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     

Evaluation of a method for setting confidence intervals on the common mean of two normal populations

Let X₁, , X₂, …, X be distributed N(µ, σ² _x), let Y₁, Y₂, …, Y"_n be distributed N(µ, σ² _y), and let X , X , … X_m, Y₁, Y₂, …, Y_n be mutually independent. In this paper a method for setting confidence intervals on the common mean µ is proposed and evaluated.     

Small samples confidencce intervals on common mean of two normal distributions variances

Let X₁,X₂,…,X_m be distributed normally with mean μ and variance σ² _X; Let Y₁,Y₂,…,Y_n be distributed normally with mean μ and variance σ² _Y; let X₁,X₂,…,X_m,Y₁,Y₂,…,Y_n be jointly independent. There have been several papers written concerning point estimation of μ for this problem, but very little is available in the literature concerning confidence intervals on the common mean μ. In this paper a method is proposed that results in a confidence interval with confidence coefficient essentially equal to a prescribed value 1 - α. The method is evaluated and compnred with other methods through the expected length of the confidence interval.     

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.     

Prediction Intervals for Future Order Statistics from a Generalized Modified Weibull

Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X₍₁₎, X₍₂₎, …, X_(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.     

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 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 stochastic Newton-Raphson method

A stochastic approximation procedure of the Robbins-Monro type is considered. The original idea behind the Newton-Raphson method is used as follows. Given n approximations X₁,…, X_n with observations Y₁,…, Y_n, a least squares line is fitted to the points (X_m, Y_m),…, (X_n, Y_n) where m<n may depend on n. The (n+1)st approximation is taken to be the intersection of the least squares line with y=0. A variation of the resulting process is studied. It is shown that this process yields a strongly consistent sequence of estimates which is asymptotically normal with minimal asymptotic variance.     

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.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

On Thresholds of Moving Averages with Given On-Target Significant Levels

Let X ₁, X ₂,… be a sequence of independent and identically distributed random variables, and let Y _n , n = K, K + 1, K + 2,… be the corresponding backward moving average of order K. At epoch n ≥ K, the process Y _n will be off target by the input X _n if it exceeds a threshold. By introducing a two-state Markov chain, we define a level of significance (1 ? a)% to be the percentage of times that the moving average process stays on target. We establish a technique to evaluate, or estimate, a threshold, to guarantee that {Y _n } will stay (1 ? a)% of times on target, for a given (1 ? a)%. It is proved that if the distribution of the inputs is exponential or normal, then the threshold will be a linear function in the mean of the distribution of inputs μ _X . The slope and intercept of the line, in each case, are specified. It is also observed that for the gamma inputs, the threshold is merely linear in the reciprocal of the scale parameter. These linear relationships can be easily applied to estimate the desired thresholds by samples from the inputs.     

On the bootstrap quantile-treatment-effect test

Let {X ₁, …, X _n } and {Y ₁, …, Y _m } be two samples of independent and identically distributed observations with common continuous cumulative distribution functions F(x)=P(X≤x) and G(y)=P(Y≤y), respectively. In this article, we would like to test the no quantile treatment effect hypothesis H ₀: F=G. We develop a bootstrap quantile-treatment-effect test procedure for testing H ₀ under the location-scale shift model. Our test procedure avoids the calculation of the check function (which is non-differentiable at the origin and makes solving the quantile effects difficult in typical quantile regression analysis). The limiting null distribution of the test procedure is derived and the procedure is shown to be consistent against a broad family of alternatives. Simulation studies show that our proposed test procedure attains its type I error rate close to the pre-chosen significance level even for small sample sizes. Our test procedure is illustrated with two real data sets on the lifetimes of guinea pigs from a treatment-control experiment.     

Estimation of the derivative of a density and related functions

Let Y₁,…,Y _n, (Y₁ <Y₂<…<Y _n) be the order statistics of a random sample from a distribution F with density f on the realline. This paper gives a class of estimators of the derivativef'(x) of the density f at points x for which f has

a continuoussecond derivative. These estimators are based on spacings inthe order statistics Y_j+kn -y _j j = 1,…,n-k_n,k_n<n.     

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.     

CHARACTERIZATIONS OF SHIFT-INVARIANT DISTRIBUTIONS BASED ON SUMMATION MODULO ONE

Let X₁Y_1,…, Y_n be independent random variables. We characterize the distributions of X and Y_j satisfying the equation {X+Y₁++Y_n}=_dX, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, either X is uniformly distributed on [0,1), or Y_j has.a shifted lattice distribution and X is shift-invariant. We also give a characterization of shift-invariant distributions. Finally, we consider some special cases of this equation.     

Concomitants in a Sequence of Independent Nonidentically Distributed Random Vectors

ABSTRACT

Concomitants of order statistics are considered for the situation in which the random vectors (X ₁, Y ₁), (X ₂, Y ₂),…, (X _n , Y _n ) are independent but otherwise arbitrarily distributed. The joint and marginal distributions of the concomitants of order statistics and stochastic comparisons among the concomitants of order statistics are studied in this situation.     

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.     

Moments of Order Statistics from Weibull Distribution in the Presence of Multiple Outliers

In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X₁, X₂, …, X_{n ? p} are independent with density function f(x) while the remaining, p observations (outliers) X_{n ? p + 1}, …, X_n are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.     

Mean intergrated squared error properties and optimal kernels when estimating a diatribution function

Let X₁,X₂,… X_n be a sample of independent identically distributed (i.i.d)random variables having an unknown absolutely continuous distribution function f with density f the twofold aim of his paper consists in, firstly deriving asymptotic expressions of the mean intergrated squared error (MISE) of a kernel estimator of F when f is either assumed to be continuous everywhere or problem of finding optimal kernels in these two cases is studied in detail.