Restricted most powerful Bayesian tests for linear models

Uniformly most powerful Bayesian tests (UMPBTs) are a new class of Bayesian tests in which null hypotheses are rejected if their Bayes factor exceeds a specified threshold. The alternative hypotheses in UMPBTs are defined to maximize the probability that the null hypothesis is rejected. Here, we generalize the notion of UMPBTs by restricting the class of alternative hypotheses over which this maximization is performed, resulting in restricted most powerful Bayesian tests (RMPBTs). We then derive RMPBTs for linear models by restricting alternative hypotheses to g priors. For linear models, the rejection regions of RMPBTs coincide with those of usual frequentist F‐tests, provided that the evidence thresholds for the RMPBTs are appropriately matched to the size of the classical tests. This correspondence supplies default Bayes factors for many common tests of linear hypotheses. We illustrate the use of RMPBTs for ANOVA tests and t‐tests and compare their performance in numerical studies.     

Bayesian point null hypothesis testing via the posterior likelihood ratio

This paper gives an exposition of the use of the posterior likelihood ratio for testing point null hypotheses in a fully Bayesian framework. Connections between the frequentist P-value and the posterior distribution of the likelihood ratio are used to interpret and calibrate P-values in a Bayesian context, and examples are given to show the use of simple posterior simulation methods to provide Bayesian tests of common hypotheses.     

Blending Bayesian and Classical Tools to Define Optimal Sample-Size-Dependent Significance Levels

ABSTRACT

This article argues that researchers do not need to completely abandon the p-value, the best-known significance index, but should instead stop using significance levels that do not depend on sample sizes. A testing procedure is developed using a mixture of frequentist and Bayesian tools, with a significance level that is a function of sample size, obtained from a generalized form of the Neyman–Pearson Lemma that minimizes a linear combination of α, the probability of rejecting a true null hypothesis, and β, the probability of failing to reject a false null, instead of fixing α and minimizing β. The resulting hypothesis tests do not violate the Likelihood Principle and do not require any constraints on the dimensionalities of the sample space and parameter space. The procedure includes an ordering of the entire sample space and uses predictive probability (density) functions, allowing for testing of both simple and compound hypotheses. Accessible examples are presented to highlight specific characteristics of the new tests.     

Improved likelihood ratio tests and pearson chi-square tests for independence in two dimensional contingency tables

When an I×J contingency table has many cells having very small frequencies, the usual chi-square approximation to the upper tail of the likelihood ratio goodness-of-fit statistic, G² and Pearson chi-square statistic, X², for testing independence, are not satisfactory. In this paper we consider the problem of adjusting G² and X². Suitable adjustments are suggested on the basis of analytical investigation of asymptotic bias terms for G² and X². A Monte Carlo simulation is performed for several tables to assess the adjustments of G² and X² in order to obtain a closer approximation to the nominal level of significance.     

Admissible tests for the mean with additional information

Let X be a po-normal random vector with unknown µ and unknown covariance matrix ∑ and let X be partitioned as X = (X^′ ₍₁₎, …, X^′ _(r))′ where X_(j)is a subvector of X with dimension p_jsuch that ∑^r _j=1Pj = P0. Some admissible tests are derived for testing H₀: μ = 0 versus H₁: μ ¦0 based on a sample drawn from the whole vector X of dimension p and r additional samples drawn from X₍₁₎, X₍₂₎, …, X(r) respectively, All (r+1) samples are assumed to be independent. The distribution of some of the tests' statistics involved are also derived.     

Uniformly most powerful unbiased test in univariate linear calibration

In this paper, we deal with the hypothesis testing problems for the univariate linear calibration, where a normally distributed response variable and an explanatory variable are involved, and the observations of the response variable corresponding to known values of the explanatory variable are used for making inferences concerning a single unknown value of the explanatory variable. The uniformly most powerful unbiased tests for both one-sided and two-sided hypotheses are constructed and verified. The power behaviour of the proposed tests is numerically compared with that of the existing method, and simulations show that the proposed tests make the powers improved.     

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.     

Tests for the skewness parameter of two-piece double exponential distribution in the presence of nuisance parameters

In this paper, tests for the skewness parameter of the two-piece double exponential distribution are derived when the location parameter is unknown. Classical tests like Neyman structure test and likelihood ratio test (LRT), that are generally used to test hypotheses in the presence of nuisance parameters, are not feasible for this distribution since the exact distributions of the test statistics become very complicated. As an alternative, we identify a set of statistics that are ancillary for the location parameter. When the scale parameter is known, Neyman–Pearson's lemma is used, and when the scale parameter is unknown, the LRT is applied to the joint density function of ancillary statistics, in order to obtain a test for the skewness parameter of the distribution. Test for symmetry of the distribution can be deduced as a special case. It is found that power of the proposed tests for symmetry is only marginally less than the power of corresponding classical optimum tests when the location parameter is known, especially for moderate and large sample sizes.     

On hypothesis tests for disk-type intensities of spatial poisson processes

We consider the problem of testing whether realizations of a spatial Poisson process come from a circular intensity model or from an elliptical intensity model. We describe the behavior of the Generalized Likelihood Ratio and Wald tests, in the asymptotics of large samples. The power functions are studied under local alternatives and results of numerical simulations comparing them to the Neyman–Pearson envelope are presented.     

