A matching prior based on the modified profile likelihood for the common mean in multiple log-normal distributions

Abstract

In this paper, we develop a matching prior for the common mean in several log-normal distributions. For this problem, assigning priors appropriately for the common log-normal mean is challenging owing to the presence of nuisance parameters. Matching priors, which are priors that match the posterior probabilities of certain regions within their frequentist coverage probabilities, are commonly used in this problem. However, a closed form posterior under the derived first order matching prior is not available; further, the second order matching prior is difficult to be derived in this problem. Thus, alternatively, we derive a matching prior based on a modification of the profile likelihood. Simulation studies show that this proposed prior meets the target coverage probabilities very well even for small sample sizes. Finally, we present a real example.

Appendices

Appendix 1: Derivation of two group reference prior

We derived the two group reference prior for the parameter grouping $\{\theta ,(\sigma _1,\cdots ,\sigma _k) \}$ by the algorithm of Berger and Bernardo (1992). The compact subsets were taken to be Cartesian products of sets of the form

$$\begin{aligned} \theta \in [a_1,b_1], \sigma _1 \in [a_2,b_2], \cdots , \sigma _k \in [a_{k+1},b_{k+1}]. \end{aligned}$$

(35)

In the limit $a_1$ will tend to $-\infty$, $a_i, i=2,\cdots ,k$ will tend to 0, and $b_i, i=1,\cdots ,k$ will tend to $\infty$. For the derivation of the reference prior, from the Fisher information (2),

$$\begin{aligned} h_1= & {} 2\left( \prod _{i=1}^k n_i\sigma _i^{-4}\right) \left[ \sum _{i=1}^k n_i \prod _{j\ne i}^k\sigma _j^2(\sigma _j^2+2)\right] \left[ \prod _{i=1}^k n_i^{-1}\left( 1+{2\over \sigma _i^2}\right) ^{-1} \right] ,\\ h_2= & {} \prod _{i=1}^k n_i\left( 1+{2\over \sigma _i^2}\right) . \end{aligned}$$

Here, and below, a subscripted Q denotes a function that is constant and does not depend on any parameters but any Q may depend on the ranges of the parameters.

Step 1. Note that

$$\begin{aligned} \int _{a_{k+1}}^{b_{k+1}}\cdots \int _{a_2}^{b_2} h_{2}^{1/2} d\sigma _1 \cdots d\sigma _k = Q_1\prod _{i=1}^n n_i^{1\over 2} \end{aligned}$$

It follows that

$$\begin{aligned} \pi _{2}^l (\sigma _1,\cdots ,\sigma _k \vert \theta )= Q_1^{-1}\prod _{i=1}^k \left( 1+{2\over \sigma _i^2}\right) ^{1\over 2}. \end{aligned}$$

Step 2. Now

$$\begin{aligned} E^l \{ \log h_1 \vert \sigma _1,\cdots ,\sigma _k \}= & {} \int _{a_{k+1}}^{b_{k+1}} \cdots \int _{a_2}^{b_2} (\log h_1) \pi _{2}^l (\sigma _1,\cdots ,\sigma _k \vert \theta )d\sigma _1 \cdots d\sigma _k\\= & {} Q_2. \end{aligned}$$

It follows that

$$\begin{aligned} \pi _1^l (\theta )= \exp [E^l \{\log h_1 \vert \sigma _1,\cdots ,\sigma _k\}/2] =\exp \{Q_{2}/2\}. \end{aligned}$$

Therefore the reference prior is

$$\begin{aligned} \pi (\theta ,\sigma _1\cdots ,\sigma _k)= & {} \lim _{l\rightarrow \infty } { {\pi _2^l (\sigma _1,\cdots ,\sigma _k \vert \theta )\pi _1^l (\theta )} \over {\pi _2^l (\sigma _{10},\cdots ,\sigma _{k0} \vert \theta _{0})\pi _1^l (\theta _{0})} }\nonumber \\\propto & {} \prod _{i=1}^k \left( 1+{2\over \sigma _i^2}\right) ^{1\over 2}, \end{aligned}$$

(36)

where $\theta _{0}$ and $(\sigma _{10}\cdots ,\sigma _{k0})$ are an inner point of the interval $(-\infty ,\infty )$ and the interval $(0,\infty )$, respectively.

Appendix 2: Markov Chain Monte Carlo numerical integration

We evaluate the frequentist coverage probability by investigating the credible interval of the marginal posteriors density of $\theta$ under Jeffreys’ prior in (20). Since no closed form posterior is available, the posterior quantiles are obtained via application of the Markov Chain Monte Carlo numerical integration. We provide some of the implementation details below.

