Nonparametric estimation of mean residual functions

Situations frequently arise in practice in which mean residual life (mrl) functions must be ordered. For example, in a clinical trial of three experiments, let e (1), e (2) and e (3) be the mrl functions, respectively, for the disease groups under the standard and experimental treatments, and for the disease-free group. The well-documented mrl functions e (1) and e (3) can be used to generate a better estimate for e (2) under the mrl restriction e (1) < or = e (2) < or = e (3). In this paper we propose nonparametric estimators of the mean residual life function where both upper and lower bounds are given. Small and large sample properties of the estimators are explored. Simulation study shows that the proposed estimators have uniformly smaller mean squared error compared to the unrestricted empirical mrl functions. The proposed estimators are illustrated using a real data set from a cancer clinical trial study.     

Classical estimation of hazard rate and mean residual life functions of Pareto distribution

Abstract

In this paper we find the maximum likelihood estimates (MLEs) of hazard rate and mean residual life functions (MRLF) of Pareto distribution, their asymptotic non degenerate distribution, exact distribution and moments. We also discuss the uniformly minimum variance unbiased estimate (UMVUE) of hazard rate function and MRLF. Finally, two numerical examples with simulated data and real data set, are presented to illustrate the proposed estimates.     

Semiparametric regression for the mean and rate functions of recurrent events

The counting process with the Cox-type intensity function has been commonly used to analyse recurrent event data. This model essentially assumes that the underlying counting process is a time-transformed Poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process. Recently, Pepe and Cai, and Lawless and co-workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the Poisson-type assumption. In this paper, we provide a rigorous justification of such robust procedures through modern empirical process theory. Furthermore, we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean–rate model. The advantages of the robust procedures are demonstrated through simulation studies. An illustration with multiple-infection data taken from a clinical study on chronic granulomatous disease is also provided.     

Self‐consistent estimation of mean response functions and their derivatives

Many methods have been developed for the nonparametric estimation of a mean response function, but most of these methods do not lend themselves to simultaneous estimation of the mean response function and its derivatives. Recovering derivatives is important for analyzing human growth data, studying physical systems described by differential equations, and characterizing nanoparticles from scattering data. In this article the authors propose a new compound estimator that synthesizes information from numerous pointwise estimators indexed by a discrete set. Unlike spline and kernel smooths, the compound estimator is infinitely differentiable; unlike local regression smooths, the compound estimator is self‐consistent in that its derivatives estimate the derivatives of the mean response function. The authors show that the compound estimator and its derivatives can attain essentially optimal convergence rates in consistency. The authors also provide a filtration and extrapolation enhancement for finite samples, and the authors assess the empirical performance of the compound estimator and its derivatives via a simulation study and an application to real data. The Canadian Journal of Statistics 39: 280–299; 2011 © 2011 Statistical Society of Canada     

Data-adaptive estimation of the treatment-specific mean

An important problem in epidemiology and medical research is the estimation of the causal effect of a treatment action at a single point in time on the mean of an outcome, possibly within strata of the target population defined by a subset of the baseline covariates. Current approaches to this problem are based on marginal structural models, i.e. parametric models for the marginal distribution of counterfactual outcomes as a function of treatment and effect modifiers. The various estimators developed in this context furthermore each depend on a high-dimensional nuisance parameter whose estimation currently also relies on parametric models. Since misspecification of any of these models can lead to severely biased estimates of causal effects, the dependence of current methods on such parametric models represents a major limitation.     

Interval estimation of the common mean

Brown and Cohen (1974) considered the problem of interval estimation of the common mean of two normal populations based on independent random samples. They showed that if we take the usual confidence interval using the first sample only and centre it around an appropriate combined estimate of the common mean the resulting interval would contain the true value with higher probability. They also gave a sufficient condition which such a point estimate should satisfy. Bhattacharya and Shah (1978) showed that the estimates satisfying this condition are nearly identical to the mean of the first sample. In this paper we obtain a stronger sufficient condition which is satisfied by many point estimates when the size of the second sample exceeds ten.     

On the estimation of restricted mean

In sampling from a continuous distribution with unknown mean μ and variance σ² the problem of estimation of μ, when it is known that μ∈(a, ∞) (or μ∈(-∞, b)), is considered. The estimators proposed here lie in the interval (a, ∞) (or (-∞, b)) almost surely. The performance of these estimators is compared to that of some known estimators in the case of sampling from a normal, exponential and a weighted difference of two independent chi-square distributions.     

