An asymptotic distribution-free selection procedure for a two-way layout problem

This paper deals with an asymptotic distribution-free subset selection procedure for a two-way layout problem. The treatment effect with the largest unknown value is of interest to us. The block effect is a nuisance parameter in this problem. The proposed procedure is based on the Hodges-Lehmann estimators of location parameters. The asymptotic relative efficiency of the proposed procedure with the normal means procedure is evaluated. It is shown that the proposed procedure has a high efficiency.     

On a fisherian detour of the step-down procedure for manova

The well known step-down procedure for MANOVA given by J. Roy (1958) can be modified by combining the step-down tests using a B-optimal combination method such as Fisher's. The Fisherian detour of the stepwise MANOVA is shown to be asymptotically equivalent to the likelihood ratio test.     

A note on the effect of simple equicorrelation in detecting a spurious multivariate observation

This paper examines the robustness of the multivariate version of Grubs' (1950) procedure for detecting an outlier in a sample of n independent observations against equicorrelation of the observations. It is shown that the robustness of the univariate test to equicorrelation extends to the multivariate test in that the distribution of the maximum squared radii-test for a multivariate oulier in identical for both the independent and siaply equicorrelated data models.     

ON EMPIRICAL BAYES STOPPING TIMES

We consider the empirical Bayes decision theory where the component problems are the optimal fixed sample size decision problem and a sequential decision problem. With these components, an empirical Bayes decision procedure selects both a stopping rule function and a terminal decision rule function. Empirical Bayes stopping rules are constructed for each case and the asymptotic behaviours are investigated.     

Least-squares tests of time-series,invention effects with and without autocorrelations

Gottman's version of the Mann and Wald asymptotic test for intervention effects in time-series data is presented as a useful small sample procedure. A Monte Carlo simulaltion is conducted to evaluate the procedure for controlling Type I errors with varying values of autoregressive coefficients. Results indicate the procedure works better than Gottman's work originally indicated. However, in some cases error rates can be unacceptably high. Procedures for evaluating changes in level in the presence of autocorrelation and slope are suggested and evaluated.     

Error rates in classification consisting of discrete and continuous variables in the presence of covariates

We consider classifying an object based on mixed continuous and discrete variables between two populations. Mixed discrete and continuous covariates with identical means in both populations are amongst the variables. Under the location model with homogeneous location specific conditional dispersion matrices for both populations, the Bayes rule is given. Classification is implemented by a plug-in version of the Bayes rule with full covariate adjustment. An asymptotic expansion of the overall expected error of the procedure is derived. Our findings generalize several classical results.     

Sequential estimation of the mean of a first-order autoregressive process

This paper considers the problem of sequential point estimation, under an appropriate loss function, of the location parameter when the errors form an autoregressive process with unknown scale and autoregressive parameters, A sequential procedure is developed and an asymptotic second order expansion is provided for the difference between expected stopping time and the optimal fixed sample size procedure. Also, the asymptotic normality of the stopping time is proved. Though the procedure Is asymptotically risk efficient, it. Is not clear whether it has bounded regret.     

Selecting logistic populations using the sample medians

This paper is concerned primarily with subset selection procedures based on the sample mediansof logistic populations. A procedure is given which chooses a nonempty subset from among kindependent logistic populations, having a common known variance, so that the populations with thelargest location parameter is contained in the subset with a pre‐specified probability. Theconstants required to apply the median procedure with small sample sizes (≤= 19) are tabulated and can also be used to construct simultaneous confidence intervals. Asymptotic formulae are provided for application with larger sample sizes. It is shown that, under certain situations, rules based on the median are substantially more efficient than analogous procedures based either on sample means or on the sum of joint ranks.     

