A closer examination on some parametric alternatives to the ANOVA F-test

In experiments, the classical (ANOVA) F-test is often used to test the omnibus null-hypothesis μ₁ = μ₂ ... = μ _j = ... = μ _n (all n population means are equal) in a one-way ANOVA design, even when one or more basic assumptions are being violated. In the first part of this article, we will briefly discuss the consequences of the different types of violations of the basic assumptions (dependent measurements, non-normality, heteroscedasticity) on the validity of the F-test. Secondly, we will present a simulation experiment, designed to compare the type I-error and power properties of both the F-test and some of its parametric adaptations: the Brown & Forsythe F^*-test and Welch’s V_w-test. It is concluded that the Welch V_w-test offers acceptable control over the type I-error rate in combination with (very) high power in most of the experimental conditions. Therefore, its use is highly recommended when one or more basic assumptions are being violated. In general, the use of the Brown & Forsythe F^*-test cannot be recommended on power considerations unless the design is balanced and the homoscedasticity assumption holds.     

Dropping observations without affecting posterior and predictive distributions

Suppose in a distribution problem, the sample information W is split into two pieces W ₁ and W ₂, and the parameters involved are split into two sets, π containing the parameters of interest, and θ containing nuisance parameters. It is shown that, under certain conditions, the posterior distribution of π does not depend on the data W ₂, which can thus be ignored. This also has consequences for the predictive distribution of future (or missing) observations. In fact, under similar conditions, the predictive distributions using W or just W ₁ are identical.     

THE EFFECTS OF CORRELATION AMONG OBSERVATIONS ON THE ACCURACY OF APPROXIMATING THE DISTRIBUTION OF SAMPLE MEAN BY ITS ASYMPTOTIC DISTRIBUTION1

In this paper, we investigate the effects of correlation among observations on the accuracy of approximating the distribution of sample mean by its asymptotic distribution. The accuracy is investigated by the Berry-Esseen bound (BEB), which gives an upper bound on the error of approximation of the distribution function of the sample mean from its asymptotic distribution for independent observations. For a given sample size (n₀) the BEB is obtained when the observations are independent. Let this be BEB. We then find the sample size (n_*) required to have BEB below BEB₀, when the observations are dependent. Comparison of n_* with n₀ reveals the effects of correlation among observations on the accuracy of the asymptotic distribution as an approximation. It is shown that the effects of correlation among observations are not appreciable if the correlation is moderate to small but it can be severe for extreme correlations.     

On testing more IFRA ordering-II

ABSTRACT

Suppose F and G are two life distribution functions. It is said that F is more IFRA (increasing failure rate average) than G (written by F ? _*G) if G^{? 1}F(x) is star-shaped on (0, ∞). In this paper, the problem of testing H₀: F = _*G against H₁: F ? _*G and F ≠ _*G is considered in both cases when G is known and when G is unknown. We propose a new test based on U-statistics and obtain the asymptotic distribution of the test statistics. The new test is compared with some well-known tests in the literature. In addition, we apply our test to a real data set in the context of reliability.     

A series of group divisible designs

Bose and Shrikhande C19763 proved that if D(m, k, ?) is a Baer subdesign of another SBIBD D₁ (v₁, k₁ ?), k₁>k, then it also contains a complementary subdesign D^* which is symmetric GDD, D^* (v^*, k^*; ?-1, ?; m, n). Utilising this, we give a necessary condition for a SBIBD D to be a Baer subdesign of D₁ and also give the parameters. Some GD designs are constructed.     

Truncation of the Bechhofer-Kiefer-Sobel sequential procedure for selecting the multinomial event which has the largest probability

In this article we study the effect of truncation on the performance of an open vector-at-a-time sequential sampling procedure (P^* _B) proposed by Bechhofer, Kiefer and Sobel , for selecting the multinomial event which has the largest probability. The performance of the truncated version (P^{* B} _T) is compared to that of the original basic procedure (P^* _B). The performance characteristics studied include the probability of a correct selection, the expected number of vector-observations (n) to terminate sampling, and the variance of n. Both procedures guarantee the specified probability of a correct selection. Exact results and Monte Carlo sampling results are obtained. It is shown that P^* _{B T}is far superior to P^* _B in terms of E{n} and Var{n}, particularly when the event probabilities are equal.The performance of P^* _{B T} is also compared to that of a closed vector-at-a-time sequential sampling procedure proposed for the same problem by Ramey and Alam; this procedure has here to fore been claimed to be the best one for this problem. It is shown that p^* _{B T} is superior to the Ramey-Alam procedure for most of the specifications of practical interest.     

