Kolmogorov-type tests of goodness-of-fit against stochastic ordering

This article addresses the problem of testing the null hypothesis H₀ that a random sample of size n is from a distribution with the completely specified continuous cumulative distribution function F_n(x). Kolmogorov-type tests for H₀ are based on the statistics C⁺ _n = Sup[F_n(x)?F₀(x)] and C^? _n=Sup[F₀(x)?F_n(x)], where F_n(x) is an empirical distribution function. Let F(x) be the true cumulative distribution function, and consider the ordered alternative H₁: F(x)≥F₀(x) for all x and with strict inequality for some x. Although it is natural to reject H₀ and accept H₁ if C ⁺ _n is large, this article shows that a test that is superior in some ways rejects F₀ and accepts H₁ if C^mdash _n is small. Properties of the two tests are compared based on theoretical results and simulated results.     

On the Markov property of order statistics

The order statistics from a sample of size n≥3 from a discrete distribution form a Markov chain if and only if the parent distribution is supported by one or two points. More generally, a necessary and sufficient condition for the order statistics to form a Markov chain for (n≥3) is that there does not exist any atom x₀ of the parent distribution F satisfying F(x₀-)>0 and F(x₀)<1. To derive this result a formula for the joint distribution of order statistics is proved, which is of an interest on its own. Many exponential characterizations implicitly assume the Markov property. The corresponding putative geometric characterizations cannot then be reasonably expected to obtain. Some illustrative geometric characterizations are discussed.     

Rate of consistency of one sample tests of location

Let X₁,…,X_n be a sample from a population with continuous distribution function F(x?θ) such that F(x)+F(-x)=1 and 0<F(x)<1, x?R¹. It is shown that the power- function of a monotone test of H: θ=θ₀ against K: θ>θ₀ cannot tend to 1 as θ?θ₀ → ∞ more than n times faster than the tails of F tend to 0. Some standard as well as robust tests are considered with respect to this rate of convergence.     

COMPARISON OF THREE NON-PARAMETRIC TESTS FOR BIVARIATE INTERCHANGEABILITY1

Given a random sample(X₁, Y₁), …,(X_n, Y_n) from a bivariate (BV) absolutely continuous c.d.f. H (x, y), we consider rank tests for the null hypothesis of interchangeability H₀: H(x, y). Three linear rank test statistics, Wilcoxon (W_N), sum of squared ranks (SSR_N) and Savage (S_N), are described in Section 1. In Section 2, asymptotic relative efficiency (ARE) comparisons of the three types of tests are made for Morgenstern (Plackett, 1965) and Moran (1969)BV alternatives with marginal distributions satisfying G(x) = F(x/θ) for some θ≠ 1. Both gamma and lognormal marginal distributions are used.     

Estimation of the derivative of a density and related functions

Let Y₁,…,Y _n, (Y₁ <Y₂<…<Y _n) be the order statistics of a random sample from a distribution F with density f on the realline. This paper gives a class of estimators of the derivativef'(x) of the density f at points x for which f has

a continuoussecond derivative. These estimators are based on spacings inthe order statistics Y_j+kn -y _j j = 1,…,n-k_n,k_n<n.     

Goodness-of-fit test for density estimation

It is often necessary to test whether X,…, X_n are from a certain density f(x) or not. Most test statistics such as the Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling statistics are based on the empirical distribution function F(x). In this paper we suggest a test statistic based on the integrated squared error of the kernel density estimator. We derive the asymptotic distribution of the statistic under the null and alternative hypothesis. Some simulation results for power comparisons are also given.     

Exact percentage points for the Kolmogorov test on truncated versions of known continuous distributions with unknown truncation parameters

In this paper we deal with Kolmogorov-Smirnov testson uniformity with completely or partly unknown limits. Tables of exact percentage points are presented or referred using the wellknown determinant formula given by Steck (1971). It is shown that these tables also give the percentage points for the analogous statistics of the test on truncated versions of known continuous distributions with completely or partly unknown truncation limits. We will give some examples of these applications. Among these are the tests on exponentiality and on Pareto distribution with known shape parameter and unknown lower terminal.     

