Non-conjugate prior distribution assessment for multivariate normal sampling

Elicitation methods are proposed for quantifying expert opinion about a multivariate normal sampling model. The natural conjugate prior family imposes a relationship between the mean vector and the covariance matrix that can portray an expert's opinion poorly. Instead we assume that opinions about the mean and the covariance are independent and suggest innovative forms of question which enable the expert to quantify separately his or her opinion about each of these parameters. Prior opinion about the mean vector is modelled by a multivariate normal distribution and about the covariance matrix by both an inverse Wishart distribution and a generalized inverse-Wishart (GIW) distribution. To construct the latter, results are developed that give insight into the GIW parameters and their interrelationships. Certain of the elicitation methods exploit unconditional assessments as fully as possible, since these can reflect an expert's beliefs more accurately than conditional assessments. Methods are illustrated through an example.     

Prior distribution elicitation for generalized linear and piecewise-linear models

An elicitation method is proposed for quantifying subjective opinion about the regression coefficients of a generalized linear model. Opinion between a continuous predictor variable and the dependent variable is modelled by a piecewise-linear function, giving a flexible model that can represent a wide variety of opinion. To quantify his or her opinions, the expert uses an interactive computer program, performing assessment tasks that involve drawing graphs and bar-charts to specify medians and other quantiles. Opinion about the regression coefficients is represented by a multivariate normal distribution whose parameters are determined from the assessments. It is practical to use the procedure with models containing a large number of parameters. This is illustrated through practical examples and the benefit from using prior knowledge is examined through cross-validation.     

Geeta distribution and its properties

A new discrete distribution defined over all the positive integers and with the name of Geeta distribution is described. It is L-shaped like the logarithmic series distribution, Yule distribution and the discrete Pareto distribution but is far more versatile than them as it has two parameters. It belongs to the classes of location parameter distributions, modified power series distributions, Lagrange series distributions and exponential distributions. Its mean fi, variance a2 and two recurrence formulae for higher central moments are obtained. Convolution theorem and variations in the model with changes in the parameters have been considered. ML estimators, MVU estimators and estimators based of mean and variance and on mean and first frequency have been derived.     

Testing for a trend in exponential means: A two-moment approximation to the null distribution of the likelihood-ratio statistic

The likelihood-ratio test statistic for testing homogeneity of exponential means with an ordered alternative has a rather complex null distribution. Expressions for the mean and variance of its null distribution are derived, and the accuracy of a two-moment chi-squared approximation is studied. The coefficients needed to implement the approximation are tabled. The application of these results in testing for a constant versus a nondecreasing intensity in a nonhomogeneous Poisson process is also discussed.     

The empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the MLE method

Most of the samples in the real world are from the normal distributions with unknown mean and variance, for which it is common to assume a conjugate normal-inverse-gamma prior. We calculate the empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the Maximum Likelihood Estimation (MLE) method in two theorems. After that, we illustrate the two theorems for the monthly simple returns of the Shanghai Stock Exchange Composite Index.     

Bayesian bootstrap methods for developmental toxicity studies

Estimates of mean response for a developmental toxicity study are developed using the techniques of Bayesian bootstrap. Using this method, a joint posterior distribution of mean response is simulated, providing a means for determining estimated variance and confidence statements. The approach allows for effects on litter size to be taken into consideration in the estimation of mean response. In addition a method is given for the incorporation of prior information into the analysis. The prior information may be information about mean response and about the litter size distribution as well. Results are compared with likelihood based estimates.     

A generalized normal distribution

Undoubtedly, the normal distribution is the most popular distribution in statistics. In this paper, we introduce a natural generalization of the normal distribution and provide a comprehensive treatment of its mathematical properties. We derive expressions for the nth moment, the nth central moment, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, and the asymptotic distribution of the extreme order statistics. We also discuss estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix.     

The discrepancy of P-values and posterior probability: Two-sided hypothesis of variance of normal distribution with unknown mean

This article is concerned with the comparison of P-value and Bayesian measure in point null hypothesis for the variance of Normal distribution with unknown mean. First, using fixed prior for test parameter, the posterior probability is obtained and compared with the P-value when an appropriate prior is used for the mean parameter. In the second, lower bounds of the posterior probability of H₀ under a reasonable class of prior are compared with the P-value. It has been shown that even in the presence of nuisance parameters, these two approaches can lead to different results in the statistical inference.     

Multivariate normal mean–variance mixture distribution based on Birnbaum–Saunders distribution

A multivariate normal mean–variance mixture based on a Birnbaum–Saunders (NMVMBS) distribution is introduced and several properties of this new distribution are discussed. A new robust non-Gaussian ARCH-type model is proposed in which there exists a relation between the variance of the observations, and the marginal distributions are NMVMBS. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of this normal mean–variance mixture distribution is given. A simulation study and some real data are used to demonstrate the modelling strength of this new model.     

A new generalized normal distribution: Properties and applications

Abstract

The most commonly studied generalized normal distribution is the well-known skew-normal by Azzalini. In this paper, a new generalized normal distribution is defined and studied. The distribution is unimodal and it can be skewed right or left. The relationships between the parameters and the mean, variance, skewness, and kurtosis are discussed. It is observed that the new distribution has a much wider range of skewness and kurtosis than the skew-normal distribution. The method of maximum likelihood is proposed to estimate the distribution parameters. Two real data sets are applied to illustrate the flexibility of the distribution.     

Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java.     

Bayesian analysis of the linear regression model with an edgeworth series prior distribution

The purpose of the present investigation 1s to observe the effect of departure from normahty of the prior distribution of regresslon parameters on the Bayman analysis of a h e a r regresslon model Assuming an Edgeworth serles prior distribution for the regresslon coefficients and gamma prior for the disturbances precision, the expressions for the posterlor distribution, posterlor mean and Bayes risk under a quadratic loss function are obtalned The results of a numerical evaluation are also analyzed     

Importance resampling for the smoothed bootstrap

Alternative methods of estimating properties of unknown distributions include the bootstrap and the smoothed bootstrap. In the standard bootstrap setting, Johns (1988) introduced an importance resam¬pling procedure that results in more accurate approximation to the bootstrap estimate of a distribution function or a quantile. With a suitable “exponential tilting” similar to that used by Johns, we derived a smoothed version of importance resampling in the framework of the smoothed bootstrap. Smoothed importance resampling procedures were developed for the estimation of distribution functions of the Studentized mean, the Studentized variance, and the correlation coefficient. Implementation of these procedures are presented via simulation results which concentrate on the problem of estimation of distribution functions of the Studentized mean and Studentized variance for different sample sizes and various pre-specified smoothing bandwidths for the normal data; additional simulations were conducted for the estimation of quantiles of the distribution of the Studentized mean under an optimal smoothing bandwidth when the original data were simulated from three different parent populations: lognormal, t(3) and t(10). These results suggest that in cases where it is advantageous to use the smoothed bootstrap rather than the standard bootstrap, the amount of resampling necessary might be substantially reduced by the use of importance resampling methods and the efficiency gains depend on the bandwidth used in the kernel density estimation.     

Bayesian Conditional Mean Estimation in Log‐Normal Linear Regression Models with Finite Quadratic Expected Loss

Log‐normal linear regression models are popular in many fields of research. Bayesian estimation of the conditional mean of the dependent variable is problematic as many choices of the prior for the variance (on the log‐scale) lead to posterior distributions with no finite moments. We propose a generalized inverse Gaussian prior for this variance and derive the conditions on the prior parameters that yield posterior distributions of the conditional mean of the dependent variable with finite moments up to a pre‐specified order. The conditions depend on one of the three parameters of the suggested prior; the other two have an influence on inferences for small and medium sample sizes. A second goal of this paper is to discuss how to choose these parameters according to different criteria including the optimization of frequentist properties of posterior means.     

Elicitation

There are various situations in which it may be important to obtain expert opinion about some unknown quantity or quantities. But it is not enough simply to ask the expert for an estimate of the unknown quantity: we also need to know how far from that estimate the true value might be. Tony O'Hagan describes the process of elicitation: the formulation of the expert's knowledge in the form of a probability distribution.     

The absorption distribution and the q-binomial theorem

Approximations for the mean and variance for the absorption distribution are found by means of a transformation of ordinary binomial to Gaussian or q-binomial coefficients.     

A New Procedure to Monitor the Mean of a Quality Characteristic

The Shewhart, Bonferroni-adjustment, and analysis of means (ANOM) control charts are typically applied to monitor the mean of a quality characteristic. The Shewhart and Bonferroni procedure are utilized to recognize special causes in production process, where the control limits are constructed by assuming normal distribution for known parameters (mean and standard deviation), and approximately normal distribution regarding to unknown parameters. The ANOM method is an alternative to the analysis of variance method. It can be used to establish the mean control charts by applying equicorrelated multivariate non central t distribution. In this article, we establish new control charts, in phases I and II monitoring, based on normal and t distributions having as a cause a known (or unknown) parameter (standard deviation). Our proposed methods are at least as effective as the classical Shewhart methods and have some advantages.     

A Bayesian Approach to the Estimation of Expected Cell Counts by Using Log-Linear Models

ABSTRACT

In this article, Bayesian estimation of the expected cell counts for log-linear models is considered. The prior specified for log-linear parameters is used to determine a prior for expected cell counts, by means of the family and parameters of prior distributions. This approach is more cost-effective than working directly with cell counts because converting prior information into a prior distribution on the log-linear parameters is easier than that of on the expected cell counts. While proceeding from the prior on log-linear parameters to the prior of the expected cell counts, we faced with a singularity problem of variance matrix of the prior distribution, and added a new precision parameter to solve the problem. A numerical example is also given to illustrate the usage of the new parameter.     

Change Point Detection in The Skew-Normal Model Parameters

Bayesian inference under the skew-normal family of distributions is discussed using an arbitrary proper prior for the skewness parameter. In particular, we review some results when a skew-normal prior distribution is considered. Considering this particular prior, we provide a stochastic representation of the posterior of the skewness parameter. Moreover, we obtain analytical expressions for the posterior mean and variance of the skewness parameter. The ultimate goal is to consider these results to one change point identification in the parameters of the location-scale skew-normal model. Some Latin American emerging market datasets are used to illustrate the methodology developed in this work.