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.     

Tests of symmetry based on transformed empirical processes

A probability distribution function F is said to be symmetric when 1 ‐ F(x) ‐ F(‐x) = 0 for all x∈ R. Given a sequence of alternatives contiguous to a certain symmetric F₀, the authors are concerned with testing for the null hypothesis of symmetry. The proposed tests are consistent against any nonsymmetric alternative, and their power with respect to the given sequence can easily be optimized. The tests are constructed by means of transformed empirical processes with an adequate selection of the underlying isometry, and the optimum power is obtained by suitably choosing the score functions. The test statistics are very easy to compute and their asymptotic distributions are simple.     

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.     

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.     

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.     

A one-sided test for testino homogeneity of scale parameters against simple ordered alternative

In an earlier paper the authors (1997) extended the results of Hayter (1990) to the two parameter exponential probability model. This paper addressee the extention to the scale parameter case under location-scale probability model. Consider k (k≧3) treatments or competing firms such that an observation from with treatment or firm follows a distribution with cumulative distribution function (cdf) F_i(x)=F[(x-μ_i)/Q_i], where F(·) is any absolutely continuous cdf, i=1,…,k. We propose a test to test the null hypothesis H₀:θ₁=…=θ_k against the simple ordered alternative H₁:θ₁≦…≦θ_k, with at least one strict inequality, using the data X_i,j, i=1,…k; j=1,…,n₁. Two methods to compute the critical points of the proposed test have been demonstrated by talking k two parameter exponential distributions. The test procedure also allows us to construct simultaneous one sided confidence intervals (SOCIs) for the ordered pairwise ratios θ_j/θ_i, 1≦i<j≦k. Statistical simulation revealed that: 9i) actual sizes of the critical points are almost conservative and (ii) power of the proposed test relative to some existing tests is higher.     

Exponential Family Techniques for the Lognormal Left Tail

Let X be lognormal(μ,σ²) with density f(x); let θ > 0 and define . We study properties of the exponentially tilted density (Esscher transform) f_θ(x) = e^?θxf(x)/L(θ), in particular its moments, its asymptotic form as θ→∞ and asymptotics for the saddlepoint θ(x) determined by . The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of the sum of lognormals S_n=X₁+?+X_n: a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf F_n(x) and the pdf f_n(x) of S_n are given in a range of values of σ²,n and x motivated by portfolio value‐at‐risk calculations.     

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.     

An exact test for nbue class of survival functions

Let F(x) be a life distribution. An exact test is given for testing H₀ F is exponential, versusH₁Fε NBUE (NWUE); along with a table of critical values for n=5(l)80, and n=80(5)65. An asymptotic test is made available for large values of n, where the standardized normal table can be used for testing.     

An asymptotic sequential test based on confidence sequences

In the context of a translation parameter family of distributions F₀(x) = F(x-θ) an asymptotic sequential test of H₀: θ ≤ -△ versus H₁: θ ≥ △ developed. The test is based on confidence sequences. In the special case where F is a specified normal distribution the proposed test is uniformly at least as efficient (in the sense of Rechanter (1960)) as the Wald sequention probibilty ratio test.     

