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.     

L'effet de l'erreur d'arrondi sur l'estimateur d'une densité par la méthode du noyau

In a model for rounded data suppose that the random sample X₁,.,.,X_n,. i.i.d., is transformed into an observed random sample X_1Δ,.,.,X_nΔ, where X_iΔ = 2vΔ if X_i, ∈ (2vΔ - Δ, 2vΔ + Δ), for i = 1,.,.,n. We show that the precision Δ of the observations has an important effect on the shape of the kernel density estimator, and we identify important points for the graphical display of this estimator. We examine the IMSE criteria to find the optimal window under the rounded-data model.     

Two-Sample Tests Based on the Integrated Empirical Process

Abstract

Let X ₁, …, X _m and Y ₁, …, Y _n be independent random variables, where X ₁, …, X _m are i.i.d. with continuous distribution function (df) F, and Y ₁, …, Y _n are i.i.d. with continuous df G. For testing the hypothesis H ₀: F = G, we introduce and study analogues of the celebrated Kolmogorov–Smirnov and one- and two-sided Cramér-von Mises statistics that are functionals of a suitably integrated two-sample empirical process. Furthermore, we characterize those distributions for which the new tests are locally Bahadur optimal within the setting of shift alternatives.     

Consistent deconvolution in density estimation

Suppose we have n observations from X = Y + Z, where Z is a noise component with known distribution, and Y has an unknown density f. When the characteristic function of Z is nonzero almost everywhere, we show that it is possible to construct a density estimate f_n such that for all f, Iim_n| |=0.     

Bivariate extreme value distributions

Let (X_i, Y_i), i = 1, 2,…, n, be n independent observations from a bivariate population and let X_(n) = max X_i and Y_(n) = max Y_i. This article gives a necessary and sufficient condition for the weak convergence of the distribution function of (X_(n), Y_(n)) to a nondegenerate distribution.     

On multivariate variable-kernel density estimates for time series

Let X₁ be a strictly stationary multiple time series with values in R^d and with a common density f. Let X₁,.,.,X_n, be n consecutive observations of X₁. Let k = k_n, be a sequence of positive integers, and let H_ni be the distance from X_i to its kth nearest neighbour among X_j, j i. The multivariate variable-kernel estimate f_n, of f is defined by where K is a given density. The complete convergence of f_n, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.     

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.     

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 ?}.     

Correlation estimation in Downton's bivariate exponential distribution using incomplete samples

Suppose (X, Y) has a Downton's bivariate exponential distribution with correlation ρ. For a random sample of size n from (X, Y), let X _r:n be the rth X-order statistic and Y _[r:n] be its concomitant. We investigate estimators of ρ when all the parameters are unknown and the available data is an incomplete bivariate sample made up of (i) all the Y-values and the ranks of associated X-values, i.e. (i, Y _[i:n]), 1≤i≤n, and (ii) a Type II right-censored bivariate sample consisting of (X _i:n , Y _[i:n]), 1≤i≤r<n. In both setups, we use simulation to examine the bias and mean square errors of several estimators of ρ and obtain their estimated relative efficiencies. The preferred estimator under (i) is a function of the sample correlation of (Y _i:n , Y _[i:n]) values, and under (ii), a method of moments estimator involving the regression function is preferred.     

Nonparametric estimation of survival functions for incomplete observations when the life time distribution is proportionally related ot the censoring time distribution

Let X₁, X₂…,X_n be a random sample from [ILM0001] and let Y₁, …,Y_n be a random sample from [ILM0002]. Then instead of observing a complete sample X₁,…X_n, we can only observe the pairs Z_i. = min(X_i.,Y_i) and [ILM0003] In this paper, we consider estimation of survival function [ILM0004] when [ILM0005], where β is an unknown positive real number.

     

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     

Sequential Fixed-Width Confidence Interval for a Function of the Parameters

Let X₁…, X_m and Y₁…, Y_n be two independent sequences of i.i.d. random variables with distribution functions F_x(.|θ) and F_y(. | φ) respectively. Let g(θ, φ) be a real-valued function of the unknown parameters θ and φ. The purpose of this paper is to suggest a sequential procedure which gives a fixed-width confidence interval for g(θ, φ) so that the coverage probability is approximately α (preas-signed). Certain asymptotic optimality properties of the sequential procedure are established. A Monte Carlo study is presented.     

Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles

Let (X, Y) be a bivariate random vector whose distribution function H(x, y) belongs to the class of bivariate extreme-value distributions. If F₁ and F₂ are the marginals of X and Y, then H(x, y) = C{F₁(x),F₂(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F₁(X)}/{log F₁(X)F₂(Y)} and W = C{F₁{(X),F₂(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is présentés. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{F₁(X),F₂(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.     

Orderings for series and parallel systems comprising heterogeneous exponentiated Weibull-geometric components

Let X₁, …, X_n be independent random variables with X_i ～ EWG(α, β, λ_i, p_i), i = 1, …, n, and Y₁, …, Y_n be another set of independent random variables with Y_i ～ EWG(α, β, γ_i, q_i), i = 1, …, n. The results established here are developed in two directions. First, under conditions p₁ = ??? = p_n = q₁ = ??? = q_n = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ₁ = ??? = λ_n = γ₁ = ??? = γ_n = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p₁, …, p_n) and (q₁, …, q_n). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.     

A new class of bivariate lifetime distributions

This paper introduces a new class of bivariate lifetime distributions. Let {X_i}_{i ? 1} and {Y_i}_{i ? 1} be two independent sequences of independent and identically distributed positive valued random variables. Define T₁ = min?(X₁, …, X_M) and T₂ = min?(Y₁, …, Y_N), where (M, N) has a discrete bivariate phase-type distribution, independent of {X_i}_{i ? 1} and {Y_i}_{i ? 1}. The joint survival function of (T₁, T₂) is studied.     

Nonparametric Estimation of a Regression Function: Limiting Distribution2

Consider the regression model Y_i= g(x_i) + e_i, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x₀ < x₁ < … < x_n≤ 1 are chosen so that max_1≤i≤n(x_i-x_{i- 1}) = 0(n^-1), and where {e_i} are i.i.d. with Ee₁= 0 and Var e₁ - s?². In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g_1n of Cheng & Lin (1979), g_2n of Clark (1977), and g_3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &_in, i = 1, 2, 3, and that of the isotonic estimator g**_n are considered.     

Comparing Fisher information in record values and their concomitants with random observations

In a sequence of bivariate random variables {(X _i , Y _i ), i≥1} from a continuous distribution with a real parameter θ, general comparison results between the amount of Fisher information about θ contained in the sequence of the first n records and their concomitants, and the desired information in an i.i.d. sample of size n from the parent distribution are established. Some relationships between reliability properties and the proposed criteria are obtained in situations in which the univariate counterpart of the underlying bivariate family belongs to location, scale or shape families. It is also shown that in some classes of bivariate families, the concerned information property is equivalent to that of its univariate counterpart. The proposed procedure is illustrated by considering several examples.     

ON ASYMPTOTIC OPTIMALITY OF SOME TESTS FOR EXPONENTIAL DISTRIBUTIONS

The probability density function (pdf) of a two parameter exponential distribution is given by f(x; p, s?) =s?^-1 exp {-(x - ρ)/s?} for x≥ρ and 0 elsewhere, where 0 < ρ < ∞ and 0 < s?∞. Suppose we have k independent random samples where the ith sample is drawn from the ith population having the pdf f(x; ρ_i, s?_i), 0 < ρ_i < ∞, 0 < s?_i < s?_i < and f(x; ρ, s?) is as given above. Let X_i1 < X_i2 <… < X_iri denote the first r_i order statistics in a random sample of size n_i, drawn from the ith population with pdf f(x; ρ_i, s?_i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρ_i, s?_i, i = 1, 2,…, k based on the above observations are asymptotically optimal in the sense of Bahadur efficiency. Our results are similar to those for normal distributions.     

Nonparametric statistical procedures for the changepoint problem

Let X₁,…,X_r?1,X_r,X_r+1,…,X_n be independent, continuous random variables such that X_i, i = 1,…,r, has distribution function F(x), and X_i, i = r+1,…,n, has distribution function F(x?Δ), with -∞ <Δ< ∞. When the integer r is unknown, this is refered to as a change point problem with at most one change. The unknown parameter Δ represents the magnitude of the change and r is called the changepoint. In this paper we present a general review discussion of several nonparametric approaches for making inferences about r and Δ.