
%    \\
%    $ (C)$&  & $\widehat{f}_{C}$ & 0.1087270 &0.0633246 &0.0521351 \\
%    &IG & $\widetilde{f}_{TS}$  & \textbf{0.0762176} &\textbf{0.0460224} &\textbf{0.0422681}  \\
%    &  & $\widetilde{f}_{JLN}$ & 0.1120883  &0.0709170 &0.0630050 \\
%    \\
%     &  & $\widehat{f}_{C}$ &0.0962075 & 0.0339195 & 0.0168980\\
%    & BP& $\widetilde{f}_{TS}$ & 0.0925232 &0.0462616  & 0.0223108 \\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.0801466}  &\textbf{0.0283054} & \textbf{0.0167536}\\
%    %\\
%%    $(D)$  &  & $\widehat{f}_{C}$   &    & & \\
%%    &  IG &$\widetilde{f}_{TS}$ &  &  & \\
%%    &  & $\widetilde{f}_{JLN}$ & &    & \\
%%    \\
%%     &  & $\widehat{f}_{C}$ & &  & \\
%%    & BP& $\widetilde{f}_{TS}$ &  & & \\
%%    &  & $\widetilde{f}_{JLN}$ &  & & \\
%    \hline
%\end{tabular}
%\end{table}
%\begin{table}[h!]
%\footnotesize
%\caption{Empirical ${\rm IV}$ values based on 500 replications for the previous considered pmf's.}\label{tabiv}
%\begin{tabular}{lllllll}
%    \hline
%    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
%    $f$& Kernel & Estimator & $n=50$ & $n=200$ & $n=500$ \\
%    \hline
%    \hline
%    $ (A)$  &  & $\widehat{f}_{C}$ &0.0178377 &0.0039471  & 0.0019735\\
%    & IG& $\widetilde{f}_{TS}$ &0.0253947 & 0.0057555  &0.0028777\\
%    &  & $\widetilde{f}_{JLN}$ & \textbf{0.0172661} & \textbf{0.0025959}&\textbf{0.0012965} \\
%    \\
%     &  & $\widehat{f}_{C}$ &0.0279763 &0.0139881 & 0.0063582\\
%    & BP& $\widetilde{f}_{TS}$ &0.0435317 & 0.0217658 &0.0098935 \\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.0212143}  &\textbf{0.0106071} & \textbf{0.0053035}\\
%    \\
%    $ (B)$&  & $\widehat{f}_{C}$ & 0.0006485  & 2.016082e-05 & 7.076967e-05 \\
%    & IG& $\widetilde{f}_{TS}$ & 0.0008666 & 2.282389e-05  &9.412477e-05  \\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.0006074}  &\textbf{1.121584e-05}&\textbf{6.022064e-05}  \\
%    \\
%     &  & $\widehat{f}_{C}$ &0.0007530 &0.0001082  &9.530546e-05 \\
%    & BP& $\widetilde{f}_{TS}$ &0.0013701  &0.0002016 &0.0001339 \\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.0004022} &\textbf{0.0001063} & \textbf{7.652086e-05}\\
%    \\
%    $ (C)$&  & $\widehat{f}_{C}$ &0.0165242  &0.0067936 &0.0024625\\
%    &IG & $\widetilde{f}_{TS}$  & 0.0259142 &0.0085878 &0.0020851  \\
%    &  & $\widetilde{f}_{JLN}$ &\textbf{0.0120625} & \textbf{0.0048777}&\textbf{0.0011844} \\
%    \\
%     &  & $\widehat{f}_{C}$  &0.0209582 &0.0099987 &0.0055389 \\
%    & BP& $\widetilde{f}_{TS}$  &\textbf{0.0133923} &0.0077139&\textbf{0.0044169 }\\
%    &  & $\widetilde{f}_{JLN}$  &0.0178828 &\textbf{0.0057491} &0.0044646 \\
%    %\\
%%    $(D)$  &  & $\widehat{f}_{C}$     &  & & \\
%%    &  IG &$\widetilde{f}_{TS}$ &   & & \\
%%    &  & $\widetilde{f}_{JLN}$ & & & \\
%%    \\
%%     &  & $\widehat{f}_{C}$ & &  & \\
%%    & BP& $\widetilde{f}_{TS}$ & & &  & \\
%%    &  & $\widetilde{f}_{JLN}$  & & & \\
%    \hline
%\end{tabular}
%\end{table}
%\newpage
% From Tables \ref{tabise}, \ref{tabisb} and \ref{tabiv} , we can observe immediately that:
%\begin{enumerate}
%   \item the means of ${\rm ISE}$ and ${\rm ISB}$ based on 500 replications decrease as sample size $n$ increases for the all estimators;
%%   \item the mean of the UCV bandwidths decrease as sample size $n$ increases as for the ${\rm ISE}$ and ${\rm ISB}$;
% \item the TS-IG, TS-BP, JLN-IG and JLN-BP kernel estimators outperform the standard IG and BP kernel estimators in the senses of ${\rm ISE}$ and ${\rm ISB}$ for all sample sizes;
%  \item the JLN kernel estimator based on BP kernel in general  performs better than the others competitors in the term of ${\rm ISE}$;
%  \item the JLN kernel estimator based on BP kernel works better than the others competitors in the sense of
% ${\rm ISB}$.
