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

Abstract

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.

Appendix: proofs of propositions

Proof of Proposition 2

Since one has

$$\begin{aligned} \textrm{Bias}[{\widehat{f}}_n({\varvec{x}})]=p_{d}({\varvec{x}};\widehat{\varvec{\theta }}_n){\mathbb {E}}[{\widehat{w}}_n({\varvec{x}})]-f({\varvec{x}}) \text{ and } \textrm{var}[{\widehat{f}}_n({\varvec{x}})]=[p_{d}({\varvec{x}};\widehat{\varvec{\theta }}_n)]^2 \textrm{var} [{\widehat{w}}_n({\varvec{x}})], \end{aligned}$$

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

$$\begin{aligned} {\mathbb {E}} \left[ {{\widehat{w}}}_n({\varvec{x}})\right]&= {\mathbb {E}}\left[ {\textbf{G}}_{{\varvec{x}},{\textbf{h}},\ell }({\textbf{X}}_{1})/p_{d}({\textbf{X}}_1;\widehat{\varvec{\theta }}_n)\right] \\&=\int _{[0,\infty )^d}{\textbf{G}}_{{\varvec{x}},{\textbf{h}},\ell }({\textbf{u}})\left[ p_{d}({\textbf{u}};\widehat{\varvec{\theta }}_n)\right] ^{-1}f({\textbf{u}})d{\textbf{u}} ={\mathbb {E}}\left[ w\left( {{{\mathcal {G}}}}_{{\varvec{x}},{\textbf{h}},\ell }\right) \right] \\&= w({\varvec{x}})+\left[ \displaystyle \sum _{r=1}^{d-\ell }h_r \frac{\partial w}{\partial x_r}({\varvec{x}})+\displaystyle \sum _{r=1}^{d-\ell }\frac{1}{2}\right. \\&\quad \left. \left( x_r h_r+2h_r^2 \right) \frac{\partial ^2 w}{\partial x_r^2}({\varvec{x}})+\frac{1}{2}\displaystyle \sum _{s=1}^{\ell }x_s h_s \frac{\partial ^2 w}{\partial x_s^2}({\varvec{x}})\right] +\left( 1+o\left\{ \displaystyle \sum _{j=1}^{d}h_j^2 \right\} \right) , \end{aligned}$$

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:

$$\begin{aligned} \textrm{var} \left[ {{\widehat{w}}}_n({\varvec{x}})\right]&= \frac{1}{n}\textrm{var}\left[ {\textbf{G}}_{{\varvec{x}},{\textbf{h}},\ell }({\textbf{X}}_{1})/p_{d}({\textbf{X}}_1;\widehat{\varvec{\theta }}_n)\right] \\&=\frac{1}{n}\left[ \int _{[0,\infty )^d}{\textbf{G}}^2_{{\varvec{x}},{\textbf{h}},\ell }({\textbf{u}})[p_{d}({\textbf{u}};\widehat{\varvec{\theta }}_n)]^{-2}f({\textbf{u}})d{\textbf{u}}+O(1)\right] \\&=\frac{1}{n} \int _{[0,\infty )^d}{\textbf{G}}^2_{{\varvec{x}},{\textbf{h}},\ell }({\textbf{u}})[p_{d}({\textbf{u}};\widehat{\varvec{\theta }}_n)]^{-2}\begin{pmatrix}f({\varvec{x}})+({\varvec{x}}-{\textbf{u}})^T\nabla f({\varvec{x}})\\ +\frac{1}{2}({\varvec{x}}-{\textbf{u}})^T{{{\mathcal {H}}}}f({\varvec{x}}) ({\varvec{x}}-{\textbf{u}})\\ +o\left[ \left( ||{\varvec{x}}-{\textbf{u}}||^2\right) \right] \end{pmatrix} d{\textbf{u}}\\&=\frac{1}{n}f({\varvec{x}})[p_{d}({\varvec{x}};\widehat{\varvec{\theta }}_n)]^{-2}||{\textbf{G}}_{{\varvec{x}},{\textbf{h}},\ell }||_2^2++o\left( n^{-1}\displaystyle \prod _{j=1}^{d}h_j^{-1/2}\right) \\&=\frac{1}{n}f({\varvec{x}})[p_{d}({\varvec{x}};\widehat{\varvec{\theta }}_n)]^{-2}\displaystyle \prod _{k=1}^{d-\ell }\left( \frac{\Gamma (1+2x_k/h_k)}{2^{1+2x_k/h_k}\Gamma (1+x_k/h_k)}h_{k}^{-1}\right) \\&\prod _{s \in {\mathbb {I}}^{}_{2}}\left( \frac{\Gamma (1+\lambda _{s}^2/2)}{2^{1+\lambda _{s}^2/2}\Gamma (1+\lambda _{s}^2/4)}h_{s}^{-1}\right) \\&\quad \quad \prod _{j \in {\mathbb {I}}_{2}^c}\left( \frac{1}{2\pi ^{1/2}}h_j^{-1/2}x_j^{-1/2}\right) +o\left( n^{-1}\displaystyle \prod _{j=1}^{d}h_j^{-1/2}\right) \\&=\frac{1}{n}f({\varvec{x}})[p_{d}({\varvec{x}};\widehat{\varvec{\theta }}_n)]^{-2}\displaystyle \prod _{k \in {\mathbb {I}}^{}_{1}}\left( \frac{\Gamma (1+2\lambda _{k})}{2^{1+2\lambda _{k}}\Gamma (1+\lambda _{k})}h_{k}^{-1}\right) \\& \quad \prod _{s \in {\mathbb {I}}^{}_{2}}\left( \frac{\Gamma (1+\lambda _{s}^2/2)}{2^{1+\lambda _{s}^2/2}\Gamma (1+\lambda _{s}^2/4)}h_{s}^{-1}\right) \nonumber \\&\quad \prod _{j \in {\mathbb {I}}_{}^c}\left( \frac{1}{2\pi ^{1/2}}h_j^{-1/2}x_j^{-1/2}\right) +o\left( n^{-1}\displaystyle \prod _{j=1}^{d}h_j^{-1/2}\right) , \end{aligned}$$

