Experimental determination of the power functions of the two-sample rank tests of WILCOXON, VAN DER WAERDEN and TERRY by Monte Carlo techniques

In this paper the authors present the results of a sampling experiment to determine the power functions of the two-sample rank tests of WILCOXON, VAN DER WAERDEN and TERRY against shift alternatives for normal parent distributions, based on 2000 trials for each alternative. The sample sizes considered are m = n = 6 and m = n = 10. The powers of the three rank tests are compared with the power of the STUDENT t-test and with each other. The results indicate that in small samples (i) the power of the WILCOXON test is not much smaller than the power of the t-test and (ii) the normal scores tests are only slightly superior to the WILCOXON test, if at all.     

Een Monte-Carlo-bepaling van overschrijdingskansen, in verband met een keuzetest

A Monte-Carlo method for a test of significance, applied to points on a lattice, in connection with a vocational preference test, by C. A. G, Nass.
Appendix by Constance van Eeden.
A periodical rectangular lattice, with a period of k.m, is considered. Thus there are N = k.m points on the lattice, repeated in the two perpendicular directions. Two points are said to be "connected" if they are adjacent in a straight or diagonal way. Thus, if k and m 3, every point is connected with 8 other pooints. Out of the N points of the lattice, n points are selected and the total number of connections x, of all possible pairs of those n points is considered for a vocational preference test with k = m = 9, N = 81, n = 10. The problem is to test whether the sum y = x₁+…+ x*** from a sample of h values of x, is significantly small, under the hypothesis that in the h cases the n points are selected at random with equal chance. A Monte-Carlo sample of 100 values of x was taken, using random numbers. For h = 1, the problem was solved by the determination of P( y x₁), assuming that y is taken at random from the 101 values of x, supplied by the Monte-Carlo sample and x₁ for fixed values of x₁. For h - 2, a similar solution is given. For greater values of h, Student's two-sample test, with correction for continuity is suggested. For h = 2 the results of Student's test are compared with those of the solution mentioned above.
In the appendix a summary is given of results found by P.A.P Moran and P. V. Krishna Iyer for some closely related problems. Further some results concerning exact distributions, moments and asymptotic distributions for C. A. G. Nass' problem are given. The proofs of these results may be found in a paper by C. van Eeden and A. R. Bloemena (1959).     

Duration response measurement error

The impact of response measurement error in duration data is investigated using small parameter asymptotic approximations and compared with the effect of hazard function heterogeneity. The approximations lead to a specification test to detect measurement error which is shown to be related to the class of Information Matrix tests. It is shown that in a commonly used class of models the test statistic is exactly pivotal. The second order asymptotic properties of the alternative forms of the test statistic are derived and the quality of the approximations and the performance of the test are investigated via Monte Carlo experimentation.     

Approximations of the eigenvalues of the covariance matrix of a first-order autoregressive process

This paper provides explicit estimates of the eigenvalues of the covariance matrix of an autoregressive process of order one. Also explicit error bounds are established in closed form. Typically, such an error bound is given by $\text{ε}{}_{\mathrm{k}}\text{= (}\text{4}\text{(n+1)}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{ρ}{}^2\text{sin}\text{(}\text{kπ}\text{(n+1)}\text{)}$, so that the approximations improve as the size of the matrix increases. In other words, the accuracy of the approximations increases as direct computations become more costly.     

On the statistical evaluation of hit patterns

Part A of this paper describes a statistical test procedure for examining the hitting accuracy of missiles by the evaluation of hit patterns. The procedure is illustrated by a numerical example which is supplemented by a sensitivity analysis.Part B presents the theoretical background of the test procedure. In this connection approximate methods are sometimes employed, mainly to simplify practical applications. Special emphasis is placed on an examination of the size of errors in the approximations used, and on an appraisal of the power of the test.     

关于含四个未知函数的费马型丢番图函数方程的非平凡解的研究

本文利用亚纯函数值分布论的思想方法,对费马型丢番图函数方程fn1(z)+fn2(z)+fn3(z)+fn4(z)=1的非平凡解的状况进行研究,得到如下结果:假设函数方程fn1(z)+fn2(z)+fn3(z)+fn4(z)=1存在非常数的亚纯函数解f1,f2,f3,f4令τn:=|f31 f32 f33 f34 f21f′1 f22f′3 f23f′3 f24f′4L1(f1)L1(f2)L1(f3)L1(f4)L2(f1)L2(f2)L2(f3)L2(f4)|及T*(r)=T(r,f1)+T(r,f2)+T(r,f3)+T(r,f4),这里L(1fm)=(n-1)fmf′m2+f m2f″m,L(2fm)=(n-1)(n-2)f′m3+3(n-1)fmf′mf″m+f m2+fm''(m=1,2,3,4)。那么,(1)当n≥17时,τ(nz)是整函数;(2)当n≥13时,m(r,τn)=S*(r);这里S*(r)=O{T*(r)(}r→∞,rE),E是R+的线性测度有限的某子集。     