A combined p-value test for multiple hypothesis testing

Tests that combine p-values, such as Fisher's product test, are popular to test the global null hypothesis H₀ that each of n component null hypotheses, H₁,…,H_n, is true versus the alternative that at least one of H₁,…,H_n is false, since they are more powerful than classical multiple tests such as the Bonferroni test and the Simes tests. Recent modifications of Fisher's product test, popular in the analysis of large scale genetic studies include the truncated product method (TPM) of Zaykin et al. (2002), the rank truncated product (RTP) test of Dudbridge and Koeleman (2003) and more recently, a permutation based test—the adaptive rank truncated product (ARTP) method of Yu et al. (2009). The TPM and RTP methods require users' specification of a truncation point. The ARTP method improves the performance of the RTP method by optimizing selection of the truncation point over a set of pre-specified candidate points. In this paper we extend the ARTP by proposing to use all the possible truncation points {1,…,n} as the candidate truncation points. Furthermore, we derive the theoretical probability distribution of the test statistic under the global null hypothesis H₀. Simulations are conducted to compare the performance of the proposed test with the Bonferroni test, the Simes test, the RTP test, and Fisher's product test. The simulation results show that the proposed test has higher power than the Bonferroni test and the Simes test, as well as the RTP method. It is also significantly more powerful than Fisher's product test when the number of truly false hypotheses is small relative to the total number of hypotheses, and has comparable power to Fisher's product test otherwise.     

Linear representation of M-estimates in linear models

Consider the linear regression model, y_i = x_iβ₀ + e_i, i = l,…,n, and an M-estimate β of β_o obtained by minimizing Σρ(y_i — x_iβ), where ρ is a convex function. Let S_n = ΣX_iX_iX_i and r_n = S_n^½ (β — β₀) — S_n ² Σx_ih(e_i), where, with a suitable choice of h(.), the expression Σ x_ix(e,) provides a linear representation of β. Bahadur (1966) obtained the order of r_n as n→ ∞ when β_o is a one-dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of r_n as n → ∞ for a general M-estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them.     

New estimators of distribution functions

ABSTRACT

This article considers the estimation of a distribution function F_X(x) based on a random sample X₁, X₂, …, X_n when the sample is suspected to come from a close-by distribution F₀(x). The new estimators, namely the preliminary test (PTE) and Stein-type estimator (SE) are defined and compared with the “empirical distribution function” (edf) under local departure. In this case, we show that Stein-type estimators are superior to edf and PTE is superior to edf when it is close to F₀(x). As a by-product similar estimators are proposed for population quantiles.     

Some competitors of the mood test for the two-sample scale problem

Let X_l,…,X_n (Y_l,…,Y_m) be a random sample from an absolutely continuous distribution with distribution function F(G).A class of distribution-free tests based on U-statistics is proposed for testing the equality of F and G against the alternative that X's are more dispersed then Y's. Let 2 ? C ? n and 2 ? d ? m be two fixed integers. Let ?_c,d(X_il,…,X_ic ; Y_jl,…,X_jd)=1(-1)when max as well as min of {X_il,…,X_ic ; Y_jl,…,Y_jd } are some X_i's (Y_j's)and zero oterwise. Let S_c,d be the U-statistic corresponding to ?_c,d.In case of equal sample sizes, S22 is equivalent to Mood's Statistic.Large values of S_c,d are significant and these tests are quite efficient     

Invariant tests for the equality of k scale parameters under spherical symmetry

We consider tests for scale parameters when the underlying distribution belongs to the class of spherically symmetric laws. A (nx1) random vector x has a spherically symmetric distribution if the distribution of x is identical to the distribution of Px for all (n×n) orthogonal matrices P. Using the principle of invariance we show that the usual normal-theory tests are not only invariant tests but are also exactly robust with respect to this class of spherically symmetric laws.     

A modified Kolmogorov-Smirnov test for the inverse gaussian density with unknown parameters

The Kolmogorov-Smirnov (KS) test is an empirical distribution function (EDF) based goodness-of-fit test that requires the underlying hypothesized density to be continuous and completely specified. When the parameters are unknown and must be estimated from the data, standard tables of the KS test statistic are not valid. Approximate upper tail percentage points of the KS statistic for the inverse Gaussian (IG) distribution with unknown parameters are tabled in this paper.

