Composite transformation models: A fiducial perspective

This paper discusses inferences for the parameters of a transformation model in the presence of a scalar nuisance parameter that describes the shape of the error distribution. The development is from the point of view of conditional inference and thus is an attempt to extend the classical fiducial (or structural inference) argument. For known shape parameter it is straightforward to derive a fiducial distribution of the transformation parameters from which confidence points can be obtained. For unknown shape parameter, the paper discusses a certain average of these fiducial distributions. The weights used in this averaging process are naturally induced by the action of the underlying group of transformations and correspond to a noninformative prior for the nuisance parameter. This results in a confidence distribution for the transformation parameters which in some cases has good frequentist properties. The method is illustrated by some examples.     

Natural (Non‐)Informative Priors for Skew‐symmetric Distributions

In this paper, we present an innovative method for constructing proper priors for the skewness (shape) parameter in the skew‐symmetric family of distributions. The proposed method is based on assigning a prior distribution on the perturbation effect of the shape parameter, which is quantified in terms of the total variation distance. We discuss strategies to translate prior beliefs about the asymmetry of the data into an informative prior distribution of this class. We show via a Monte Carlo simulation study that our non‐informative priors induce posterior distributions with good frequentist properties, similar to those of the Jeffreys prior. Our informative priors yield better results than their competitors from the literature. We also propose a scale‐invariant and location‐invariant prior structure for models with unknown location and scale parameters and provide sufficient conditions for the propriety of the corresponding posterior distribution. Illustrative examples are presented using simulated and real data.     

A Matching Prior for the Shape Parameter of the Skew‐Normal Distribution

Abstract. This paper deals with the issue of performing a default Bayesian analysis on the shape parameter of the skew‐normal distribution. Our approach is based on a suitable pseudo‐likelihood function and a matching prior distribution for this parameter, when location (or regression) and scale parameters are unknown. This approach is important for both theoretical and practical reasons. From a theoretical perspective, it is shown that the proposed matching prior is proper thus inducing a proper posterior distribution for the shape parameter, also when the likelihood is monotone. From the practical perspective, the proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals.     

Quasi-Likelihood Regression with Multiple Indices and Smooth Link and Variance Functions

Abstract.  A flexible semi-parametric regression model is proposed for modelling the relationship between a response and multivariate predictor variables. The proposed multiple-index model includes smooth unknown link and variance functions that are estimated non-parametrically. Data-adaptive methods for automatic smoothing parameter selection and for the choice of the number of indices M are considered. This model adapts to complex data structures and provides efficient adaptive estimation through the variance function component in the sense that the asymptotic distribution is the same as if the non-parametric components are known. We develop iterative estimation schemes, which include a constrained projection method for the case where the regression parameter vectors are mutually orthogonal. The proposed methods are illustrated with the analysis of data from a growth bioassay and a reproduction experiment with medflies. Asymptotic properties of the estimated model components are also obtained.     

Maximum Likelihood Estimation for the 4-Parameter Beta Distribution

This paper addresses the problem of obtaining maximum likelihood estimates for the parameters of the Pearson Type I distribution (beta distribution with unknown end points and shape parameters). Since they do not seem to have appeared in the literature, the likelihood equations and the information matrix are derived. The regularity conditions which ensure asymptotic normality and efficiency are examined, and some apparent conflicts in the literature are noted. To ensure regularity, the shape parameters must be greater than two, giving an (assymmetrical) bell-shaped distribution with high contact in the tails. A numerical investigation was carried out to explore the bias and variance of the maximum likelihood estimates and their dependence on sample size. The numerical study indicated that only for large samples (n ≥ 1000) does the bias in the estimates become small and does the Cramér-Rao bound give a good approximation for their variance. The likelihood function has a global maximum which corresponds to parameter estimates that are inadmissable. Useful parameter estimates can be obtained at a local maximum, which is sometimes difficult to locate when the sample size is small.     

On the Choice of a Prior for Bayesian D-Optimal Designs for the Logistic Regression Model with a Single Predictor

The Bayesian design approach accounts for uncertainty of the parameter values on which optimal design depends, but Bayesian designs themselves depend on the choice of a prior distribution for the parameter values. This article investigates Bayesian D-optimal designs for two-parameter logistic models, using numerical search. We show three things: (1) a prior with large variance leads to a design that remains highly efficient under other priors, (2) uniform and normal priors lead to equally efficient designs, and (3) designs with four or five equidistant equally weighted design points are highly efficient relative to the Bayesian D-optimal designs.     

