\def\papertitle{Adaptive gamma-BSPE kernel density estimatimation for nonnegative heavy tailed data}
\def\paperauthorA{ZIANE Yasmina}
\def\paperauthorB{ZOUGAB Nabil}
\def\paperauthorC{ADJABI Smail}
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\title{\papertitle}
\affiliation{\paperauthorA, \paperauthorB, \paperauthorC}	% This command replaces \name{The DAFx Crew}
{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
}

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\begin{abstract}
In this work, we consider the nonparametric estimation of the probability density function for nonnegative heavy-tailed (HT) data.
 The objective is first to propose a new estimator that will combine two regions of observations (high and low density), while associating
  to the high density region a gamma kernel and to the low density region a BS-PE kernel. Then, to compare the proposed estimator with the
  classical estimator in order to evaluate the performance of the new estimator. The choice of bandwidth is investigated by adopting the
  popular cross-validation technique and two variants of bayesian approach. Finally, the performances of the proposed estimator and the classical
   estimator are illustrated by a simulation study and real data.
\end{abstract}
\hspace{-0.5cm}\textbf{Keywords}:
Bayesian bandwidth selector, \textrm{BS-PE} kernel, Cross validation, Gamma kernel, Heavy tailed data, Kernel density estimation,
 MCMC method.

\section{Introduction}
In this work, we are interested in the estimation of the heavy tailed data density with non-negative support \cite{ZAZ2015} and \cite{ZZA2018}. This data type requires special methods because of their specific characteristics which are slow decay to zero and the rare observations in the tail.
As the parametric methods do not meet the characteristics of this data type, the nonparametric kernel method is proposed.
The efficiency of the latter depends on the choice of its two parameters, the kernel $K$ and the smoothing parameter $h$.
The most used kernels in the literature are the symmetric kernels such as the Gaussian kernel and the Epanechnicov kernel for unbounded
support densities. However, when we want to estimate densities with unbounded support, the classical kernel estimator becomes non
consistent, because of edge effects. This problem is due to the use of symmetric kernels which assign a weight outside the support when
the smoothing is taken into account near the edge. To address this problem, several authors have proposed a new family of the asymmetric kernel.
See \cite{chen2000} (gamma and modified gamma kernels), \cite{scai2004} (inverse and reciprocal inverse Gaussian kernels), \cite{jin2003} (lognormal and Birnbaum-Saunders (BS) kernels)
and recently \cite{marchant2013} proposed Generalized Birnbaum-Saunders (GBS) kernel for estimating densities with nonnegative
support, that includes BS-power-exponential (BS-PE) and BS-Student (BS-t) kernels, it is proposed for analyzing nonnegative Heavy Tailed (HT) data.\\
The performance of the associated kernel density estimator depends crucially on the smoothing parameter which controls the smoothing quality of the
estimator. Classical methods have been proposed for the smoothing parameter choice, the family of cross-validations, they are interesting in
practice because they are guided only by the observations. However, the drawback of these methods is that they tend to provide under or
oversmoothed estimators when the data are small or medium size or when we want to estimate complex functions. So, to deal with this problem, the Bayesian
approach has been proposed.\\
In this work, we based on the idea of \cite{ZZA2021}, where they proposed a subdivision of the HT dataset into two subsets (two regions) with low
and high density (Low Density Region (LDR) and High Density Region (HDR)), and associated to each region a smoothing parameter ($h_{LDR}$ and $h_{HDR}$).
We propose an estimator composed of two different kernels gamma and BSPE, where the gamma kernel is associated with the high density region and the BSPE
kernel associated with the high density region (see also \cite{markovich2016}). The new Gamma-BSPE kernel density has two smoothing parameters (bandwidths) that will be selected by the
adaptive Bayesian approach. A comparative study is conducted with the work of \cite{ZZA2021}, where they considered a single BSPE kernel for both regions.\\
The paper is structured as follows. Section 2, presents the classical BSPE kernel estimator. In section 3, we introduce the new gamma-BSPE kernel estimator.
In section 4, we give the procedure proposed for derived the adaptive bandwidths. Simulation studies and application of real data are presented
in Section 5 and 6. Section 7 concludes the paper.
\section{The classical BS-PE Kernel estimator}
Given a random sample $X_1,\ldots , X_n$, the {\rm BS-PE} kernel estimator of an unknown pdf $f$ with nonnegative support is given by
\begin{equation}\label{BS-PEdensity}
  \begin{array}{ccl}
     \hat{f}_{BS-PE,h}(x) & = & \frac{1}{n}\sum\limits_{i=1}^{n}K_{x,h}(X_{i}) \\
      & = & \frac{\nu}{n2^{\frac{1}{2\nu}}\Gamma(\frac{1}{2\nu})\sqrt{4h}}\sum\limits_{i=1}^{n} \left(\frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right)\exp\left(\frac{-1}{2h^{\nu}}\left(\frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right)^{\nu}\right),
   \end{array}
\end{equation}
where $x>0$ is the point where the density is estimated, $h>0$ is a smoothing parameter and $\nu >0$ is a fixed parameter.\\
The expression of the bias and variance for $\hat{f}_{BS-PE}(x)$ are derived by Marchant et al. (2013). The asymptotic bias when
$h\rightarrow 0$ is given by
\begin{equation*}
  Bias(\hat{f}_{BS-PE}(x))=\frac{hu_{1}(g)}{2}\left(xf'(x)+x^2f''(x)\right)+o(h),
\end{equation*}
\begin{equation*}
  Var(\hat{f}_{BS-PE}(x))=\frac{c^2}{c_{g^2}nh^{1/2}x}f(x)+o\left(\frac{1}{nh^{1/2}}\right)
\end{equation*}
where,
$u_1(g)=\frac{2^{\frac{1}{\nu}}\Gamma(\frac{3}{2\nu})}{\Gamma(\frac{1}{2\nu})}$, $c^2=\frac{\nu}{2^{\frac{1}{2\nu}}\Gamma(\frac{1}{2\nu})}$ and $c_{g^2}=\frac{\nu}{\Gamma(\frac{1}{2\nu})}$


