A note on hazard rates of order statistics

Let F_k:m be the cumulative disribution function of the kth order statistic in a sample of size n from a distribution

F(x) with density function f(x).The primary objective of this paper is to show that F_k+1mis IHR(increasing hazard rate) if F_km(x)is IHH and that F_k-1:n(x)is DHR.(decreasing hazard rate) if F_km(x) is DHR.     

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 Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

Asymptotic Properties of Random Weighted Empirical Distribution Function

This article studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X₁, X₂, ???, X_n is a sequence of independent random variables, and this sequence is not required to be identically distributed. Denote the empirical distribution function of the sequence by F_n(x). Based on the random weighting method and F_n(x), the random weighted empirical distribution function H_n(x) is constructed and the asymptotic properties of H_n are discussed. Under weak conditions, the Glivenko–Cantelli theorem and the central limit theorem for the random weighted empirical distribution function are obtained. The obtained results have also been applied to study the distribution functions of random errors of multiple sensors.     

Characterization of weak convergence for smoothed empirical and quantile processes under ϕ-mixing

Let X_n, n ⩾ 1 be a sequence of ϕ-mixing random variables having a smooth common distribution function F. The smoothed empirical distribution function is obtained by integrating a kernel type density estimator. In this paper we provide necessary and sufficient conditions for the central limit theorem to hold for smoothed empirical distribution functions and smoothed sample quantiles. Also, necessary and sufficient conditions are given for weak convergence of the smoothed empirical process and the smoothed uniform quantile process.     

On a Berry-Esséen theorem for a Studentized jackknife L-estimate

Consider a linear function of order statistics (“L-estimate”) which can be expressed as a statistical function T(F_n) based on the sample cumulative distribution function F_n. Let T*(F_n) be the corresponding jackknifed version of T(F_n), and let V²_n be the jackknife estimate of the asymptotic variance of n ^1/2T(F_n) or n ^1/2T*(F_n). In this paper, we provide a Berry-Esséen rate of the normal approximation for a Studentized jackknife L-estimate n^1/2[T*(F_n) - T(F)]/V_n, where T(F) is the basic functional associated with the L-estimate.     

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 ?}.     

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.     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.     

On characterization of the exponential distribution by spacings

Let X be a non-negative random variable with cumulative probability distribution function F. Suppse X₁, X₂, ..., X_n be a random sample of size n from F and X_i,n is the i-th smallest order statistics. We define the standardized spacings D_r,n=(n-r) (X_r+1,n-X_r,n), 1≤r≤n, with D_O,n=nX_1,n and D_n,n=0. Characterizations of the exponential distribution are given by considering the expectation and hazard rates of D_r,n.     

Edgeworth expansions for nonparametric distribution estimation with applications

In this paper, we will investigate the nonparametric estimation of the distribution function F of an absolutely continuous random variable. Two methods are analyzed: the first one based on the empirical distribution function, expressed in terms of i.i.d. lattice random variables and, secondly, the kernel method, which involves nonlattice random vectors dependent on the sample size n; this latter procedure produces a smooth distribution estimator that will be explicitly corrected to reduce the effect of bias or variance. For both methods, the non-Studentized and Studentized statistics are considered as well as their bootstrap counterparts and asymptotic expansions are constructed to approximate their distribution functions via the Edgeworth expansion techniques. On this basis, we will obtain confidence intervals for F(x) and state the coverage error order achieved in each case.     

Characterizations of the exponential distribution by some properties of the record values

A sequence {X_n, n≥1} of independent and identically distributed random variables with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max {X₁, X₂, ..., X_j?1}. We define L(n)=min {j|j>L(n?1), X_j>X_L(n?1)}, with L(0)=1. Let Z_n,m=X_L(n)?X_L(m), n>m≥0. Some characterizations of the exponential distribution are considered in terms of Z_n,m and X_L(m).     

