Assurance test and its equivalent truncated sequential test

Abstract

In order to save more test cost, assurance test and its equivalent truncated sequential test are studied. In a commonly used case, the operating characteristic (OC) function and expected test time (ETT) function of an assurance test are derived in a concise way. Equivalent test and relative concepts are defined. The procedures to construct a near equivalent truncated sequential test of an assurance test are established. Computation studies show that the near equivalent truncated sequential tests proposed in this paper keep almost the same OC curves with the assurance tests respectively. However, they can save the ETTs dramatically. In fact, the results show that the near equivalent truncated sequential tests can save around 50% of ETTs than the assurance tests respectively.     

Optimal group sequential designs constrained on both overall and stage one error rates

Abstract

Optimized group sequential designs proposed in the literature have designs minimizing average sample size with respect to a prior distribution of treatment effect with overall type I and type II error rates well-controlled (i.e., at final stage). The optimized asymmetric group sequential designs that we present here additionally consider constrains on stopping probabilities at stage one: probability of stopping for futility at stage one when no drug effect exists as well as the probability of rejection when the maximum effect size is true at stage one so that accountability of group sequential design is ensured from the first stage throughout.     

Detecting Abrupt Leaks in Blended Underground Storage Tanks

We suggest and compare two multiple change-point algorithms (segmentation and sequential) for accurate detection of the onset of abrupt leaks in blended underground storage tanks. We apply these algorithms to two simulated scenarios to demonstrate the advantages of the sequential algorithm, and then we apply the sequential algorithm to the Cary blended site data. In addition, we obtain a confidence set for the locations of the change points conditional on the number of change points by inverting the related hypothesis test.     

Parallel analysis approach for determining dimensionality in canonical correlation analysis

ABSTRACT

Canonical correlations are maximized correlation coefficients indicating the relationships between pairs of canonical variates that are linear combinations of the two sets of original variables. The number of non-zero canonical correlations in a population is called its dimensionality. Parallel analysis (PA) is an empirical method for determining the number of principal components or factors that should be retained in factor analysis. An example is given to illustrate for adapting proposed procedures based on PA and bootstrap modified PA to the context of canonical correlation analysis (CCA). The performances of the proposed procedures are evaluated in a simulation study by their comparison with traditional sequential test procedures with respect to the under-, correct- and over-determination of dimensionality in CCA.     

Asymptotic optimality of double sequential mixture likelihood ratio test

In this article, we propose to use the weighted expected sample size (WESS) to evaluate the overall performance of sequential test plans on a finite set of parameters. Motivated by minimizing the WESS to control the expected sample sizes, we develop the method of double sequential mixture likelihood ratio test (2-SMLRT) for one-sided composite hypotheses. It is proved that the 2-SMLRT is asymptotically optimal on and its stopping time is finite under some conditions. The 2-SMLRT is general and includes the sequential probability ratio test (SPRT) and the double sequential probability ratio test (2-SPRT) as special cases. Simulation studies show that compared with the SPRT and 2-SPRT, the 2-SMLRT has smaller WESS and relative mean index with less or comparable expected sample sizes when the null hypothesis or alternative hypothesis holds.     

An Algorithm Based on Singular Spectrum Analysis for Change-Point Detection

Abstract

This paper is devoted to application of the singular-spectrum analysis to sequential detection of changes in time series. An algorithm of change-point detection in time series, based on sequential application of the singular-spectrum analysis is developed and studied. The algorithm is applied to different data sets and extensively studied numerically. For specific models, several numerical approximations to the error probabilities and the power function of the algorithm are obtained. Numerical comparisons with other methods are given.     

Sequential Procedures for Selecting the Most Probable Multinomial Cell When a Nuisance Cell Is Present

Abstract

This article considers the problem of selecting the most probable cell in a multinomial distribution in the presence of a nuisance cell. Two open sequential procedures are proposed and studied. One is a two-stage procedure and the other a multistage procedure.     

THE OPTIMAL TEST FOR SELECTING THE GREATER OF TWO BINOMIAL PROBABILITIES

The efficiency of a sequential test is related to the “importance” of the trials within the test. This relationship is used to find the optimal test for selecting the greater of two binomial probabilities, p_α and p_b, namely, the stopping rule is “gambler's ruin” and the optimal discipline when p_α+p_b≤ 1 (≥ 1) is play-the-winner (loser), i.e. an α-trial which results in a success is followed by an α-trial (b-trial) whereas an α-trial which results in a failure is followed by α b-trid (α-trial) and correspondingly for b-trials.     