%\begin{center}
%\begin{tabular}{|c|c|c|c|}
%  \hline
%  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
%  BS-PE kernel &$u_1(g)$ & $c^2$ & $c_{g^2}$ \\
%  \hline
%  &$\frac{2^{\frac{1}{\nu}}\Gamma(\frac{3}{2\nu})}{\Gamma(\frac{1}{2\nu})}$ & $\frac{\nu}{2^{\frac{1}{2\nu}}\Gamma(\frac{1}{2\nu})}$ & $\frac{\nu}{\Gamma(\frac{1}{2\nu})}$ \\
%  \hline
%\end{tabular}
%\end{center}

\section{The gamma-BSPE Kernel estimator}
In this section, we present a new density estimator for heavy tailed data which is flexible on the domain near the zero boundary and
that estimate the heavy tail of the distribution, the latter is based on: dividing the observations into two regions, namely the low-density
 region ({\rm LDR}) and high-density region ({\rm HDR}), and assigning two different bandwidths to these two regions \cite{ZZA2021}.
  We also propose to combine two asymmetric gamma and BS-PE kernels (\cite{chen2000} and \cite{marchant2013}) as follows:  associate a gamma kernel for the
   high-density region ({\rm HDR}) (near bord) and BS-PE kernel for the low-density region ({\rm LDR}).\\