Bayes estimation and prediction of the two-parameter gamma distribution

In this article, the Bayes estimates of two-parameter gamma distribution are considered. It is well known that the Bayes estimators of the two-parameter gamma distribution do not have compact form. In this paper, it is assumed that the scale parameter has a gamma prior and the shape parameter has any log-concave prior, and they are independently distributed. Under the above priors, we use Gibbs sampling technique to generate samples from the posterior density function. Based on the generated samples, we can compute the Bayes estimates of the unknown parameters and can also construct HPD credible intervals. We also compute the approximate Bayes estimates using Lindley's approximation under the assumption of gamma priors of the shape parameter. Monte Carlo simulations are performed to compare the performances of the Bayes estimators with the classical estimators. One data analysis is performed for illustrative purposes. We further discuss the Bayesian prediction of future observation based on the observed sample and it is seen that the Gibbs sampling technique can be used quite effectively for estimating the posterior predictive density and also for constructing predictive intervals of the order statistics from the future sample.     

Estimation of parameters in DNA mixture analysis

In [7], a Bayesian network for analysis of mixed traces of DNA was presented using gamma distributions for modelling peak sizes in the electropherogram. It was demonstrated that the analysis was sensitive to the choice of a variance factor and hence this should be adapted to any new trace analysed. In this paper, we discuss how the variance parameter can be estimated by maximum likelihood to achieve this. The unknown proportions of DNA from each contributor can similarly be estimated by maximum likelihood jointly with the variance parameter. Furthermore, we discuss how to incorporate prior knowledge about the parameters in a Bayesian analysis. The proposed estimation methods are illustrated through a few examples of applications for calculating evidential value in casework and for mixture deconvolution.     

Bayesian analysis for partially complete time and type of failure data

In this paper, we consider the Bayesian analysis of competing risks data, when the data are partially complete in both time and type of failures. It is assumed that the latent cause of failures have independent Weibull distributions with the common shape parameter, but different scale parameters. When the shape parameter is known, it is assumed that the scale parameters have Beta–Gamma priors. In this case, the Bayes estimates and the associated credible intervals can be obtained in explicit forms. When the shape parameter is also unknown, it is assumed that it has a very flexible log-concave prior density functions. When the common shape parameter is unknown, the Bayes estimates of the unknown parameters and the associated credible intervals cannot be obtained in explicit forms. We propose to use Markov Chain Monte Carlo sampling technique to compute Bayes estimates and also to compute associated credible intervals. We further consider the case when the covariates are also present. The analysis of two competing risks data sets, one with covariates and the other without covariates, have been performed for illustrative purposes. It is observed that the proposed model is very flexible, and the method is very easy to implement in practice.     

Bayesian predictive power: choice of prior and some recommendations for its use as probability of success in drug development

Bayesian predictive power, the expectation of the power function with respect to a prior distribution for the true underlying effect size, is routinely used in drug development to quantify the probability of success of a clinical trial. Choosing the prior is crucial for the properties and interpretability of Bayesian predictive power. We review recommendations on the choice of prior for Bayesian predictive power and explore its features as a function of the prior. The density of power values induced by a given prior is derived analytically and its shape characterized. We find that for a typical clinical trial scenario, this density has a u‐shape very similar, but not equal, to a β‐distribution. Alternative priors are discussed, and practical recommendations to assess the sensitivity of Bayesian predictive power to its input parameters are provided. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of hybrid censored competing risks data

In this paper, we consider the analysis of hybrid censored competing risks data, based on Cox's latent failure time model assumptions. It is assumed that lifetime distributions of latent causes of failure follow Weibull distribution with the same shape parameter, but different scale parameters. Maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving a one-dimensional optimization problem, and we propose a fixed-point type algorithm to solve this optimization problem. Approximate MLEs have been proposed based on Taylor series expansion, and they have explicit expressions. Bayesian inference of the unknown parameters are obtained based on the assumption that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameters have Beta–Gamma priors. We propose to use Markov Chain Monte Carlo samples to compute Bayes estimates and also to construct highest posterior density credible intervals. Monte Carlo simulations are performed to investigate the performances of the different estimators, and two data sets have been analysed for illustrative purposes.     

