SELECTING THE m POPULATIONS WITH LARGEST MEANS FROM k NORMAL POPULATIONS WITH UNKNOWN VARIANCES

This paper gives a two-sample procedure for selecting the m populations with the largest means from k normal populations with unknown variances. The method is a generalization of a recent work by Ofosu [1973] and hence should find wider practical applications. The experimenter takes an initial sample of preset size N₀ from each population and computes an unbiased estimate of its variance. From this estimate he determines the second sample size for the population according to a table presented for this purpose. The populations associated with the m largest overall sample means will be selected. The procedure is shown to satisfy a confidence requirement similar to that of Ofosu.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

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.     

Max-min multiple comparison procedure for comparing several dose levels with a zero dose control

The comparison of increasing doses of a compound to a zero dose control is of interest in medical and toxicological studies. Assume that the mean dose effects are non-decreasing among the non-zero doses of the compound. A simple procedure that modifies Dunnett's procedure is proposed to construct simultaneous confidence intervals for pairwise comparisons of each dose group with the zero dose control by utilizing the ordering of the means. The simultaneous lower bounds and upper bounds by the new procedure are monotone, which is not the case with Dunnett's procedure. This is useful to categorize dose levels. The expected gains of the new procedure over Dunnett's procedure are studied. The procedure is shown by real data to compare well with its predecessor.     

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 note on successive comparisons between several ordered variances

ABSTRACT

In this article, a procedure for comparisons between k (k ? 3) successive populations with respect to the variance is proposed when it is reasonable to assume that variances satisfy simple ordering. Critical constants required for the implementation of the proposed procedure are computed numerically and selected values of the computed critical constants are tabulated. The proposed procedure for normal distribution is extended for making comparisons between successive exponential populations with respect to scale parameter. A comparison between the proposed procedure and its existing competitor procedures is carried out, using Monte Carlo simulation. Finally, a numerical example is given to illustrate the proposed procedure.     

Non-orthogonal designs for measuring dispersion effects in sequential factor screening experiments using search linear models

“Dispersion” effects are considered in addition to “Location” effects of factors in the inferential procedure of sequential factor screening experiments with m factors each at two levels under search linear models. Search designs in measuring "Dispersion" and "Location" effects of factors are presented for both stage one and stage two of factor screening experiments with 4 ≤ m ≤ 10.     

Subset selection of the best treatment via many-one tests

The method of Gupta (1956, 1965) was developed to select a subset from k normal populations that contains the best populations with given probability. This paper shows a duality between the general goal of selecting a subset for the best population and many-one tests. A population should be regarded as ‘candidate’ for the best population and thus retained in the subset if the samples from the other populations are not significantly better. Based on this ‘idea’ a general selection procedure is proposed using many-one tests for the comparison of each population against the remaining ones.     

Estimation of population mean and variance in flock management: a ranked set sampling approach in a finite population setting

Ranked set sampling is a sampling technique that provides substantial cost efficiency in experiments where a quick, inexpensive ranking procedure is available to rank the units prior to formal, expensive and precise measurements. Although the theoretical properties and relative efficiencies of this approach with respect to simple random sampling have been extensively studied in the literature for the infinite population setting, the use of ranked set sampling methods has not yet been explored widely for finite populations. The purpose of this study is to use sheep population data from the Research Farm at Ataturk University, Erzurum, Turkey, to demonstrate the practical benefits of ranked set sampling procedures relative to the more commonly used simple random sampling estimation of the population mean and variance in a finite population. It is shown that the ranked set sample mean remains unbiased for the population mean as is the case for the infinite population, but the variance estimators are unbiased only with use of the finite population correction factor. Both mean and variance estimators provide substantial improvement over their simple random sample counterparts.     

Unbiased tests for some one-sided testing problems

Unbiased tests are found for various testing problems. In the first model considered we test homogeneity of k + 1 independent one-parameter exponential family populations vs. the tree-top ordering alternative. The tree-top alternative is appropriate for one-sided comparisons for treatments with a control. In the next set of models normality is assumed. In one such model k independent populations have different unknown means but have an unknown common variance. An independent estimate of the variance exists. We test homogeneity of means against the alternative of no homogeneity. We also consider the alternative of an ordering of the means as well as the tree-top ordering. The final model considered is when we take a random sample from a multivariate normal population with unknown mean vector and an unknown covariance matrix of the intraclass type. We test the hypothesis that the mean vector is the zero vector against the one-sided alternative that each mean is nonnegative (with at least one positive).     

THE COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN 2r FACTORIAL EXPERIMENTS AFTER PRELIMINARY HYPOTHESIS TESTING

We consider a 2^r factorial experiment with at least two replicates. Our aim is to find a confidence interval for θ, a specified linear combination of the regression parameters (for the model written as a regression, with factor levels coded as ?1 and 1). We suppose that preliminary hypothesis tests are carried out sequentially, beginning with the rth‐order interaction. After these preliminary hypothesis tests, a confidence interval for θ with nominal coverage 1 ?α is constructed under the assumption that the selected model had been given to us a priori. We describe a new efficient Monte Carlo method, which employs conditioning for variance reduction, for estimating the minimum coverage probability of the resulting confidence interval. The application of this method is demonstrated in the context of a 2³ factorial experiment with two replicates and a particular contrast θ of interest. The preliminary hypothesis tests consist of the following two‐step procedure. We first test the null hypothesis that the third‐order interaction is zero against the alternative hypothesis that it is non‐zero. If this null hypothesis is accepted, we assume that this interaction is zero and proceed to the second step; otherwise, we stop. In the second step, for each of the second‐order interactions we test the null hypothesis that the interaction is zero against the alternative hypothesis that it is non‐zero. If this null hypothesis is accepted, we assume that this interaction is zero. The resulting confidence interval, with nominal coverage probability 0.95, has a minimum coverage probability that is, to a good approximation, 0.464. This shows that this confidence interval is completely inadequate.     