A Two-stage minimax procedure with screening for selecting the largest normal mean

The problem of selecting the normal population with the largest population mean when the populations have a common known variance is considered. A two-stage procedure is proposed which guarantees the same probability requirement using the indifference-zone approach as does the single-stage procedure of Bechhofer (1954). The two-stage procedure has the highly desirable property that the expected total number of observations required by the procedure is always less than the total number of observations required by the corresponding single-stage procedure, regardless of the configuration of the population means. The saving in expected total number of observations can be substantial, particularly when the configuration of the population means is favorable to the experimenter. The saving is accomplished by screening out “non-contending” populations in the first stage, and concentrating sampling only on “contending” populations in the second stage.

The two-stage procedure can be regarded as a composite one which uses a screening subset-type approach (Gupta (1956), (1965)) in the first stage, and an indifference-zone approach (Bechhofer (1954)) applied to all populations retained in the selected sub-set in the second stage. Constants to implement the procedure for various k and P? are provided, as are calculations giving the saving in expected total sample size if the two-stage procedure is used in place of the corresponding single-stage procedure.     

A two-stage minimax procedure with screening for selecting the largest normal mean (ii): an improved pcs lower bound and associated tables

This paper is a follow-up to an earlier article by the authors in which they proposed a two-stage procedure with screening to select the normal population with the largest population mean when the populations have a common known variance. The two-stage procedure has the highly desirable property that the expected total number of observations required by the procedure is always less than the total number of observations required by the corresponding single-stage procedure of Bechhofer (1954), regardless of the configuration of the population means. The present paper contains new results which make possible the more efficient implementation of the two-stage procedure. Tables for this purpose are given, and the improvements achieved (which are substantial) are assessed.     

On a conjecture concerning the least favorable configuration of certain two-stage selection procedures

Given k normal populations with unknown means and a common known variance a two-stage procedure p₁ with screening in the first stage to find the population with the largest mean using the indifference-zone approach is under concern. It was proposed and studied previously by Cohen (1959), Alam (1970) and Tamhane and Bechhofer (1977, 1979). But up to now a conjecture concerning the least favorable parameter configuration of p₁remained unproved for k ≥ 3. In this paper we give a non-standard proof of the conjecture in case of k = 3 for p₁. which (under minor changes) works also for a simplified version of p₁,. Besides, a counterexample is provided to show that another (more intuitive) method of proof fails to work.     

Recursive procedures for multiple decisions: finite time memory and stepwise maximum likelihood procedure

A version of the multiple decsion problem is studied in which the procedure is based only on the current observation and the previous decision. A necessary and sufficient condition for inconsistency of the stepwise maximum likelihood procedure is shown to be the boundedness of the likelihood ratios. In the case of consistency the (typically slow) rate of convergence to zero of the error probabilities is determined.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

Some comments on two-stage selection procedures

In this paper we study the procedures of Dudewicz and Dalal ( 1975 ), and the modifications suggested by Rinott ( 1978 ), for selecting the largest mean from k normal populations with unknown variances. We look at the case k = 2 in detail, because there is an optimal allocation scheme here. We do not really allocate the total number of samples into two groups, but we estimate this optimal sample size, as well, so as to guarantee the probability of correct selection (written as P(CS)) at least P^?, 1/2 < P^? < 1 . We prove that the procedure of Rinott is “asymptotically in-efficient” (to be defined below) in the sense of Chow and Robbins ( 1965 ) for any k  2. Next, we propose two-stage procedures having all the properties of Rinott's procedure, together with the property of “asymptotic efficiency” - which is highly desirable.     

Asymptotic expansions for sequential confidence levels in inverse linear regression

Non-linear renewal theory is used to derive second order asymptotic expansions for the coverage probability of a fixed-width sequential confidence interval for an unknown parameter xin the inverse linear regression model. These expansions are obtained for a two-stage sequential procedure, proposed by Perng and Tong (1974) for the construction of a confidence interval for x.     