Sequential order statistics from dependent random variables

Abstract

This paper provides an extension for “sequential order statistics” (SOS) introduced by Kamps. It is called “developed sequential order statistics” (DSOS) and is useful for describing lifetimes of engineering systems when component lifetimes are dependent. Explicit expressions for the joint density function, the marginal distributions and the means of DSOS are derived. Under the well known “conditional proportional hazard rate” (CPHR) model and the Gumbel families of copulas for dependency among component lifetimes, some findings are reported. For example, it is proved that the joint density functions of DSOS and SOS have the same structure. Various illustrative examples are also given.     

ASYMPTOTIC EXPANSIONS APPLIED TO F-TEST CRITERIA

Using asymptotic expansions of the Kummer hypergeometric function, the sequential. F-test criterion is evaluated asymptotically in terms of the sample size. The continuation region inequalities for the test are inverted and expressed in terms of known test criteria. A rapidly converging algorithm for carrying out the sequential procedure is provided. This makes the F-test easier for the practitioner to use. Almost sure finite termination of the sequential. F-test is asserted by appealing to the continuation inequalities and a heuristic asymptotic expansion of the test criterion. Average stopping times of the sequential procedure for a variety of population means and population number configurations are tabulated. The computer symbolic manipulation program MAPLE was used to derive some formulae.     

Optimal design of variable acceptance sampling plans for mixture distribution

ABSTRACT

The paper investigates the design of single and sequential variable acceptance sampling plans for a mixture distribution. Mixture distributions are seen in many practical problems such as life testing experiments of electronic components and clinical trials. The sampling plans for this kind of situations are not well addressed in the literature. We first propose a single sampling plan for a distribution which is a mixture of two exponential distributions. An optimization problem which minimizes the total cost of testing at given producer's and consumer's risks is solved to obtain the plan parameters. Two different sequential sampling plans are also defined and plan parameters are obtained by solving corresponding optimization problems. Finally, a case study, a simulation study and a sensitivity analysis are presented to illustrate our sampling plans.     

Incomplete block designs for asymmetric parallel line assays

Abstract

A method of construction of A-optimal binary block designs for asymmetrical parallel line assays, i.e., the assays in which the number doses for standard and test preparation are unequal has been considered. The method is illustrated with examples. Two cases of this method have been considered. In the first case, designs obtained are of equal replications of the doses. In the second case, designs with unequal replications are obtained.     

Point estimation after early stopping in a repeated measures trial

When the individual measurements are statistically independent, the maximum likelihood estimator calculated at the end of a sequential procedure overestimates the underlying effect. There are many clinical trials in which we are interested in comparing changes in responses between two treatment groups sequentially. Lee and DeMets (1991, JASA 86, 757–762) proposed a group sequential method for comparing rates of change when a response variable is measured for eaeh patient at successive follow-up visits. They assumed that the response follows the linear mixed effects model and derived the asymptotic joint distribution of the sequentially computed statistics. In this article, we consider the maximum likelihood estimator (MLE), the median unbiased estimator (MUE) and the midpoint of a 100(1-α)% confidence interval as point estimators for the rate of change in the linear mixed effects model, and investigate their properties by Monte Carlo simulation.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.     

THE SIMPLE LINEAR CALIBRATION PROBLEM AS AN OPTIMAL EXPERIMENTAL DESIGN

ABSTRACT

The aim of this paper is to consider the linear calibration problem through an optimal design approach, to evaluate the approximate variance of the calibrating value and provide approximate confidence intervals. The sequential approach through Stochastic Approximation is also applied to obtain a D-optimal design is considered.     

Sequential Fixed-Width Confidence Interval for a Function of the Parameters

Let X₁…, X_m and Y₁…, Y_n be two independent sequences of i.i.d. random variables with distribution functions F_x(.|θ) and F_y(. | φ) respectively. Let g(θ, φ) be a real-valued function of the unknown parameters θ and φ. The purpose of this paper is to suggest a sequential procedure which gives a fixed-width confidence interval for g(θ, φ) so that the coverage probability is approximately α (preas-signed). Certain asymptotic optimality properties of the sequential procedure are established. A Monte Carlo study is presented.     

Robustness Properties of the Pitman–Morgan Test

Abstract

The classical Pitman–Morgan test is known to be optimal for testing equality of the variances of components of a bivariate normal vector. We first show that it is also optimal for a generalized model involving the matrix spherical distribution. Then we discuss and demonstrate, both analytically and empirically, that it is nonrobust, i.e., its type I error control is inexact both asymptotically and in moderate size bivariate random samples.