Towards a theory of point optimal testing

This paper puts the case for the inclusion of point optimal tests in the econometrician's repertoire. They do not suit every testing situation but the current evidence, which is reviewed here, indicates that they can have extremely useful Small-sample power properties. As well as being most powerful at a nominated point in the alternative hypothesis parameter space, they may also have optimum power at a number of other points and indeed be uniformly most powerful when such a test exists. Point optimal tests can also be used to trace out the maxemum attainable power envelope for a given testing problem, thus providing a benchmark against which test procedures can be evaluated. In some cases, point optimal tests can be constructed from tests of simple null hypothesis against a simple alternative. For a wide range of models of interst to econometricians, this paper shows how one can check whether a point optimal test can be constructed in this way. When it cannot, one may wish to consider approximately point optimal tests. As an illustration, the approach is applied to the non-nested problem of testing for AR(1) distrubances against MA(1) distrubances in the linear regression model.     

Using point optimal test of a simple null hypothesis for testing a composite null hypothesis via maximized Monte Carlo approach

King’s Point Optimal (PO) test of a simple null hypothesis is useful in a number of ways, for example it can be used to trace the power envelope against which existing tests can be compared. However, this test cannot always be constructed when testing a composite null hypothesis. It is suggested in the literature that approximate PO (APO) tests can overcome this problem, but they also have some drawbacks. This paper investigates if King’s PO test can be used for testing a composite null in the presence of nuisance parameters via a maximized Monte Carlo (MMC) approach, with encouraging results.     

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.     

On optimal tests for separate hypotheses and conditional probability integral transformations

Consider the problem of testing the composite null hypothesis that a random sample X₁,…,X_n is from a parent which is a member of a particular continuous parametric family of distributions against an alternative that it is from a separate family of distributions. It is shown here that in many cases a uniformly most powerful similar (UMPS) test exists for this problem, and, moreover, that this test is equivalent to a uniformly most powerful invariant (UMPI) test. It is also seen in the method of proof used that the UMPS test statistic Is a function of the statistics U₁,…,U_n?k obtained by the conditional probability integral transformations (CPIT), and thus that no Information Is lost by these transformations, It is also shown that these optimal tests have power that is a nonotone function of the null hypothesis class of distributions, so that, for example, if one additional parameter for the distribution is assumed known, then the power of the test can not lecrease. It Is shown that the statistics U₁, …, U_n?k are independent of the complete sufficient statistic, and that these statistics have important invariance properties. Two examples at given. The UMPS tests for testing the two-parameter uniform family against the two-parameter exponential family, and for testing one truncation parameter distribution against another one are derived.     

A Comparison of Unit-Root Test Criteria

During the past 15 years, the ordinary least squares estimator and the corresponding pivotal statistic have been widely used for testing the unit-root hypothesis in autoregressive processes. Recently, several new criteria, based on maximum likelihood estimators and weighted symmetric estimators, have been proposed. In this article, we describe several different test criteria. Results from a Monte Carlo study that compares the power of the different criteria indicate that the new tests are more powerful against the stationary alternative. Of the procedures studied, the weighted symmetric estimator and the unconditional maximum likelihood estimator provide the most powerful tests against the stationary alternative. As an illustration, the weekly series of one-month treasury-bill rates is analyzed.     

TESTING FOR MOVING AVERAGE REGRESSION DISTURBANCES

This paper considers a locally optimal procedure for testing for first order moving average disturbances in the linear regression model. For this hypothesis testing problem, the Durbin-Watson test is shown to be approximately locally best invariant while the new test is most powerful invariant in a given neighbourhood of the alternative hypothesis. Two versions of the test procedure are recommended for general use; one for problems involving positively correlated disturbances and one for negatively correlated disturbances. An empirical comparison of powers shows the clear superiority of the recommended tests over the Durbin-Watson test. Selected bounds for the tests' significance points are tabulated.     

An Empirical Likelihood Ratio Test for Normality

The empirical likelihood ratio (ELR) test for the problem of testing for normality is derived in this article. The sampling properties of the ELR test and four other commonly used tests are provided and analyzed using the Monte Carlo simulation technique. The power comparisons against a wide range of alternative distributions show that the ELR test is the most powerful of these tests in certain situations.     

