Extended tables of the distribution of friedman’ s s-statistic in the two-way layout

Friedman’s (1937, 1940) S-statistic is designed to test the hypothesis that there is no treatment effect in a randomized-block design with k treatments and n blocks. In this paper we give tables of the null distribution of S for k = 5, n = 6(1)8, and for k = 6, n = 2(1)6. Computational details are discussed.     

Modified kolmogorov-smirnov tests of goodness of fit

The Kolmogorov-Smirnov (K–S) one-sided and two-sided tests of goodness of fit based on the test statistics D⁺ _n D^? _n and D_n are equivalent to tests based on taking the cumulative probability of the i–th order statistic of a sample of size n to be (i–.5)/n. Modified test statistics C⁺ _n, C^? _n and C_n are obtained by taking the cumulative probability to be i/(n+l). More generally, the cumula-tive probability may be taken to be (i?δ)/(n+l?2δ), as suggested by Blom (1958), where 0 less than or equal δ less than or equal .5. Critical values of the test statis-tics can be found by interpolating inversely in tables of the proba-bility integrals obtained by setting a=l/(n+l?2δ) in an expression given by Pyke (1959). Critical values for the D's (corresponding to δ=.5) have been tabulated to 5DP by Miller (1956) for n=1(1)100. The authors have made analogous tabulations for the C's (corresponding to δ=0) [previously tabulated by Durbin (1969) for n=1(1)60(2)100] and for the test statistics E⁺ _n, E^? _n and E_n corresponding to δ f.3. They have also made a Monte Carlo comparison of the power of the modified tests with that of the K–S test for several hypothetical distributions. In a number of cases, the power of the modified tests is greater than that of the K–S test, especially when the standard deviation is greater under the alternative than under the null hypo-thesis.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

A weighted bootstrap approximation for comparing the error distributions in nonparametric regression

Several procedures have been proposed for testing the equality of error distributions in two or more nonparametric regression models. Here we deal with methods based on comparing estimators of the cumulative distribution function (CDF) of the errors in each population to an estimator of the common CDF under the null hypothesis. The null distribution of the associated test statistics has been approximated by means of a smooth bootstrap (SB) estimator. This paper proposes to approximate their null distribution through a weighted bootstrap. It is shown that it produces a consistent estimator. The finite sample performance of this approximation is assessed by means of a simulation study, where it is also compared to the SB. This study reveals that, from a computational point of view, the proposed approximation is more efficient than the one provided by the SB.     

A NEW APPROXIMATION FOR FISHER'S Z

As the sample size increases, the coefficient of skewness of the Fisher's transformation z= tanh^-1r, of the correlation coefficient decreases much more rapidly than the excess of its kurtosis. Hence, the distribution of standardized z can be approximated more accurately in terms of the t distribution with matching kurtosis than by the unit normal distribution. This t distribution can, in turn be subjected to Wallace's approximation resulting in a new normal approximation for the Fisher's z transform. This approximation, which can be used to estimate the probabilities, as well as the percentiles, compares favorably in both accuracy and simplicity, with the two best earlier approximations, namely, those due to Ruben (1966) and Kraemer (1974). Fisher (1921) suggested approximating distribution of the variance stabilizing transform z=(1/2) log ((1 +r)/(1r)) of the correlation coefficient r by the normal distribution with mean = (1/2) log ((1 + p)/(lp)) and variance =l/(n3). This approximation is generally recognized as being remarkably accurate when ||Gr| is moderate but not so accurate when ||Gr| is large, even when n is not small (David (1938)). Among various alternatives to Fisher's approximation, the normalizing transformation due to Ruben (1966) and a t approximation due to Kraemer (1973), are interesting on the grounds of novelty, accuracy and/or aesthetics. If r?= r/√ (1r²) and r?|Gr = |Gr/√(1|Gr²), then Ruben (1966) showed that (1) g_n (r,|Gr) ={(2n5)/2}^1/2r?r{(2n3)/2}^1/2r?|GR, {1 + (1/2)(r?r²+r?|Gr²)}^1/2 is approximately unit normal. Kraemer (1973) suggests approximating (2) t_n (r, |Gr) = (r|GR¹) √ (n2), √(11r²) √(1|Gr²) by a Student's t variable with (n2) degrees of freedom, where after considering various valid choices for |Gr¹ she recommends taking |Gr¹= |Gr*, the median of r given n and |Gr.     

