On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron's bootstrap. Asymptotic properties of these generalised bootstrapped empirical distributions are obtained, under mild conditions by a martingale approach. We also obtain some new results on the weak convergence of the empirical semi-Markov processes. We apply these general results in several statistical problems such as the construction of confidence bands and the goodness-of-fit tests where the limiting distributions are derived under the null hypothesis. Finally, we introduce the quantile estimators and their bootstrapped versions in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.     

ASYMPTOTIC PROPERTIES FOR MAXIMUM LIKELIHOOD ESTIMATORS FOR RELIABILITY AND FAILURE RATES OF MARKOV CHAINS

ABSTRACT

In this paper, we shall study a homogeneous ergodic, finite state, Markov chain with unknown transition probability matrix. Starting from the well known maximum likelihood estimator of transition probability matrix, we define estimators of reliability and its measurements. Our aim is to show that these estimators are uniformly strongly consistent and converge in distribution to normal random variables. The construction of the confidence intervals for availability, reliability, and failure rates are also given. Finally we shall give a numerical example for illustration and comparing our results with the usual empirical estimator results.     

On the Variances and Convariances of the Duration State Sizes of Semi-Markov Systems

In the present article we study the asymptotic behaviour of the sequence of vectors with components expectations, variances, and covariances of the state sizes of a semi-Markov system. In this respect, we transform the semi-Markov system into a Markov system with a different though equivalent state space and relate the sequence of the transition probabilities of the respective Markov system as functions of the parameters of the semi-Markov system. Also, we study the asymptotic behavior of the sequence of vectors with components variances and covariances of the duration state sizes, of the related Markov system, under perturbation of the transition probability matrices. We use the results in the study of asymptotic behavior under perturbation of the sequence of vectors with components variances and covariances of the semi-Markov system.     

Parameter Estimation in Conditional Heteroscedastic Models

Abstract

We study asymptotics of parameter estimates in conditional heteroscedastic models. The estimators considered are those obtained by minimizing certain functionals and those obtained by solving estimation equations. We establish consistency and derive asymptotic limit laws of the estimators. Condition under which the limit law is normal is studied. Further, bootstrap for these estimators is discussed. The limiting distribution of the estimators is not necessary always normal, and we present a real data example to illustrate this.     

Estimation of Survival Function in Cox Regression Model Under Random Censoring From Both Sides

In this article we present three types of parametric–non parametric estimators for conditional survival function in Cox proportional hazards regression model when the lifetime of interest is subjected to random censorship from both sides. We prove consistency and asymptotic normality of estimators.     

Conditional Quantile Estimation for Truncated and Associated Data

Abstract

In survival or reliability studies, it is common to have data which are not only incomplete but weakly dependent too. Random truncation and censoring are two common forms of such data when they are neither independent nor strongly mixing but rather associated. The focus of this paper is on estimating conditional distribution and conditional quantile functions for randomly left truncated data satisfying association condition. We aim at deriving strong uniform consistency rates and asymptotic normality for the estimators and thereby, extend to association case some results stated under iid and α-mixing hypotheses. The performance of the quantile function estimator is evaluated on simulated data sets.     

Asymptotic normality of kernel estimators based upon incomplete data

In this paper, we are concerned with nonparametric estimation of the density and the failure rate functions of a random variable X which is at risk of being censored. First, we establish the asymptotic normality of a kernel density estimator in a general censoring setup. Then, we apply our result in order to derive the asymptotic normality of both the density and the failure rate estimators in the cases of right, twice and doubly censored data. Finally, the performance and the asymptotic Gaussian behaviour of the studied estimators, based on either doubly or twice censored data, are illustrated through a simulation study.     

Nonparametric reliability estimation of semi-Markov processes

We define estimators of reliability and availability of such systems and show that they are uniformly strongly consistent and that each of them converges weakly to a normal random variable. The result of Baxter and Li (Scand. J. Statist. 21 (1994) 277) for an alternated renewal process appears as a particular case of our result. A method for constructing confidence intervals is also given. This is made possible by use of recent results of Ouhbi and Limnios (Appl. Stochast. Mod. Data Anal. 12 (4) (1996) 209; C. R. Acad. Sci. Paris 325 (1997) 921) establishing asymptotic properties of the estimators of the semi-Markov kernel and the Markov renewal matrix, respectively.     

A resampling method by perturbing the estimating functions for quantile regression with missing data

In this article, we propose a resampling method based on perturbing the estimating functions to compute the asymptotic variances of quantile regression estimators under missing at random condition. We prove that the conditional distributions of the resampling estimators are asymptotically equivalent to the distributions of quantile regression estimators. Our method can deal with complex situations, where the response and part of covariates are missing. Numerical results based on simulated and real data are provided under several designs.     

Bernstein conditional density estimation with application to conditional distribution and regression functions

In this paper we propose a smooth nonparametric estimation for the conditional probability density function based on a Bernstein polynomial representation. Our estimator can be written as a finite mixture of beta densities with data-driven weights. Using the Bernstein estimator of the conditional density function, we derive new estimators for the distribution function and conditional mean. We establish the asymptotic properties of the proposed estimators, by proving their asymptotic normality and by providing their asymptotic bias and variance. Simulation results suggest that the proposed estimators can outperform the Nadaraya–Watson estimator and, in some specific setups, the local linear kernel estimators. Finally, we use our estimators for modeling the income in Italy, conditional on year from 1951 to 1998, and have another look at the well known Old Faithful Geyser data.     