\subsection*{Gamma kernel}
The gamma kernel is nonnegative and possess good boundary properties for wide class of densities, it is given by:
\begin{equation}\label{GAMMA}
  K_{Gam(x,h)}(y)=\frac{y^{\frac{x}{h}}}{\Gamma(1+\frac{x}{h})h^{1+\frac{x}{h}}}\exp\left(-\frac{y}{h}\right)\mathbf{1}_{\{0\leq x< \infty\}}(y);
\end{equation}
where $\Gamma(y)=\int_{0}^{\infty}t^{y-1}\exp(-t)dt$ is the classical gamma function with $y>0$, and $\mathbf{1}_{\{0\leq x< \infty\}}$ denote the indicator function.\\
The classical gamma kernel estimator of an unknown pdf $f$ with nonnegative support is given by
\begin{equation}\label{fgamma}
  \hat{f}_{Gam,h}(x)=\frac{1}{n}\sum\limits_{i=1}^{n} \frac{X_i^{\frac{x}{h}}}{\Gamma(1+\frac{x}{h})h^{1+\frac{x}{h}}}\exp\left(-\frac{X_i}{h}\right)\mathbf{1}_{\{0\leq x< \infty\}}(X_i)
\end{equation}
\subsection*{gamma-BSPE kernel estimator}
%The BS-PE kernel introduced by Marchant et al. (2013) and Apply by several autors like ZIANE et al(2015), ZOUGAB et (2016),ZIANE et al (2018)... The BS-PE kernel with
%fixed parameter $\nu >0$ is given by
%\begin{equation*}
%  K_{(x,h)}(y)=\frac{\nu}{2^{\frac{1}{2\nu}}\Gamma(\frac{1}{2\nu})\sqrt{4h}}\left(\frac{1}{\sqrt{xh}}+\sqrt{\frac{x}{y^3}}\right)\exp\left(\frac{-1}{2h^\nu}\left(\frac{y}{x}+\frac{x}{y}-2\right)\right),\ y>0.
%\end{equation*}
%where $x>0$ is the target while $h >0$ is the bandwidth and (·) is the gamma function
After the subdivision of the data set into two subsets, we present the estimator associated with this subdivision by associating different kernels to the two
regions, gamma kernel for the HDR region and BSPE for the LDR region. The gamma-BSPE Kernel estimator is given by:
%\begin{equation}\label{fGamBSPE}
%  \hat{f}_{h^{(0)}, h^{(1)}}(x)=\frac{1}{n}\sum_{j=1}^{n}\left\{ I_{j} K_{Gam(x,h^{(1)})}(x_{j}) + (1-I_{j})K_{BS-PE(x,h^{(0)})}(x_{j})\right\}.
%\end{equation}
\begin{eqnarray}\label{fGamBSPE}
% \nonumber to remove numbering (before each equation)
  \hat{f}_{h^{(0)}, h^{(1)}}(x) &=& \frac{1}{n}\sum_{j=1}^{n}\left\{ I_{j} K_{x,h^{(1)}}(x_{j}) + (1-I_{j})K_{x,h^{(0)}}(x_{j})\right\},\nonumber \\
   &=& \frac{1}{n}\sum_{j=1}^{n}\left\{ I_{j} K_{Gam(x,h^{(1)})}(x_{j}) + (1-I_{j})K_{BS-PE(x,h^{(0)})}(x_{j})\right\}.
\end{eqnarray}


where
$$I_{j}=\left\{
             \begin{array}{ll}
               1, & if ~~~~~~x_{j} \in S_{(HDR)},~~ j=1, \ldots, n; \\
               0, & else.
             \end{array}
           \right.
$$\\
$S_{(HDR)}$: the observations of the high-density region ({\rm HDR}), $S_{(LDR)}$: the observations of the low-density region ({\rm LDR})\\
and $h^{(1)}$ denotes the bandwidth assigned to the observations of $S_{(HDR)}$, and $h^{(0)}$ the bandwidth assigned to the observations of $S_{(LDR)}$.
\section{Adaptive Bayesian bandwidth selection}
In this section, we derive the variable Bayesian bandwidths at each subset ($S_{(HDR)}$ and $S_{(LDR)}$)  (Bayesian adaptive approach) for
 Equation (\ref{fGamBSPE}) in the kernel density estimation context, with positive support using the gamma-BSPE kernels. We treat $h^{(1)}$ 
 and $h^{(0)}$ as random quantities with prior distributions $\pi_{1}(\cdot)$ and $\pi_{0}(\cdot)$. As proposed by\cite{ZAZ2015}, we
  assume that the variable bandwidths $h^{(1)}$ and $h^{(0)}$ has a prior distributions with parameters $\alpha$, $\beta$ and $\nu=2$ ;
   this prior is defined by
\begin{equation}\label{prior0}
     \pi(h^{(0)})=\frac{\nu}{\Gamma(\alpha)\beta^{\alpha}}\frac{1}{(h^{(0)})^{\alpha\nu+1}}\exp\left(\frac{-1}{\beta (h^{(0)})^{\nu}}\right),\ \ h^{(0)}>0
\end{equation}
and
\begin{equation}\label{prior1}
     \pi(h^{(1})=\frac{\nu}{\Gamma(\alpha)\beta^{\alpha}}\frac{1}{(h^{(1)})^{\alpha\nu+1}}\exp\left(\frac{-1}{\beta (h^{(1)})^{\nu}}\right),\ \ h^{(1)}>0
\end{equation}
The posterior of $h^{(1)}$ and $h^{(0)}$ for given $\{x_1, x_2, \ldots , x_n\}$ is
\begin{equation}\label{Posterior}
  \hat{\pi}(h^{(1)}, h^{(0)}|x_{1},x_{2},\ldots x_{n})\propto\left\{\prod_{i=1}^{n}\hat{f}_{h^{(0)}, h^{(1)}}(x_i)\right\}\pi(h^{(0)})\pi(h^{(1)})
\end{equation}
Under the squared error loss, the Bayes estimator of the smoothing parameters $h^{(1)}$ and $h^{(0)}$ is
the mean of the posterior density, given by:
\begin{equation}\label{hsquared}
  \left(\hat{h}^{(1)},\hat{h}^{(0)}\right)=\int\int (h^{(1)},h^{(0)})\hat{\pi}(h^{(1)},h^{(0)}|x_{1},x_{2},\ldots x_{n})dh^{(1)} dh^{(0)}.
\end{equation}

