Approximations to the powers of the likelihood ratio tests: the loop ordering and slippage alternatives

Approximations to the power functions of the likelihood ratio tests of homogeneity of normal means against the simple loop ordering at slippage alternatives are considered. If a researcher knows which mean is smallest and which is largest, but does not know how the other means are ordered, then a simple loop ordering is appropriate. The accuracy of the several moment approximations are studied for the case of known variances and it is found that for powers in the range typically of interest, the two-moment approximation seems quite adequate. Approximations based on mixtures of noncentral F variables are developed for the case of unknown variances. The critical values of the test statistics are also tabulated for selected levels of significance.     

BURR CRITICAL VALUE APPROXIMATIONS FOR TESTS OF AUTOCORRELATION AND HETEROSCEDASTICITY

The paper evaluates the accuracy of Burr approximations of critical values and p-values for test a of autocorrelation and heteroscedasticity in the linear regression model.     

Testing for changes using permutations of U-statistics

The critical values for various tests based on U-statistics to detect a possible change are obtained through permutations of the observations. We obtain the same approximations for the permutated U-statistics under the no change null hypothesis as well as under the exactly one change alternative. The results are used to show that the simulated critical values are asymptotically valid under the null hypothesis and the tests reject with the probability tending to one under the alternative.     

Large sample power of absolute moment tests

Sample kurtosis is a member of the large class of absolute moment tests of normality. We compare kurtosis to other absolute moment tests to determine which are the most powerful at detecting long‐tailed symmetric departures from normality for large samples. The large sample power of the tests is calculated using Geary's (1947) approximations of the moments of the test statistics. Using the system of Gram-Charlier symmetric distributions as alternatives, the most power is obtained using a moment in the range 2.5 ‐ 3.5.     

Assessing Lund's critical values for testing for outliers in linear regression models

This note assesses Lund's critical bounds as approximations to the critical values of a size-a test for detecting single outlier in a multiple linear regression model. A Monte Carlo experiment shows that the empirical sizes are very close to the nominal sizes, and the regressors have little effect on the empirical sizes. Lund's tables are revised to higher accuracy.     

Saddlepoint approximations for some models of circular data

In this article we provide saddlepoint approximations for some important models of circular data. The particularity of these saddlepoint approximations is that they do not require solving the saddlepoint equation iteratively, so their evaluation is immediate. We first give very accurate approximations to P-values, critical values and power functions for some optimal tests regarding the concentration parameter under wrapped symmetric α-stable and circular normal models. Then, we consider an approximation to the distribution of a projection of the two-dimensional Pearson random walk with exponential step sizes.     

Rejoinder

Abstract

In this article several formulae for the approximation of the critical values for tests on the actual values of the process capability indices CPL, CPU, and C_pk are provided. These formulae are based on different approximations of the percentiles of the noncentral t distribution and their performance is evaluated by comparing the values assessed through them from the exact critical values, for several significance levels, test values, and sample sizes. As supported by the obtained results, some of the presented techniques constitute valuable tools in situations where the exact critical values of the tests are not available, since one may approximate them readily and rather accurately through them.     

A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.     

Distributions of lrt associated with two-parameter exponential distributions

In this paper, an exact distribution of the likelihood ratio criterion for testing the equality of p two-parameter exponential distributions is obtained for unequal sample sizes in a computational form. A useful asymptotic expansion of the distribution is also obtained up to the order of n^-4 with the second term of the order of n^-3 and so can be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n where n is the combined sample size. In fact the first term alone which is a single beta distribution provides a powerful approximation for moderately large values of n.     

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.     

First moment approximations for order statistics from the extreme value distribution

Approximate expressions of the first moment of the order statistics of standard extreme value distributions are proposed. We compare different previously given approximations with the exact values. The results show that the approximation given here fits the exact values better than previously given models.     

Saddlepoint p-values for a class of tests for comparing competing risks with censored data

One of the general problems in clinical trials and mortality rates is the comparison of competing risks. Most of the test statistics used for independent and dependent risks with censored data belong to the class of weighted linear rank tests in its multivariate version. In this paper, we introduce the saddlepoint approximations as accurate and fast approximations for the exact p-values of this class of tests instead of the asymptotic and permutation simulated calculations. Real data examples and extensive simulation studies showed the accuracy and stability performance of the saddlepoint approximations over different scenarios of lifetime distributions, sample sizes and censoring.     

Durbin–Watson and Generalized Durbin–Watson Tests for Autocorrelations and Randomness

The Durbin–Watson (DW) test for lag 1 autocorrelation has been generalized (DWG) to test for autocorrelations at higher lags. This includes the Wallis test for lag 4 autocorrelation. These tests are also applicable to test for the important hypothesis of randomness. It is found that for small sample sizes a normal distribution or a scaled beta distribution by matching the first two moments approximates well the null distribution of the DW and DWG statistics. The approximations seem to be adequate even when the samples are from nonnormal distributions. These approximations require the first two moments of these statistics. The expressions of these moments are derived.     

