Likelihood ratio test for simultaneous testing of the mean and the variance of a normal distribution

In this paper, we consider the simultaneous testing of the mean and the variance of a normal distribution. The exact distribution of the likelihood ratio test statistic is obtained, which is not available in the literature. The critical points of the exact test are reported. We also consider some of the other exact and asymptotic tests. The powers of these tests are compared using the Monte Carlo simulations.     

Maximum Average-Power (MAP) Tests

The objective of this article is to propose and study frequentist tests that have maximum average power, averaging with respect to some specified weight function. First, some relationships between these tests, called maximum average-power (MAP) tests, and most powerful or uniformly most powerful tests are presented. Second, the existence of a maximum average-power test for any hypothesis testing problem is shown. Third, an MAP test for any hypothesis testing problem with a simple null hypothesis is constructed, including some interesting classical examples. Fourth, an MAP test for a hypothesis testing problem with a composite null hypothesis is discussed. From any one-parameter exponential family, a commonly used UMPU test is shown to be also an MAP test with respect to a rich class of weight functions. Finally, some remarks are given to conclude the article.     

The Slippage Problem for the Circular Normal Distribution

The slippage problem occurs when an unspecified observation in a given random sample is from a distribution other than that for all the remaining observations. This paper studies the problem in terms of the 'slip' in the mean direction of a circular normal distribution. The slippage problem is first treated as a multiple decision problem with a prior which is invariant under the permutations of the hypotheses. The probabilities of accepting the various hypotheses for the Bayes rule with respect to this prior are explicitly obtained. The likelihood ratio tests for this slippage problem, for the cases when the mean directions are both known and unknown, are shown to be easily computable. The tests are illustrated through two well-known datasets. The performances of a range of tests are compared using extensive simulation.     

UMPU tests based on sequential order statistics

Sequential order statistics with conditional proportional hazard rates form a regular exponential family in the model parameters. This finding is used to establish uniformly most powerful unbiased (UMPU) tests for a variety of hypotheses.     

Small sample equivalence tests for exponentiality

We consider small sample equivalence tests for exponentialy. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require much larger sample sizes, in comparison with classical “difference tests,” to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of small sample equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. We perform small sample simulation studies to assess the bias of our various tests and show that all of the Bayes posteriors we consider are integrable. Our simulation studies show that the saddlepoint-approximated UMPU method performs remarkably well for small sample sizes and is the only method that consistently exhibits an empirical significance level close to the nominal 5% level.     

On Tests for Equicorrelation Coefficient of a Standard Symmetric Multivariate Normal Distribution

Sampson (1976, 1978) has considered applications of the standard symmetric multivariate normal (SSMN) distribution and the estimation of its equi-correlation coefficient, ρ. Tests for ρ are considered here. The likelihood ratio test suffers from several theoretical and practical shortcomings. We propose the locally most powerful (LMP) test which is globally (one-sided) unbiased, very simple to compute and is based on the best natural unbiased estimator of ρ. Exact null and non-null distributions of the test statistic are presented and percentage points are given. Statistical curvature (Efron, 1975) indicates that its performance improves with mk (sample size × dimension) while exact power computations show that even for reasonably small values of mk the performance is quite encouraging. Recalling Brown's (1971) cautions we establish by local comparison with the LMP similar test for ρ in the SMN (Rao, 1973) distribution, that here the additional information on the mean and variance is quite worthwhile.     

Tests for symmetry of two-piece normal distribution

In this paper, classical optimum tests for symmetry of two-piece normal distribution is derived. Uniformly most powerful one-sided test for the skewness parameter is obtained when the location and scale parameters are known and is compared with sequential probability ratio test. An ad-hoc test for symmetry and likelihood ratio test for symmetry for large samples, can be found in literature for this distribution. But in this paper, we derive exact likelihood ratio test for symmetry, when location parameter is known. The exact power of the test is evaluated for different sample sizes.     

A Nonrandomized,Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably.

The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon.

In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ₀, for a fixed threshold value γ₀, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test.

Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.     

A comparison of some conditional and unconditional exact tests for 2x2 contingency tables

In 1935, R.A. Fisher published his well-known “exact” test for 2x2 contingency tables. This test is based on the conditional distribution of a cell entry when the rows and columns marginal totals are held fixed. Tocher (1950) and Lehmann (1959) showed that Fisher s test, when supplemented by randomization, is uniformly most powerful among all the unbiased tests UMPU). However, since all the practical tests for 2x2 tables are nonrandomized - and therefore biased the UMPU test is not necessarily more powerful than other tests of the same or lower size. Inthis work, the two-sided Fisher exact test and the UMPU test are compared with six nonrandomized unconditional exact tests with respect to their power. In both the two-binomial and double dichotomy models, the UMPU test is often less powerful than some of the unconditional tests of the same (or even lower) size. Thus, the assertion that the Tocher-Lehmann modification of Fisher's conditional test is the optimal test for 2x2 tables is unjustified.     

Upper Bound for p-Value of the Test of Multivariate Normal Ordered Mean Vectors Against all Alternatives

Consider the problem of testing the isotonic of several p-variate normal mean vectors against all alternatives. It is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown. In the present paper, a test statistic is proposed for this problem of testing. A reformulation of the test statistic is given based on the orthogonal projections on the closed convex cones and then the upper bound for p-value of the test statistic is computed.     

