Comparison of renewal function estimators for censored data

The computation of the renewal function when the distribution function is completely known has received much attention in the literature. However, in many cases the form of the distribution function is unknown and has to be estimated nonparametrically. A nonparametric estimator for the renewal function for complete data was suggested by Frees (1986). In many cases, however, censoring of the lifetime might occur. We shall present parametric and nonparametric estimators of the renewal function based on censored data. In a simulation study we compare the nonparametric estimators with parametric estimators for the Weibull and lognormal distribution. The study suggests that the nonparametric estimator is a viable alternative to the parametric estimators when the lifetime distribution is unknown. Also, the nonparametric estimator is computationally simpler than the parametric estimator.     

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.     

Approximate Bayesian analysis of doubly censored samples from mixture of two Weibull distributions

The purpose of the paper is to estimate the parameters of the two-component mixture of Weibull distribution under doubly censored samples using Bayesian approach. The choice of Weibull distribution is made due to its (i) capability to model failure time data from engineering, medical and biological sciences (ii) added advantages over the well-known lifetime distributions such as exponential, Raleigh, lognormal and gamma distribution in terms of flexibility, increasing and decreasing hazard rate and closed-form distribution function and hazard rate. The proposed two-component mixture of Weibull distribution is even more flexible than its conventional form. However, the estimation of the parameters from the proposed mixture is more complex. Further, we have assumed couple of loss functions under non informative prior for the Bayesian analysis of the parameters from the mixture model. As the resultant Bayes estimators and associated posterior risks cannot be derived in the closed form, we have used the importance sampling and Lindley’s approximation to obtain the approximate estimates for the parameters of the mixture model. The comparison between the performances of approximation techniques has been made on the basis of simulation study and real-life data analysis. The importance sampling is found to be better than Lindley’s approximation as it gives better estimation for shape and mixing parameters of the mixture model and computations under this technique are much easier/shorter than those under Lindley’s approximation.     

A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.

     

A note on the asymptotic relative efficiency of estimators from the additive hazards model

In this note, the asymptotic variance formulas are explicitly derived and compared between the parametric and semiparametric estimators of a regression parameter and survival probability under the additive hazards model. To obtain explicit formulas, it is assumed that the covariate term including a regression coefficient follows a gamma distribution and the baseline hazard function is constant. The results show that the semiparametric estimator of the regression coefficient parameter is fully efficient relative to the parametric counterpart when the survival time and a covariate are independent, as in the proportional hazards model. Relative to a more realistic case of the parametric additive hazards model with a Weibull baseline, the loss of efficiency of the semiparametric estimator of survival probability is moderate.     

Divergence based robust estimation of the tail index through an exponential regression model

The extreme value theory is very popular in applied sciences including finance, economics, hydrology and many other disciplines. In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction characterized by its tail index; there are three broad classes of tails—the Pareto type, the Weibull type and the Gumbel type. The simplest and most common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has a high bias; it is also highly non-robust in presence of outliers with respect to the assumed model. There have been some recent attempts to produce asymptotically unbiased or robust alternative to the Hill estimator; however all the robust alternatives work for any one type of tail. This paper proposes a new general estimator of the tail index that is both robust and has smaller bias under all the three tail types compared to the existing robust estimators. This essentially produces a robust generalization of the estimator proposed by Matthys and Beirlant (Stat Sin 13:853–880, 2003) under the same model approximation through a suitable exponential regression framework using the density power divergence. The robustness properties of the estimator are derived in the paper along with an extensive simulation study. A method for bias correction is also proposed with application to some real data examples.     

Weibull Regression for Lifetimes Measured with Error

Models are considered in which true lifetimes are generated by a Weibull regression model and measured lifetimes are determined from the true times by certain measurement error models. Adjusted estimators are obtained under one parametric specification. The bias properties of these estimators and standard estimators are compared both theoretically, using small measurement error asymptotics, and by simulation. The standard estimators of regression coefficients, other than the intercept, are bias-robust. The adjusted estimator of the shape parameter removes the bias of the standard estimator.     

