Robustness of R-Chart to Non Normality

This study addresses the appropriate d ₃ values for constructing range control charts (R-charts) when the distributions of the processes are the uniform, triangular, exponential, and Erlang. Comparisons of the range charts are based on Type I error probabilities obtained using simulations. The results reveal that inappropriate use of the d ₃ values strongly affected the performance of the R-charts. Practitioners should be more careful in selecting suitable coefficients when using R-charts methods to process data. The distribution of the processes must be examined before the coefficients are chosen.     

Planning Life Tests Based on Progressively Type-I Grouped Censored Data from the Weibull Distribution

In this article, we apply the simulated annealing algorithm to determine optimally spaced inspection times for the two-parameter Weibull distribution for any given progressive Type-I grouped censoring plan. We examine how the asymptotic relative efficiencies of the estimates are affected by the position of the monitoring points and the number of monitoring points used. A comparison of different inspection plans is made that will enable the user to select a plan for a specified quality goal. Using the same algorithm, we can also determine an optimal progressive Type-I grouped censoring plan when the inspection times and the expected proportions of total failures in the experiment are pre-fixed. Finally, we discuss the sample size and the acceptance constant of the progressively Type-I grouped censored reliability sampling plan when the optimal inspection times are used.     

A Multiple Three-Decision Procedures for Comparing Several Treatments with a Control Under Heteroscedasticity

In this article, the design-oriented two-stage multiple three-decision procedure is proposed to classify a set of normal populations with respect to a control under heteroscedasticity. The statistical tables of percentage points and the power-related design constants, to implement our new two-stage procedure, are given. Sometimes when the sample for the second stage is not available, the one-stage data analysis procedure is proposed. Classifying a treatment better than control when it is actually worse (and vice versa) is known as type III error. Both the two-stage and one-stage procedures control the type III error rate at a specified level. The relationship between the two-stage and one-stage procedures is discussed. Finally, the application of the proposed procedures is illustrated with an example.     

A Three-Stage Bayesian Adaptive Phase I/II Design and Simulation Studies

Instead of using traditional separate phase I and II trials, in this article, we propose using a parallel three-stage phase I/II design, incorporating a dose expansion approach to flexibly evaluate the safety and efficacy of dose levels, and to select the optimal dose. In the proposed design, both the toxicity and efficacy responses are binary endpoints. A 3+3-based procedure is used for initial period of dose escalation at stage 1; at this level, the dose can be expanded to stage 2 for exploratory efficacy studies of phase IIa, while simultaneously, the safety testing can advance to a higher dose level. A beta-binomial model is used to model the efficacy responses. There are two placebo-controlled randomization interim monitoring analyses at stage 2 to select the promising doses to be recommended to stage 3 for further efficacy studies of phase IIb. An adaptive randomization approach is used to assign more patients to doses with higher efficacy levels at stage 3. We examine the properties of the proposed design through extensive simulation studies by using R programming language, and also compare the new design with the conventional design and a competing adaptive Bayesian design. The simulation results show that our design can efficiently assign more patients to doses with higher efficacy levels and is superior to the two competing designs in terms of total sample size reduction.     

Notes on the two-stage procedure under an incomplete block two-period crossover design

Abstract

Under an incomplete block crossover design with two periods, we derive the least-squares estimators for the period effect, treatment effects and carry-over effects in explicit formulae based on within-patient differences. Using the commonly-used strategy of searching a base model for making inferences in regression analysis, we define a two-stage test procedure in studying treatment effects. On the basis of Monte Carlo simulation, we evaluate the performance of the two-stage procedure for hypothesis testing, point and interval estimation of treatment effects in a variety of situations. We note that use of the two-stage procedure can be potentially misleading and hence one should not apply a test procedure to exclusively determine whether he/she needs to account for the carry-over effect in studying treatment effects. We use the double-blind crossover trial comparing two different doses of formoterol with placebo on the forced expiratory volume in 1 second (FEV₁) readings to illustrate the use of the two-stage procedure, as well as the distinction between use of two-stage procedure and the approach with assuming no carry-over effects based on one's subjective knowledge.     

Testing hypotheses in truncated samples by means of divergence statistics

In this paper we consider the problem of testing hypotheses in parametric models, when only the first r (of n) ordered observations are known.Using divergence measures, a procedure to test statistical hypotheses is proposed, Replacing the parameters by suitable estimators in the expresion of the divergence measure, the test statistics are obtained.Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators for truncated samples are considered.Applications of these results in testing statistical hypotheses, on the basis of truncated data, are presented.The small sample behavior of the proposed test statistics is analyzed in particular cases.A comparative study of power values is carried out by computer simulation.     

Bayesian life test sampling plans for the two parameter exponential distribution

In this paper we consider the determination of Bayesian life test acceptance sampling plans for finite lots when the underlying lifetime distribution is the two parameter exponential. It is assumed that the prior distribution is the natural conjugate prior, that the costs associated with the actions accept and reject are known functions of the lifetimes of the items, and that the cost of testing a sample is proportional to the duration of the test. Type 2 censored sampling is considered where a sample of size n is observed only until the rth failure occurs and the decision of whether to accept or reject the remainder of the lot is made on the basis of the r observed lifetimes. Obtaining the optimal sample size and the optimal censoring number are difficult problems when the location parameter of the distribution is restricted to be non-negative. The case when the positivity restriction on the location parameter is removed has been investigated. An example is provided for illustration.     

Approximate mrpp p-values obtained from four exact moments

A F0RTRAN-77 subroutine for a general version of multi-response permutation procedures (MRPP) is described. The exact four moments are employed in conjunction with the Pearson type I, type III, and type VI distributions to calculate the associated P-values.     

Asymptotic equivalence between two empirical mean residual life processes

There are two mean residual life estimates for right censored data. One is based on the Kaplan-Meier estimate, the other, based on the Susarla-Van Ryzin estimate for survival function. In this paper, we define the empirical mean residual life process for right censored data and show that the two empirical mean residual life processes based on the Kaplan-Meier and Susarla-Van Ryzin estimates are asymptotically equivalent uniformly on an interval under some conditions. Also we discuss the case which the asymptotic equivalence might fail.     

Sharp Bounds for Generalized Order Statistics via Logarithmic Moments

ABSTRACT

We present sharp bounds for expectations of generalized order statistics with random indices. The bounds are expressed in terms of logarithmic moments E X ^a (log max {1, X}) ^b of the underlying observation X. They are attainable and provide characterizations of some non trivial distributions. No restrictions are imposed on the parameters of the generalized order statistics model.