Testing for superadditivity of the mean value function of a non-homogeneous poisson process

Consider a non-homogeneous Poisson process, N(t), with mean value function Λ(t) and intensity function λ(t). A conditional test of the hypothesis that the process is homogeneous, versus alternatives for which Λ(t) is superadditive, was proposed by Hollander and Proschan (1974). A new test for superadditivity of Λ(t), which is based on a linear combination of the occurrence times of the process N{t) is suggested in this paper. Though this test has the same Pitman efficiency as the Hollander-Proschan test, it is shown by Monte-Carlo simulation that our test has more power for many important alternatives. Tables for the exact null distribution of the test statistic have been given.     

On a life testing problem involving the two parameter exponential distribution

For a hypothesis testing problem involving the location and scale parameters of an exponential distribution, Perng (1977) proposed a test procedure based on the first r out of n observed failure times. In this paper the likelihood ratio test is determined, critical values are provided and the asymptotic null distribution is determined. An alternate test based on an F statistic is also proposed and the critical regions and power functions of the procedures are compared.     

Moments of some J-shaped distributions

This paper concerns a family of univariate distributions suggested by Topp & Leone in 1955. Topp & Leone provided no motivation for this new family and by way of properties they derived only the first four integer-order moments, i.e. E(Xn) for n=1, r 2, r 3, r 4 . In this paper we provide a motivation for the family of distributions and derive explicit algebraic expressions for: (1) hazard rate function; (2) E(Xn) when n ± 1 is any integer; (3) E(Xn) for n=1, r 2, r … r , r 10 , and (4) E[{X-E(X)} n] , n=2, r 3, r 4 . We also give an expression for the characteristic function and discuss issues on estimation and simulation. The main calculations of this paper use properties of the Gauss hypergeometric function.     

A note on spearman's footrule

We obtain exact tables of the null distribution of Spearman's footrule for sample sizes n = 4(1)40 by using a certain Markov chain property, and we investigate the adequacy of approximations to the distribution.     

面板数据双因素误差回归模型序列相关和随机效应的联合LM检验

对面板数据双因素误差回归模型构造了检验序列相关和随机效应的一种联合LM检验,发现该LM统计量也是检验联合假设H0：σμ^2=λ=0的Baltagi-Li LM统计量和检验假设H0：σv^2=λ=0的Breusch-Pagan-LM统计量之和。当面板数据的个体数N充分大时,该联合LM统计量的渐近分布是χ^2（3）分布;无论双因素误差面板数据回归模型的剩余误差项是AR（1）过程还是MA（1）过程,联合LM检验是相同的,即对随机效应和一阶序列相关的联合LM检验是独立于序列相关的形式。     

The weighted compounding of probabilities from three independent hypothesis tests

Tail probabilities from three independent hypothesis tests can be combined to form a test statistic of the form P₁,P₂ ^θ2,P₃ ^θ3.The null distribution of the combined test statistic is presented and critical values for α=0.01 and 0.05 are provided.The power of this test is discussed for the special case ofthree independent F-tests.     

A Generalization of Ahmad's Class of Mann–Whitney–Wilcoxon Statistics

In the spirit of the recent work of Ahmad (1996) this paper introduces another class of Mann–Whitney–Wilcoxon test statistics. The test statistic compares the r th and s th powers of the tail probabilities of the underlying probability distributions. The choice of r + s = 4 improves the Pitman efficiency for uniform, exponential, lognormal and normal distributions and keeps the same efficiency as the Mann–Whitney–Wilcoxon test for logistic and double exponential distributions. The two-sample test is modified for the one-sample problem with symmetric underlying distribution.     

A simple nonparametric test for bivariate symmetry about a line

Over the years many researchers have dealt with testing the hypotheses of symmetry in univariate and multivariate distributions in the parametric and nonparametric setup. In a multivariate setup, there are several formulations of symmetry, for example, symmetry about an axis, joint symmetry, marginal symmetry, radial symmetry, symmetry about a known point, spherical symmetry, and elliptical symmetry among others. In this paper, for the bivariate case, we formulate a concept of symmetry about a straight line passing through the origin in a plane and accordingly develop a simple nonparametric test for testing the hypothesis of symmetry about a straight line. The proposed test is based on a measure of deviance between observed counts of bivariate samples in suitably defined pairs of sets. The exact null distribution and non-null distribution, for specified classes of alternatives, of the test statistics are obtained. The null distribution is tabulated for sample size from n=5 up to n=30. The null mean, null variance and the asymptotic null distributions of the proposed test statistics are also obtained. The empirical power of the proposed test is evaluated by simulating samples from the suitable class of bivariate distributions. The empirical findings suggest that the test performs reasonably well against various classes of asymmetric bivariate distributions. Further, it is advocated that the basic idea developed in this work can be easily adopted to test the hypotheses of exchangeability of bivariate random variables and also bivariate symmetry about a given axis which have been considered by several authors in the past.     

