Adaptive chi-square tests

The minimax hypothesis-testing problem is considered. Let H₀: f=f₀≡1 and H₁ consist of smooth densities. It is shown that the optimal, in the minimax sense, order of distinguishing is attained by a procedure based on simultaneous use ofχ ²-tests corresponding to the growing number of intervals of grouping of the sample. Bibliography: 16 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 150–166. Translated by S. Yu. Pilyugin.     

Semi-aligned rank tests

Summary The distribution-free test based on semi-aligned rankings for no treatment effects in a two-way layout, with unequal number of replications in each cell is considered. The asymptotic χ-square distribution of the test statistic under the null hypothesis is derived. The Pitman asymptotic relative efficiency of the test (i) based on semi-aligned rankings with respect to the test (ii) based on within-block rankings, is shown to be larger than one as the number of blocks tends to infinity. Also the asymptotic properties of linear rank statistics (i) and (ii) are investigated and the asymptotic relative efficiency of the test (i) with respect to the test (ii) is again shown to be larger than one.     

Robust slippage tests

Summary The robust slippage testing problems ofk+1 approximately known simple hypotheses are formulated as the slippage testing problems ofk+1 composite hypotheses. It is shown that if there is a representativek+1-tuple (called a least favorable slippage tuple) of simple hypotheses, then maximin tests are given by the slippage analogues of the Neyman-Pearson tests for this tuple. Thek-sample case is treated concerning this subject. In the general situations that there does not exist any least favorable slippage tuple, a method for constructing tests is proposed and applied to the case that composite hypotheses are described in terms of certain capacities (ε-contamination, total variation). The variants of the derived tests are also suggested.     

Rank tests for regression and k-sample rank tests under random censorship

The uncensored observations and the censored ones are ranked separately. A test statistic is obtained by suitably combining the rank statistics corresponding to these two groups. This approach leads to simple tests that closely resemble those for the uncensored situation.     

Goodness-of-fit tests for copulas

This paper defines two distribution free goodness-of-fit test statistics for copulas. It states their asymptotic distributions under some composite parametric assumptions in an independent identically distributed framework. A short simulation study is provided to assess their power performances.     

Non-parametricc-sample tests with regression

For testing that several regression lines are concurrent, a class of rank score tests is proposed. This class includes the usual Wilcoxon and normal scores type of tests. The performance of the proposed tests is shown to be the same as that of rank score tests in the ordinaryc-sample problem.     

On locally optimal nonparametric tests

Summary There is an abundancy of problems in which no parametric model realistically describes the situation and in which, accordingly, we have to resort to nonparametric methods. As the numerical problems connected with nonparametric tests are becoming less and less important, rank tests, permutation tests and the like are becoming more and more part of the standard armatory of applied statisticians. The lack of tabulated critical values, for instance, should no longer be a serious objection against the use of permutation tests in practice; cf. Edgington (1987).The rationale underlying permutation and rank tests has been outlined in quite a number of text books and papers; cf. Fraser (1957), Lehmann (1959), Hájek-Sidák (1967) or Witting (1970). Roughly speaking, permutation tests are constructibel if the data can be condensed by means of a sufficient and complete statistic allowing for the proper kind of conditioning. Rank tests arise if the underlying problem is invariant with respect to (w.r.t.) a large group of transformations which leads to a maximal invariant statistic consisting of (signed) ranks.Most practical nonparametric problems, however, are too complex to be tractable by just one of those approaches. Many of them, however, can be handled by a combination of both techniques. In this paper we outline the logic underlying that combined reduction method and apply it to construct locally most powerful tests. Moreover, we discuss what we label Hoeffding's transfer problem, i.e. the uniformity aspect of locally most powerful tests with respect to the starting point at the boundary. We are concentrating on the discussion of the nonparametric two-sample location and scale problem. Further important problems are mentioned in Section III.This is a written account of an invited lecture delivered by the third author on occasion of the 14th Symposium über Operations Research, Ulm, September 6–8, 1989.     

What are comparison tests?

<正>When the series have been known to be convergent or divergent,and we want to determine whether the new series converges or diverges.At this situation,we can use the comparison tests to compare a given series with a series that is known.For example:When we see the new series     

Conditional tests for elliptical symmetry

In this paper, we suggest the conditional test procedures for testing elliptical symmetry of multivariate distribution. The conditional tests are exactly valid if the symmetric center and the shape matrix are given and are asymptotically valid if they are unknowns to be estimated. The equivalence, in the large sample sense, between the conditional tests and their unconditional counterparts is established. The power behavior of the tests under global as well as local alternatives is investigated theoretically. A small simulation study is performed.     