Bivariate Weibull regression model based on censored samples

The most natural parametric distribution to consider is the Weibull model because it allows for both the proportional hazard model and accelerated failure time model. In this paper, we propose a new bivariate Weibull regression model based on censored samples with common covariates. There are some interesting biometrical applications which motivate to study bivariate Weibull regression model in this particular situation. We obtain maximum likelihood estimators for the parameters in the model and test the significance of the regression parameters in the model. We present a simulation study based on 1000 samples and also obtain the power of the test statistics.     

Robust Mean Estimation Under a Possibly Incorrect Log-Normality Assumption

Nonparametric and parametric estimators are combined to minimize the mean squared error among their linear combinations. The combined estimator is consistent and for large sample sizes has a smaller mean squared error than the nonparametric estimator when the parametric assumption is violated. If the parametric assumption holds, the combined estimator has a smaller MSE than the parametric estimator. Our simulation examples focus on mean estimation when data may follow a lognormal distribution, or can be a mixture with an exponential or a uniform distribution. Motivating examples illustrate possible application areas.     

Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

Bayesian estimation and prediction with multiply Type-II censored samples of sequential order statistics from one- and two-parameter exponential distributions

Based on multiply Type-II censored samples of sequential order statistics, Bayesian estimators are derived for the parameters of one- and two-parameter exponential distributions. In the one-parameter set-up, the posterior density is obtained under the assumption that the prior distribution is given by an inverse Gamma distribution, and the Bayes estimator with respect to squared error loss is calculated. Its performance is illustrated by a numerical example and compared with two non-Bayesian estimators, namely the BLUE and the approximate maximum likelihood estimator (AMLE). Moreover, prediction of future failure times is considered. Minimum risk equivariant estimators and predictors are deduced from the given results. Finally, similar results are presented for the two-parameter situation.     

Performance of coefficient of variation estimators in ranked set sampling

In this paper, we propose and evaluate the performance of different parametric and nonparametric estimators for the population coefficient of variation considering Ranked Set Sampling (RSS) under normal distribution. The performance of the proposed estimators was assessed based on the bias and relative efficiency provided by a Monte Carlo simulation study. An application in anthropometric measurements data from a human population is also presented. The results showed that the proposed estimators via RSS present an expressively lower mean squared error when compared to the usual estimator, obtained via Simple Random Sampling. Also, it was verified the superiority of the maximum likelihood estimator, given the necessary assumptions of normality and perfect ranking are met.     

On a Correction of the Scale MLE for a Two-Parameter Exponential Distribution Under Progressive Type-I Censoring

In this article, we present a corrected version of the maximum likelihood estimator (MLE) of the scale parameter with progressively Type-I censored data from a two-parameter exponential distribution. Furthermore, we propose a bias correction of both the location and scale MLE. The properties of the estimates are analyzed by a simulation study which also illustrates the effect of the correction. Moreover, the presented estimators are applied to two data sets. Finally, it is shown that the correction of the scale estimator is also necessary for other distributions with a finite left endpoint of support (e.g., three-parameter Weibull distributions).     

An improper form of Weibull distribution for competing risks analysis with Bayesian approach

ABSTRACT

In survival analysis, individuals may fail due to multiple causes of failure called competing risks setting. Parametric models such as Weibull model are not improper that ignore the assumption of multiple failure times. In this study, a novel extension of Weibull distribution is proposed which is improper and then can incorporate to the competing risks framework. This model includes the original Weibull model before a pre-specified time point and an exponential form for the tail of the time axis. A Bayesian approach is used for parameter estimation. A simulation study is performed to evaluate the proposed model. The conducted simulation study showed identifiability and appropriate convergence of the proposed model. The proposed model and the 3-parameter Gompertz model, another improper parametric distribution, are fitted to the acute lymphoblastic leukemia dataset.     