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

$$\begin{aligned} N({\textbf{h}}_{i}\mid {\textbf{X}}_{i})&= \left( \frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{n-1}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)} \prod _{\ell =1}^{d} G_{\rho (X_{i\ell },h_{i\ell }),h_{i\ell }}(X_{j\ell })\right) \nonumber \\&\quad \left( \prod _{\ell =1}^{d} \frac{\beta _{\ell }^{\alpha }}{\Gamma (\alpha )}h_{i\ell }^{-\alpha -1}\exp (-\beta _{\ell }/h_{i\ell })\right) \nonumber \\&=\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{(n-1)[\Gamma (\alpha )]^{d}}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)} \nonumber \\&\prod _{\ell =1}^{d} \frac{G_{\rho (X_{i\ell },h_{i\ell }),h_{i\ell }}(X_{j\ell })}{\beta _{\ell }^{-\alpha }h_{i\ell }^{\alpha +1}}\exp (-\beta _{\ell }/h_{i\ell }). \end{aligned}$$

(17)

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

$$\begin{aligned}&\frac{G_{\rho (X_{ik};h_{ik}),h_{ik}}(X_{jk})}{\beta _{k}^{-\alpha }h_{ik}^{\alpha +1}}\exp (-\beta _{k}/h_{ik})\nonumber \\&\quad =\frac{X_{jk}^{(X_{ik}/2h_{ik})^{2}}\exp (-X_{jk}/h_{ik})}{h_{ik}^{1+(X_{ik}/2h_{ik})^{2}} \Gamma [1+(X_{ik}/2h_{ik})^{2}]\beta _{k}^{-\alpha }h_{ik}^{\alpha +1}}\exp (-\beta _{k}/h_{ik})\nonumber \\&\quad \simeq \frac{X_{jk}^{\lambda _{ik}}\exp [-(X_{jk}+ \beta _{k})/h_{ik}]}{h_{ik}^{\lambda _{ik}+\alpha +2}\beta _{k}^{-\alpha }\Gamma (1+\lambda _{ik})}\nonumber \\&\quad =\frac{ \Gamma (\lambda _{ik}+ \alpha +1) X_{jk}^{\lambda _{ik}}}{\beta _{k}^{-\alpha }\Gamma (\lambda _{ik}+1)(X_{jk}+\beta _{k})^{\lambda _{ik}+\alpha +1}} \times \frac{(X_{jk}+\beta _{k})^{\lambda _{ik}+\alpha +1}\exp [-(X_{jk}+ \beta _{k})/h_{ik}]}{h_{ik}^{\lambda _{ik}+\alpha +2}\Gamma (\lambda _{ik}+\alpha +1)}\nonumber \\&\quad =A_{ijk}\,IG_{\lambda _{ik}+\alpha +1,X_{jk}+ \beta _{k}}(h_{ik}), \end{aligned}$$