Adaptive tests of regression functions via multiscale generalized likelihood ratios

Many applications of nonparametric tests based on curve estimation involve selecting a smoothing parameter. The author proposes an adaptive test that combines several generalized likelihood ratio tests in order to get power performance nearly equal to whichever of the component tests is best. She derives the asymptotic joint distribution of the component tests and that of the proposed test under the null hypothesis. She also develops a simple method of selecting the smoothing parameters for the proposed test and presents two approximate methods for obtaining its P‐value. Finally, she evaluates the proposed test through simulations and illustrates its application to a set of real data.     

Bayesian inference method for model validation and confidence extrapolation

This paper presents a Bayesian-hypothesis-testing-based methodology for model validation and confidence extrapolation under uncertainty, using limited test data. An explicit expression of the Bayes factor is derived for the interval hypothesis testing. The interval method is compared with the Bayesian point null hypothesis testing approach. The Bayesian network with Markov Chain Monte Carlo simulation and Gibbs sampling is explored for extrapolating the inference from the validated domain at the component level to the untested domain at the system level. The effect of the number of experiments on the confidence in the model validation decision is investigated. The probabilities of Type I and Type II errors in decision-making during the model validation and confidence extrapolation are quantified. The proposed methodologies are applied to a structural mechanics problem. Numerical results demonstrate that the Bayesian methodology provides a quantitative approach to facilitate rational decisions in model validation and confidence extrapolation under uncertainty.     

Default Bayesian goodness-of-fit tests for the skew-normal model

In this paper we propose a series of goodness-of-fit tests for the family of skew-normal models when all parameters are unknown. As the null distributions of the considered test statistics depend only on asymmetry parameter, we used a default and proper prior on skewness parameter leading to the prior predictive p-value advocated by G. Box. Goodness-of-fit tests, here proposed, depend only on sample size and exhibit full agreement between nominal and actual size. They also have good power against local alternative models which also account for asymmetry in the data.     

Pseudo‐likelihood ratio tests for semiparametric multivariate copula model selection

The authors propose pseudo‐likelihood ratio tests for selecting semiparametric multivariate copula models in which the marginal distributions are unspecified, but the copula function is parameterized and can be misspecified. For the comparison of two models, the tests differ depending on whether the two copulas are generalized nonnested or generalized nested. For more than two models, the procedure is built on the reality check test of White (2000). Unlike White (2000), however, the test statistic is automatically standardized for generalized nonnested models (with the benchmark) and ignores generalized nested models asymptotically. The authors illustrate their approach with American insurance claim data.     

Bootstrap tests for time varying cointegration

This article proposes wild and the independent and identically distibuted (i.i.d.) parametric bootstrap implementations of the time-varying cointegration test of Bierens and Martins (2010 Bierens, H. J., Martins, L. F. (2010). Time varying cointegration. Econometric Theory 26:1453–1490.[Crossref], [Web of Science ®] , [Google Scholar]). The bootstrap statistics and the original likelihood ratio test share the same first-order asymptotic null distribution. Monte Carlo results suggest that the bootstrap approximation to the finite-sample distribution is very accurate, in particular for the wild bootstrap case. The tests are applied to study the purchasing power parity hypothesis for twelve Organisation for Economic Cooperation and Development (OECD) countries and we only find evidence of a constant long-term equilibrium for the U.S.–U.K. relationship.     

Bayesian hypothesis testing for selected regression coefficients

ABSTRACT

We develop new Bayesian regression tests for prespecified regression coefficients. Simple, closed forms of the Bayes factors are derived that depend only on the regression t-statistic and F-statistic and the usual associated t and F distributions. The priors that allow those forms are simple and also meaningful, requiring minimal but practically important subjective inputs.     

Nonparametric tests for the mean of a non-negative population

We construct level-α tests for testing the null hypothesis that the mean of a non-negative population falls below a prespecified nominal value. These tests make no assumption about the distribution function other than that it be supported on [0,∞). Simple tests are derived based on either the sample mean or the sample product. The nonparametric likelihood ratio test is also discussed in this context. We also derive the uniformly most powerful monotone (UMP) tests for special cases.     

Some extensions of zero-inflated models and Bayesian tests for them

Zero-inflated power series distribution is commonly used for modelling count data with extra zeros. Inflation at point zero has been investigated and several tests for zero inflation have been examined. However sometimes, inflation occurs at a point apart from zero. In this case, we say inflation occurs at an arbitrary point j. The j-inflation has been discussed less than zero inflation. In this paper, inflation at an arbitrary point j is studied with more details and a Bayesian test for detecting inflation at point j is presented. The Bayesian method is extended to inflation at arbitrary points i and j. The relationship between the distribution for inflation at point j, inflation at points i and j and missing value imputation is studied. It is shown how to obtain a proper estimate of the population variance if a mean-imputed missing at random data set is used. Some simulation studies are conducted and the proposed Bayesian test is applied on two real data sets.     

Bayes factor in testing precise hypotheses

Substitution of a mixed prior distribution by a continuous one for the point null hypothesis testing problem is discussed. Conditions are established in order to approximate the Bayes factors for the two problems. Besides, trough this approximation an assignation of priorprobabilities is suggested.     

Likelihood ratio tests for fuzzy hypotheses testing

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which hypotheses are imprecise. In this paper, we recall and redefine some concepts about fuzzy hypotheses testing, and then we introduce the likelihood ratio test for fuzzy hypotheses testing. Finally, we give some applied examples.