HAZARD RATE ESTIMATION FOR CENSORED DATA BY WAVELET METHODS

ABSTRACT

We study the estimation of a hazard rate function based on censored data by non-linear wavelet method. We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based hazard rate estimators under randomly censored data. We show this MISE formula, when the underlying hazard rate function and censoring distribution function are only piecewise smooth, has the same expansion as analogous kernel estimators, a feature not available for the kernel estimators. In addition, we establish an asymptotic normality of the nonlinear wavelet estimator.     

Nonparametric regression estimation of conditional tails: the random covariate case

We present families of nonparametric estimators for the conditional tail index of a Pareto-type distribution in the presence of random covariates. These families are constructed from locally weighted sums of power transformations of excesses over a high threshold. The asymptotic properties of the proposed estimators are derived under some assumptions on the conditional response distribution, the weight function and the density function of the covariates. We also introduce bias-corrected versions of the estimators for the conditional tail index, and propose in this context a consistent estimator for the second-order tail parameter. The finite sample performance of some specific examples from our classes of estimators is illustrated with a small simulation experiment.     

Estimation of entropy-type integral functionals

ABSTRACT

Entropy-type integral functionals of densities are widely used in mathematical statistics, information theory, and computer science. Examples include measures of closeness between distributions (e.g., density power divergence) and uncertainty characteristics for a random variable (e.g., Rényi entropy). In this paper, we study U-statistic estimators for a class of such functionals. The estimators are based on ε-close vector observations in the corresponding independent and identically distributed samples. We prove asymptotic properties of the estimators (consistency and asymptotic normality) under mild integrability and smoothness conditions for the densities. The results can be applied in diverse problems in mathematical statistics and computer science (e.g., distribution identification problems, approximate matching for random databases, two-sample problems).     

Implications of Model Misspecification in Robust Tests for Recurrent Events

Chronic disease processes often feature transient recurrent adverse clinical events. Treatment comparisons in clinical trials of such disorders must be based on valid and efficient methods of analysis. We discuss robust strategies for testing treatment effects with recurrent events using methods based on marginal rate functions, partially conditional rate functions, and methods based on marginal failure time models. While all three approaches lead to valid tests of the null hypothesis when robust variance estimates are used, they differ in power. Moreover, some approaches lead to estimators of treatment effect which are more easily interpreted than others. To investigate this, we derive the limiting value of estimators of treatment effect from marginal failure time models and illustrate their dependence on features of the underlying point process, as well as the censoring mechanism. Through simulation, we show that methods based on marginal failure time distributions are shown to be sensitive to treatment effects delaying the occurrence of the very first recurrences. Methods based on marginal or partially conditional rate functions perform well in situations where treatment effects persist or in settings where the aim is to summarizee long-term data on efficacy.     

On robust forecasting in dynamic vector time series models

In this article, robust estimation and prediction in multivariate autoregressive models with exogenous variables (VARX) are considered. The conditional least squares (CLS) estimators are known to be non-robust when outliers occur. To obtain robust estimators, the method introduced in Duchesne [2005. Robust and powerful serial correlation tests with new robust estimates in ARX models. J. Time Ser. Anal. 26, 49–81] and Bou Hamad and Duchesne [2005. On robust diagnostics at individual lags using RA-ARX estimators. In: Duchesne, P., Rémillard, B. (Eds.), Statistical Modeling and Analysis for Complex Data Problems. Springer, New York] is generalized for VARX models. The asymptotic distribution of the new estimators is studied and from this is obtained in particular the asymptotic covariance matrix of the robust estimators. Classical conditional prediction intervals normally rely on estimators such as the usual non-robust CLS estimators. In the presence of outliers, such as additive outliers, these classical predictions can be severely biased. More generally, the occurrence of outliers may invalidate the usual conditional prediction intervals. Consequently, the new robust methodology is used to develop robust conditional prediction intervals which take into account parameter estimation uncertainty. In a simulation study, we investigate the finite sample properties of the robust prediction intervals under several scenarios for the occurrence of the outliers, and the new intervals are compared to non-robust intervals based on classical CLS estimators.     

Most Likely Transformations

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterization of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterization thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood‐based estimation for conditional transformation models that allows the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.     

Note on estimation of reliability under bivariate pareto stress-strength model

In this paper, we discuss the problem of estimating reliability (R) of a component based on maximum likelihood estimators (MLEs). The reliability of a component is given byR=P[Y<X]. Here X is a random strength of a component subjected to a random stress(Y) and (X,Y) follow a bivariate pareto(BVP) distribution. We obtain an asymptotic normal(AN) distribution of MLE of the reliability(R).     

Pareto Poisson–Lindley distribution with applications

A new lifetime distribution is introduced based on compounding Pareto and Poisson–Lindley distributions. Several statistical properties of the distribution are established, including behavior of the probability density function and the failure rate function, heavy- and long-right tailedness, moments, the Laplace transform, quantiles, order statistics, moments of residual lifetime, conditional moments, conditional moment generating function, stress–strength parameter, Rényi entropy and Song's measure. We get maximum-likelihood estimators of the distribution parameters and investigate the asymptotic distribution of the estimators via Fisher's information matrix. Applications of the distribution using three real data sets are presented and it is shown that the distribution fits better than other related distributions in practical uses.