We cannot derive an analytical expression as the estimate of $\left(\hat{h}^{(1)},\hat{h}^{(0)}\right)$ from the formula (\ref{Posterior}) and (\ref{hsquared}). However, we propose using the Markov Chain
Monte Carlo method (MCMC) for the approximation. We use a randamwalk metropolis algorithm to sample $\left\{h^{(1)}, h^{(0)}\right\}$ and the sampling algorithm is
briefly described below :
\begin{description}
  \item[Step 01] Initialize $\textbf{h}_{(0)}$, where $\textbf{h}=\left(h^{(1)}, h^{(0)}\right)$.
  \item[Step 02] For $i \in \{1,\ldots, M\}$,
  \begin{description}
    \item[a)] Generate $\tilde{\textbf{h}}\thicksim$ truncate Normal $(\textbf{h}_{(i-1)},\sigma^{2})$.
    \item[b)] Calculate the acceptance probability
    $\alpha=min\{1, \frac{\pi(\tilde{\textbf{h}}/x)}{\pi(\textbf{h}_{(i-1)}/x)}\frac{truncate\ Normal(\textbf{h}_{(i-1)},\sigma^{2})}{truncate\ Normal (\tilde{\textbf{h}},\sigma^{2})}\}$.
    \begin{equation*}
      \textbf{h}_{(i)}=\left\{
                         \begin{array}{ll}
                           \tilde{\textbf{h}}, & \hbox{$\mu < \alpha$,\ $\mu\sim\mathrm{U}_{[0, 1]}$;} \\
                           \textbf{h}_{(i-1)}, & \hbox{else.}
                         \end{array}
                       \right.
    \end{equation*}

  \end{description}
  \item[Step 03] $i=i+1$ and go to step 2.
\end{description}
Reject $(\textbf{h}_{(0)},\textbf{h}_{(1)},\ldots,\textbf{h}_{(M_{0})})$ which represents burn-in period, and estimate $\textbf{h}$ by\\
\begin{equation*}
    \hat{\textbf{h}}= \frac{1}{M-M_{0}}\sum_{i=M_{0}+1}^{M} \textbf{h}_{(i)}.
\end{equation*}

%The popular convergence diagnostics are due to [7] and [19]. In this work,
%we use the convergence diagnostic of [20] that is the Batch Mean Standard Error
%(BMSE) and the Simulation Inefficiency Factor (SIF).