The non-null distributions in anoya

Under the traditional assumptions, any entry in ANOVA interpreted to include all Linear model analyses] is equivalent in disiributien to a quadratic form Q=[μ₁+σ₁Z₁]²+…+ [μ_ν+σ_νZ_ν]²]wherein Z₁..Z_ν are independent standard normal variables. Test statisics in ANOVE are distributed as ratio R of two depenbent such quadretic forms. The non-null distribution of R is a mixture of null distributions; the mixing variable is an easy generalitatlon of the Poisson variable. Fast algorithms yield the power function in both fixed and random effects models in AVOVA to user-specified accuracy.     

Empirical bayes tests in a positive exponential family with partial information on priors

We investigate an empirical Bayes testing problem in a positive exponential family having pdf f{x/θ)=c(θ)u(x) exp(?x/θ), x>0, θ>0. It is assumed that θ is in some known compact interval [C₁, C₂]. The value C₁ is used in the construction of the proposed empirical Bayes test δ^* _n. The asymptotic optimality and rate of convergence of its associated Bayes risk is studied. It is shown that under the assumption that θ is in [C₁, C₂] δ^* _n is asymptotically optimal at a rate of convergence of order O(n^?1/n n). Also, δ^* _n is robust in the sense that δ^* _n still possesses the asymptotic optimality even the assumption that "C₁≦θ≦C₂ may not hold.     

On the strong Kotz approximation of Dirichlet random vectors

Let (X ₁, X ₂) be a bivariate L _p -norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters α₁,α₂. If p=α₁=α₂=2, then (X ₁, X ₂) is a spherical random vector. The estimation of the conditional distribution of Z _u *:=X ₂ | X ₁>u for u large is of some interest in statistical applications. When (X ₁, X ₂) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Z _u * can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Z _u :=X ₂| X ₁=u. In this paper, we are interested in conditional limit results in terms of convergence of the density functions considering a d-dimensional LpGSD random vector. Stating our results for the bivariate setup, we show that the density function of Z _u * and Z _u can be approximated by the density function of a Kotz type I LpGSD distribution, provided that the associated random radius has distribution function in the Gumbel max-domain of attraction. Further, we present two applications concerning the asymptotic behaviour of concomitants of order statistics of bivariate Dirichlet samples and the estimation of the conditional quantile function.     

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.     

Estimators of location based on Kolmogorov-Smirnov-type statistics

Two classes of estimators of a location parameter ø₀ are proposed, based on a nonnegative functional H₁^* of the pair (D₁^ø_N, G^ø_N), where and where F_N is the sample distribution function. The estimators of the first class are defined as a value of ø minimizing H₁^*; the estimators of the second class are linearized versions of those of the first. The asymptotic distribution of the estimators is derived, and it is shown that the Kolmogorov-Smirnov statistic, the signed linear rank statistics, and the Cramérvon Mises statistics are special cases of such functionals H₁^*;. These estimators are closely related to the estimators of a shift in the two-sample case, proposed and studied by Boulanger in B2 (pp. 271–284).     

Win-probabilities for comparing two binary outcomes

This article considers the problem of choosing between two treatments that have binary outcomes with unknown success probabilities p₁ and p₂. The choice is based upon the information provided by two observations X₁ ～ B(n₁, p₁) and X₂ ～ B(n₂, p₂) from independent binomial distributions. Standard approaches to this problem utilize basic statistical inference methodologies such as hypothesis tests and confidence intervals for the difference p₁ ? p₂ of the success probabilities. However, in this article the analysis of win-probabilities is considered. If X*₁ represents a potential future observation from Treatment 1 while X*₂ represents a potential future observation from Treatment 2, win-probabilities are defined in terms of the comparisons of X*₁ and X*₂. These win-probabilities provide a direct assessment of the relative advantages and disadvantages of choosing either treatment for one future application, and their interpretation can be combined with other factors such as costs, side-effects, and the availabilities of the two treatments. In this article, it is shown how confidence intervals for the win-probabilities can be constructed, and examples of their use are provided. Computer code for the implementation of this new methodology is available from the authors.     