A study of the power of the KS test for the IG distribution indicates that the test is able todiscriminate between the IG distribution and distributions such as the uniform and exponentialdistributions that are very different in shape, but is relatively unable to discriminate between the IG distribution and distributions that are similar in shape such as the lognormal and Weibull distributions. In modeling settings the former distinction is typically more important to make than the latter distinction.     

Repeated chi-square testing

A sequentialized version of the x²; goodness of fit test, called repeated x,²; test, is introduced. The form of the asymptotic distribution of the repeated x² test statistic is given under the null hypothesis as well as under local alternatives. For various numbers of cells Monte Carlo results are given for critical values, power and distribution of stopping time. Finally, the perfor-mance of the repeated and the fixed sample x² test are compared.     

On the bootstrap quantile-treatment-effect test

Let {X ₁, …, X _n } and {Y ₁, …, Y _m } be two samples of independent and identically distributed observations with common continuous cumulative distribution functions F(x)=P(X≤x) and G(y)=P(Y≤y), respectively. In this article, we would like to test the no quantile treatment effect hypothesis H ₀: F=G. We develop a bootstrap quantile-treatment-effect test procedure for testing H ₀ under the location-scale shift model. Our test procedure avoids the calculation of the check function (which is non-differentiable at the origin and makes solving the quantile effects difficult in typical quantile regression analysis). The limiting null distribution of the test procedure is derived and the procedure is shown to be consistent against a broad family of alternatives. Simulation studies show that our proposed test procedure attains its type I error rate close to the pre-chosen significance level even for small sample sizes. Our test procedure is illustrated with two real data sets on the lifetimes of guinea pigs from a treatment-control experiment.     

Density estimation for samples satisfying a certain absolute regularity condition

Let {ξ_i} be an absolutely regular sequence of identically distributed random variables having common density function f(x). Let H_k(x,y) (k=1, 2,…) be a sequence of Borel-measurable functions and f_n(x)=n^?1(H_n(x,ξ₁)+…+H_n(x,ξ_n)) the empirical density function. In this paper, the asymptotic property of the probability P(sup_x|f_n(x)?f(x)|>ε) (n→∞) is studied.     

Large sample theory of the Langevin distribution

The Langevin (or von Mises-Fisher) distribution of random vector x on the unit sphere ω_q in $\text{R}$^q has a density proportional to exp κμ'x where μ'x is the scalar product of x with the unit modal vector μ and κ?0 is a concentration parameter. This paper studies estimation and tests for a wide variety of situations when the sample sizes are large. Geometrically simple test statistics are given for many sample problems even when the populations may have unequal concentration parameters.     

The number of records within a random interval of the current record value

LetX ₁,X ₂, … be a sequence of i.i.d. random variables with some continuous distribution functionF. LetX(n) be then-th record value associated with this sequence and μ _n ⁻ , μ _n ⁺ be the variables that count the number of record values belonging to the random intervals(f−(X(n)), X(n)), (X(n), f+(X(n))), wheref−, f+ are two continuous functions satisfyingf−(x)<x, f+(x)>x. Properties of μ _n ⁻ , μ _n ⁺ are studied in the present paper. Some statistical applications connected with these variables are also provided.     

A computational investigation of conover's kolmogorov-smirnov test for discrete distributions

The tabled significance values of the Kolmogorov-Smirnov goodness-of-fit statistic determined for continuous underlying distributions are conservative for applications involving discrete underlying distributions. Conover (1972) proposed an efficient method for computing the exact significance level of the Kolmogorov-Smirnov test for discrete distributions; however, he warned against its use for large sample sizes because “the calculations become too difficult.”

In this work we explore the relationship between sample size and the computational effectiveness of Conover's formulas, where “computational effectiveness” is taken to mean the accuracy attained with a fixed precision of machine arithmetic. The nature of the difficulties in calculations is pointed out. It is indicated that, despite these difficulties, Conover's method of computing the Kolmogorov-Smirnov significance level for discrete distributions can still be a useful tool for a wide range of sample sizes.