\section{Simulation study}
In this section, we examine and compare the performances of the adaptive badwidth approach for \big(gamma-BSPE kernel estimator and BSPE kernel estimator
proposed in \cite{ZZA2021}\big), with global bandwidth approach \big(bayesian global and classical UCV method \big), by using several nonnegative
heavy tailed distributions.\\
The optimal bandwidth selected by classical method {\rm UCV} was obtained by :
\begin{equation*}
  h_{UCV}=\arg \min\limits_{h} UCV(h),
\end{equation*}
where :
\begin{eqnarray*}
{\rm UCV}(h) &=&\int \hat{f}_{\{h^{(0)}, h^{(1)}\}}^{2}(x)dx-\frac{2}{n}%
\sum\limits_{i=1}^{n}\hat{f}_{\{h^{(0)}, h^{(1)},i\}}(X_{i})
%&=&\int \left(\frac{1}{n}\sum_{i=1}^{n}K_{x,h}(X_{i})\right)^{2}dx-\frac{2}{n(n-1)}%
%\sum\limits_{i=1}^{n}\sum\limits_{i\neq j}K_{X_{i},h}(X_{j})
\end{eqnarray*}%
and $\hat{f}_{\{h^{(0)}, h^{(1)},i\}}$ being computed as $\hat{f}_{\{h^{(0)}, h^{(1)}\}}$ by excluding $X_{i}$.\\
We consider the target densities labeled \textbf{D1}, \textbf{D2} and \textbf{D3}. Functional forms of these densities are given by table \ref{tab1}.\\
\begin{table}
\scriptsize
%\begin{center}
\caption{Distributions in simulation study.}\label{tab1}
\begin{tabular}{cccc}
  \hline
  & Distribution & Density & Parameters \\
  \hline\\
   \textbf{D1} & \emph{lognormal}($\mu,\sigma$) & $f_{1}(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(\frac{-1}{2\sigma^{2}}\left(\ln(x)-\mu\right)^2\right).$ & $(\mu,\sigma)=(1,1)$ \\\\
  \textbf{D2} & \emph{Burr}($k,r$) & $\frac{kx^{k-1}}{(1+rx^{k})^{r+1}}$ & $(k,r)=(3,1)$ \\\\
  \textbf{D3} & Mixture of $p$\emph{Gamma}$(\alpha_{1})$ & $ p \times \frac{x^{\alpha_{1}-1}\exp(-x)}{\Gamma(\alpha_{1})}+p \times \frac{x^{\alpha_{2}-1}\exp(-x)}{\Gamma(\alpha_{2})}$ & $(\alpha_{1},\alpha_{2}, p)=(2.5,10,0.5)$ \\
    & and $p$\emph{Gamma}$(\alpha_{2})$ &  &\\
    \hline
\end{tabular}
%\end{center}
\end{table}
This comparison is based on the data simulated from \textbf{D1}, \textbf{D2} and \textbf{D3} and five samples sizes $n = 10$, $25$,
$50$, $100$ and $n = 500$, using $Nsim = 100$ replications. We examine the performances of these methods via integrated
square error ({\rm ISE}) criterion, defined by :
\begin{equation}\label{ISE}
  ISE= \int \{\hat{f}_{h}(x)-f(x)\}^{2} dx
\end{equation}
Table 2 presents the average ISE ($\overline{ISE}$) and the average of bandwidth $h$ ($\overline{h}$) based on 100 replications for the estimators of \textbf{D1}, \textbf{D2}
 and \textbf{D3}. The burn-in period contains $M_0=1500$ iterations
and the following $M=3000$ iterations were recorded. From Table 2, we observe that:

\begin{itemize}
  \item [$\bullet$] For all estimators, the means of $ISE$ and $h$ based on 100 replications decrease as sample size $n$ increases.
  \item [$\bullet$] For all sample sizes and models considered, the adaptive Bayesian approach (BSPE and Gamma-BSPE) outperforms
  the global Bayesian approach and the classical UCV method.
  \item [$\bullet$] We notice that, mean h associated with the high density region (HDR) is smaller than the mean h associated with
   the low density region (HDR) for both adaptive Bayesian approaches (BSPE and Gamma-BSPE).
  \item [$\bullet$] A comparison between the UCV and global bayesian approaches. We notice that for almost all the considered models,
   the global bayesian approach is better than the UCV for small sample sizes, but for medium and large sample sizes, the UCV works better.
   \item [$\bullet$] The adaptive Bayesian approach with two different kernels (gamma for HDR and BSPE for LDR), outperforms the adaptive Bayesian
    approach with the same kernel (BSPE for both HDR and LDR regions), for models D1 and D3 for almost all sizes considered. Contrary to the D2 model,
     where the adaptive Bayesian approach with the same kernel (BSPE) is better.
\end{itemize}