A One Sided Test Based on Sample Quasi Ranges

Abstract

Let the data from the ith treatment/population follow a distribution with cumulative distribution function (cdf) F _i (x) = F[(x ? μ _i )/θ _i ], i = 1,…, k (k ≥ 2). Here μ _i (?∞ < μ _i  < ∞) is the location parameter, θ _i (θ _i  > 0) is the scale parameter and F(?) is any absolutely continuous cdf, i.e., F _i (?) is a member of location-scale family, i = 1,…, k. In this paper, we propose a class of tests to test the null hypothesis H ₀ ? θ₁ = · = θ _k against the simple ordered alternative H _A  ? θ₁ ≤ · ≤ θ _k with at least one strict inequality. In literature, use of sample quasi range as a measure of dispersion has been advocated for small sample size or sample contaminated by outliers [see David, H. A. (1981). Order Statistics. 2nd ed. New York: John Wiley, Sec. 7.4]. Let X _i1,…, X _in be a random sample of size n from the population π _i and R _ir  = X _i:n?r  ? X _i:r+1, r = 0, 1,…, [n/2] ? 1 be the sample quasi range corresponding to this random sample, where X _i:j represents the jth order statistic in the ith sample, j = 1,…, n; i = 1,…, k and [x] is the greatest integer less than or equal to x. The proposed class of tests, for the general location scale setup, is based on the statistic W _r  = max_1≤i<j≤k (R _jr /R _ir ). The test is reject H ₀ for large values of W _r . The construction of a three-decision procedure and simultaneous one-sided lower confidence bounds for the ratios, θ _j /θ _i , 1 ≤ i < j ≤ k, have also been discussed with the help of the critical constants of the test statistic W _r . Applications of the proposed class of tests to two parameter exponential and uniform probability models have been discussed separately with necessary tables. Comparisons of some members of our class with the tests of Gill and Dhawan [Gill A. N., Dhawan A. K. (1999). A One-sided test for testing homogeneity of scale parameters against ordered alternative. Commun. Stat. – Theory and Methods 28(10):2417–2439] and Kochar and Gupta [Kochar, S. C., Gupta, R. P. (1985). A class of distribution-free tests for testing homogeneity of variances against ordered alternatives. In: Dykstra, R. et al., ed. Proceedings of the Conference on Advances in Order Restricted Statistical Inference at Iowa city. Springer Verlag, pp. 169–183], in terms of simulated power, are also presented.     

Admissible tests for the mean with additional information

Let X be a po-normal random vector with unknown µ and unknown covariance matrix ∑ and let X be partitioned as X = (X^′ ₍₁₎, …, X^′ _(r))′ where X_(j)is a subvector of X with dimension p_jsuch that ∑^r _j=1Pj = P0. Some admissible tests are derived for testing H₀: μ = 0 versus H₁: μ ¦0 based on a sample drawn from the whole vector X of dimension p and r additional samples drawn from X₍₁₎, X₍₂₎, …, X(r) respectively, All (r+1) samples are assumed to be independent. The distribution of some of the tests' statistics involved are also derived.     

Changepoint Problems Under Random Censorship

Let X ₁, …, X _[nθ], X _{[nθ] + 1}, …, X _n be a sequence of independent random variables (the “lifetimes”) such that X _j ? F ₁ for 1 ≤ j ≤ [nθ] and X _j ? F ₂ for [nθ] + 1 ≤ j ≤ n, with F ₁ ≠ F ₂ unknown. In this paper we investigate an estimator θ _n for the changepoint θ if the X's are subject to censoring. The rate of almost sure convergence of θ _n to θ is established and a test for the hypothesis θ = 0, i.e. “no change”, is proposed.     

A measure of general functional dependence between two continuous variables

Existing measures in the literature that are specifically concerned with testing and measuring independence between two continuous variables are all based on examining the definition of independence, i.e., F_XY(x, y) = F_X(x)F_Y(y). A new measure is constructed uniquely in this paper that uses the absolute value of first difference on adjacent ranks of one variable with respect to the other. This measure captures the degree of functional dependence attributable to the amount of randomness and the complexity of the underlying bivariate dependence structure in a commensurate way that existing coefficients are incapable of. As a test statistic of independence, this measure is shown to have comparable or better power than existing statistics against a wide range of alternative hypotheses that consist of functional and multivalued relational dependence with additive noise.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

On large deviation asymptotics of some tests in time series

Let {X_t} be a Gaussian stationary process with spectral density f_θ(λ). The problem considered is that of testing a simple hypothesis H₀:θ=θ₀ against the alternative A:θ≠θ₀. For this we investigate the Bahadur efficiency of the likelihood ratio, Rao, modified Wald and Wald tests. The Bahadur efficiency is based on the large deviation theory. Then it is shown that the asymptotics of the above tests are identical up to second-order in a certain sense. We show that this result makes a sharp contrast with the ordinary higher-order asymptotic theory for tests.     

Lois limites pour les statistiques d'ordre dans le cas non identiquement distribue

Let X₁, X₂,… be a sequence of independent random variables with distribution functions F₁, where 1 ≤ i ≤ n, and for each n ≥ 1 let X_1,n ≤… ≤ X_n,n denote the order statistics of the first n random variables. Under suitable hypotheses about the F₁, we characterize the limit distribution functions H(x) for which P(X_k,n ? a_nx + b_n) → H(x), where a_n > 0 and b_n are real constants. We consider the cases where κ = κ(n) satisfies √n {κ(n)/n — λ} → 0 and √n {κ(n)/n — λ} → ∞ separately.