Second order minimax estimation of the mean

In this study we consider the problem of the improvement of the sample mean in the second order minimax estimation sense for a mean belonging to an unrestricted mean parameter space R⁺${\mathbb{R}}^{+}$. We solve this problem for the class of natural exponential families (NEF's) whose variance functions (VF's) are regular at zero and at infinity. Such a class of VF's (or NEF's) is huge and contains (among others): Polynomial VF's (e.g., quadratic VF's in the Morris class, cubic VF's in the Letac&Mora class and VF's in the Hinde–Demétrio class); VF's belonging to the Tweedie class with power VF's, VF's belonging to the Babel class and many others. Moreover, we show that if the canonical parameter space of the corresponding NEF is R$\mathbb{R}$ (which is obviously the case if the support of the NEF is bounded), then the sample mean as an estimator of the mean cannot be further improved. This work presents an original constructive methodology and provides with constructive tools enabling to obtain explicit forms of the second order minimax estimators as well as the forms of the related weight functions. Our work establishes a substantial generalization of the results obtained so far in the literature. Illustrations of the resulting methods are provided and a simulation-based analysis is presented for the negative binomial case.     

Sequential estimation of the mean of NEF-PVF distributions

Let F = {F₀: 0 ϵ Θ} denote the class of natural exponential family of distributions having power variance function, (NEF-PVF). We consider the problem of sequentially estimating the mean μ of F₀ ϵ F, based on i.i.d. observations from F₀. We propose an appropriate sequential estimation procedure under a combined loss of estimation error and sampling cost. We provide expansion for the regret R_a and study its asymptotic properties. We show that R_a = cv²(μ) + o(1) as a → ∞, where c > 0 is a known constant and v(μ) denotes the coefficient of variation of F₀.     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

Maximum likelihood estimation of treatment effects for samples subject to cregression to the mean

A general maximum likelihood approach for estimating the effects of treatments applied to samples subject to regression to the mean is outlined. Models may be specified in terms of three factors: whether the treatment effect is multiplicative or additive, whether the treatment group is above or below some truncation point and the type of sample involved. The way in which solutions may be obtained for all 16 models so defined is described.     

Bayesian estimation of the mean of an autoregressive process

The marginal posterior probability density function (pdf) for the mean of a stationary pth order Gaussian autoregressive process is derived using the conditional likelihood function. While the posterior pdf provides a small sample analysis, the pdf is not well known and must be analyzed numerically. This is relatively easy since it is a function of only one variable. Two sets of examples are presented. The first set involves synthetic data generated by computer, and the second set deals with energy expenditure data on a bum patient.     

Improved estimation of the mean vector for student-t model

Improved James-Stein type estimation of the mean vector μ of a multovaroate Student-t population of dimension p with ν degrees of freedom is considered. In addition to the sample data, uncertain prior information on the value of the mean vector, in the form of a null hypothesis, is used for the estiamtion. The usual maximum liklihood estimator((mle) of μ is obtained and a test statistic for testing H₀:μ=μ₀ is derived. Based on the mle of μ and the tes statistic the preliminary test estimator (PTE), Stein-type shrinkage estimator (SE) and positive-rule shrinkage esiimator (PRSE) are defined. The bias and the quadratic risk of the estimators are evaiuated. The relative performances of the estimators are mvestigated by analyzing the risks under different condltlons It is observed that the FRSE dommates over he other three estimators, regardless of the vaiidity of the null hypothesis and the value ν.     

Bayes estimation of a langevin mean direction

A Langevin distribution with two parameters (mean direction and concentration parameter) has been extensively used for modeling and analyzing problems related to directional data. In this article, we examine the estimation problem for the mean direction. Bayes estimators are derived with respect to a conjugate as well as the Jeffreys’ priors. Further in case of unknown concentration parameter, other priors are also chosen. An extensive analysis of risk behavior of Bayes estimators is carried out with the help of simulations.     

Nonparametric estimation of survival functions for doubly censored observations when the lifetime hazard rate is proportionally related to the censoring hazard rates

Let T, X and Y be non-negative random variables, where T is the time of occurrence of an event of interest, X and Y being the lefl and right censoring variables respectively.

In this paper we propose a nonparametric estimator of the survival function, ST, when T, X and Y are supposed to be independent and their corresponding hazard rates are proportionally related. In this way, our results extend Ebrahimi's work (1985) to the doubly censored data case.     

Sequential estimation of the mean of a first-order autoregressive process

This paper considers the problem of sequential point estimation, under an appropriate loss function, of the location parameter when the errors form an autoregressive process with unknown scale and autoregressive parameters, A sequential procedure is developed and an asymptotic second order expansion is provided for the difference between expected stopping time and the optimal fixed sample size procedure. Also, the asymptotic normality of the stopping time is proved. Though the procedure Is asymptotically risk efficient, it. Is not clear whether it has bounded regret.