Estimation of a scale parameter in mixture models with unknown location

Estimation of the scale parameter in mixture models with unknown location is considered under Stein's loss. Under certain conditions, the inadmissibility of the “usual” estimator is established by exhibiting better estimators. In addition, robust improvements are found for a specified submodel of the original model. The results are applied to mixtures of normal distributions and mixtures of exponential distributions. Improved estimators of the variance of a normal distribution are shown to be robust under any scale mixture of normals having variance greater than the variance of that normal distribution. In particular, Stein's (Ann. Inst. Statist. Math. 16 (1964) 155) and Brewster's and Zidek's (Ann. Statist. 2 (1974) 21) estimators obtained under the normal model are robust under the t model, for arbitrary degrees of freedom, and under the double-exponential model. Improved estimators for the variance of a t distribution with unknown and arbitrary degrees of freedom are also given. In addition, improved estimators for the scale parameter of the multivariate Lomax distribution (which arises as a certain mixture of exponential distributions) are derived and the robustness of Zidek's (Ann. Statist. 1 (1973) 264) and Brewster's (Ann. Statist. 2 (1974) 553) estimators of the scale parameter of an exponential distribution is established under a class of modified Lomax distributions.     

Estimating the distribution function of symmetric pairs

We discuss three classes of bivariate symmetry models and study the estimation of their distribution functions (DFs). Under radial symmetry, an estimator based on the mean of the empirical and survival DFs is considered. For exchangeable symmetry, an estimator based on the mean of the empirical DF and its exchangeable image is presented. At their intersection, we define radial exchangeability and study estimation of its DF. The symmetrized estimators coincide with the non parametric maximum likelihood estimators of the DF under each model. We obtain their mean and variance and state their asymptotic normality. The relative efficiency of the estimators for the bivariate normal distribution is obtained.     

A comparison of dependence function estimators in multivariate extremes

Various nonparametric and parametric estimators of extremal dependence have been proposed in the literature. Nonparametric methods commonly suffer from the curse of dimensionality and have been mostly implemented in extreme-value studies up to three dimensions, whereas parametric models can tackle higher-dimensional settings. In this paper, we assess, through a vast and systematic simulation study, the performance of classical and recently proposed estimators in multivariate settings. In particular, we first investigate the performance of nonparametric methods and then compare them with classical parametric approaches under symmetric and asymmetric dependence structures within the commonly used logistic family. We also explore two different ways to make nonparametric estimators satisfy the necessary dependence function shape constraints, finding a general improvement in estimator performance either (i) by substituting the estimator with its greatest convex minorant, developing a computational tool to implement this method for dimensions \(D\ge 2\) or (ii) by projecting the estimator onto a subspace of dependence functions satisfying such constraints and taking advantage of Bernstein–Bézier polynomials. Implementing the convex minorant method leads to better estimator performance as the dimensionality increases.     

Parameter estimation in an alternating repair model

The Weibull intensity is often used in survival analysis of technical products, it is an extremely important intensity to characterize the probabilistic behavior of a large number of real phenomena. This intensity is especially useful as a failure model analyzing the reliability of different types of systems. In this paper we develop statistical methods for an alternating repair model using Weibull intensity. The maximum likelihood estimator is considered for determining the estimations of the model parameters. The distribution of the life times after perfect repairs and imperfect repairs are obtained. The estimation of the Fisher information matrix is given. Simultaneous confidence regions based on the likelihood ratio statistics are developed for the estimators of the shape and scale parameter of the Weibull intensity. Finally simulation study will be given.     

Local Box–Cox transformation on time-varying parametric models for smoothing estimation of conditional CDF with longitudinal data

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation approaches without any structural assumptions may lead to inadequate and numerically unstable estimators in practice. We propose in this paper a nonparametric approach based on time-varying parametric models for estimating the conditional distribution functions with a longitudinal sample. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model after local Box–Cox transformation. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through a simulation study. Application and simulation results show that smoothing estimation from time-variant parametric models outperforms the existing kernel smoothing estimator by producing narrower pointwise bootstrap confidence band and smaller root mean squared error.     

Empirical bayes shrinkage estimation of reliability in the exponential distribution

In this paper we propose two empirical Bayes shrinkage estimators for the reliability of the exponential distribution and study their properties. Under the uniform prior distribution and the inverted gamma prior distribution these estimators are developed and compared with a preliminary test estimator and with a shrinkage testimator in terms of mean squared error. The proposed empirical Bayes shrinkage estimator under the inverted gamma prior distribution is shown to be preferable to the preliminary test estimator and the shrinkage testimator when the prior value of mean life is clsoe to the true mean life.