The inverse gaussian distribution: Some properties and characterizations

This article presents some structural properties of the inverse Gaussian distribution, together with several new characterizations based on constancy of regression of suitable functions on the sum of n independent identically distributed random variables. A decomposition of the statistic λσ (X^?1_i?X^?1) into n - 1 independent chi-squared random variables, each with one degree of freedom, is given when n is of the form 2^r.     

Bounds on Bivariate Distribution Functions with Given Margins and Known Values at Several Points

Let H(x, y) be a continuous bivariate distribution function with known marginal distribution functions F(x) and G(y). Suppose the values of H are given at several points, H(x _i , y _i ) = θ _i , i = 1, 2,…, n. We first discuss conditions for the existence of a distribution satisfying these conditions, and present a procedure for checking if such a distribution exists. We then consider finding lower and upper bounds for such distributions. These bounds may be used to establish bounds on the values of Spearman's ρ and Kendall's τ. For n = 2, we present necessary and sufficient conditions for existence of such a distribution function and derive best-possible upper and lower bounds for H(x, y). As shown by a counter-example, these bounds need not be proper distribution functions, and we find conditions for these bounds to be (proper) distribution functions. We also present some results for the general case, where the values of H(x, y) are known at more than two points. In view of the simplification in notation, our results are presented in terms of copulas, but they may easily be expressed in terms of distribution functions.     

On the Exponential Decay Rate of the Tail of a Discrete Probability Distribution

Abstract

We give a sufficient condition for the exponential decay of the tail of a discrete probability distribution π = (π _n ) _n≥0 in the sense that lim _n→∞(1/n) log∑ _i>n π _i  = ?θ with 0 < θ < ∞. We focus on analytic properties of the probability generating function of a discrete probability distribution, especially, the radius of convergence and the number of poles on the circle of convergence. Furthermore, we give an example of an M/G/1 type Markov chain such that the tail of its stationary distribution does not decay exponentially.     

Some New Results on Likelihood Ratio Orderings for Spacings of Heterogeneous Exponential Random Variables

Let X ₁, X ₂,…, X _n be independent exponential random variables with X _i having failure rate λ _i for i = 1,…, n. Denote by D _i:n  = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…, n, where X _0:n ≡ 0. It is shown that if λ _n+1 ≤ [≥] λ _k for k = 1,…, n then D _n:n  ≤ _lr D _n+1:n+1 and D _1:n  ≤ _lr D _2:n+1 [D _2:n+1 ≤ _lr D _2:n ], and that if λ _i  + λ _j  ≥ λ _k for all distinct i,j, and k then D _n?1:n  ≤ _lr D _n:n and D _n:n+1 ≤ _lr D _n:n , where ≤ _lr denotes the likelihood ratio order. We also prove that D _1:n  ≤ _lr D _2:n for n ≥ 2 and D _2:3 ≤ _lr D _3:3 for all λ _i 's.     

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.     

Stochastic Comparisons and Dependence of Spacings from Two Samples of Exponential Random Variables

Let X ₁,X ₂,…,X _n be independent exponential random variables such that X _i has hazard rate λ for i = 1,…,p and X _j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D _i:n (λ, λ*) = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…,n, where X _0:n ≡ 0. It is shown that the spacings (D _1,n ,D _2,n ,…,D _n:n ) are MTP₂, strengthening one result of Khaledi and Kochar (2000), and that (D _1:n (λ₂, λ*),…,D _n:n (λ₂, λ*)) ≤ _lr (D _1:n (λ₁, λ*),…,D _n:n (λ₁, λ*)) for λ₁ ≤ λ* ≤ λ₂, where ≤ _lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ₁ < λ₂.     

Constants on a Uniform Berry-Esseen Bound on Some Borel Sets in ℝ k via Stein's Method

For each n, k ∈ ?, let Y _i  = (Y _i1, Y _i2,…, Y _ik ), 1 ≤ i ≤ n be independent random vectors in ? ^k with finite third moments and Y _ij are independent for all j = 1, 2,…, k. In this article, we use the Stein's technique to find constants in uniform bounds for multidimensional Berry-Esseen inequality on a closed sphere, a half plane and a rectangular set.     

