Asymptotic Relative Efficiencies Of Tests Research Articles

Rank tests and regression rank score tests in measurement error models

The rank and regression rank score tests of linear hypothesis in the linear regression model are modified for measurement error models. The modified tests are still distribution free. Some tests of linear subhypotheses are invariant to the nuisance parameter, others are based on the aligned ranks using the R -estimators. The asymptotic relative efficiencies of tests with respect to tests in models without measurement errors are evaluated. The simulation study illustrates the powers of the tests.

Theoretical efficiency comparisons of independence tests based on multivariate versions of Spearman’s rho

Schmid and Schmidt (Stat Probab Lett 77:407–416, 2007) recently described multivariate extensions to the population and sample versions of Spearman’s rank correlation coefficient. In this paper, the asymptotic relative efficiency of tests for multivariate independence based on the latter measures are explicitly computed under many distributional scenarios based on copula models, which complement findings made by Stepanova (Math Methods Stat 12:197–217, 2003). New Spearman-type statistics based on the so-called Mobius decomposition of the empirical copula process and of the survival copula process are also investigated in the light of their asymptotic relative efficiencies.

Asymptotic relative efficiency of tests at the boundary of regular statistical models

Consider a sequence of tests which are asymptotically equivalent to some score tests for a regular model. In the present paper we address the question what's their power and asymptotic relative efficiency (ARE) at the boundary of the statistical model when the Fisher information becomes infinite but still local asymptotic normality (LAN) holds. The boundary models are frequently called almost regular. It is shown that almost regular models and efficient procedures for the almost regular case are completely different from the regular situation. The ARE of regular tests collapses at the boundary. A specific example is the two-sample life time regression model given by Weibull distributions. Here, the Wilcoxon two-sample and Savage rank tests (log-rank test) have ARE=0 at the boundary. The results are based on a uniform LAN-expansion for master models specified by tangents. This part is of own interest.

GMM tests for the Katz family of distributions

Generalized method of moments (GMM) is used to develop tests for discriminating discrete distributions among the two-parameter family of Katz distributions. Relationships involving moments are exploited to obtain identifying and over-identifying restrictions. The asymptotic relative efficiencies of tests based on GMM are analyzed using the local power approach and the approximate Bahadur efficiency. The paper also gives results of Monte Carlo experiments designed to check the validity of the theoretical findings and to shed light on the small sample properties of the proposed tests. Extensions of the results to compound Poisson alternative hypotheses are discussed.

Bounds on AREs of Tests Following Box-Cox Transformations

Bounds on the asymptotic relative efficiency (ARE) of the Box-Cox transformed two-sample $t$-test to the ordinary $t$-test are obtained under local alternatives. It is shown that the ARE is at least 1 for location-shift models. In the case of scale-shift models, a similar bound applies provided the limiting value of the estimated power transformation is greater than 1. If instead the Box-Cox transformed $t$-test is compared against the ordinary $t$-test applied to the log-transformed data, then the ARE is bounded below by 1 for all scale-shift models, regardless of the limiting value of the power transformation. The results extend naturally to the multisample $F$-test. A small simulation study to evaluate the validity of the asymptotic results for finite-sample sizes is also reported.

A class of tests for testing an increasing failure-rate-average distribution with randomly right-censored data

For testing exponentiality versus (nonexponential) increasing failure rate average (IFRA) alternatives (which are nonexponential) using the randomly right censored data, a class of test statistics based on a functional of the Kaplan-Meier estimator is proposed. The asymptotic relative efficiencies of tests from this class with respect to other test statistics are derived. The efficiency loss due to censoring is studied. The proposed tests are applied to published survival data. >

The asymptotic relative efficiency of score tests in a generalized linear model with surrogate covariates

SUMMARY In many epidemiological studies, the exposure variable of interest cannot be measured directly. The classical approaches to errors in variables in regression do not extend easily to the nonlinear models commonly used in epidemiologic al research. Furthermore, the traditional additive measurement error model cannot adequately represent many surrogate relationships. By considering the effect of using surrogate independent variables on the efficient score statistic, some of the difficulties inherent in the estimation problem may be avoided. For the null hypothesis of no association, a simple and flexible procedure can be used to calculate the optimal score test. The asymptotic relative efficiency of this test to the test based upon the true exposures is derived. The optimal test is also compared to the naive procedure of substituting the surrogate into the score test for the true exposure.