(18)

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

$$\begin{aligned}&\frac{G_{\rho (X_{i\ell };h_{i\ell }),h_{i\ell }}(X_{j\ell })}{\beta _{\ell }^{-\alpha }h_{i\ell }^{\alpha +1}}\exp (-\beta _{\ell }/h_{i\ell })\nonumber \\&\quad =\frac{X_{j\ell }^{(X_{i\ell }/h_{i\ell })-1}\exp (-X_{j\ell }/h_{i\ell })}{h_{i\ell }^{X_{i\ell }/h_{i\ell }} \Gamma (X_{i\ell }/h_{i\ell })\beta _{\ell }^{-\alpha }h_{i\ell }^{\alpha +1}}\exp (-\beta _{\ell }/h_{i\ell })\nonumber \\&\quad =\frac{X_{j\ell }^{(X_{i\ell }/h_{i\ell })-1}\exp (-X_{j\ell }/h_{i\ell })}{h_{i\ell }^{X_{i\ell }/h_{i\ell }} (X_{i\ell }/h_{i\ell })^{-1}\Gamma (1+X_{i\ell }/h_{i\ell })\beta _{\ell }^{-\alpha }h_{i\ell }^{\alpha +1}}\exp (-\beta _{\ell }/h_{i\ell })\nonumber \\&\quad =\frac{X_{j\ell }^{-1}}{\beta _{\ell }^{-\alpha }X_{i\ell }^{-1}}\frac{\exp [-(X_{j\ell }+ \beta _{\ell }-X_{i\ell }\log X_{j\ell })/h_{i\ell }]}{h_{i\ell }^{(X_{i\ell }/h_{i\ell })+\alpha +2}\sqrt{2\pi }(X_{i\ell }/h_{i\ell })^{(X_{i\ell }/h_{i\ell })+1/2}\exp (-X_{i\ell }/h_{i\ell })}\nonumber \\&\quad =\frac{X_{j\ell }^{-1}\Gamma (\alpha +1/2)}{\beta _{\ell }^{-\alpha }X_{i\ell }^{-1/2}\sqrt{2\pi }[C_{ij\ell }(\beta _\ell )]^{\alpha +1/2}} \times \frac{[C_{ij\ell }(\beta _\ell )]^{\alpha +1/2}\exp (-C_{ij\ell }(\beta _\ell )/h_{i\ell })}{h_{i\ell }^{\alpha +3/2}\Gamma (\alpha +1/2)}\nonumber \\&\quad =B_{ij\ell }(\alpha ,\beta _\ell )\,IG_{\alpha +1/2,C_{ij\ell }(\beta _\ell )}(h_{i\ell }), \end{aligned}$$

(19)

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

$$\begin{aligned} N({{\textbf {h}}}_{i}\mid {{\textbf {X}}}_{i})&=\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{(n-1)[\Gamma (\alpha )]^{d}}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)} \nonumber \\&\quad \left( \prod _{k \in {\mathbb {I}}_{{\textbf{X}}_i}}A_{ijk}(\alpha ,\beta _k)\,IG_{\lambda _{ik}+\alpha +1,X_{jk}+\beta _{k}}(h_{ik})\right) \nonumber \\&\quad \quad \quad \times \left( \prod _{\ell \in {\mathbb {I}}^{c}_{{\textbf{X}}_i}}B_{ij\ell }(\alpha ,\beta _\ell )\,IG_{\alpha +1/2,S_{ij\ell }}(h_{i\ell })\right) . \end{aligned}$$

(20)