New Developments of the Taguchi Quality Selection Model

The classical Taguchi quality selection model considers the case there is one input characteristic and one output characteristic, and it aims to select the best input mean to minimize the loss of quality. In paper we consider much more general situation: there are n input characteristics and moutput characteristics. For three different cases, namely n < m, n=m and n > m, we develop different techniques to determine the best input means to minimize the loss of quality. It is very interesting to see that the linear programming appears naturally and the well-known simplex algorithm has been used to solve the problems.     

Some approximations of n-copulas

We consider several approximations to n-copulas: the checkmin, checkerboard, Bernstein, and shuffle of min approximations. The checkerboard, Bernstein, and shuffle of min approximations have been studied in the n = 2 case. We investigate these constructions in arbitrary finite dimensions and consider some of the ways in which they converge or fail to converge to the original copula.     

Exact distribution of Durbin's distribution-free test statistic for Balanced Incomplete Block Designs, and comparison with the Chi-square and F approximation

Durbin's distribution-free rank test can be used to test the null hypothesis that there are no differences among the treatments in a Balanced Incomplete Block Design. Until now only the chi-square and F approximations for the test statistic were known. In this paper the exact distribution has been given for 15 designs and comparison is made with the chi-square and F approximation.     

Tests for multiple change points under ordered alternatives

We consider the problem of testing the null hypothesis of no change against the alternative of multiple change points in a series of independent observations when the changes are in the same direction. We extend the tests of Terpstra (1952), Jonckheere (1954) and Puri (1965) to the change point setup. We obtain the asymptotic null distribution of the proposed tests. We also give approximations for their limiting critical values and tables of their finite sample Monte Carlo critical values. The results of Monte Carlo power studies conducted to compare the proposed tests with some competitors are reported. This research was supported by research grant SS045 of Kuwait University. Acknowledgments. We wish to thank the two referees for their comments and suggestions which have significantly improved the presentation. We are particularly thankful to one of the referees for suggesting the test statistics T_n1 ^* (k) and T_n2 ^* (k).     

A comparison of several approximations to the power function of the likelihood ratio test in covariance structure analysis*

In this paper we compare alternative asymptotic approximations to the power of the likelihood ratio test used in covariance structure analysis for testing the fit of a model. Alternative expressions for the noncentrality parameter (ncp) lead to different approximations to the power function. It appears that for alternative covariance matrices close to the null hypothesis, the alternative ncp's lead to similar values, while for alternative covariance matrices far from H_o the different expressions for the ncp can conflict substantively. Monte Carlo evidence shows that the ncp proposed in Satorra and Saris (1985) gives the most accurate power approximations.     

A LIMIT THEOREM FOR BERNOULLI RV'S AND FELLER'S SHOE PROBLEM

Consider n sets of objects, each set consisting of m distinct types (for instance n place settings each made up of m distinct dishes and silverware pieces.) s items are drawn at random from the mn items. The distribution of the number of complete sets (each consisting of all m items) in the sample of s is asymptotically Poisson distributed with parameter (a /m )^m if s = an ^1–1 and n →∞. This fact can be interpreted in terms of a certain limit theorem for a sequence of i.i.d Bernoulli rv's.     

Generalized two-sample U-statistics for clustered data

In this paper we investigate two-sample U -statistics in the case of clusters of repeated measurements observed on individuals from independent populations. The observations on the i -th individual in the first population are denoted by     , 1 ≤  i  ≤  m , and those on the k -th individual in the second population are denoted by     , 1 ≤  k  ≤  n . Given the kernel φ ( x ,  y ), we define the generalized two-sample U -statistic by
We derive the asymptotic distribution of U _{m , n} for large sample sizes. As an application we study the generalized Mann–Whitney–Wilcoxon rank sum test for clustered data.     

Bootstrap inference in systems of single equation error correction models

We analyze OLS-based tests of long-run relationships, weak exogeneity and short-run dynamics in conditional error correction models. Unweighted sums of single equation test statistics are used for hypothesis testing in pooled systems. When model errors are (conditionally) heteroskedastic tests of weak exogeneity and short run dynamics are affected by nuisance parameters. Similarly, on the pooled level the advocated test statistics are no longer pivotal in presence of cross-sectional error correlation. We prove that the wild bootstrap provides asymptotically valid critical values under both conditional heteroskedasticity and cross-sectional error correlation. A Monte-Carlo study reveals that in small samples the bootstrap outperforms first-order asymptotic approximations in terms of the empirical size even if the asymptotic distribution of the test statistic does not depend on nuisance parameters. Opposite to feasible GLS methods the approach does not require any estimate of cross-sectional correlation and copes with time-varying patterns of contemporaneous error correlation.     