On distribution function estimation with partially rank-ordered set samples: estimating mercury level in fish using length frequency data

We study the non-parametric estimation of a continuous distribution function F based on the partially rank-ordered set (PROS) sampling design. A PROS sampling design first selects a random sample from the underlying population and uses judgement ranking to rank them into partially ordered sets, without measuring the variable of interest. The final measurements are then obtained from one of the partially ordered sets. Considering an imperfect PROS sampling procedure, we first develop the empirical distribution function (EDF) estimator of F and study its theoretical properties. Then, we consider the problem of estimating F, where the underlying distribution is assumed to be symmetric. We also find a unique admissible estimator of F within the class of nondecreasing step functions with jumps at observed values and show the inadmissibility of the EDF. In addition, we introduce a smooth estimator of F and discuss its theoretical properties. Finally, we expand on various numerical illustrations of our results via several simulation studies and a real data application and show the advantages of PROS estimates over their counterparts under the simple random and ranked set sampling designs.     

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     

A note on a strong renewal theorem

Let {S_n, n ≥ 1} be a sequence of partial sums of independent and identically distributed non-negative random variables with a common distribution function F. Let F belong to the domain of attraction of a stable law with exponent α, 0 < α < 1. Suppose H(t) = ? N(t), t ? 0, where N(t) = max(n : S_n ≥ t). Under some additional assumptions on F, the difference between H(t) and its asymptotic value as t → ∞ is estimated.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

On a conjecture of Kakosyan,Klebanov and Melamed

Suppose X₁, X₂, ..., X_m is a random sample of size m from a population with probability density function f(x), x>0 and let X_1,m<...m,m be the corresponding order statistics. We assume m as an integer valued random variable with P(m=k)=p(1?p)^k?1, k=1, 2, ... and 0 and n X_1,n for fixed n characterizes the exponential distribution. In this paper we prove that under the assumption of monotone hazard rate the identical distribution of and (n?r+1) (X_r,n?X_r?1,n) for some fixed r and n with 1≤r≤n, n≥2, X_0,n=0, characterizes the exponential distribution. Under the assumption of monotone hazard rate the conjecture of Kakosyan, Klebanov and Melamed follows from the above result with r=1.     

A zero-one law for large order statistics

Fix r ≥ 1, and let {M_nr} be the rth largest of {X₁,X₂,…X_n}, where X₁,X₂,… is a sequence of i.i.d. random variables with distribution function F. It is proved that P[M_nr ≤ u_n i.o.] = 0 or 1 according as the series Σ^∞_n=3Fⁿ(u_n)(log log n)^r/n converges or diverges, for any real sequence {u_n} such that n{1 -F(u_n)} is nondecreasing and divergent. This generalizes a result of Bamdorff-Nielsen (1961) in the case r = 1.     

Prediction of record values

Given a number of record values from independent and identically distributed random variables with a continuous distribution function F, our aim is to predict future record values under suitable assumptions on the tail of F. In this paper, we are primarily concerned with finding reasonable tolerance regions for future record values. Two methods are proposed. The first one deals with the case where we observe only record values. The second one makes use of the information brought by the complete sample.     

Finite mixtures of natural exponential families

Let μ be a positive measure concentrated on R⁺ generating a natural exponential family (NEF) F with quadratic variance function V_F(m), m being the mean parameter of F. It is shown that v(dx) = (γ+x)μ(γ ≥ 0) (γ ≥ 0) generates a NEF G whose variance function is of the form l(m)Δ+cΔ(m), where l(m) is an affine function of m, Δ(m) is a polynomial in m (the mean of G) of degree 2, and c is a constant. The family G turns out to be a finite mixture of F and its length-biased family. We also examine the cases when F has cubic variance function and show that for suitable choices of γ the family G has variance function of the form P(m) + Q(m)m where P, Q are polynomials in m of degree m2 while Δ is an affine function of m. Finally we extend the idea to two dimensions by considering a bivariate Poisson and bivariate gamma mixture distribution.