A comparison of selection procedures for selecting the best normal population under heterogeneity of variance

This study examines the comparative probabilities of making a correct selection when using the means procedure (M), the medians procedure (D) and the rank-sum procedure (S) to correctly select the normal population with the largest mean under heterogeneity of variance. The comparison is conducted by using Monte-Carlo simulation techniques for 3, 4, and 5 normal populations under the condition that equal sample sizes are taken from each population. The population means and standard deviations are assumed to be equally-spaced. Two types of heterogeneity of variance are considered: (1) associating larger means with larger variances, and (2) associating larger means with smaller variances.     

Approximations for a fixed sample size selection procedure for negative binomial populations

A large sample approximation of the least favorable configuration for a fixed sample size selection procedure for negative binomial populations is proposed. A normal approximation of the selection procedure is also presented. Optimal sample sizes required to be drawn from each population and the bounds for the sample sizes are tabulated. Sample sizes obtained using the approximate least favorable configuration are compared with those obtained using the exact least favorable configuration. Alternate form of the normal approximation to the probability of correct selection is also presented. The relation between the required sample size and the number of populations involved is studied.     

Determination of sample size for selecting the smallest of k poisson population means

A procedure for selecting a Poisson population with smallest mean is considered using an indifference zone approach. The objective is to determine the smallest sample size n required from k ≥ 2 populations in order to attain the desired probability of correct selection. Since the means procedure is not consistent with respect to the difference or ratio alone, two distance measures are used simultaneously to overcome the difficulty in obtaining the smallest probability of correct selection that is greater than some specified limit. The constants required to determine n are computed and tabulated. The asymptotic results are derived using a normal approximation. A comparison with the exact results indicates that the proposed approximation works well. Only in the extreme cases small increases in n are observed. An example of industrial accident data is used to illustrate this procedure.     

A subset selection procedure for multinomial distributions

A subset selection procedure is developed for selecting a subset containing the multinomial population that has the highest value of a certain linear combination of the multinomial cell probabilities; such population is called the ‘best’. The multivariate normal large sample approximation to the multinomial distribution is used to derive expressions for the probability of a correct selection, and for the threshold constant involved in the procedure. The procedure guarantees that the probability of a correct selection is at least at a pre-assigned level. The proposed procedure is an extension of Gupta and Sobel's [14] selection procedure for binomials and of Bakir's [2] restrictive selection procedure for multinomials. One illustration of the procedure concerns population income mobility in four countries: Peru, Russia, South Africa and the USA. Analysis indicates that Russia and Peru fall in the selected subset containing the best population with respect to income mobility from poverty to a higher-income status. The procedure is also applied to data concerning grade distribution for students in a certain freshman class.     

A Note on “Modified Tukey's Control Chart”

Tercero-Gomez et al. [Communications in Statistics - Simulation and Computation, 2012, 41(9), pp. 1566–1579] modified the Tukey's chart to improve the insensitivity of signaling mean shifts when the population is skewly distributed. They examine the efficiency for several skewed populations with the modified Tukey's chart (MTCC). The parameters of the MTCC were intuitively calibrated to fit several gamma distributions without following a specific mathematical procedure. Moreover, they performed no optimization for specific distributions. This study optimizes the MTCC with a statistical model for each skewed population and reexamines its efficiency. A numerical comparison shows that the optimal MTCC is sensitive to detect the shift of the means.     

On estimating the mean of the selected population with unknown variance

This paper compares four estimators of the mean of the selected population from two normal populations with unknown means and common but unknown variance. The selection procedure is that the population yielding the largest sample mean is selected. The four estimators considered are invariant under both location and scale transformations. The bias and mean square errors of the four estimators are computed and compared. The conclusions are close to those reported by Dahiya ‘1974’, even for small sample sizes     

A restricted subset selection procedure for Weibull populations

Consider k(k ≥ 2) two-parameter Weibull populations. We want to select a subset of the populations not exceeding m in size such that the subset contains at least ? of the t best populations. We have proposed a procedure which uses either the maximum likelihood estimators or ‘simplified’ linear estimators of the parameters. The estimators are based on type II censored data. The ranking of the populations is done by comparing their reliabilities at a certain fixed time. In selected cases the constants for the procedure are tabulated using Monte Carlo methods.     

Testing for zero-inflation in count series: application to occupational health

Count data series with extra zeros relative to a Poisson distribution are common in many biomedical applications. A score test is presented to assess whether the zero-inflation problem is significant to warrant the analysis by the more complex zero-inflated Poisson autoregression model. The score test is implemented as a computer program in the Splus platform. For illustration, the test procedure is applied to a workplace injury series where many zero counts are observed due to the heterogeneity in injury risk and the dynamic population involved.     

Dynamic principal component analysis with missing values

Dynamic principal component analysis (DPCA), also known as frequency domain principal component analysis, has been developed by Brillinger [Time Series: Data Analysis and Theory, Vol. 36, SIAM, 1981] to decompose multivariate time-series data into a few principal component series. A primary advantage of DPCA is its capability of extracting essential components from the data by reflecting the serial dependence of them. It is also used to estimate the common component in a dynamic factor model, which is frequently used in econometrics. However, its beneficial property cannot be utilized when missing values are present, which should not be simply ignored when estimating the spectral density matrix in the DPCA procedure. Based on a novel combination of conventional DPCA and self-consistency concept, we propose a DPCA method when missing values are present. We demonstrate the advantage of the proposed method over some existing imputation methods through the Monte Carlo experiments and real data analysis.