Testing whether F is “more IFRA than is G”

For testing the null hypothesis that two life distributions F and G are equal versus the alternative hypothesis that F is more increasing failure rate average than is G, a class of tests is proposed. The properties of tests in this class such as unbiasedness, consistency and asymptotic normality are established. Monte Carlo estimates of power are obtained for small sample sizes. The asymptotic relative efficiency of tests in the class with respect to Hollander, Park and Proschan's (1986) test is shown to be reasonably high.

The Asymptotic Efficiency of Tests Using Misclassified Data in Contingency Tables

A simple expression is given for the approximate upper bound of the asymptotic relative efficiency of tests between nested loglinear models using misclassified data versus those using data with no classification errors. This efficiency depends on the probabilities of data being misclassified into the wrong cells of the contingency table. An example demonstrates that the loss of efficiency due to misclassification can be substantial.

A class of location-scale nonparametric tests

SUMMARY A class of two-sample nonparametric tests for location and scale shift, simultaneously, is proposed. The asymptotic distribution of the proposed class is obtained both under the hypothesis and alternative. Using this result it is shown how the asymptotic relative efficiency of tests in the proposed class can be obtained. F2(x) = F(a2x +4f2), respectively. The purpose of this paper is to study a class of two-sample nonparametric test statistics which can be used to test HO: Fl(x) = F2(x) = F(x) for all real x, versus the alternative Ha:acc * a2 or /h t /h' or both. The proposed statistics will be expressed as functions of two nonparametric statistics, one sensitive to scale shift and the other to location shift. Lepage (1971) proposed a statistic which is a combination of the Wilcoxon and Ansari- Bradley statistics. Furthermore, he showed that since the Wilcoxon and Ansari-Bradley statistics are uncorrelated under Ho, his proposed statistic has a limiting central chi- squared distribution with two degrees of freedom. Thus the critical values needed to carry out an approximate test of Ho can be obtained from a standard chi-squared table. The question regarding the asymptotic power behaviour of Lepage's statistic is of interest but is not, however, discussed by Lepage. Thus the asymptotic relative efficiency of the proposed statistics will also be discussed, as a means of comparing such statistics.

The Average Critical Value Method and the Asymptotic Relative Efficiency of Tests

The Average Critical Value Method and the Asymptotic Relative Efficiency of Tests

The average critical value method and the asymptotic relative efficiency of tests

The average critical value method and the asymptotic relative efficiency of tests

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SummaryTwo test statistics t and b, testing equality of probabilities pi of success in k different series against the hypothesis of trend with given numbers gi (weights) of the series, are compared. The first teststatistic, due to C. van Ee den en J, Hemelrijk [1] is with ni1 equal to the number of successes in ni trials and Defining trend by it appears that the teststatistic t gives rise to a test consistent for the complete set of alternatives τ≠ 0.The other teststatistic is b gives rise to a test which is not consistent for the general set of alternatives τ≠ 0 but for a rather important subset of these alternatives, i.e. those alternatives which show a lineair trend.Neither of the tests in necessarily unbiased. The asymptotic relative efficiency of test t with respect to b is equal to or lower than unity. (Equal in casen1= n2=…= nk with b = t;in this case b is also consistent for the set of alternatives τ≠ 0). The variances of the teststatistics can be estimated with Both tests are based on the approximately normal distribution of the teststatistics. To judge this approximation the 3rd and 4th cumulants of the distributions of the statistics are evaluated in terms of number of elements of a binomial distribution.It is concluded that in case of possible non‐lineair relationships the teststatistic t is preferable as it gives rise to a consistent, designfree test. In case a lineair relationship has to be tested against the hypothesis of no trend the teststatistic b has to be prefierred, especially if the number of trials in the series are very different. An example is discussed.

The asymptotic relative efficiencies of tests and the derivatives of their power functions

Abstract For comparing two consistent tests of a simple null hypothesis H 0 : θ = θ0 against a given alternative hypothesis H 0 : θ = θ1, the measure most frequently used is the asymptotic relative efficiency (ARE), due to Pitman (1948). The ARE is defined as the limit of the reciprocal of the ratio of sample sizes required to attain the same power. The limit is taken as sample sizes tend to infinity and simultaneously θ1 → θ0, this being necessary to keep the powers of the tests bounded away from 1.