% \end{enumerate}
%
%\section{Application to air pollution data}
%This section illustrates the performances of MBC-IG and MBC-BP kernel density estimators for ${\rm O}_{3}$ data. These data concern the study of the daily tropospheric ozone concentrations (in ppb=ppm$\times$1000) observed in New York during May-September, 1973, provided by the New York State Department of Conservation; see, e.g., \cite{nada} and \cite{lvbs}. These data have been also analyzed recently by \cite{slzm}. Note that in our study, we have devised these data by 10. Table \ref{tabdesc} provide the summary statistics these real data. We can observe that the considered data are nonnegative. Then, the IG and BP kernels are appropriate for analyzing them.
%\begin{table}[htbp]
%\begin{center}
%\caption{Descriptive of the analyzed  ${\rm O}_{3}$ data.}\label{tabdesc}
%\begin{tabular}{ccccccc}
%\hline
%Data set & $n$ & Max & Min & Median & Mean & SD \\
%\hline
%${\rm O}_{3}$  &  116 &  16.80 & 0.10 & 3.15 & 4.21 & 3.29 \\
%\hline
%\end{tabular}
%\end{center}
%\end{table}
%
%
% %\begin{figure}[H]
%% \centering
%% \includegraphics[width=11cm,height=11cm]{data1}
%% \caption{Histogram of ${\rm O}_{3}$ data.}\label{o3}
%% \end{figure}
%
% Now, we apply the TS-IG, TS-BP, JLN-IG and JLN-BP kernel estimators to estimate the density for ${\rm O}_{3}$ data. For TS-IG and TS-BP kernel estimator, the value of $a$ is fixed at 0.5. For comparison, the standard IG and BP kernel estimators are also employed for estimating the density of  considered real data. The corresponding bandwidths for the estimators
%are chosen using the LSCV procedure. The density estimates are plotted in Figure \ref{estimationdata}. We can see that the smoothing of the MBC-IG and MBC-BP kernel estimators is diffident, in comparison to corresponding standard IG and BP kernel estimators, which tend to under smooth the density of these data.
%\begin{figure}[h!]
%\begin{tabular}{ll}
%%\includegraphics[width=5cm,height=5cm]{at.eps}&
%\includegraphics[width=8cm,height=7cm]{Real-Data-InverseGamma.eps}&
%\includegraphics[width=8cm,height=7cm]{Real-Data-BP.eps}\\
%\includegraphics[width=8cm,height=7cm]{HistData.eps}&
%\includegraphics[width=8cm,height=7cm]{Real-Data2-InverseGamma.eps}
%%\includegraphics[width=8cm,height=7cm]{Real-Data2-BP.eps}\\
%
%%\includegraphics[width=8cm,height=7cm]{geometric_n=500.eps}\\
%%\includegraphics[width=8cm,height=7cm]{melangepoisson_n=50.eps}&
%%\includegraphics[width=8cm,height=7cm]{melangepoisson_n=500.eps}
%\end{tabular}
%\caption{The estimation of O3 data with n = 116 using the classical IG kernel density estimator, the IG-TS estimator, the IG-JLN kernel density estimator (left) and using the classical BP kernel density estimator, the BP-TS estimator, the BP-JLN kernel density estimator (right).}\label{estimationdata}
%\end{figure}
\section{conclusions}
This paper has extended the application of the multiplicative bias correction (MBC) approaches for Inverse Gamma (IG) and Beta Prime (BP) kernels in order to estimate densities with nonnegative data. As in several papers, \cite{h10} and \cite{hs14}, we have demonstrated that these two approaches of MBC improve the order of magnitude in bias from $O(h)$ to $O(h^{2})$. The performances of the MBC-IG and MBC-BP kernel estimators (TS-IG, JLN-IG, TS-BP and JLN-BP kernel estimators) with the least square cross validation (LSCV) bandwidth selectors are investigated.
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\end{document}
