Percentage points of the statistics for testing hypotheses on mean vectors of multivariate normal distributions with missing observations

A problem of testing of hypotheses on the mean vector of a multivariate normal distribution with unknown and positive definite covariance matrix is considered when a sample with a special, though not unusual, pattern of missing observations from that population is available. The approximate percentage points of the test statistic are obtained and their accuracy has been checked by comparing them with some exact percentage points which are calculated for complete samples and some special incomplete samples. The approximate percentage points are in good agreement with exact percentage points. The above work is extended to the problem of testing the hypothesis of equality of two mean vectors of two multivariate normal distributions with the same, unknown covariance matrix     

Model-based clustering of multivariate skew data with circular components and missing values

Motivated by classification issues that arise in marine studies, we propose a latent-class mixture model for the unsupervised classification of incomplete quadrivariate data with two linear and two circular components. The model integrates bivariate circular densities and bivariate skew normal densities to capture the association between toroidal clusters of bivariate circular observations and planar clusters of bivariate linear observations. Maximum-likelihood estimation of the model is facilitated by an expectation maximization (EM) algorithm that treats unknown class membership and missing values as different sources of incomplete information. The model is exploited on hourly observations of wind speed and direction and wave height and direction to identify a number of sea regimes, which represent specific distributional shapes that the data take under environmental latent conditions.     

Estimation of parameters for a Birnbaum–Saunders regression model with censored data

Little work has been published on the analysis of censored data for the Birnbaum–Saunders distribution (BISA). In this article, we implement the EM algorithm to fit a regression model with censored data when the failure times follow the BISA. Three approaches to implement the E-Step of the EM algorithm are considered. In two of these implementations, the M-Step is attained by an iterative least-squares procedure. The algorithm is exemplified with a single explanatory variable in the model.     

Implementing particle swarm optimization algorithm to estimate the mixture of two Weibull parameters with censored data

We present the maximum likelihood estimation (MLE) via particle swarm optimization (PSO) algorithm to estimate the mixture of two Weibull parameters with complete and multiple censored data. A simulation study is conducted to assess the performance of the MLE via PSO algorithm, quasi-Newton method and expectation-maximization (EM) algorithm for different parameter settings and sample sizes in both uncensored and censored cases. The simulation results showed that the PSO algorithm outperforms the quasi-Newton method and the EM algorithm in most cases regarding bias and root mean square errors. Two numerical examples are used to demonstrate the performance of our proposed method.     

Bayes Analysis and Comparison of Accelerated Weibull and Accelerated Birnbaum–Saunders Models

Several models are proposed in the literature for modeling fatigue data resulting from materials subject to cyclic stress and strain. Accelerated Weibull and accelerated Birnbaum–Saunders distributions are most commonly used models. Whereas the accelerated Weibull model is easier compared to accelerated Birnbaum–Saunders, it fails to represent the situation equally well. The present article focuses on Bayes analysis of the two models and provides a comparison based on some important Bayesian tools. Model compatibility study using predictive simulation ideas is preceded by the said comparison. Throughout, the posterior simulations are carried out by Markov chain Monte Carlo procedure.     

Estimation under modified Weibull distribution based on right censored generalized order statistics

In this paper, the maximum likelihood (ML) and Bayes, by using Markov chain Monte Carlo (MCMC), methods are considered to estimate the parameters of three-parameter modified Weibull distribution (MWD(β, τ, λ)) based on a right censored sample of generalized order statistics (gos). Simulation experiments are conducted to demonstrate the efficiency of the proposed methods. Some comparisons are carried out between the ML and Bayes methods by computing the mean squared errors (MSEs), Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates to illustrate the paper. Three real data sets from Weibull(α, β) distribution are introduced and analyzed using the MWD(β, τ, λ) and also using the Weibull(α, β) distribution. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic, {AIC and BIC} to emphasize that the MWD(β, τ, λ) fits the data better than the other distribution. All parameters are estimated based on type-II censored sample, censored upper record values and progressively type-II censored sample which are generated from the real data sets.     

A generating algorithm for linear trend-free and nearly linear trend-free block designs

The theory and properties of trend-free (TF) and nearly trend-free (NTF) block designs are wel1 developed. Applications have been hampered because a methodology for design construction has not been available.

This article begins with a short review of concepts and properties of TF and NTF block designs. The major contribution is provision of an algorithm for the construction of linear and nearly linear TF block designs. The algorithm is incorporated in a computer program in FORTRAN 77 provided in an appendix for the IBM PC or compatible microcomputer, a program adaptable also to other computers. Three sets of block designs generated by the program are given as examples.

A numerical example of analysis of a linear trend-free balanced incomplete block design is provided.     

The robust economic statistical design of the Hotelling's T2 chart

ABSTRACT

Economic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T² chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.     

A time-varying measurement error model for age of onset of a psychiatric diagnosis: applied to first depressive episode diagnosed by the Composite International Diagnostic Interview (CIDI)

A substantial degree of uncertainty exists surrounding the reconstruction of events based on memory recall. This form of measurement error affects the performance of structured interviews such as the Composite International Diagnostic Interview (CIDI), an important tool to assess mental health in the community. Measurement error probably explains the discrepancy in estimates between longitudinal studies with repeated assessments (the gold-standard), yielding approximately constant rates of depression, versus cross-sectional studies which often find increasing rates closer in time to the interview. Repeated assessments of current status (or recent history) are more reliable than reconstruction of a person's psychiatric history based on a single interview. In this paper, we demonstrate a method of estimating a time-varying measurement error distribution in the age of onset of an initial depressive episode, as diagnosed by the CIDI, based on an assumption regarding age-specific incidence rates. High-dimensional non-parametric estimation is achieved by the EM-algorithm with smoothing. The method is applied to data from a Norwegian mental health survey in 2000. The measurement error distribution changes dramatically from 1980 to 2000, with increasing variance and greater bias further away in time from the interview. Some influence of the measurement error on already published results is found.     

Reliability models for categorical data

As assumed hypothetical consensus category corresponding to a case being classified provides a basis for assessment of reliability of judges. Equivalent judges are characterised by the joint probability distribution of the judge assignment and the consensus category. Estimates of the conditional probabilities of judge assignment given consensus category and of consensus category given judge assignments are indices of reliability. All parameters can be estimated if data include classifications of a number of cases by 3 or more judges. Restrictive assumptions are imposed to obtain models for data from classifications by two judges. Maximum likelihood estimation is discussed and illustrated by example for the 3 or more judges case.