CUSUM Control Charts for Multivariate Poisson Distribution

A cumulative sum control chart for multivariate Poisson distribution (MP-CUSUM) is proposed. The MP-CUSUM chart is constructed based on log-likelihood ratios with in-control parameters, Θ₀, and shifts to be detected quickly, Θ₁. The average run length (ARL) values are obtained using a Markov Chain-based method. Numerical experiments show that the MP-CUSUM chart is effective in detecting parameter shifts in terms of ARL. The MP-CUSUM chart with smaller Θ₁ is more sensitive than that with greater Θ₁ to smaller shifts, but more insensitive to greater shifts. A comparison shows that the proposed MP-CUSUM chart outperforms an existing MP chart.     

Nonparametric Bayesian inference for multivariate density functions using Feller priors

Multivariate density estimation plays an important role in investigating the mechanism of high-dimensional data. This article describes a nonparametric Bayesian approach to the estimation of multivariate densities. A general procedure is proposed for constructing Feller priors for multivariate densities and their theoretical properties as nonparametric priors are established. A blocked Gibbs sampling algorithm is devised to sample from the posterior of the multivariate density. A simulation study is conducted to evaluate the performance of the procedure.     

Calculating the power or sample size for the logistic and proportional hazards models

An algorithm is presented for calculating the power for the logistic and proportional hazards models in which some of the covariates are discrete and the remainders are multivariate normal. The mean and covariance matrix of the multivariate normal covariates may depend on the discrete covariates.

The algorithm, which finds the power of the Wald test, uses the result that the information matrix can be calculated using univariate numerical integration even when there are several continuous covariates. The algorithm is checked using simulation and in certain situations gives more accurate results than current methods which are based on simple formulae. The algorithm is used to explore properties of these models, in particular, the power gain from a prognostic covariate in the analysis of a clinical trial or observational study. The methods can be extended to determine power for other generalized linear models.     

Minimally informative distributions with given rank correlation for use in uncertainty analysis

Minimum information bivariate distributions with uniform marginals and a specified rank correlation are studied in this paper. These distributions play an important role in a particular way of modeling dependent random variables which has been used in the computer code UNICORN for carrying out uncertainty analyses. It is shown that these minimum information distributions have a particular form which makes simulation of conditional distributions very simple. Approximations to the continuous distributions are discussed and explicit formulae are determined. Finally a relation is discussed to DAD theorems, and a numerical algorithm is given (which has geometric rate of covergence) for determining the minimum information distributions.     

On a system of nonhomogeneous components sharing a common frailty

The components of a reliability system subjected to a common random environment usually have dependent lifetimes. This paper studies the stochastic properties of such a system with lifetimes of the components following multivariate frailty models and multivariate mixed proportional reversed hazard rate (PRHR) models, respectively. Through doing stochastic comparison, we devote to throwing a new light on how the random environment affects the number of working components of a reliability system and on assessing the performance of a k-out-of-n system.     

Testing the equality of several covariance matrices

Using a new approach based on Meijer G-functions and computer simulation, we numerically compute the exact null distribution of the modified-likelihood ratio statistic used to test the hypothesis that several covariances matrices of normal distributions are equal. Small samples of different sizes are considered, and for the case of two matrices, we introduce a new test based on determinants, with the null distribution of its criterion also fully computable. Comparisons with published results show the accuracy of our approach, which is proved to be more flexible and adaptable to different cases.     

On designing an acceptance sampling plan for the Pareto lifetime model

In this paper, the design of reliability sampling plans for the Pareto lifetime model under progressive Type-II right censoring is considered. Sampling plans are derived using the decision theoretic approach with a suitable loss or cost function that consists of sampling cost, rejection cost, and acceptance cost. The decision rule is based on the estimated reliability function. Plans are constructed within the Bayesian context using the natural conjugate prior. Simulations for evaluating the Bayes risk are carried out and the optimal sampling plans are reported for various sample sizes, observed number of failures and removal probabilities.     

A Bayesian approach to non-parametric monotone function estimation

Summary.  The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.