{\tiny\begin{table}
\begin{center}
 \caption{Average $ISE$ ($\overline{ISE}$) (with average $h$ ($\overline{h}$) brackets) based on 100 replications for D1, D2 and D3 distributions}
\begin{tabular}{cccccc}
\hline
Density     & $n$ & $\overline{ISE}_{UCV} $   & $\overline{ISE}_{Bayes-global} $ & $\overline{ISE}_{Bayes-Adap_{BSPE}} $ & $\overline{ISE}_{Bayes-Adap_{Gam-BSPE}} $ \\
            &     &  $(\overline{h}_{UCV})$   & $(\overline{h}_{Bayes-global})$  & $(\overline{h}_{HDR},\overline{h}_{LDR})$        & $(\overline{h}_{HDR},\overline{h}_{LDR})$      \\
            \hline
            & 10  & 0.07298109      & 0.03339965                   & 0.03116670                 & \textbf{0.02446340}                 \\
            &     & (1.29691400)    & (0.41601620)                 & (0.41572240,0.43166210)    & (0.42456180, 0.43005550)    \\
            & 50  & 0.01992840      & \textbf{0.00867429}                   & 0.00944265                 & 0.00944339               \\
\textbf{D1} &     & (0.66709700)    & (0.56194740)                 & (0.21892850, 0.35730160)   & (0.15519320, 0.35793270)    \\
            & 100 & 0.00812029      & 0.013134670                  & 0.00545582                 & \textbf{0.00481987}               \\
            &     & (0.14878390)    & (1.23848300)                 & (0.18921140, 0.26361370)   & (0.12187420, 0.26551030)\\
            & 250 & 0.00464116      & 0.01115482                   & 0.00421646                 & \textbf{0.00226334}  \\
            &     & (0.09514660)    & (1.09769077)                 & (0.90787800, 0.25401930)   & (0.07194098, 0.24740547)\\\\
            & 10  & 0.18070430      & 0.09793666                   & \textbf{0.08833576}                 & 0.09152228                                    \\
            &     & (0.63006541)    & (0.47316392)                 & (0.42567424, 0.42613821)   & (0.35051259, 0.41623607)                     \\
            & 50  & 0.03882595      & 0.02937818                   & \textbf{0.01933364}                 & 0.02062382                                   \\
\textbf{D2} &     & (0.06591049)    & (0.23530874)                 & (0.33260233, 0.33640982)   & (0.28967948, 0.34092399)                      \\
            & 100 &  0.04370892     & 0.01372667                   & \textbf{0.00810524}                 &  0.01026966                                   \\
            &     & (0.02410700)    & (0.20874785)                 & (0.24873629, 0.29847000)   & (0.21676625, 0.29696059)                    \\
            & 250 &  0.02080307     & 0.00948743                   & 0.00499513                  & 0.00684447                                  \\
            &     & (0.018870869)   & (0.18647112)                 & (0.18162649, 0.1989421610)   & (0.16955425, 0.18021157)          \\\\
            & 10  & 0.0232216915    & 0.02009696                   & 0.01920234                                 & \textbf{0.01701824 }                                    \\
            &     & (1.167033916)   & (0.80816964)                 & (0.36240572, 0.92421295)                  & (0.37011558, 0.40727976)                     \\
\textbf{}   & 50  & 0.02737774      & 0.00871603                   & 0.00807053                                 & \textbf{0.00781293 }                                    \\
\textbf{D3} &     & (1.09355602)    & (0.15824070)                 & (0.15292585, 0.29172937)                  & (0.15211303,0.27504394)                     \\
            & 100 & 0.015952670     & 0.00477119                   & 0.00370813                                & \textbf{0.00364616}                                   \\
            &     & (0.08969938)    & (0.16248214)                 & (0.10336488, 0.29077918)                 &  (0.10264756, 0.26801841)                                               \\
            & 250 & 0.00968110      & 0.00428933                   & 0.00238951                                & \textbf{0.00237538}                                \\
            &     & (0.03910971)    & (0.12370519)                 & (0.07658197, 0.19733326)                 & (0.09667585, 0.16960317) \\
            \hline
\end{tabular}
\end{center}\label{ISE}
\end{table}}
The comparison is also given in Figure \ref{esti-den12} and \ref{esti-den3}. That presents the plots of the pdf estimates for D1, D2 and D3, with UCV and bayesian method for the
 choice of bandwidth parameter. The results are given for sample size $n = 200$ and for one replication. We can observe that the smoothing quality is satisfactory
  for the adaptive Bayesian approach, practically for the three considered models. The adaptive Bayesian Gamma-BSPE approach, reproduces well the bimodality of the D3 model.
  We also notice that the smoothing quality by the classical UCV approach is poor for the D2 model.

