Two-moment approximations to some null distributions in order-restricted inference: Unequal sample sizes

Bartholomew's statistics for testing homogeneity of normal means with ordered alternatives have null distributions which are mixtures of chi-squared or beta distributions according as the variances are known or not. If the sample sizes are not equal, the mixing coefficients can be difficult to compute. For a simple order and a simple tree ordering, approximations to the significance levels of these tests have been developed which are based on patterns in the weight sets. However, for a moderate or large number of means, these approximations can be tedious to implement. Employing the same approach that was used in the development of these approximations, two-moment chisquared and beta approximations are derived for these significance levels. Approximations are also developed for the testing situation in which the order restriction is the null hypothesis. Numerical studies show that in each of the cases the two-moment approximation is quite satisfactory for most practical purposes.     

The power functions of the likelihood ratio tests for a simple tree ordering in normal means: unequal weights

Likelihood ratio tests are considered for two testing situations; testing for the homogeneity of k normal means against the alternative restricted by a simple tree ordering trend and testing the null hypothesis that the means satisfy the trend against all alternatives. Exact expressions are given for the power functions for k = 3 and 4 and unequal sample sizes, both for the case of known and unknown population variances, and approximations are discussed for larger k. Also, Bartholomew’s conjectures concerning minimal and maximal powers are investigated for the case of equal and unequal sample sizes. The power formulas are used to compute powers for a numerical example.     

The one-way analysis of variance with ordered alternatives: A modification of bartholomew's Ē2 Test

Tests of homogeneity of normal means with the alternative restricted by an ordering on the means are considered. The simply ordered case, μ₁ ≤ μ₂ ≤ ··· ≤ μ_k, and the simple tree ordering, μ₁ ≤ μ_j, for; j= 2, 3,…, k, are emphasized. A modification of the likelihood-ratio test is proposed which is asymptotically equivalent to it but is more robust to violations of the hypothesized orderings. The new test has power at the points satisfying the hypothesized ordering which is similar to that of the likelihood-ratio test provided the degrees of freedom are not too small. The modified test is shown to be unbiased and consistent.     

Isotonic rules for selecting good exponential populations

The problem of selecting exponential populations better than a control under a simple ordering prior is investigated. Based on some prior information, it is appropriate to set lower bounds for the concerned parameters. The information about the lower bounds of the concerned parameters is taken into account to derive isotonic selection rules for the control known case. An isotonic selection rule for the control unknown case is also proposed. A criterion is proposed to evaluate the performance of the selection rules. Simulation comparisons among the performances of several selection rules are carried out. The simulation results indicate that for the control known case, the new proposed selection rules perform better than some earlier existing selection rules.     

On Mixture Failure Rates Ordering

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. This is due to the fact that the mixture failure rate is “bent down” compared with the corresponding unconditional expectation of the baseline failure rate, which was proved previously for some specific cases. We generalize this result and discuss the “weakest populations are dying first” property, which leads to the change in the failure rate shape. We also consider the problem of mixture failure rate ordering for the ordered mixing distributions. Two types of stochastic ordering are analyzed: ordering in the likelihood ratio sense and ordering in variances when the means are equal.     

On estimation and testing arising from order restricted balanced mixed model

The randomized complete block design is one of the most widely used experimental designs to systematically control the variability arising from known nuisance sources. The balanced mixed effects model is usually appropriate for such an experiment when the blocks used in the experiment are randomly chosen. In applications with k increasing or decreasing treatment levels, there is sometimes prior knowledge about the ordering of the treatment effects. The most commonly seen orderings include simple ordering, simple tree ordering and umbrella orderings with known or unknown peaks. A natural question is how to incorporate the prior ordering information in estimating the parameters in a balanced mixed effects model so that the estimated treatment effects are consistent with the prior information and the estimated variances of the block effects and experiment errors are nonnegative. In this paper we derive the maximum likelihood estimators of the parameters in a balanced mixed model subject to any partial ordering of the treatment effects, which includes the usual maximum likelihood estimators as a special case. An example is provided to illustrate the results.     