Power studies of tests for uniformity,II

Results from a power study of six statistics for testing that a sample is from a uniform distribution on the unit interval (0,1) are reported. The test statistics are all well-known and each of them was originally proposed because they should have high power against some alternative distributions. The tests considered are the Pearson probability product test, the Neyman smooth test, the Sukhatme test, the Durbin-Kolmogorov test, the Kuiper test, and the Sherman test. Results are given for each of these tests against each of four classes of alternatives. Also, the most powerful test against each member of the first three alternatives is obtained, and the powers of these tests are given for the same sample sizes as for the six general "omnibus" test statistics. These values constitute a "power envelope" against which all tests can be compared. The Neyman smooth tests with 2nd and 4th degree polynomials are found to have good power and are recommended as general tests for uniformity.     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

The Use of Posterior Predictive P-Values in Testing Goodness-of-Fit

In this article, we introduce two goodness-of-fit tests for testing normality through the concept of the posterior predictive p-value. The discrepancy variables selected are the Kolmogorov-Smirnov (KS) and Berk-Jones (BJ) statistics and the prior chosen is Jeffreys’ prior. The constructed posterior predictive p-values are shown to be distributed independently of the unknown parameters under the null hypothesis, thus they can be taken as the test statistics. It emerges from the simulation that the new tests are more powerful than the corresponding classical tests against most of the alternatives concerned.     

Most mean powerful invariant test for testing two-dimensional parameter spaces

This paper investigates the application of the most mean powerful invariant test to the problem of testing for joint MA(1)–MA(4) disturbances against joint AR(1)–AR(4) disturbances in the linear regression model. The most mean powerful invariant test was introduced by Begum and King (Most mean powerful invariant test of a composite null against a composite alternative. Comp. Statist. Data Analysis, 2004, forthcoming) and is based on the generalized Neyman–Pearson lemma which provides an optimal test of certain composite hypotheses. The most mean powerful invariant test can be computationally intensive. Previous applications have only involved testing problems whose null hypotheses, after reduction through invariance arguments, are one dimensional. This is the first application involving null and alternative hypotheses which are two dimensional. A Monte Carlo experiment was conducted to assess the small sample performance of the test with encouraging results. The increase in dimension does increase significantly the computational effort required to apply the test.     

Some tests for uniformity of circular distributions powerful against multimodal alternatives

The author introduces new statistics suited for testing uniformity of circular distributions and powerful against multimodal alternatives. One of them has a simple expression in terms of the geometric mean of the sample of chord lengths. The others belong to a family indexed by a continuous parameter. The asymptotic distributions under the null hypothesis are derived. We compare the power of the new tests against Stephens's alternatives with those of Ajne, Watson, and Hermans‐Rasson's tests. Some of the new tests are the most powerful when the alternative has three or four modes. A heuristic justification of this feature is given. An application to the analysis of archaeological data is provided. The Canadian Journal of Statistics 38:80–96; 2010 © 2010 Statistical Society of Canada     

Bayesian testing of an exponential point null hypothesis

The problem of testing a point null hypothesis involving an exponential mean is The problem of testing a point null hypothesis involving an exponential mean is usual interpretation of P-values as evidence against precise hypotheses is faulty. As in Berger and Delampady (1986) and Berger and Sellke (1987), lower bounds on Bayesian measures of evidence over wide classes of priors are found emphasizing the conflict between posterior probabilities and P-values. A hierarchical Bayes approach is also considered as an alternative to computing lower bounds and “automatic” Bayesian significance tests which further illustrates the point that P-values are highly misleading measures of evidence for tests of point null hypotheses.     

ON SOME OPTIMALITY PROPERTIES OF FISHER-RAO SCORE FUNCTION IN TESTING AND ESTIMATION

The score function is associated with some optimality features in statistical inference. This review article looks on the central role of the score in testing and estimation. The maximization of the power in testing and the quest for efficiency in estimation lead to score as a guiding principle. In hypothesis testing, the locally most powerful test statistic is the score test or a transformation of it. In estimation, the optimal estimating function is the score. The same link can be made in the case of nuisance parameters: the optimal test function should have maximum correlation with the score of the parameter of primary interest. We complement this result by showing that the same criterion should be satisfied in the estimation problem as well.     