Variance Shifts,Structural Breaks,and Stationarity Tests

This article considers the problem of testing the null hypothesis of stochastic stationarity in time series characterized by variance shifts at some (known or unknown) point in the sample. It is shown that existing stationarity tests can be severely biased in the presence of such shifts, either oversized or undersized, with associated spurious power gains or losses, depending on the values of the breakpoint parameter and on the ratio of the prebreak to postbreak variance. Under the assumption of a serially independent Gaussian error term with known break date and known variance ratio, a locally best invariant (LBI) test of the null hypothesis of stationarity in the presence of variance shifts is then derived. Both the test statistic and its asymptotic null distribution depend on the breakpoint parameter and also, in general, on the variance ratio. Modifications of the LBI test statistic are proposed for which the limiting distribution is independent of such nuisance parameters and belongs to the family of Cramér–von Mises distributions. One such modification is particularly appealing in that it is simultaneously exact invariant to variance shifts and to structural breaks in the slope and/or level of the series. Monte Carlo simulations demonstrate that the power loss from using our modified statistics in place of the LBI statistic is not large, even in the neighborhood of the null hypothesis, and particularly for series with shifts in the slope and/or level. The tests are extended to cover the cases of weakly dependent error processes and unknown breakpoints. The implementation of the tests are illustrated using output, inflation, and exchange rate data series.     

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.     

Wavelet thresholding via a Bayesian approach

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be difficult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.     

A prior for the variance in hierarchical models

The choice of prior distributions for the variances can be important and quite difficult in Bayesian hierarchical and variance component models. For situations where little prior information is available, a ‘nonin-formative’ type prior is usually chosen. ‘Noninformative’ priors have been discussed by many authors and used in many contexts. However, care must be taken using these prior distributions as many are improper and thus, can lead to improper posterior distributions. Additionally, in small samples, these priors can be ‘informative’. In this paper, we investigate a proper ‘vague’ prior, the uniform shrinkage prior (Strawder-man 1971; Christiansen & Morris 1997). We discuss its properties and show how posterior distributions for common hierarchical models using this prior lead to proper posterior distributions. We also illustrate the attractive frequentist properties of this prior for a normal hierarchical model including testing and estimation. To conclude, we generalize this prior to the multivariate situation of a covariance matrix.     

Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.     

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.     

Selecting more reliable weibull populations than a control

This paper deals with the selection of Weibull populations that are more reliable than a control population at some specified time. In the case when the shape parameters are known, a locally optimal selection rule is derived. From this rule, a modified one is proposed for the case when the shape parameter is unknown but has a known prior distribution. Simulation study shows that this modified rule is quite robust.     

Bayesian Analysis of a Generalized Poisson Distribution

A generalized form of the Poisson Distribution with two parameters will be estimated by the Bayesian technique. When one of the parameters is known, several important parametric functions will be estimated and a numerical comparison with estimates obtained by the methods of maximum likelihood and unbiased minimum variance will be drawn. The simplicity of the posterior distribution of the unknown parameter enables us to construct exact probability intervals, and to devise a statistic to test the homogeneity of several populations. When the two parameters are unknown, dependent priors are being considered. Although the posterior distributions are sensitive to the choice of the prior, the posterior estimates are very stable and we use the Pearson system of curves to construct approximate posterior confidence limits for the parameters.     

A Conway Maxwell Poisson type generalization of the negative hypergeometric distribution

Abstract

Negative hypergeometric distribution arises as a waiting time distribution when we sample without replacement from a finite population. It has applications in many areas such as inspection sampling and estimation of wildlife populations. However, as is well known, the negative hypergeometric distribution is over-dispersed in the sense that its variance is greater than the mean. To make it more flexible and versatile, we propose a modified version of negative hypergeometric distribution called COM-Negative Hypergeometric distribution (COM-NH) by introducing a shape parameter as in the COM-Poisson and COMP-Binomial distributions. It is shown that under some limiting conditions, COM-NH approaches to a distribution that we call the COM-Negative binomial (COMP-NB), which in turn, approaches to the COM Poisson distribution. For the proposed model, we investigate the dispersion characteristics and shape of the probability mass function for different combinations of parameters. We also develop statistical inference for this model including parameter estimation and hypothesis tests. In particular, we investigate some properties such as bias, MSE, and coverage probabilities of the maximum likelihood estimators for its parameters by Monte Carlo simulation and likelihood ratio test to assess shape parameter of the underlying model. We present illustrative data to provide discussion.