Sample size selection in clinical trials when population means are subject to a partial order: one-sided ordered alternatives

The statistical methodology under order restriction is very mathematical and complex. Thus, we provide a brief methodological background of order-restricted likelihood ratio tests for the normal theoretical case for the basic understanding of its applications, and relegate more technical details to the appendices. For data analysis, algorithms for computing the order-restricted estimates and computation of p-values are described. A two-step procedure is presented for obtaining the sample size in clinical trials when the minimum power, say 0.80 or 0.90 is specified, and the normal means satisfy an order restriction. Using this approach will result in reduction of 14-24% in the sample size required when one-sided ordered alternatives are used, as illustrated by several examples.     

Exact and Approximate Distributions of the Maximum Likelihood Estimate in a Simple Factor Analysis Model

We study a factor analysis model with two normally distributed observations and one factor. In the case when the errors have equal variance, the maximum likelihood estimate of the factor loading is given in closed form. Exact and approximate distributions of the maximum likelihood estimate are considered. The exact distribution function is given in a complex form that involves the incomplete Beta function. Approximations to the distribution function are given for the cases of large sample sizes and small error variances. The accuracy of the approximations is discussed     

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.     

Convex Ordering for Bivariate Distributions

An attempt is made to extend well-known univariate notion of convex ordering to bivariate case. A convex ordered family for bivariate distributions is then introduced and its properties are examined.     

On the distribution of the maximum of bivariate normal random variables with general means and variances

A great amount of effort has been devoted to achieving exact expressions for moments of order statistics of independent normal random variables, as well as the dependent case with the same correlation coefficients, means and variances. It does not seem as if there are handy formulae for the order statistics of even the simple bivariate normal random variables when the means and variances are allowed to be different. In this paper we give an explicit formula for the Lanl ace-Stielties Transform of the maximum of bivariate normal random variables by which we obtain formulae for the first two moments in the standard way.     

Comparing Survival Times for Treatments with Those for a Control Under Proportional Hazards

Inferences for survival curves based on right censored data are studied for situations in which it is believed that the treatments have survival times at least as large as the control or at least as small as the control. Testing homogeneity with the appropriate order restricted alternative and testing the order restriction as the null hypothesis are considered. Under a proportional hazards model, the ordering on the survival curves corresponds to an ordering on the regression coefficients. Approximate likelihood methods, which are obtained by applying order restricted procedures to the estimates of the regression coefficients, and ordered analogues to the log rank test, which are based on the score statistics, are considered. Mau's (1988) test, which does not require proportional hazards, is extended to this ordering on the survival curves. Using Monte Carlo techniques, the type I error rates are found to be close to the nominal level and the powers of these tests are compared. Other order restrictions on the survival curves are discussed briefly.     

A STOCHASTIC ORDERING FOR RANDOM VARIABLES WITH APPLICATIONS

A new univariate stochastic ordering is introduced. Some characterization results for such an ordering are stated. It is proved that the ordering is an integral stochastic ordering, obtaining a maximal generator. By means of this generator, the main properties of the ordering are deduced. A method for introducing univariate stochastic orderings, suggested by the new ordering, is analysed. Relationships with other stochastic orderings are also developed. To conclude, an example of an application of the new ordering to the field of medicine is proposed.     

A maxmin linear test of normal means and its application to lachin’s data

For stochastic ordering tests for normal distributions there exist two well known types of tests. One of them is based on the maximum likelihood ratio principle, the other is the most stringent somewhere most powerful test of Schaafsma and Smid(for a comprehensive treatment see Robertson, Wright and Dykstra(1988), for the latter test also Shi and Kudo(1987)). All these tests are in general numerically tedious. Wei, Lachin(1984)and particularly Lachin(1992)formulate a simple and easily computable test. However, it is not known so far for which sort of ordered alternatives his test is optimal

In this paper it is shown that his procedure is a maxmin test for reasonable subalternatives, provided the covariance matrix has nonnegative row sums. If this property is violated then his procedure can be altered in such a manner that the resul ting test again is a maxmin test. An example is glven where the modified procedure even in the least favourable case leads to a nontrifling increase in power. The fact that Lachins test resp. the modified version are maxmin tests on appropriate subalternatives amounts to the property that they are maxmin tests on subhypotheses which are relevant in practical applications.     