On hypothesis testing for Poisson processes: Regular case

ABSTRACT

We consider the problem of hypothesis testing in the situation when the firsthypothesis is simple and the second one is local one-sided composite. We describe the choice of the thresholds and the power functions of the Score Function test, of the General Likelihood Ratio test, of the Wald test, and of two Bayes tests in the situation when the intensity function of the observed inhomogeneous Poisson process is smooth with respect to the parameter. It is shown that almost all these tests are asymptotically uniformly most powerful. The results of numerical simulations are presented.     

Hypotheses Testing: Poisson Versus Self-exciting

Abstract.  We consider the problem of hypotheses testing with the basic simple hypothesis: observed sequence of points corresponds to the stationary Poisson process with known intensity. The alternatives are stationary self-exciting point processes. We consider one-sided parametric and one-sided non-parametric composite alternatives and construct locally asymptotically uniformly most powerful tests. The results of numerical simulations of the tests are presented.     

Testing nonparametric statistical functionals with applications to rank tests

The present paper discusses how nonparametric tests can be deduced from statistical functionals. Efficient and asymptotically most powerful maximin tests are derived. Their power function is calculated under implicit alternatives given by the functional for one – and two – sample testing problems. It is shown that the asymptotic power function does not depend on the special implicit direction of the alternatives but only on quantities of the functional. The present approach offers a nonparametric principle how to construct common rank tests as the Wilcoxon test, the log rank test, and the median test from special two-sample functionals. In addition it is shown that studentized permutation tests yield asymptotically valid tests for certain extended null hypotheses given by functionals which are strictly larger than the common i.i.d. null hypothesis. As example tests concerning the von Mises functional and the Wilcoxon two-sample test are treated.     

Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis

This paper introduces a general framework for testing hypotheses about the structure of the mean function of complex functional processes. Important particular cases of the proposed framework are as follows: (1) testing the null hypothesis that the mean of a functional process is parametric against a general alternative modelled by penalized splines; and (2) testing the null hypothesis that the means of two possibly correlated functional processes are equal or differ by only a simple parametric function. A global pseudo‐likelihood ratio test is proposed, and its asymptotic distribution is derived. The size and power properties of the test are confirmed in realistic simulation scenarios. Finite‐sample power results indicate that the proposed test is much more powerful than competing alternatives. Methods are applied to testing the equality between the means of normalized δ‐power of sleep electroencephalograms of subjects with sleep‐disordered breathing and matched controls.     

On testing the number of components in finite mixture models with known relevant component distributions

The limiting distribution of the log-likelihood-ratio statistic for testing the number of components in finite mixture models can be very complex. We propose two alternative methods. One method is generalized from a locally most powerful test. The test statistic is asymptotically normal, but its asymptotic variance depends on the true null distribution. Another method is to use a bootstrap log-likelihood-ratio statistic which has a uniform limiting distribution in [0,1]. When tested against local alternatives, both methods have the same power asymptotically. Simulation results indicate that the asymptotic results become applicable when the sample size reaches 200 for the bootstrap log-likelihood-ratio test, but the generalized locally most powerful test needs larger sample sizes. In addition, the asymptotic variance of the locally most powerful test statistic must be estimated from the data. The bootstrap method avoids this problem, but needs more computational effort. The user may choose the bootstrap method and let the computer do the extra work, or choose the locally most powerful test and spend quite some time to derive the asymptotic variance for the given model.     

On equivalence of two variances of a bivariate normal vector

A test for assessing the equivalence of two variances of a bivariate normal vector is constructed. It is uniformly more powerful than the two one-sided tests procedure and the power improvement is substantial. Numerical studies show that it has a type I error close to the test level at most boundary points of the null hypothesis space. One can apply this test to paired difference experiments or 2×2 crossover designs to compare the variances of two populations with two correlated samples. The application of this test on bioequivalence in variability is presented. We point out that bioequivalence in intra-variability implies bioequivalence in variability, however, the latter has a larger power.