On the maxima of heterogeneous gamma variables

In this article, we establish some new results on stochastic comparisons of the maxima of two heterogenous gamma variables with different shape and scale parameters. Let X₁ and X₂ [X*₁ and X*₂] be two independent gamma variables with X_i?[X*_i] having shape parameter r_i?[r*_i] and scale parameter λ_i?[λ*_i], i = 1, 2. It is shown that the likelihood ratio order holds between the maxima, X_{2: 2} and X*_{2: 2} when λ₁ = λ*₁ ? λ₂ = λ*₂ and r₁ ? r*₁ ? r₂ = r*₂. We also prove that, if r_i, r*_i ∈ (0, 1], (r₁, r₂) majorizes (r*₁, r₂*), and (λ₁, λ₂) is p-larger than (λ*₁, λ₂*), then X_{2: 2} is larger than X*_{2: 2} in the sense of the hazard rate order [dispersive order]. Some numerical examples are provided to illustrate the main results. The new results established here strengthen and generalize some of the results known in the literature.     

Optimality of the posterior predictive p-value based on the posterior. Odds

Thompson (1997) considered a wide definition of p-value and found the Baves p-value for testing a ooint null hypothesis H₀: θ= θ₀ versus H₁: θ ≠ θ₀. In this paper, the general case of testing H₀: θ ∈ ?₀ versus H₁: θ ∈ ?^c ₀ is studied. A generalization of the concept of p-value is given, and it is proved that the posterior predictive p-value based on the posterior odds is (asymptotically) a Bayes p-value. Finally, it is suggested that this posterior predictive p-value could be used as a reference p-value     

Generation of multivariate distributions by vertical density representation

Troutt (1991,1993) proposed the idea of the vertical density representation (VDR) based on Box-Millar method. Kotz, Fang and Liang (1997) provided a systematic study on the multivariate vertical density representation (MVDR). Suppose that we want to generate a random vector X[d]Rⁿthat has a density function ?(x). The key point of using the MVDR is to generate the uniform distribution on [D]_?(v) = {x :?(x) = v} for any v > 0 which is the surface in RⁿIn this paper we use the conditional distribution method to generate the uniform distribution on a domain or on some surface and based on it we proposed an alternative version of the MVDR(type 2 MVDR), by which one can transfer the problem of generating a random vector X with given density f to one of generating (X, X_n+i) that follows the uniform distribution on a region in Rⁿ⁺¹defined by ?. Several examples indicate that the proposed method is quite practical.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

On An Intriguing Distributional Identity

For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω₁ → Ω₂ a continuous, strictly increasing function, such that Ω₁∩Ω₂?(a, b), but otherwise arbitrary, we establish that the random variables F(X) ? F(g(X)) and F(g^{? 1}(X)) ? F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U ? ψ(U) = ^dψ^{? 1}(U) ? U for U ～ U(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.     

The Likelihood Ratio Test with the Box–Cox Transformation for the Normal Mixture Problem: Power and Sample Size Study

Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ₂ ? μ₁)/σ, where μ₁ and μ₂ are the component means and σ² is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ^?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)² × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.     

Nonlinear regression in a trivariate elliptical distribution via a linear combination of order statistics

In this paper, by assuming that (X, Y ₁, Y ₂)^T has a trivariate elliptical distribution, we derive the exact joint distribution of X and a linear combination of order statistics from (Y ₁, Y ₂)^T and show that it is a mixture of unified bivariate skew-elliptical distributions. We then derive the corresponding marginal and conditional distributions for the special case of t kernel. We also present these results for an exchangeable case with t kernel and illustrate the established results with an air-pollution data.     

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.