Precise Large Deviations for Sums of Independent Random Variables with Consistently Varying Tails

In this article, we investigate the precise large deviations for a sum of independent but not identical distributed random variables. {X _n , n ≥ 1} are independent non-negative random variables with distribution functions {F _n , n ≥ 1}. We assume that the average of right tails of distribution functions F _n is equivalent to some distribution function F with consistently varying tails. In applications, we apply our main results to a realistic example (Pareto-type distribution) and obtain a specific result.     

Nonparametric Prediction Intervals for Future Order Statistics in a Proportional Hazard Model

Consider k independent random samples with different sample sizes such that the ith sample comes from the cumulative distribution function (cdf) F _i  = 1 ? (1 ? F)^{α _i} , where α _i is a known positive constant and F is an absolutely continuous cdf. Also, suppose that we have observed the maximum and minimum of the first k samples. This article shows how one can construct the nonparametric prediction intervals for the order statistics of the future samples on the basis of these information. Three schemes are studied and in each case exact expressions for the prediction coefficients of prediction intervals are derived. Numerical computations are given for illustrating the results. Also, a comparison study is done while the complete samples are available.     

Asymptotic Distribution for Symmetric Spacing Statistics

A new procedure for testing the H ₀: μ₁ = ··· = μ _k against the alternative H _u :μ₁ ≥ ··· ≥μ _r  ≤ ··· ≤ μ _k with at least one strict inequality, where μ _i is the location parameter of the ith two-parameter exponential distribution, i = 1,…, k, is proposed. Exact critical constants are computed using a recursive integration algorithm. Tables containing these critical constants are provided to facilitate the implementation of the proposed test procedure. Simultaneous confidence intervals for certain contrasts of the location parameters are derived by inverting the proposed test statistic. In comparison to existing tests, it is shown, by a simulation study, that the new test statistic is more powerful in detecting U-shaped alternatives when the samples are derived from exponential distributions. As an extension, the use of the critical constants for comparing Pareto distribution parameters is discussed.     

Integral Representation of a Distribution Associated with a Pure Birth Process

ABSTRACT

This article deals with a distribution associated with a pure birth process starting with no individuals, with birth rates λ _n  = λ for n = 0, 2,…, m ? 1 and λ _n  = μ for n ≥ m. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions, and others. Using this representation, the kth factorial moments of the distribution is obtained. Some other forms of this distribution are also given.     

A Strong Invariance Theorem of the Tail Empirical Copula Processes

We study the behavior of bivariate empirical copula process 𝔾 _n (·, ·) on pavements [0, k _n /n]² of [0, 1]², where k _n is a sequence of positive constants fulfilling some conditions. We provide a upper bound for the strong approximation of 𝔾 _n (·, ·) by a Gaussian process when k _n /n↘γ as n → ∞, where 0 ≤ γ ≤1.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

DISTRIBUTIONS IN R WITH ROTATIONAL SYMMETRIES1

Let X ∈ R be a random vector with a distribution which is invariant under rotations within the subspaces V_j (dim V_j. = q_j) whose direct sum is R. The large sample distributions of the eigenvalues and vectors of M_n= n^-1Σⁿ_l x_ix_i are studied. In particular it is shown that several eigenvalue results of Anderson & Stephens (1972) for uniformly distributed unit vectors hold more generally.     

Empirical Bayes Testing for the Mean Lifetime of Exponential Distributions: Unequal Sample Sizes Case

This paper deals with an empirical Bayes testing problem for the mean lifetimes of exponential distributions with unequal sample sizes. We study a method to construct empirical Bayes tests {δ* _nl + 1,n }^∞ _n = 1 for the sequence of the testing problems. The asymptotic optimality of {δ* _nl + 1,n }^∞ _n = 1 is studied. It is shown that the sequence of empirical Bayes tests {δ* _nl + 1,n }^∞ _n = 1 is asymptotically optimal, and its associated sequence of regrets converges to zero at a rate (ln n)^4M?1/n, where M is an upper bound of sample sizes.