Multiple combined gamma kernel estimations for nonnegative data with Bayesian adaptive bandwidths

The modified (or second version) gamma kernel of Chen [Probability density function estimation using gamma kernels, Annals of the Institute of Statistical Mathematics 52 (2000), pp. 471–480] should not be automatically preferred to the standard (or first version) gamma kernel, especially for univariate convex densities with a pole at the origin. In the multivariate case, multiple combined gamma kernels, defined as a product of univariate standard and modified ones, are here introduced for nonparametric and semiparametric smoothing of unknown orthant densities with support \([0,\infty )^d\). Asymptotical properties of these multivariate associated kernel estimators are established. Bayesian estimation of adaptive bandwidth vectors using multiple pure combined gamma smoothers, and in semiparametric setup, are exactly derived under the usual quadratic function. The simulation results and four illustrations on real datasets reveal very interesting advantages of the proposed combined approach for nonparametric smoothing, compare to both pure standard and pure modified gamma kernel versions, and under integrated squared error and average log-likelihood criteria.

We are specially grateful to an Associate Editor for his valuable comments that significantly improved the paper. For the second coauthor, this work is supported by the EIPHI Graduate School (contract ANR-17-EURE-0002). The first two authors dedicate this paper to Professor Blaise Somé for his 70th birthday.

Authors and Affiliations

1. Laboratoire Sciences et Techniques, Université Thomas SANKARA, 12 BP 417 Ouagadougou 12, Ouagadougou, Burkina Faso

   Sobom M. Somé

2. Laboratoire d’Analyse Numérique d’Informatique et de BIOmathématique, Université Joseph KI-ZERBO, 03 BP 7021, Ouagadougou, Burkina Faso

   Sobom M. Somé

3. Laboratoire de Mathématiques de Besançon UMR 6623 CNRS-UFC, Université Bourgogne Franche-Comté, 16 route de Gray, 25030, Besançon, cedex, France

   Célestin C. Kokonendji

4. Laboratoire de mathématiques et connexes de Bangui, Université de Bangui, B.P. 908 Avenue des Martyrs, Bangui, Centrafrique

   Célestin C. Kokonendji

5. Research Unit LaMOS, University of Bejaia, route de Targa-Ouzemour, 06000, Bejaia, Algeria

   Smail Adjabi

6. Department of Economics and Statistics, University of Mauritius, Reduit, 80837, Mauritius

   Naushad A. Mamode Khan

7. Département de Mathématiques, Faculté des Sciences et Sciences Appliquées, Université de Bouira, Le Pôle Universitaire, 10000, Bouira, Algeria

   Said Beddek

Corresponding author

Correspondence to Sobom M. Somé.

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Appendix: proofs of propositions

Proof of Proposition 2

Since one has

it is enough to calculate \({\mathbb {E}}[{\widehat{w}}_n({\varvec{x}})]\) and \(\textrm{var}[{\widehat{w}}_n({\varvec{x}})]\) using \({\widehat{w}}_n({\varvec{x}})=n^{-1}\sum _{i=1}^n {\textbf{G}}_{{\varvec{x}},{\textbf{h}},\ell }({\textbf{X}}_i)/p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)\) for all \({\varvec{x}}\in {\mathbb {T}}_d^+\) and \({\textbf{G}}_{{\varvec{x}},{\textbf{h}},\ell }=\left( \prod _{s=1}^{d-\ell }G_{x_{s},h_{s}}\right) \left( \prod _{r=1}^{\ell }G_{\rho (x_{r};h_{r}),h_{r}}\right) \) from (6). Indeed, one successively has

which leads to the result of \(\textrm{Bias}[{\widehat{f}}_n({\varvec{x}})]\).

About the variance term, f being bounded leads to \({\mathbb {E}}\left[ {\textbf{G}}_{{\varvec{x}},{\textbf{h}},\ell }({\textbf{X}}_{j})\right] =O(1)\). Also, we denote by \(\nabla f({\varvec{x}})\) and \({\mathcal {H}}f({\varvec{x}})\) the gradient vector and the corresponding Hessian matrix of the function f at \({\varvec{x}}\), respectively. It successively follows:

and the desired result of \(\textrm{var}[{\widehat{f}}_n({\varvec{x}})]\) is therefore deduced. \(\square \)

Proof of Proposition 3

(i) Let us represent \(\pi ({\textbf{h}}_{i}\mid {\textbf{X}}_{i})\) of (14) as the ratio of \(N({\textbf{h}}_{i}\mid {\textbf{X}}_{i}):={\widehat{f}}_{n,{\textbf{h}}_i,-i}({\textbf{X}}_i) \pi ({\textbf{h}}_{i})\) and \(\int _{[0, \infty )^{d}}N({\textbf{h}}_{i}\mid {\textbf{X}}_{i})d{\textbf{h}}_{i}\). From (13) and (16) the numerator is first equal to

