Comparing Two Population Means and Variances: A Parametric Robust Way

Abstract

This article introduces a parametric robust way of comparing two population means and two population variances. With large samples the comparison of two means, under model misspecification, is lesser a problem, for, the validity of inference is protected by the central limit theorem. However, the assumption of normality is generally required, so that the inference for the ratio of two variances can be carried out by the familiar F statistic. A parametric robust approach that is insensitive to the distributional assumption will be proposed here. More specifically, it will be demonstrated that the normal likelihood function can be adjusted for asymptotically valid inferences for all underlying distributions with finite fourth moments. The normal likelihood function, on the other hand, is itself robust for the comparison of two means so that no adjustment is needed.     

A computational investigation of conover's kolmogorov-smirnov test for discrete distributions

The tabled significance values of the Kolmogorov-Smirnov goodness-of-fit statistic determined for continuous underlying distributions are conservative for applications involving discrete underlying distributions. Conover (1972) proposed an efficient method for computing the exact significance level of the Kolmogorov-Smirnov test for discrete distributions; however, he warned against its use for large sample sizes because “the calculations become too difficult.”

In this work we explore the relationship between sample size and the computational effectiveness of Conover's formulas, where “computational effectiveness” is taken to mean the accuracy attained with a fixed precision of machine arithmetic. The nature of the difficulties in calculations is pointed out. It is indicated that, despite these difficulties, Conover's method of computing the Kolmogorov-Smirnov significance level for discrete distributions can still be a useful tool for a wide range of sample sizes.     

Designing two factor experiments for selecting interactions

This paper considers the problem of devising a single stage procedure for selecting the treatment combination associated with the largest interaction in a two-factor rfx× c experiment having independent normally distributed observations with common known variance. The intuitive procedure based on the best linear unbiased estimators of the population interactions is employed. Initially an indifference zone formulation is used; the problem of determining the least favorable configuration is reduced to a nonlinear programming problem with log concave objective function and a convex polytope as feasible region. A solution technique is introduced in the context of an illustrative example. The problem is also considered using a preferred population formulation; this approach requires a strengthened version of the indifference zone probability requirement. It is shown that the same sample size guarantees this requirement as does the earlier one.     

Procedures for improving the mle for ordered binomial parameters

The problem is that of estimating the probabilities of m independent binomial random variables when their probabilities are known to be nondecreasing and the loss function is squared error. In the cases where the m.l.e. is inadmissible (essentially when the total number of trials is 7 or more) we present a method for modifying the m.l.e. to get a better estimator. The method requires a series of changes. At each step we alter the action taken by the m.l.e. on each of three, appropriately chosen, points in the sample space.     

Evaluating the impact of utility company billing plans on residential energy consumption

Many utility companies offer their customers the choice of participation in an average payment plan, which enables them to pay a fixed sum for their utility bill each month (with final settlement at the end of the billing year), instead of the conventional “pay as you go” billing procedure. Because customers on average payment plans are protected from paying large bills during peak energy-use seasons and because the information about monthly energy use and its cost is perhaps less salient to them, it was hypothesized that customers on the average payment plan would use more electricity than customers not on the plan. Using a nonequivalent control group design, the electricity consumption of a selection of customers of two utility companies (Ns = 475 and 74) was examined. The results showed that there was no evidence to support the hypothesis. Since the logic of hypothesis testing does not permit the ready acceptance of the null hypothesis, several procedural, methodological, and statistical points were made to buttress the conclusion that the average payment plans had no effect on electricity consumption.     

A class of Shrunken estimators for normal mean

This paper presents a class of estimators for the mean of a normal population and determines the conditions on characterizing scalars under which the class of estimators uniformly dominates over the conventional sample mean according to the mean-square-error criterion.     

Comparing migrants to non-migrants: The case of Dutch migration to New Zealand

We analyse post-war Dutch migration to New Zealand. We document that history, reflect on analytical and econometric modelling and then combine a sample of Dutch migrants in New Zealand with a representative sample of Dutch in The Netherlands to estimate wage equations and the determinants of the migration decision. We use the results for ex post evaluation of the migration decision.Joop Hartog is Fellow of IZA (Bonn), CESifo (München), AIAS and Tinbergen Institute (Amsterdam). Work on this project was begun when Hartog was Erskine Visitor at the University of Canterbury, Christchurch New Zealand. Rainer Winkelmann is Fellow of CEPR and IZA (Bonn). This paper is produced as part of a CEPR research programme on European Migration from Economic Analysis to Policy Response, supported by a grant from the Commission of the European Communities under its Human Capital and Mobility Programme (no. ERBCHRXCT940515).It was presented at the European Society for Population Economics meetings in Bonn, June 2000. We gratefully acknowledge comments by Justin Lee Tobias, Jacques Poot, Ed Vytlacil, an anonymous referee and Paul Schultz. Responsible editor: T. Paul Schultz.     

Screening procedures using quadratic forms

In quality control, a performance variable having a two-sided specification limit is common for assessing lot quality. Sometimes it is difficult or impossible to measure the performance variable directly; for example, when testing is destructive, expensive, or when the performance variable is related to the lifetime of the product. However, it may happen that there are several concomitant variables which are easily measured and which correlate highly with the variable of interest. Thus, one may use measurements on these variables to select or screen product which will have a high conditional probability of meeting product specification. We consider this situation when all variables have a joint multivariate normal distribution and the specification limits on the performance variable are two-sided.     

On a multiple endpoints testing problem

In a clinical trial comparing drug with placebo, where there are multiple primary endpoints, we consider testing problems where an efficacious drug effect can be claimed only if statistical significance is demonstrated at the nominal level for all endpoints. Under the assumption that the data are multivariate normal, the multiple endpoint-testing problem is formulated. The usual testing procedure involves testing each endpoint separately at the same significance level using two-sample t-tests, and claiming drug efficacy only if each t-statistic is significant. In this paper we investigate properties of this procedure. We show that it is identical to both an intersection union test and the likelihood ratio test. A simple expression for the p-value is given. The level and power function are studied; it is shown that the test may be conservative and that it is biased. Computable bounds for the power function are established.