For Jeffreys’ prior, the joint posterior distribution of $\theta ,\sigma _2,\ldots ,\sigma _{k-1}$ and $\sigma _k$ given $\mathbf x$ is

$$\begin{aligned} \pi (\theta ,\sigma _1,\ldots ,\sigma _k \vert \mathbf{x})\propto & {} \left( \prod _{i=1}^k \sigma _i^{-n_i-2}\right) \left[ \sum _{i=1}^k n_i \prod _{j\ne i}^k\sigma _j^2(\sigma _j^2+2)\right] ^{1\over 2}\nonumber \\&\quad \times \, \exp \left\{ - \sum _{i=1}^{k}{1\over 2\sigma _i^2} {\sum _{j=1}^{n_i}\left( \log x_{ij}-\theta +{\sigma _i^2\over 2}\right) ^2}\right\} . \end{aligned}$$

This results in full conditional distributions as follows:

$$\begin{aligned} (\theta \vert \sigma _1,\ldots ,\sigma _k, \mathbf{x})\propto & {} \exp \left\{ - {1\over 2}{ \left( {n_1\over \sigma _1^2}+\cdots +{n_k\over \sigma _k^2}\right) \left[ \theta -g(\sigma _1,\ldots ,\sigma _k)\right] ^2} \right\} , \end{aligned}$$

(37)

$$\begin{aligned} (\sigma _i \vert \theta ,\sigma _{j, j\ne i=1,\ldots ,k}, \mathbf{x})\propto & {} \sigma _i^{-n_i-2} \exp \left\{ - {S_i^2\over 2\sigma _i^2}\right\} \nonumber \\&\times \,\left[ \sum _{i=1}^k n_i \prod _{j\ne i}^k\sigma _j^2(\sigma _j^2+2)\right] ^{1\over 2} \exp \left\{ - {n_i ({\bar{x}}_i -\theta +{\sigma _i^2\over 2})^2\over 2\sigma _i^2} \right\} ,\nonumber \\ \end{aligned}$$

(38)

where ${\bar{x}}_i=\sum _{j=1}^{n_i}\log x_{ij}/n_i$, $S_i^2=\sum _{j=1}^{n_i}(\log x_{ij}-{\bar{x}}_i)^2$ and $g(\sigma _1,\ldots ,\sigma _k)$ is given by

$$\begin{aligned} { {n_1\over \sigma _1^2}\left( {\bar{x}}_1+{\sigma _1^2\over 2}\right) +\cdots +{n_k\over \sigma _k^2}\left( {\bar{x}}_k+{\sigma _k^2\over 2}\right) \over {n_1\over \sigma _1^2}+\cdots +{n_k\over \sigma _k^2} }. \end{aligned}$$

For the reference prior, the conditional density of $\theta$ is the same as the case of Jeffreys’ prior, and the conditional densities of $\sigma _i$ are different as follows.

$$\begin{aligned} (\sigma _i \vert \theta ,\sigma _{j, j\ne i=1,\ldots ,k}, \mathbf{x})\propto & {} \sigma _i^{-n_i} \exp \left\{ - {S_i^2\over 2\sigma _i^2}\right\} \nonumber \\&\quad \times \,\left( 1+{2\over \sigma _i^2}\right) ^{1\over 2} \exp \left\{ - {n_i ({\bar{x}}_i -\theta +{\sigma _i^2\over 2})^2\over 2\sigma _i^2} \right\} . \end{aligned}$$

(39)

For the conditional distributions of $\sigma _i, i=1,\ldots ,k$, considering the rest being non-standard, the Metropolis–Hastings algorithm is used to generate samples along the lines of Chib and Greenberg (1995).

In each case, we computed the 5th and the 95th posterior quantilies from a sample of size 50,000 (discarding the first 30,000) and repeated the iterations 20,000 times to estimate the coverage probability.

Appendix 3: Propriety of posterior distribution

For Jeffreys’ prior, the joint posterior for $\theta , \sigma _1,\ldots ,\sigma _{k-1}$ and $\sigma _k$ given $\mathbf{x}$ is

$$\begin{aligned} \pi (\theta ,\sigma _1,\ldots ,\sigma _k \vert \mathbf{x})\propto & {} \left( \prod _{i=1}^k \sigma _i^{-n_i-2}\right) \left[ \sum _{i=1}^k n_i \prod _{j\ne i}^k\sigma _j^2(\sigma _j^2+2)\right] ^{1\over 2}\nonumber \\&\quad \times \,\exp \left\{ - \sum _{i=1}^{k}{1\over 2\sigma _i^2} {\sum _{j=1}^{n_i}\left( \log x_{ij}-\theta +{\sigma _i^2\over 2}\right) ^2}\right\} . \end{aligned}$$

(40)

First, we integrate with respect to $\theta$ from (40). Then,

$$\begin{aligned}&\pi (\sigma _1,\ldots ,\sigma _k \vert \mathbf{x})\nonumber \\&\quad \propto \left( \prod _{i=1}^k \sigma _i^{-n_i-2}\right) \left[ \sum _{i=1}^k n_i \prod _{j\ne i}^k\sigma _j^2(\sigma _j^2+2)\right] ^{1\over 2}\nonumber \\&\quad \quad \times \,\left( {n_1\over \sigma _1^2}+\cdots +{n_k\over \sigma _k^2}\right) ^{-{1\over 2}} \exp \left\{ -\sum _{i=1}^k{S_i^2\over 2\sigma _i^2}\right\} \exp \left\{ -g(\sigma _1,\ldots ,\sigma _k)\right\} \nonumber \\&\quad \le \left( \prod _{i=1}^k \sigma _i^{-n_i-2}\right) \left[ \sum _{i=1}^k n_i \prod _{j\ne i}^k\sigma _j^2(\sigma _j^2+2)\right] ^{1\over 2}\nonumber \\&\quad \quad \times \,\left( {n_1\over \sigma _1^2}+\cdots +{n_k\over \sigma _k^2}\right) ^{-{1\over 2}} \exp \left\{ -\sum _{i=1}^k{S_i^2\over 2\sigma _i^2}\right\} \equiv \pi '(\sigma _1,\ldots ,\sigma _k \vert \mathbf{x}), \end{aligned}$$

