Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

Some reliability properties of order statistics

Let X_i:j denote the i^th order statistic of a random sample of size j from a continuous life distribution. We show that if X_k:n, is IFR, IFRA, NBU, or DMRL, so are X_k+1:n, X_k+1:n?1 and X_k+1:n+1. Further we show that, in the first three cases, X_k+1:n+2 also shares the corresponding property if k ≤ (n+3)/2. We also present dual results for DFR, DFRA and NWU classes.     

CRITICAL VALUES FOR INFERENCE ABOUT NORMAL DISPERSION1

Let F_N(.) be the density function of X²_N. Values of C₁/N, i= 1, 2, satisfying the twin conditions Pr (C₁≤X²_N≤C₂)=1-α and the conditional expectation of X²_N given C₁≤X²_N≤C₂ is N are tabulated for α=.2, .1, .05, .01, .005, .001, N=1(1)20(2)50(5)150(10)350. The second condition may be replaced by the condition f_N+2(C₁)=f_N+2V(C₂). The author has with him a bigger table giving C₁ and C₂ for α=.2, .1, .05, .01, .005, .001, N=1(1)350 to three decimals (to three significant digits, if some decimals are not significant). Several applications are mentioned. A practical application that is perhaps not obvious is to test whether two or more counts are distributed as independent Poisson variables. The new simple formulae used in the construction of the table are given and should prove useful in obtaining accurate values for omitted entries and in increasing the accuracy of entries.     

Weak and strong uniform consistency rates of kernel density estimates for randomly censored data

Let X₁,., X_n, be i.i.d. random variables with distribution function F, and let Y₁,.,.,Y_n be i.i.d. with distribution function G. For i = 1, 2,.,., n set δ_i, = 1 if X_i ≤ Y_i, and 0 otherwise, and X_i, = min{X_i, K_i}. A kernel-type density estimate of f, the density function of F w.r.t. Lebesgue measure on the Borel o-field, based on the censored data (δ_i, X_i), i = 1,.,.,n, is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher-order differentiability assumption on f. A procedure for relaxing such assumptions is also proposed.     

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 ?}.     

A Note on the Calculation of Pr{X 1 < X 2 < ··· < Xk }

Suppose that X_i are independent random variables, and that X_i has cdf F_i (x), 1 ≤ i ≤ k. Many statistical problems involve the probability Pr{X ₁ < X ₂ < ··· < X_k }. In this note a numerical method is proposed for computing this probability.     

Uniform asymptotic linearity of a process based on a signed rank statistic

Consider the process with, cf. (1.2) on page 265 in B1, X₁, …, X_N a sample from a distribution F and, for i = 1, …, N, R${}_{\text{|x}{}_{\text{1}}\text{, - q}{}_{\text{1}}\text{ø|}}$ , the rank of |X₁ - q₁ø| among |X₁ - q₁ø|, …, |X_N - q_Nø|. It is shown that, under certain regularity conditions on F and on the constants p_i and q_i, T^ø_N(t) is asymptotically approximately a linear function of ø uniformly in t and in ø for |ø| ≤ C. The special case where the p_i and the q_i, are independent of i is considered.     

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.     

A signed rank test for censored paired lifetimes

A locally most powerful signed rank test is proposed for the comparison of two independent lifetimes under the accelerated failure time model.

The test is based on N independent pairs(X_i, Y_i), i = 1, …, N: it is supposed that the shortest lifetime in each pair is observed and the experiment is stopped after r(r≤N and fixed) such lifetimes are available (type II censoring).

Actual scores of the test statistic are computed for some specific source distributions of the observations. The asymptotic distribution of the test statistic, as well as the asymptotic power and efficiency are given. The values of these efficiencies are computed for the case where the X_i follow and exponential, Weibull Gamma or Rayleigh distribution.     

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.     

Inference procedures in some bivariate exponential models under hybrid random censoring

We consider a life testing experiment in whichn units are put on test, successive lifetimes (X ₁,X ₂) of both componentsC ₁ andC ₂ are recorded and the observation is terminated either at ther-th order statistic ofY _i =Min(X _1i ,X _2i ),i=1,…,n i.e.Y _(r) or a random timeT _i whichever is reached first. This mixture of random censoring and type-II censoring schemes, we call as hybrid random censoring which is of wide use. We use this censoring scheme and obtain maximum likelihood estimation of the parameters and develop large sample tests in bivariate exponential (BVE) models proposed by Marshall-Olkin (1967), Block-Basu (1974), Freund (1961) and Preschan-Sullo (1974).     

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.     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Let X₁,…, X_n be mutually independent non-negative integer-valued random variables with probability mass functions f_i(x) > 0 for z= 0,1,…. Let E denote the event that {X₁≥X₂≥…≥X_n}. This note shows that, conditional on the event E, X_i-X_i+ 1 and X_i+ 1 are independent for all t = 1,…, k if and only if X_i (i= 1,…, k) are geometric random variables, where 1 ≤k≤n-1. The k geometric distributions can have different parameters θ_i, i= 1,…, k.     

A predictive approach for the selection of a fixed number of good treatments

This paper offers a predictive approach for the selection of a fixed number (= t) of treatments from k treatments with the goal of controlling for predictive losses. For the ith treatment, independent observations X _ij (j = 1,2,…,n) can be observed where X _ij ’s are normally distributed N(θ _i ; σ ²). The ranked values of θ _i ’s and X _i ’s are θ ₍₁₎ ≤ … ≤ θ _(k) and X _[1] ≤ … ≤ X _[k] and the selected subset S = {[k], [k? 1], … , [k ? t+1]} will be considered. This paper distinguishes between two types of loss functions. A type I loss function associated with a selected subset S is the loss in utility from the selector’s view point and is a function of θ _i with i ? S. A type II loss function associated with S measures the unfairness in the selection from candidates’ viewpoint and is a function of θ _i with i ? S. This paper shows that under mild assumptions on the loss functions S is optimal and provides the necessary formulae for choosing n so that the two types of loss can be controlled individually or simultaneously with a high probability. Predictive bounds for the losses are provided, Numerical examples support the usefulness of the predictive approach over the design of experiment approach.     

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.     

A Test of Independence Based on the Likelihood of Cut-Points

We define a test statistic C _n based on the sum of the likelihood ratio statistics for testing independence in the 2 × 2 tables defined at n sample cut-points (X _i , Y _i ). The asymptotic distribution of C _n , given the cut-points, is sum of dependent χ² variables with one degree of freedom. We use the bootstrap to obtain the distribution of C _n . We compare the performance of several tests of bivariate independence, including Pearson, Spearman, and Kendall correlations, Blum-Kiefer-Rosenblatt statistic, and C _n under several copulas and given marginal distributions.     

More bounds for moments of some rank statistics

Let X₁,…,X₇ be i.i.d. random variables with a common continuous distribution F, Two parameters, μ(F) = P(X₁ < X₅ and X₁+X₄ < X₂+X₃) and λ(F) = P(X₁+X₄ < X₂+X₃ and X₁+X₇ < X₅+X₆), which appear in the moments of some rank statistics have been studied by several authors. It is shown that the existing lower bound, 3/10 ≤ μ(F) can be improved to 3/10 < μ(F) and that no further improvement is possible. It is also shown that the existing upper bounds μ(F) ≤ (2^1/2+6)/24 ≈ 0.30893 and λ(F) ≤ 7/24 ≈ 0.29167 can be improved to [14+(2/3)^1/2]/48 ≈ 0.30868 and {7 ? [1 ? (2/3)^1/2]²/4}/24 ≈ 0.29132.     

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.