General treatment of goodness-of-fit tests based on Kullback–Leibler information

This paper introduces a general goodness-of-fit test based on the estimated Kullback–Leibler information. The test uses the Vasicek entropy estimate. Two special cases of the test for location–scale and shape families are discussed. The results are used to introduce goodness-of-fit tests for the uniform, Laplace, Weibull and beta distributions. The critical values and powers for some alternatives are obtained by simulation.     

The complementary beta distribution

The complementary beta distribution is proposed as a new distribution on the unit interval. It results from reversing the roles of the distribution and quantile functions of the beta distribution. It has some attractive properties that are complementary to those of the beta distribution. In particular, the complementary beta distribution is much more amenable than the beta distribution to exact computations involving expectations of order statistics, including L-moments. At least for a wide range of parameter values, complementary beta and beta distributions with parameters that are reciprocals of the other's parameters are good approximations to one another. We also note the position of the complementary beta distribution in a wider family of distributions defined through the same simple form for their quantile density functions.     

Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments

Approximate t-tests of single degree of freedom hypotheses in generalized least squares analyses (GLS) of mixed linear models using restricted maximum likelihood (REML) estimates of variance components have been previously developed by Giesbrecht and Burns (GB), and by Jeske and Harville (JH), using method of moment approximations for the degrees of freedom (df) for the tstatistics. This paper proposes approximate Fstatistics for tests of multiple df hypotheses using one-moment and two-moment approximations which may be viewed as extensions of the GB and JH methods. The paper focuses specifically on tests of hypotheses concerning the main-plot treatment factor in split-plot experiments with missing data. Simulation results indicate usually satisfactory control of Type I error rates.     

Evaluating the Accuracy of Small P‐Values In Genetic Association Studies Using Edgeworth Expansions

The asymptotic distributions of many classical test statistics are normal. The resulting approximations are often accurate for commonly used significance levels, 0.05 or 0.01. In genome‐wide association studies, however, the significance level can be as low as 1×10⁻⁷, and the accuracy of the p‐values can be challenging. We study the accuracies of these small p‐values are using two‐term Edgeworth expansions for three commonly used test statistics in GWAS. These tests have nuisance parameters not defined under the null hypothesis but estimable. We derive results for this general form of testing statistics using Edgeworth expansions, and find that the commonly used score test, maximin efficiency robust test and the chi‐squared test are second order accurate in the presence of the nuisance parameter, justifying the use of the p‐values obtained from these tests in the genome‐wide association studies.     

Moment inequalities for HNBRUE with hypothesis testing application

The paper deals with life distributions which are harmonic new better than renewal used in expectation (HNBRUE). The goal is to derive moment inequalities and use them for further analysis of the class HNBRUE. We establish new characterization of exponentiality versus HNBRUE. Pitman’s asymptotic relative efficiency is employed to assess the performance of the proposed test with respect to other available tests. Finally, we carried out numerical simulation to produce table for the critical values of the test.     

Moment properties of estimators for a type 1 extreme-value regression model

A regression model is considered in which the response variable has a type 1 extreme-value distribution for smallest values. Bias approximations for the maximum likelihood estimators are pivm and a bias reduction estimator for the scale parameter is proposed. The small sample moment properties of the maximum likelihood estimators are compared with the properties of the ordinary least squares estimators and the best linear unbiased estimators based on order statistics for grouped data.     

Computing Critical Values of Exact Tests by Incorporating Monte Carlo Simulations Combined with Statistical Tables

Various exact tests for statistical inference are available for powerful and accurate decision rules provided that corresponding critical values are tabulated or evaluated via Monte Carlo methods. This article introduces a novel hybrid method for computing p‐values of exact tests by combining Monte Carlo simulations and statistical tables generated a priori. To use the data from Monte Carlo generations and tabulated critical values jointly, we employ kernel density estimation within Bayesian‐type procedures. The p‐values are linked to the posterior means of quantiles. In this framework, we present relevant information from the Monte Carlo experiments via likelihood‐type functions, whereas tabulated critical values are used to reflect prior distributions. The local maximum likelihood technique is employed to compute functional forms of prior distributions from statistical tables. Empirical likelihood functions are proposed to replace parametric likelihood functions within the structure of the posterior mean calculations to provide a Bayesian‐type procedure with a distribution‐free set of assumptions. We derive the asymptotic properties of the proposed nonparametric posterior means of quantiles process. Using the theoretical propositions, we calculate the minimum number of needed Monte Carlo resamples for desired level of accuracy on the basis of distances between actual data characteristics (e.g. sample sizes) and characteristics of data used to present corresponding critical values in a table. The proposed approach makes practical applications of exact tests simple and rapid. Implementations of the proposed technique are easily carried out via the recently developed STATA and R statistical packages.