(41)

where $S_i^2=\sum _{j=1}^{n_i} (\log x_{ij} - {\bar{x}}_i)^2,$${\bar{x}}_i = \sum _{j=1}^{n_i} \log x_{ij}/n_i, i=1,\ldots ,k$ and $g(\sigma _1,\ldots ,\sigma _k)$ is a function of $\sigma _1,\ldots ,\sigma _{k-1}$ and $\sigma _k$. If $0<\sigma _i<1, i=1,\ldots ,k$ then

$$\begin{aligned} \pi '(\sigma _1,\ldots ,\sigma _k \vert \mathbf{x}) \le k_1 \left( \prod _{i=1}^k \sigma _i^{-n_i-2}\right) \exp \left\{ -\sum _{i=1}^k{S_i^2\over 2\sigma _i^2}\right\} . \end{aligned}$$

(42)

Therefore, (42) is appropriate, if $n_i+1>0, i=1,\ldots ,k$. Here, $k_1$ is a constant. Next, if $\sigma _i\ge 1, i=1,\ldots ,k$ then

$$\begin{aligned} \pi '(\sigma _1,\ldots ,\sigma _k \vert \mathbf{x}) \le k_2 \left( \prod _{i=1}^k \sigma _i^{-n_i+1}\right) \exp \left\{ -\sum _{i=1}^k{S_i^2\over 2\sigma _i^2}\right\} . \end{aligned}$$

(43)

Then, (43) is proper, if $n_i-2>0, i=1,\ldots ,k$. Here, $k_2$ is a constant. Finally, if $0<\sigma _i<1, i=1,\ldots ,l$ and $\sigma _i\ge 1, i=l+1,\ldots ,k$ then

$$\begin{aligned} \pi '(\sigma _1,\ldots ,\sigma _k \vert \mathbf{x}) \le k_3 \left( \prod _{i=1}^l \sigma _i^{-n_i-2}\right) \left( \prod _{i=l+1}^{k}\sigma _i^{-n_i+1}\right) \exp \left\{ -\sum _{i=1}^k {S_i^2\over 2\sigma _i^2}\right\} . \end{aligned}$$

(44)

Therefore the (44) is proper, if $n_i+1>0, i=1,\ldots ,l$ and $n_i-2>0, i=l+1,\ldots ,k$. Here $k_3$ is a constant.

For the reference prior, the joint posterior for $\theta , \sigma _1,\ldots ,\sigma _{k-1}$ and $\sigma _k$ given $\mathbf{x}$ is

$$\begin{aligned} \pi (\theta ,\sigma _1,\ldots ,\sigma _k \vert \mathbf{x})\propto & {} \left( \prod _{i=1}^k \sigma _i^{-n_i}\right) \left[ \prod _{i=1}^k \left( 1+{2\over \sigma _i^2}\right) ^{1\over 2}\right] \nonumber \\&\quad \times \,\exp \left\{ - \sum _{i=1}^{k}{S_i^2\over 2\sigma _i^2} -{\sum _{i=1}^{k}{n_i\over 2\sigma _i^2} \left( {\bar{x}}_{i}-\theta +{\sigma _i^2\over 2}\right) ^2}\right\} .\quad \end{aligned}$$

(45)

Then we have the following equation.

$$\begin{aligned} \pi (\theta ,\sigma _1,\ldots ,\sigma _k \vert \mathbf{x})\le & {} \left( \prod _{i=1}^k \sigma _i^{-n_i}\right) \left[ \prod _{i=1}^k \left( 1+{2\over \sigma _i^2}\right) ^{1\over 2}\right] \nonumber \\&\quad \times \,\exp \left\{ - \sum _{i=1}^{k}{S_i^2\over 2\sigma _i^2} -{{n_j\over 2\sigma _j^2} \left( {\bar{x}}_{j}-\theta +{\sigma _j^2\over 2}\right) ^2}\right\} \nonumber \\\le & {} \left( \prod _{i=1}^k \sigma _i^{-n_i}\left( 1+\sqrt{2}\sigma _i^{-1}\right) \right) \nonumber \\&\quad \times \,\exp \left\{ - \sum _{i=1}^{k}{S_i^2\over 2\sigma _i^2} -{{n_j\over 2\sigma _j^2} \left( {\bar{x}}_{j}-\theta +{\sigma _j^2\over 2}\right) ^2}\right\} . \end{aligned}$$

(46)

Therefore the (46) is proper, if $n_i-2>0, i=1,\ldots ,k$. This completes the proof. $\square$