Locally Maximin Test for the Mixture Proportion with Type-I Censoring

Mixtures of distributions arise frequently in the context of life testing experiments, where one also encounters the problem of censored data. In this article, we derive the locally most powerful (LMP) test for testing the mixing proportion in a general mixture model based on type-I censored data. We also prove the additional properties of unbiasedness and locally maximin using a novel approach. To this end, we prove an extension of a standard lemma in the testing literature (Lehmann, 1986 Lehmann , E. L. ( 1986 ). Testing Statistical Hypotheses . New York : Wiley .[Crossref] , [Google Scholar]) relating to families with monotone likelihood ratio.     

Multivariate Two-Sided Tests for Normal Mean Vectors with Unknown Covariance Matrix

A numerically stable method is developed which computes seemingly tight bounds at a small computational cost relative to the model size, when that model size is large, for the unreliability and bounds for the unreliability using, respectively, exact and bounding failure/repair continuous-time Markov chain models of fault-tolerant systems with exponential failure and repair time distributions, in which repair is deferred until some condition on the collection of failed components is satisfied, and, then, proceeds until reaching the state without failed components, with failure rates much smaller than repair rates and not too different output rates from states with deferred repair.     

Hypothesis Testing and Confidence Regions for the Mean Sojourn Time of an M/M/1 Queueing System

This article discusses testing hypotheses and confidence regions with correct levels for the mean sojourn time of an M/M/1 queueing system. The uniformly most powerful unbiased tests for three usual hypothesis testing problems are obtained and the corresponding p values are provided. Based on the duality between hypothesis tests and confidence sets, the uniformly most accurate confidence bounds are derived. A confidence interval with correct level is proposed.     

On optimal testing for the equality of equicorrelation: An example of loss in power

Sen Gupta (1988) considered a locally most powerful (LMP) test for testing nonzero values of the equicorrelation coefficient of a standard symmetric multivariate normal distribution. This paper constructs analogous tests for the symmetric multivariate normal distribution. It shows that the new test is uniformly most powerful invariant even in the presence of a nuisance parameter, σ². Further applications of LMP invariant tests to several equicorrelated populations have been considered and an extension to panel data modeling has been suggested.     

Tests for detecting overdispersion in poisson models

Collings and Margolin(1985) developed a locally most powerful unbiased test for detecting negative binomial departures from a Poisson model, when the variance is a quadratic function of the mean. Kim and Park(1992) developed a locally most powerful unbiased test, when the variance is a linear function of the mean. It is found that a different mean-variance structure of a negative binomial derives a different locally optimal test statistic.

In this paper Collings and Margolin's and Kim and Park's results are unified and extended by developing a test for overdispersion in Poisson model against Katz family of distributions, Our setup has two extensions: First, Katz family of distributions is employed as an extension of the negative binomial distribution. Second, the mean-variance structure of the mixed Poisson model is given by σ² = μ+cμ^r for arbitrary but fixed r. We derive a local score test for testing H₀ : c = 0. Superiority of a new test is proved by the asymtotic relative efficiency as well as the simulation study.     

On Parameter Inference for Step-Stress Accelerated Life Test with Geometric Distribution

In this article, we present the parameter inference in step-stress accelerated life tests under the tampered failure rate model with geometric distribution. We deal with Type-II censoring scheme involved in experimental data, and provide the maximum likelihood estimate and confidence interval of the parameters of interest. With the help of the Monte-Carlo simulation technique, a comparison of precision of the confidence limits is demonstrated for our method, the Bootstrap method, and the large-sample based procedure. The application of two industrial real datasets shows the proposed method efficiency and feasibility.     

On Threshold Choice in Hypothesis Testing for Dynamical Systems with Small Noise

We considered the problem of testing a simple hypothesis against composite one-sided alternative by the continuous time observations of diffusion process with small noise. Moreover, we propose a test which is asymptotically equivalent to the Neyman-Pearson test for local alternatives. The special choice of the threshold allows us to improve the rate of convergence of the first type error to the given value. The calculation of this threshold is based on the stochastic expansion of the test statistics and on the Edgeworth expansion of its distribution function.     

Multivariate Two-Sided Tests for Normal Mean Vectors Based on Approximations of Likelihood Ratio Test

Assuming that all components of a normal mean vector are simultaneously non negative or non positive, we consider a multivariate two-sided test for testing whether the normal mean vector is equal to zero or not. Since the likelihood ratio test is accompanied with theoretical and computational complications, we discuss two kinds of approximations of the likelihood ratio test. One is based on a conservative critical value determined by a certain inequality. The other is constructed by the approximation of the likelihood ratio test proposed by Tang et al. (1989). We compare the likelihood ratio test and two kinds of approximations through numerical examples regarding critical values and the power of the test.     

Optimal record-based statistical procedures for the two-parameter exponential distribution

There are a number of situations in which the experimental data observed are record statistics. In this paper, optimal confidence intervals as well as uniformly most powerful (MP) tests for one-sided alternatives are developed. Since a uniformly MP test for a two-sided alternative does not exist, generalized likelihood ratio and uniformly unbiased and invariant tests are derived for the two parameters of the exponential distribution based on record data. For illustrative purposes, a data set on the times between consecutive telephone calls to a company's switchboard is analysed using the proposed procedures. Finally, some open problems in this direction are pointed out.     

Power comparisons of single-sample winsorized T,trimmed T and student's T

The objective of this study is to evaluate the performance of Winsorized t which was proposed to be used as one alternative to the student's t statistic when the parent population is non-normal. The criterion we use is the power of the test. Comparative performance of the Winsorized t, trimmed t, and Studen$tCs t for normal populations and for Studen$tCs t distribution with 8, 4, and 2 degrees of freedom was investigated.