Ordering Comparison of Logarithmic Series Random Variables with their Mixtures

The problem of comparing some known distributions in various types of stochastic orderings has been of interest to many authors. In particular, several authors have been recently concerned with the comparison of Poisson, binomial, and negative binomial distributions with their respective mixtures. Incidentally, these distributions are among the four well-known distributions of the family of generalized power series distributions (GPSD's). The remaining distribution is the logarithmic series distribution. In this paper, we shall be concerned with comparing this remaining distribution of the class GPSD with its mixture in terms of various types of stochastic orderings such as the simple stochastic, likelihood ratio, uniformly more variable, convex, hazard rate and expectation orderings. Derivation of the results in this case prove to be computationally trickier than the other three. The special case when the means of the two distributions are the same is also discussed. Finally, an illustrative explicit example is provided.     

On Uniformly Optimal Networks: A Reversal of Fortune?

In this article, the general problem of comparing the performance of two communication networks is examined. The standard approach, using stochastic ordering as a metric, is reviewed, as are the mixed results on the existence of uniformly optimal networks (UONs) which have emerged from this approach. While UONs have been shown to exist for certain classes of networks, it has also been shown that no UON network exists for other classes. Results to date beg the question: Is the problem of identifying a Uniformly Optimal Network (UON) of a given size dead or alive? We reframe the investigation into UONs in terms of network signatures and the alternative metric of stochastic precedence. While the endeavor has been dead, or at least dormant, for some 20 years, the findings in the present article suggest that the question above is by no means settled. Specifically, we examine a class of networks of a particular size for which it was shown that no individual network was uniformly optimal relative to the standard metric (the uniform ordering of reliability polynomials), and we show, using the aforementioned alternative metric, that this class is totally ordered and that a uniformly optimal network exists after all. Optimality with respect to “performance per unit cost” type metrics is also discussed.     

A stein two-sample procedure for the general linear model with unequal error variances

The pronerties of the tests and confidence regions for the parameters in the classical general linear model depend upon the equality of the variances of the error terms. The level and power of tests and the confidence coefficients associated with confidence regions are vitiated when the assumption of equality is not true. Even when the error variances are equal the power of tests and the size of confidence regions depend upon the unknown common variance and hence are uncontrollable. This paper presents a two-stage procedure which yields tests and confidence regions which are completely independent of the variances of the errors and hence tests with controllable power and confidence regions of fixed controllable size are obtained.     

Rank tests for the k-sample problem with restricted alternatives

An asymptotically maximin most powerful rank test among somewhere asymptotically most powerful linear rank tests with scores generating function cf> is derived for each of the simple order alternative, the simple loop alternative and the simple tree alternative in the k-sample problem. The comparisons of the tests obtained with the rank analogues of the Bartholomew's xv tests are made in terms of local asymptotic relative efficiency. It is found that our tests are better than the rank analogues of the xk tests. Furthermore, the asymptotic equivalence of the ranking by the pooled sample to the ranking in pairs are discuss¬ed and the tests which are asymptotically equivalent to ours are given.     

Optimal Allocation of Observations for Inference on k Ordered Normal Population Means

The optimal allocation of observations when there is a natural ordering in the k normal population means is discussed. It is shown that the design which minimizes the total mean square error of the maximum likelihood estimators in the null case allocates half the observations to each of the two extreme populations. The design is obviously optimal for testing the homogeneity of means against the simple ordered alternative. It is, however, hardly acceptable for the estimation in the nonnull case. It is, therefore, shown that the observations could be allocated to the non-extreme populations according to weights which are proportional to the absolute values of the Abelson and Tukey scores at the same time keeping the minimum local power for testing the simple ordered alternative to be maximal. The design gives also the maximum minimum power, not local, for the alternative in the class of linear tests. It, of course, suffers from a small loss of efficiency for the estimation under the null case but is much better under the nonnull case than the extreme design which allocates half the observations to each of the two extreme populations. Some numerical comparisons of the mean square errors are given.