Improved estimation of scale parameters of morgenstern type bivariate uniform distribution using ranked set sampling

ABSTRACT

In this article we suggest some improved version of estimators of scale parameter of Morgenstern-type bivariate uniform distribution (MTBUD) based on the observations made on the units of the ranked set sampling regarding the study variable Y which is correlated with the auxiliary variable X, when (X, Y) follows a MTBUD. We also suggest some linear shrinkage estimators of scale parameter of Morgenstern type bivariate uniform distribution (MTBUD). Efficiency comparisons are also made in this work.     

Linear Regression Models under Conditional Independence Restrictions

Maximum likelihood estimation is investigated in the context of linear regression models under partial independence restrictions. These restrictions aim to assume a kind of completeness of a set of predictors Z in the sense that they are sufficient to explain the dependencies between an outcome Y and predictors X: ?(Y|Z, X) = ?(Y|Z), where ?(·|·) stands for the conditional distribution. From a practical point of view, the former model is particularly interesting in a double sampling scheme where Y and Z are measured together on a first sample and Z and X on a second separate sample. In that case, estimation procedures are close to those developed in the study of double‐regression by Engel & Walstra (1991) and Causeur & Dhorne (1998) . Properties of the estimators are derived in a small sample framework and in an asymptotic one, and the procedure is illustrated by an example from the food industry context.     

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.     

Estimation of R=P[Y<X] for three-parameter generalized Rayleigh distribution

Surles and Padgett [Inference for reliability and stress–strength for a scaled Burr type X distribution. Lifetime Data Anal. 2001;7:187–200] introduced a two-parameter Burr-type X distribution, which can be described as a generalized Rayleigh distribution. In this paper, we consider the estimation of the stress–strength parameter R=P[Y<X], when X and Y are both three-parameter generalized Rayleigh distributions with the same scale and locations parameters but different shape parameters. It is assumed that they are independently distributed. It is observed that the maximum-likelihood estimators (MLEs) do not exist, and we propose a modified MLE of R. We obtain the asymptotic distribution of the modified MLE of R, and it can be used to construct the asymptotic confidence interval of R. We also propose the Bayes estimate of R and the construction of the associated credible interval based on importance sampling technique. Analysis of two real data sets, (i) simulated and (ii) real, have been performed for illustrative purposes.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.     

Estimation in Partially Linear Single-Index Models with Missing Covariates

In this article, we consider a partially linear single-index model Y = g(Z ^τθ₀) + X ^τβ₀ + ? when the covariate X may be missing at random. We propose weighted estimators for the unknown parametric and nonparametric part by applying weighted estimating equations. We establish normality of the estimators of the parameters and asymptotic expansion for the estimator of the nonparametric part when the selection probabilities are unknown. Simulation studies are also conducted to illustrate the finite sample properties of these estimators.     

Dependent Right Censorship in the MOMW Distribution

In this paper, we consider paired survival data, in which pair members are subject to the same right censoring time, but they are dependent on each other. Assuming the Marshall–Olkin Multivariate Weibull distribution for the joint distribution of the lifetimes (X₁, X₂) and the censoring time X₃, we derive the joint density of the actual observed data and obtain maximum likelihood estimators, Bayes estimators and posterior regret Gamma minimax estimators of the unknown parameters under squared error loss and weighted squared error loss functions. We compare the performances of the maximum likelihood estimators and Bayes estimators numerically in terms of biases and estimated Mean Squared Error Loss.     

Multiparameter estimation in truncated power series distributions under the stein's loss

LetX ₁,…,X _p be p(≥2)independent random variables, where each X.has a distribution belonging to a one parameter truncated power series

distribution. The problem is to estimate simultaneously the unknown parameters under asymmetric loss developed by James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob. 1, 361-380). Several new classes of dominating estimators are obtained by solving a certain difference inequality.     

Generalized minimum distance estimators of a linear model with correlated errors

R^p of a linear regression model of the type Y = Xθ + ɛ, where X is the design matrix, Y the vector of the response variable and ɛ the random error vector that follows an AR(1) correlation structure. These estimators are asymptotically analyzed, by proving their strong consistency, asymptotic normality and asymptotic efficiency. In a simulation study, a better behaviour of the Mean Squared Error of the proposed estimator with respect to that of the generalized least squares estimators is observed. Received: November 16, 1998; revised version: May 10, 2000     

Estimation of a Conditional Copula and Association Measures

Abstarct. This paper is concerned with studying the dependence structure between two random variables Y ₁ and Y ₂ conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non‐parametric estimators of the ‘conditional’ copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.     

