Hubert J. Chen

Single-stage analysis of variance under heteroscedasticity

The procedures of testing the equality of normal means in the conventional analysis of variance (ANOVA) are heavily based on the assumption of the equality of the error variances. Studies have shown that the distribution of the F-test depends heavily on the unknown variances and is not robust under the violation of equal error variances. When the variances are unknown and unequal, Bishop and Dudewicz (1978) developed a design-oriented two-stage procedure for ANOVA, which requires additional samples at the second stage. In this paper we use a single-stage sampling procedure to test the null hypotheses in ANOVA models under heteroscedasticity. The single-stage procedure for ANOVA has an exact distribution and it is a data-analysis-oriented procedure. It does not require additional samples, and can reach a conclusion much earlier, save time and money. Simulation results indicate that the power of the single-stage procedure is better than the two-stage method when the initial sample size is smaller than 6, an...

Procedures for Fixed-Width Interval Estimation of the Largest Normal Mean

Abstract Suppose that we are given k(≥1) independent and normally distributed populations π1, …, πk , where πi has unknown mean μ i and unknown variance σi 2 (i = 1, …, k). Let μ[k] denote the largest of the means μ1, …, μk . A two-stage procedure is developed which provides a confidence interval for μ[k] with fixed width. Tables needed to apply the procedure are provided.

Single-stage interval estimation of the largest normal mean under heteroscedasnoty

Let there be given k(≥1) independent and normally distributed populations where πi has unknown mean μ1 and unknown variance . A single-stage sampling procedure is developed for constructing confidence intervals for the largest of the means. The method is to split up the single sample into two portions, the first consisting of n-1 observations for initial estimation and the second consisting of the remaining one for use as a spare in the final estimation. One-sided, two-sided, and asymptotically optimal confidence interval are considered. A numerical example to apply this procedure is given.

Single-Stage Multiple Comparison Procedures Under Heteroscedasticity

SYNOPTIC ABSTRACTGiven k (≥ 2) independent normal populations with unknown means and unknown (and possibly unequal) variances a single-stage sampling procedure for multiple comparisons with the largest normal mean and with a control, respectively, are proposed. The advantage of the single-stage procedure is to have design simplicity and to reach an exact solution. Simulation results indicate that the single-stage procedure appears to be a reasonable choice as compared to existing methods. Computer program and a numerical example are given.

Serum progesterone concentrations in pregnant and nonpregnant heifers and after gonadotropin releasing hormone in luteal phase heifers

Abstract Serum progesterone (P4) concentrations were quantitated in 18 Holstein heifers from days 5 to 16 after estrus in an effort to ascertain the effects of pregnancy on circulating levels of this hormone. The P4 concentration rose faster between days 5 and 10 in the pregnant heifers compared to P4 levels in both the non-pregnant heifers and the killed sperm inseminated group. It was found that serum P4 levels were significantly (P The administration of 250μg gonadotropin releasing hormone (GnRH) to 10 Holstein heifers on day 7 after estrus resulted in a significant (P

Range tests for the dispersion of several location parameters

Abstract This paper describes the use of the range statistic to test the hypothesis that a set of location parameters varies ‘little’ against the global alternative, for three different methods of measuring distance. The normal distribution is given special consideration. The paper also shows how to construct lower confidence bounds (which attain at least a given confidence level) for the three variation measures.

A range test for the equivalency of means under unequal variances

In this article, we present a range test using a two-stage sampling procedure for testing the hypothesis that the normal means are falling into a practical indifference zone. Both the level and the power associated with the proposed test are controllable and are completely independent of the unknown variances. Tables needed for implementation are given.

Simultaneous upper confidence bounds for distances from the best two-parameter exponentlal distribution

Consider k independent exponential distributions possibly with different location parameters and a common scale parameter. If the best population is defined to be the one having the largest mean or equivalently having the largest location parameter, we then derive a set of simultaneous upper confidence bounds for all distances of the means from the largest one. These bounds not only can serve as confidence intervals for all distances from the largest parameter but they also can be used to identify the best population. Relationships to ranking and selection procedures are pointed out. Cases in which scale parameters are known or unknown and samples are complete or type II censored are considered. Tables to implement this procedure are given.

On selecting a subset which contains all populations better than a control

Let π1…, πk denote k(≥ 2) populations with unknown means μ1 , …, μk and variances σ1 2 , …, σk 2 , respectively and let πo denote the control population having mean μo and variance σo 2 . It is assumed that these populations are normally distributed with correlation matrix {ρij}. The goal is to select a subset, of populations of π1 , …, πk which contains all the populations with means larger than or equal to the mean of the control one. Procedures are given for selecting such a subset so that the probability that all the populations with means larger than or equal to the mean of the control one are included in the selected subset is at least equal to a predetermined value P∗(l/k < P∗ < 1). The goal treated here is a first step screening procedure that allows the experimenter to choose a subset and withhold judgement about which one has the largest mean. Then, if the one with the largest mean is desired it can be chosen from the selected subset on the basis of cost and other considerations. Percentage poin...

A class of fixed-width confidence intervals for a ranked normal mean

Suppose that we are given k(≥ 2) independent and normally distributed populations π1, …, πk where πi has unknown mean μi and unknown variance σ2 i (i = 1, …, k). Let μ[i] (i = 1, …, k) denote the ith smallest one of μ1, …, μk. A two-stage procedure is used to construct lower and upper confidence intervals for μ[i] and then use these to obtain a class of two-sided confidence intervals on μ[i] with fixed width. For i = k, the interval given by Chen and Dudewicz (1976) is a special case. Comparison is made between the class of two-sided intervals and a symmetric interval proposed by Chen and Dudewicz (1976) for the largest mean, and it is found that for large values of k at least one of the former intervals requires a smaller total sample size. The tables needed to actually apply the procedure are provided.