Tests for multiple upper or lower outliers in an exponential sample

SUMMARY T = \[x + ... + x ]/ Sigma x (T*= \[x + ... + x ] Sigma x ) is the max k (n- k+ 1 ) (n) i k ( 1 ) (k) i imum likelihood ratio test statistic for k upper ( lower ) outliers in an exponential sample x , ..., x . The null distributions of T for k= 1,2 were given by Fisher and by Kimber 1 n k and Stevens , while those of T*(k= 1,2) were given by Lewis and Fieller . In this paper , k the simple null distributions of T and T* are found for all possible values of k, and k k percentage points are tabulated for k= 1, 2, ..., 8. In addition , we find a way of determining k, which can reduce the masking or ' swamping ' effects .     

On the distribution of a test for exponentiality based on progressively type-II right censored spacings

A test for exponentiality based on progressively Type-II right censored spacings has been proposed recently by Balakrishnan et al. (2002). They derived the asymptotic null distribution of the test statistic. In this work, we utilize the algorithm of Huffer and Lin (2001) to evaluate the exact null probabilities and the exact critical values of this test statistic.     

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.     

Average rank correlation statistics in the presence of ties

The problem of n judges ranking r objects is considered in the situation where ties are permitted. Asymtotic distributions under the null hypothesis of complete randomness in the rankings are derived for the test statistics of average rank correlations between all pairs of ranking where the rank correlations are measured either by Spearman rho or Kendall tau. The relative efficeincies of these average rank correlatins are derived using approximate Bahadur slope and limiting pitman efficiency, and in both cases the Kendall statistic is shown to be more efficient. Some interpretatins of these and related results are also given.     

Nonparametric rank-based test procedures for non-additive models in the two-way layout I. no replications

One of the major unresolved problems in the area of nonparametric statistics is the need for satisfactory rank-based test procedures for non-additive models in the two-way layout, especially when there is only one observation on each combination of the levels of the experimental factors. In this paper we consider an arbitrary non-additive model for the two-way layout with n levels of each factor. We utilize both alignment and ranking of the data together with basic properties of Latin squares to develop rank tests for interaction (non-additivity). Our technique involves first aligning within one of the main effects, ranking within the other main effects (columns and rows) and then adding the resulting ranks within “interaction bands” corresponding to orthogonal partitions of the interaction for the model, as denoted by the letters of an n × n Latin square. A Friedman-type statistic is then computed on the resulting sums. This is repeated for each of (n?1) mutually orthogonal Latin squares (thus accounting for all the interaction degrees of freedom). The resulting (n?1) Friedman-type statistics are finally combined to obtain an overall test statistic. The necessary null distribution tables for applying the proposed test for non-additivity are presented and we discuss the results of a Monte Carlo simulation study of the relative powers of this new procedure and other (parametric and nonparametric) procedures designed to detect interaction in a two-way layout with one observation per cell.     

Testing stationarity and trend stationarity against the unit root hypothesis

In this paper we propose a family of relativel simple nonparametrics tests for a unit root in a univariate time series. Almost all the tests proposed in the literature test the unit root hypothesis against the alternative that the time series involved is stationarity or trend stationary. In this paper we take the (trend) stationarity hypothesis as the null and the unit root hypothesis as the alternative. The order differnce with most of the tests proposed in the literature is that in all four cases the asymptotic null distribution is of a well-known type, namely standard Cauchy. In the first instance we propose four Cauchy tests of the stationarity hypothesis against the unit root hypothesis. Under H1 these four test statistics involved, divided by the sample size n, converge weakly to a non-central Cauchy distribution, to one, and to the product of two normal variates, respectively. Hence, the absolute values of these test statistics converge in probability to infinity 9at order n). The tests involved are therefore consistent against the unit root hypothesis. Moreover, the small sample performance of these test are compared by Monte Carlo simulations. Furthermore, we propose two additional Cauchy tests of the trend stationarity hypothesis against the alternative of a unit root with drift.