Estimation of Pr( x<y ) in the exponential case with common location parameter

This paper considers the problem of estimating the probability P = Pr(X < Y) when X and Y are independent exponential random variables with unequal scale parameters and a common location parameter. Uniformly minimum variance unbiased estimator of P is obtained. The asymptotic distribution of the maximum likelihood estimator is obtained and then the asymptotic equivalence of the two estimators is established. Performance of the two estimators for moderate sample sizes is studied by Monte Carlo simulation. An approximate interval estimator is also obtained.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Wavelet-based estimation of regression function with strong mixing errors under fixed design

We consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes B^s_{p, q}. The theory is illustrated with some numerical examples.

A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.     

Parameter estimation by contrast minimization for noisy observations of a diffusion process

We consider the estimation of unknown parameters in the drift and diffusion coefficients of a one-dimensional ergodic diffusion X when the observation Y is a discrete sampling of X with an additive noise, at times i δ, i=1,?…?, N. Assuming that the sampling interval tends to 0 while the total length-time interval tends to infinity, we prove limit theorems for functionals associated with the observations, based on local means of the sample. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent. Some examples are discussed with numerical simulations.     

A Semi-parametric Regression Model with Errors in Variables

Abstract.  In this paper, we consider a partial linear regression model with measurement errors in possibly all the variables. We use a method of moments and deconvolution to construct a new class of parametric estimators together with a non-parametric kernel estimator. Strong convergence, optimal rate of weak convergence and asymptotic normality of the estimators are investigated.     

Estimation of P{X < Y} for geometric—exponential model based on complete and censored samples

This article deals with the estimation of R = P{X < Y}, where X and Y are independent random variables from geometric and exponential distribution, respectively. For complete samples, the MLE of R, its asymptotic distribution, and confidence interval based on it are obtained. The procedure for deriving bootstrap-p confidence interval is presented. The UMVUE of R and UMVUE of its variance are derived. The Bayes estimator of R is investigated and its Lindley's approximation is obtained. A simulation study is performed in order to compare these estimators. Finally, all point estimators for right censored sample from the exponential distribution, are obtained.     

Stress-Strength Reliability of a Two-Parameter Bathtub-shaped Lifetime Distribution Based on Progressively Censored Samples

Based on progressively Type-II censored samples, this article deals with inference for the stress-strength reliability R = P(Y < X) when X and Y are two independent two-parameter bathtub-shape lifetime distributions with different scale parameters, but having the same shape parameter. Different methods for estimating the reliability are applied. The maximum likelihood estimate of R is derived. Also, its asymptotic distribution is used to construct an asymptotic confidence interval for R. Assuming that the shape parameter is known, the maximum likelihood estimator of R is obtained. Based on the exact distribution of the maximum likelihood estimator of R an exact confidence interval of that has been obtained. The uniformly minimum variance unbiased estimator are calculated for R. Bayes estimate of R and the associated credible interval are also got under the assumption of independent gamma priors. Monte Carlo simulations are performed to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose. Finally, we will generalize this distribution to the proportional hazard family with two parameters and derive various estimators in this family.     

Robust kernel estimators for additive models with dependent observations

Robust nonparametric estimators for additive regression or autoregression models under an α-mixing condition are proposed. They are based on local M-estimators or local medians with kernel weights, and their asymptotic behaviour is studied. Moreover, diese local M-estimators achieve the same univariate rate of convergence as their linear relatives.     

Bias reduction in kernel density estimation

In this paper, we propose two kernel density estimators based on a bias reduction technique. We study the properties of these estimators and compare them with Parzen–Rosenblatt's density estimator and Mokkadem, A., Pelletier, M., and Slaoui, Y. (2009, ‘The stochastic approximation method for the estimation of a multivariate probability density’, J. Statist. Plann. Inference, 139, 2459–2478) is density estimators. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators and the asymptotic MISE (Mean Integrated Squared Error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations.     

Partially functional linear varying coefficient model

In this paper, we introduce a new partially functional linear varying coefficient model, where the response is a scalar and some of the covariates are functional. By means of functional principal components analysis and local linear smoothing techniques, we obtain the estimators of coefficient functions of both function-valued variable and real-valued variables. Then the rates of convergence of the proposed estimators and the mean squared prediction error are established under some regularity conditions. Moreover, we develop a hypothesis test for the model and employ the bootstrap procedure to evaluate the null distribution of test statistic and the p-value of the test. At last, we illustrate the finite sample performance of our methods with some simulation studies and a real data application.