We consider the problem of comparing two Poisson parameters from the Bayesian perspective. Kawasaki and Miyaoka (2012b) proposed the Bayesian index θ = P(λ₁ < λ₂ | X₁, X₂) and expressed it using the hypergeometric series. In this paper, under some conditions, we give four other expressions of the Bayesian index in terms of the cumulative distribution functions of beta, F, binomial, and negative binomial distribution. Next, we investigate the relationship between the Bayesian index and the p-value of the conditional test with the null hypothesis H₀ : λ₁ ≥ λ₂ versus an alternative hypothesis H₁ : λ₁ < λ₂. Additionally, we investigate the generalized relationship between θ = P(λ₁/λ₂ < c | X₁, X₂) and the p-value of the conditional test with the null hypothesis H₀ : λ₁/λ₂ ≥ c versus the alternative H₁ : λ₁/λ₂ < c. We illustrate the utility of the Bayesian index using analyses of real data. Our finding suggests that θ can potentially be useful in an epidemiology and in a clinical trial.

We study adaptive Bayes type estimation and hybrid type estimation of both drift and volatility parameters for small diffusion processes from discrete observations. By applying adaptive maximum likelihood type estimation for small diffusion processes to the Bayesian method and by using the polynomial type large deviation inequality for the statistical random field and Ibragimov-Has’minskii-Kutoyants program, the adaptive Bayes type estimators and hybrid type estimators are obtained and we show that they have asymptotic normality and convergence of moments.

Based on the Dirichlet process as a prior, we give the Bayes estimate of the estimable parameter of an arbitrary degree, whose form is different from Yamato (1977b). From it, we derive the simple form of the limit of Bayes estimate as the parameter of the Dirichlet process tends to zero.

The weighted entropy introduced by Belis and Guiasu (1968) is viewed as a measure of uncertainty. Di Crescenzo and Longobardi (2006) proposed dynamic form of these measure namely weighted residual (WRE) and past entropies (WPE). In this paper, we extend the definition of weighted residual and past entropies to bivariate setup and obtain some of its properties. Several properties, including monotonicity and bounds of BWRE and BWRP are obtained. We also look into the problem of extending WRE and WPE for conditionally specified models. Several properties,including bounds of CWRE and CWPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function.

For the analysis of square contingency tables with ordered categories, it may be useful for applying some kinds of asymmetry model when the symmetry model does not hold. Tahata and Tomizawa (2011) considered the linear asymmetry model. In the present paper, the extended linear asymmetry model is proposed. The model indicates that the log-odds of symmetric cells are expressed as polynomial function of parameter. Also, the symmetry model is separated into two models and the relationship between test statistics is given.