Specification diagnostics based on Laguerre alternatives for econometric models of duration

Score tests of the null hypothesis of exponentially distributed durations (conditional on regressors) against alternatives in a family of approximations to arbitrary distributions for non-negative random variables are developed. The test statistics take a simple, easily calculated and interpreted form. The alternatives considered are expansions of various orders in Laguerre polynomials. An economic model generating exponential unemployment duration is presented. The statistics are applied in a look at unemployment durations in the Denver Income Maintenance Experiment data.     

Diagnostics analysis for log‐Birnbaum–Saunders regression models with censored data

This article develops influence diagnostics for log‐Birnbaum–Saunders (LBS) regression models with censored data based on case‐deletion model (CDM). The one‐step approximations of the estimates in CDM are given and case‐deletion measures are obtained. Meanwhile, it is shown that CDM is equivalent to mean shift outlier model (MSOM) in LBS regression models and an outlier test is presented based on MSOM. Furthermore, we discuss a score test for homogeneity of shape parameter in LBS regression models. Two numerical examples are given to illustrate our methodology and the properties of score test statistic are investigated through Monte Carlo simulations under different censoring percentages.     

Ex-post rational price approximations and the empirical reliability of the present-value relation

Excess volatility and regression tests have resulted in apparent rejections of the present-value relation when ex-post price approximations are employed. These approximations are based upon a sample terminal condition for prices, are not ergodic time-series, and do not result in statistics with readily calculable standard errors. Kleidon (1986a) has demonstrated that ex-post price approximations can subtly affect the reliability of certain volatility tests. We use a bootstrapped cointegration model to demonstrate some of these same effects in Mankiw, Romer and Shapiro's (1985) volatility statistics. The volatility statistics rarely have positive expected value in finite samples and still do not reject the presentvalue relation. Approximations based upon a ‘rolling’ terminal condition result in volatility statistics which have calculable large-sample errors, but even these standard errors greatly overstate the accuracy of volatility statisics in small samples. Regression tests of the present value relation are also affected by the price approximations.     

Estimators and hypothesis tests for a stochastic frontier function: A Monte Carlo analysis

This paper uses Monte Carlo experimentation to investigate the finite sample properties of the maximum likelihood (ML) and corrected ordinary least squares (COLS) estimators of the half-normal stochastic frontier production function. Results indicate substantial bias in both ML and COLS when the percentage contribution of inefficiency in the composed error (denoted by *) is small, and also that ML should be used in preference to COLS because of large mean square error advantages when * is greater than 50%. The performance of a number of tests of the existence of technical inefficiency is also investigated. The Wald and likelihood ratio (LR) tests are shown to have incorrect size. A one-sided LR test and a test of the significance of the third moment of the OLS residuals are suggested as alternatives, and are shown to have correct size, with the one-sided LR test having the better power of the two.The author would like to thank Bill Griffiths, George Battese, Howard Doran, Bill Greene and two anonymous referees for valuable comments. Any errors which remain are those of the author.     

Comparison of power for the D'Agostino and the Wilk-Shapiro test of normality for small and moderate samples

Summary Results are given of a sampling study of the sensitivities of two statistical procedures viz. the D'A gostino and the W ilk -S hapiro test of normality. Four alternative distributions were considered. For 5 small sample sizes the test of D'A gostino seems to be clearly dominated by the W ilk -S hapiro test for each of the 4 alternatives. For sample size n = 100 the recently developed approximate W ilk -S hapiro test seems to dominate the D'A gostino test for the U -shaped distribution; however, little can be said about the difference in power of these two tests for the other 3 alternatives. Random samples generated from the standard normal distribution showed the degree of approximation of the percentiles of each test procedure to be good.     

A linear approximation to the power function of a test

In this paper we obtain a linear approximation to the power function of a test that is very accurate for small sample sizes. This is especially useful for robust tests where not many power functions are available. The approximation is based on the von Mises expansion of the tail probability functional and on the Tail Area Influence Function (TAIF). The goals of the paper are, first to extend the definition of the TAIF to the case of non identically distributed random variables, defining the Partial Tail Area Influence Functions and the Vectorial Tail Area Influence Function; second, to obtain exact expressions for computing these new influence functions; and, finally, to find accurate approximations to the power function, that can be used in the case of non identically distributed random variables. We include some examples of the application of this linear approximation to tests that involve the Huber statistic and also saddlepoint tests, so proving that the approximations apply not only to simple problems but also to complex ones.