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

$$\begin{aligned}&\int _{[0, \infty )^{d}} N({{\textbf {h}}}_{i}\mid {{\textbf {X}}}_{i})\,d{{\textbf {h}}}_{i}\nonumber \\&\quad =\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{(n-1)[\Gamma (\alpha )]^{d}}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)}\nonumber \\&\quad \quad \left( \prod _{k \in {\mathbb {I}}_{{\textbf{X}}_i}}A_{ijk}(\alpha ,\beta _k)\int _{0}^{\infty }IG_{\lambda _{ik}+\alpha +1,X_{jk}+ \beta _{k}}(h_{ik})\,dh_{ik}\right) \nonumber \\&\quad \quad \times \left( \prod _{\ell \in {\mathbb {I}}^{c}_{{\textbf{X}}_i}}B_{ij\ell }(\alpha ,\beta _\ell )\int _{0}^{\infty }IG_{\alpha +1/2,C_{ij\ell }(\beta _\ell )}(h_{i\ell })\,dh_{i\ell }\right) \nonumber \\&\quad =\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{(n-1)[\Gamma (\alpha )]^{d}}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)} \left( \prod _{k \in {\mathbb {I}}_{{\textbf{X}}_i}}A_{ijk}(\alpha ,\beta _k)\right) \left( \prod _{\ell \in {\mathbb {I}}^{c}_{{\textbf{X}}_i}}B_{ij\ell }(\alpha ,\beta _\ell )\right) \nonumber \\&\quad =\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{(n-1)[\Gamma (\alpha )]^{d}}D_{i}(\alpha ,\varvec{\beta }), \end{aligned}$$

(21)

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

$$\begin{aligned}{} & {} \pi (h_{im}\mid {\textbf{X}}_{i})=\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{D_{i}(\alpha ,\varvec{\beta })}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)}\\ \quad {} & {} \left( \prod _{k \in {\mathbb {I}}_{{\textbf{X}}_i}}A_{ijk}(\alpha ,\beta _k)\right) \left( \prod _{\ell \in {\mathbb {I}}^{c}_{{\textbf{X}}_i}}B_{ij\ell }(\alpha ,\beta _\ell )\right) IG_{\alpha +1, X_{jm}+\beta _{m}}(h_{im}) \end{aligned}$$

and

$$\begin{aligned}{} & {} {\widehat{h}}_{im}={\mathbb {E}}(h_{im}\mid {\textbf{X}}_{i})=\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{D_{i}(\alpha ,\varvec{\beta })}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)}\nonumber \\ \quad {} & {} \left( \prod _{k \in {\mathbb {I}}_{{\textbf{X}}_i}}A_{ijk}(\alpha ,\beta _k)\right) \left( \prod _{\ell \in {\mathbb {I}}^{c}_{{\textbf{X}}_i}}B_{ij\ell }(\alpha ,\beta _\ell )\right) \left( \frac{X_{jm}+\beta _{m}}{\lambda _{im}+\alpha }\right) . \end{aligned}$$

(22)

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

$$\begin{aligned}&{} & {} \pi (h_{im}\mid {\textbf{X}}_{i})&=\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{D_{i}(\alpha ,\varvec{\beta })}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)}\\&&{} & {} \quad \left( \prod _{k \in {\mathbb {I}}_{{\textbf{X}}_i}}A_{ijk}(\alpha ,\beta _k)\right) \left( \prod _{\ell \in {\mathbb {I}}^{c}_{{\textbf{X}}_i}}B_{ij\ell }(\alpha ,\beta _\ell )\right) IG_{\alpha +1/2,C_{ijm}(\beta _m)}(h_{im}) \end{aligned}$$

and

$$\begin{aligned}&{} & {} {\widehat{h}}_{im}={\mathbb {E}}(h_{im}\mid {\textbf{X}}_{i})&=\frac{p_{d}({\textbf{X}}_i;\widehat{\varvec{\theta }}_n)}{D_{i}(\alpha ,\varvec{\beta })}\sum _{j=1,j\ne i}^{n} \frac{1}{p_{d}({\textbf{X}}_{j};\widehat{\varvec{\theta }}_n)}\nonumber \\ && {} & {} \quad \left( \prod _{k \in {\mathbb {I}}_{{\textbf{X}}_i}}A_{ijk}(\alpha ,\beta _k)\right) \left( \prod _{\ell \in {\mathbb {I}}^{c}_{{\textbf{X}}_i}}B_{ij\ell }(\alpha ,\beta _\ell )\right) \left( \frac{C_{ijm}(\beta _m)}{\alpha -1/2}\right) . \end{aligned}$$

(23)

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