Information theory and alternate-hypothesis tests

Lindley's concept of the information in an experiment is used to study alternate-hypothesis statistical tests. A test is considered to be a combination of a sampling rule and a decision rule which is based on the samples taken. Two information functions are formed: the information about the unknown parameter contained in the samples taken and the information contained in the decision to which these samples lead. The former quantity is at least as great as the latter, with equality if, and only if, the test is an SPRT (sequential probability ratio test) neglecting the excess over the boundary.If the average information contained in the samples taken in one alternate-hypothesis test is greater than the average information contained in the samples taken in another, then the first test has greater expected length. It is also shown that an SPRT neglecting excess is the unique alternate-hypothesis test that utilizes all the information contained in the samples taken. These results provide a proof of the standard result concerning the optimal character of the SPRT neglecting excess.The major purpose of this paper is to develop an information-theoretic analysis of alternate-hypothesis tests. By means of this approach, new results on the SPRT and alternate-hypothesis tests in general are obtained.     

Minimax tests for convex cones

Let (P : R ^p ) be a simple shift family of distributions onR ^p , and letK R ^p be a convex cone. Within the class of nonrandomized tests ofK versusR ^p K, whose acceptance regionA satisfiesA=A+K, a test with minimal bias is constructed. This minimax test is compared to a likelihood ratio type test, which is optimal with respect to a different criterion. The minimax test is mimicked in the context of linear regression and one-sided tests for covariance matrices.     

Nonparametric tests based on N-distances

This paper is devoted to goodness-of-fit and homogeneity tests based on N-distances. The work is a continuation of our research started in [2]. The power of the proposed criteria is compared with classical tests using Monte Carlo simulations. Different alternatives both in one-and multidimensional cases are investigated. Applications of N-distance statistics for testing hypotheses of symmetry (univariate case) and independence (bivariate case) are provided.     

Nonlinearity tests for bilinear systems

The purpose of this paper is to develop nonlinearity tests for open-loop bilinear systems. Lagrange multiplier tests of linear systems against a bilinear alternative are proposed. A simulation study is performed to check the validity of the asymptotic null distributions of the test statistics and to investigate the power characteristics of the tests. Two recent nonlinearity tests in the time-series context are adapted to linear systems and compared with Lagrange multiplier tests. Simulation results show that the proposed Lagrange multiplier tests are more powerful than the other tests.     

Multivariate nonparametric tests for independence

Multivariate generalizations of Bhuchongkul's bivariate rank statistics [Ann. Math. Statist.35 (1964)] have been introduced and studied in this paper for the purpose of testing mulitvariate independence. It is shown that the test statistics can be expressed as rank statistics which are easy to compute, have asymptotic normal distributions, and can detect mutual dependence in alternatives which are pairwise independent. The tests are compared to the Puri-Sen-Gokhale [[8]] tests and a normal theory test [ [1]] using Pitman efficiency.     

On generalized start-up demonstration tests

Start-up demonstration tests and various extensions have been discussed, in which a unit under the test is accepted or rejected according to some criteria. CSTF, CSCF, TSCF and TSTF are most well known start-up demonstration tests. In this paper, two kinds of more general start-up demonstration tests are introduced. CSTF, TSTF, TSCF and CSCF are all special situations of the new tests. For the new generalized start-up demonstration tests, under the assumption of independent and identically distributed trials for each test, the analytic expressions for the expectation, the probability mass function and the distribution of the test length, as well as the probability of acceptance or rejection of the unit are given. All the analyses are based on the finite Markov chain imbedding approach which avoids the complexities of the probability generating function approach and makes the results readily understood and easily extended to the non-i.i.d. cases. Furthermore, an optimal model for generalized start-up demonstration tests is proposed. Finally, a numerical example is presented to make our results more transparent, and it can demonstrate the advantages of the new tests.     

Compression tests on composite tubes

The possibility of determining the modulus of elasticity, Poisson's ratio, and compressive strength across the fibers of a unidirectional composite by means of compression tests on cross-wound tubes is examined. By analyzing the Leknitskii [3] solution for the case of a transversely isotropic cylinder it is shown that only the modulus of elasticity can be reliably determined in this way. The conclusions are confirmed by experimental data obtained by testing glass-reinforced plastic tubes.     

On multivariate signed rank tests

Summary As a test for the hypothesis of equality of means in a sample ofn matched pairs of observations from a single bivariate population in(x, y), Wilcoxon ([4], [5]) proposed a signed rank sum testU involving the differencesz=x\t-y. Tables of valuesUs have been provided forn of 50 or less and four significance levelsε. The following paper considers some generalizations of the Wilcoxon test for the case of matched bivariate or trivariate populations. These tests utilize certain properties of thep-variate normal distribution when integrated over certain regions for the cases:p=2,3.