From (3), consider the following partition \({\mathbb {I}}_{{\textbf{X}}_i}\) and \({\mathbb {I}}_{{\textbf{X}}_i}^{c}\) of \(\{1,2,...,d\}\). For \(X_{ik} \in [0,2h_{ik})\) with \(k\in {\mathbb {I}}_{{\textbf{X}}_i}\), the function \(n\mapsto \left[ X_{ik}/2h_{ik}(n)\right] ^{2}\) is bounded and then there exists a constant \(\lambda _{ik}>0\) such that \((X_{ik}/2h_{ik})^{2} \rightarrow \lambda _{ik}\) as \(n\rightarrow \infty \); see (Chen 2000, pp. 474–475). Using successively (2) and (3) with the behavior \((X_{ik}/2h_{ik})^{2}\simeq \lambda _{ik}\) as \(n\rightarrow \infty \), the term of product on \({\mathbb {I}}_{{\textbf{X}}_i}\) in (17) can be expressed as follows

with \(A_{ijk}(\alpha ,\beta _k)= [ \Gamma (\lambda _{ik}+ \alpha +1) X_{jk}^{\lambda _{ik}}]/[\beta _{k}^{-\alpha }\Gamma (\lambda _{ik}+1)(X_{jk}+\beta _{k})^{\lambda _{ik}+\alpha +1}]\) and \(IG_{\lambda _{ik}+\alpha +1,X_{jk}+\beta _{k}}(h_{ik})\) comes from (16).

Consider the largest part \({\mathbb {I}}^{c}_{{\textbf{X}}_i}=\left\{ \ell \in \{1,\ldots ,d\}~;X_{i\ell } \in [2h_{i\ell }, \infty )\right\} \). Following again (Chen 2000, pp. 474-475), we assume that for all \(X_{i\ell }\in [2h_{i\ell }, \infty )\) one has \(X_{i\ell }/h_{i\ell } \rightarrow \infty \) as \(n\rightarrow \infty \) for all \(\ell \in \{1,2,\ldots ,d\}\). From (2), (3), the Sterling formula \(\Gamma (z+1)\simeq \sqrt{2\pi }z^{z+1/2}\exp (-z)\) as \(z\rightarrow \infty \), and the well-known property \(\Gamma (z)=z^{-1}\Gamma (z+1)\) for \(z>0\), the term (17) can be successively calculated as

with \(B_{ij\ell }(\alpha ,\beta _\ell )= [X_{j\ell }^{-1}\Gamma (\alpha +1/2)]/(\beta _{\ell }^{-\alpha }X_{i\ell }^{-1/2}\sqrt{2\pi }[C_{ij\ell }(\beta _\ell )]^{\alpha +1/2})\), \(C_{ij\ell }(\beta _\ell )= X_{i\ell }\log (X_{i\ell }/X_{j\ell })+X_{j\ell }-X_{i\ell }+\beta _{\ell }\) and \(IG_{\alpha +1/2,C_{ij\ell }(\beta _\ell )}(h_{i\ell })\) is given in (16).

Combining (18) and (19), the expression \(N({{\textbf {h}}}_{i}\mid {{\textbf {X}}}_{i})\) in (17) becomes

From (20), the denominator is successively computed as follows

with \(D_{i}(\alpha ,\varvec{\beta })=p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)\sum _{j=1,j\ne i}^{n}\left( p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)\right) ^{-1}\left( \prod _{k \in {\mathbb {I}}_{{\textbf{X}}_i}}A_{ijk}(\alpha ,\beta _k)\right) \left( \prod _{\ell \in {\mathbb {I}}^{c}_{{\textbf{X}}_i}} B_{ij\ell }(\alpha ,\beta _\ell )\right) \). Finally, the ratio of (20) and (21) leads to Part (i).

(ii) We remind that the mean of the inverse gamma distribution \(\mathcal{I}\mathcal{G}(\alpha ,\beta _\ell )\) is \(\beta _\ell /(\alpha -1)\) and \({\mathbb {E}}(h_{i\ell }\mid {\textbf{X}}_{i})=\int _{0}^{\infty } h_{i\ell }\pi (h_{im}\mid {\textbf{X}}_{i})\,dh_{im}\) with \(\pi (h_{im}\mid {\textbf{X}}_{i})\) is the marginal distribution \(h_{im}\) obtained by integration of \(\pi ({\textbf{h}}_{i}\mid {\textbf{X}}_{i})\) for all components of \({\textbf{h}}_{i}\) except \(h_{im}\). Then, \(\pi (h_{im}\mid {\textbf{X}}_{i})=\int _{[0, \infty )^{d-1}}\pi ({\textbf{h}}_{i}\mid {\textbf{X}}_{i})\,d{\textbf{h}}_{i(-m)}\) where \(d{\textbf{h}}_{i(-m)}\) is the vector \(d{\textbf{h}}_{i}\) without the \(m^{th}\) component. If \(m\in {\mathbb {I}}_{{\textbf{X}}_i}\), one has

and

If \(m\in {\mathbb {I}}_{{\textbf{X}}_i}^{c}\) and \(\alpha >1/2\), one gets

and

Combining (22) and (23), we therefore get the closed expression of Part (ii). \(\square \)

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