A note on determining the number of outliers in an exponential sample by least squares procedure

In this paper, we suggest a least squares procedure for the determination of the number of upper outliers in an exponential sample by minimizing sample mean squared error. Moreover, the method can reduce the masking or “swamping” effects. In addition, we have also found that the least squares procedure is easy and simple to compute than test test procedure T _k suggested by Zhang (1998) for determining the number of upper outliers, since Zhang (1998) need to use the complicated null distribution of T _k . Moreover, we give three practical examples and a simulated example to illustrate the procedures. Further, simulation studies are given to show the advantages of the proposed method. Finally, the proposed least squares procedure can also determine the number of upper outliers in other continuous univariate distributions (for example, Pareto, Gumbel, Weibull, etc.). Received: May 10, 1999; revised version: June 5, 2000     

A unified sequential test procedure for simultaneous testing the equality of several binomial proportions to a specified standard

In this article, we propose a unified sequentially rejective test procedure for testing simultaneously the equality of several independent binomial proportions to a specified standard. The proposed test procedure is general enough to include some well-known multiple testing procedures such as the Ordinary Bonferroni procedure, Hochberg procedure and Rom procedure. It involves multiple tests of significance based on the simple binomial tests (exact or approximate) which can be easily found in many elementary standard statistics textbooks. Unlike the traditional Chi-square test of the overall hypothesis, the procedure can identify the subset of the binomial proportions, which are different from the prespecified standard with the control of the familywise type I error rate. Moreover, the power computation of the procedure is provided and the procedure is illustrated by two real examples from an ecological study and a carcinogenicity study.     

On the asymptotic distribution of the Dickey Fuller-GLS test statistic

In a very influential paper, Elliott et al. [Efficient tests for an autoregressive unit root. Econometrica. 1996;64:813–836] show that no uniformly most powerful test for the unit root testing problem exits, derive the relevant power envelope and characterize a family of point-optimal tests. As a by-product, they also propose a ‘generalized least squares (GLS) detrended’ version of the conventional Dickey–Fuller test, denoted DF-GLS, that has since then become very popular among practitioners, much more so than the point-optimal tests. In view of this, it is quite strange to find that, while conjectured in Elliott et al. [Efficient tests for an autoregressive unit root. Econometrica. 1996;64:813–836], so far there seems to be no formal proof of the asymptotic distribution of the DF-GLS test statistic. By providing three separate proofs, the current paper not only substantiates the required result, but also provides insight regarding the pros and cons of different methods of proof.     

On rules based on sample medians for selection of the largest location parameter

The problem of selection of a subset containing the largest of several location parameters is considered, and a Gupta-type selection rule based on sample medians is investigated for normal and double exponential populations. Numerical comparisons between rules based on medians and means of small samples are made for normal and contaminated normal populations, assuming the popula-tion means to be equally spaced. It appears that the rule based on sample means loses its superiority over the rule based on sample medians in case the samples are heavily contaminated. The asymptotic relative efficiency (ARE) of the medians procedure relative to the means procedure is also computed, assuming the normal means to be in a slippage configuration. The means proce-dure is found to be superior to the median procedure in the sense of ARE. As in the small sample case, the situation is reversed if the normal populations are highly contaminate.     

An empirical study of five multivariate tests for the single-factor repeated measures model

A Monte Carlo study was used to examine the Type I error and power values of five multivariate tests for the single-factor repeated measures model The performance of Hotelling's T2 and four nonparametric tests, including a chi-square and an F-test version of a rank-transform procedure, were investigated for different distributions, sample sizes, and numbers of repeated measures. The results indicated that both Hotellings T* and the F-test version of the rank-transform performed well, producing Type I error rates which were close to the nominal value. The chi-square version of the rank-transform test, on the other hand, produced inflated Type I error rates for every condition studied. The Hotelling and F-test version of the rank-transform procedure showed similar power for moderately-skewed distributions, but for strongly skewed distributions the F-test showed much better power. The performance of the other nonparametric tests depended heavily on sample size. Based on these results, the F-test version of the rank-transform procedure is recommended for the single-factor repeated measures model.