%\begin{figure}
%\begin{center}
%\begin{tabular}{cc}
%  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
%  \includegraphics[width=7cm,height=5.5cm]{lnorm(200)} & \includegraphics[width=7cm,height=5.5cm]{Burr(200)} \\
%\end{tabular}
%\end{center}
%\end{figure}

\begin{figure}
\begin{center}
\begin{tabular}{cc}
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  \includegraphics[width=6.5cm,height=6.5cm]{lnorm(200).png} & \includegraphics[width=6.5cm,height=6.5cm]{Burr(200).png} \\
  %(a) Web-traffic data & (b) vinyl chloride data \\
\end{tabular}
\caption{The estimators of heavy tailed densities \textbf{D1} and \textbf{D2}  with $n=200$, with BS-PE kernel and (global, adaptive and UCV) methods.}\label{esti-den12}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{tabular}{c}
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  \includegraphics[width=7cm,height=5.5cm]{gamma(200)}  \\
\end{tabular}
\caption{The estimators of heavy tailed densities \textbf{D3} with $n=200$, with BS-PE kernel and (global, adaptive and UCV) methods.}\label{esti-den3}
\end{center}
\end{figure}

\section{Application to real HT data}
%In this section, two real HT data sets are presented and analyzed by the proposed approach.
In this section, we illustrate the performance of the proposed estimator on two real HT data sets defined below:
\begin{itemize}
  \item \textbf{Web-traffic HT data:} These data represent the size of different web files (pdf, html, images,
video, etc.) measured in Kilo Octet from world cup (French, June 1998) server. These data
are collected for $n = 312$ queries \cite{ZZA2018}.
  \item \textbf{Vinyl chloride data:} These data present the vinyl chloride data obtained
from clean upgrading, monitoring wells in mg/L; this data set was used by \cite{Bhaumik2009}.
\end{itemize}
The table \ref{tab_real1} and \ref{tab_real3} provide the description summaries for Web-traffic and vinyl chloride data respectively.

\begin{table}
\begin{center}
\caption{Descriptive summary of the Web-traffic data set \cite{ZZA2018}.}\label{tab_real1}
\begin{tabular}{ccccccccc}
\hline
Data set & $n$ & Max & Min & Median & Mean & SD & CS & CK\\
\hline
Web-traffic &  312 & 65.613 & 0.042 & 1.362 & 4.081 & 8.520 & 5.044 &  27.897 \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Descriptive summary of the vinyl chloride data set.}\label{tab_real3}
\begin{tabular}{ccccccccc}
\hline
Data set & $n$ & Max & Min & Median & Mean & SD & CS & CK\\
\hline
vinyl chloride &  34 & 8.000 & 0.100 & 1.150 & 1.879 & 1.952 & 1.603 &   5.005 \\
\hline
\end{tabular}
\end{center}
\end{table}
Now, we apply kernel estimators to estimate
the density for trafic web and vinyl chloride data, based on different selection methods
 of the smoothing parameter (UCV, bayesian(global) and bayesian adaptive(BS-PE($\nu=2$)
 kernel and Gammma-BSPE($\nu=2$) kernel)).  The bayes variable bandwidths estimates were
 obtained with prior parameters $\alpha=2.5$ and $\beta=n^{4/5}$. The figure \ref{real_data} shows, that all the methods are capable of reproducing
the unimodality of these data. we observe that, the smoothing quality is satisfactory for almost all the considered methods.

\begin{figure}
\begin{center}
\begin{tabular}{cc}
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  \includegraphics[width=6.5cm,height=6.5cm]{trafic_web.png} & \includegraphics[width=6.5cm,height=6.5cm]{real_data3.png} \\
  (a) Web-traffic data & (b) vinyl chloride data \\
\end{tabular}
\caption{kernel estimator for Web-traffic and vinyl chloride data with sample size $n=312$ and $n=34$ respectively.} \label{real_data}
\end{center}
\end{figure}

\section{conclusions}
In this paper, we have proposed a new gamma-BSPE kernel estimator. It is based on the principle of subdividing the HT dataset
into two regions (LDR and HDR) and associating to each region the gamma and BSPE kernels. The smoothing parameter is determined
by the adaptive Bayesian approach. The simulation study showed that the adaptive bayesian approach with the gamma-BSPE kernels
and with the same BSPE kernels, performs better than the global bayesian approach and the classical UCV approach. This study also
showed that in some cases the estimator with the gamma-BSPE kernels, performs better than the estimator with the same BSPE kernels
for both regions, contrary to other cases.

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\bibitem[ZIANE et~al.(2018)]{ZZA2018}
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\end{thebibliography}




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%\bibliography{bib}
\end{document}

Markovich, L. (2016). Gamma-weibull kernel estimation of the heavy tailed densities. arXiv preprint arXiv:1604.06522.

