Sometimes pool t estimation – via shrinkage

The performance of an estimator can be improved by incorporating some additional information(s) available besides the sample information. If two censored samples are available from the same exponential distribution, it is advantageous to pool the two samples for estimating the mean life. Further, incorporating guess information facilitates accuracy borrowing by shrinkage to a guess point or interval. Both the views have been taken into consideration in the present study. The present paper proposes an estimator for the mean life time of a two parameter exponential distribution, using conditional and/or guess information on it, when the two guarantees are equal but unknown. The bias, mean square error and relative efficiency of the proposed estimator have been studied. Some theoretical results have been derived. It is observed that the proposed testimator dominates the conventional estimator in certain range of life ratio, guess life ratio and shrinkage factor. Further, it is claimed that it always fares better than the preliminary test estimator for mean life proposed by Gupta and Singh (Microelectron. Reliab., 1985, 25, 881–887).     

Goodness-of-fit tests for additive mean residual life model under right censoring

The mean residual life (MRL) measures the remaining life expectancy and is useful in actuarial studies, biological experiments and clinical trials. To assess the covariate effect, an additive MRL regression model has been proposed in the literature. In this paper, we focus on the topic of model checking. Specifically, we develop two goodness-of-fit tests to test the additive MRL model assumption. We explore the large sample properties of the test statistics and show that both of them are based on asymptotic Gaussian processes so that resampling approaches can be applied to find the rejection regions. Simulation studies indicate that our methods work reasonably well for sample sizes ranging from 50 to 200. Two empirical data sets are analyzed to illustrate the approaches.     

Fitting and testing the Marshall–Olkin extended Weibull model with randomly censored data

The random censorship model (RCM) is commonly used in biomedical science for modeling life distributions. The popular non-parametric Kaplan–Meier estimator and some semiparametric models such as Cox proportional hazard models are extensively discussed in the literature. In this paper, we propose to fit the RCM with the assumption that the actual life distribution and the censoring distribution have a proportional odds relationship. The parametric model is defined using Marshall–Olkin's extended Weibull distribution. We utilize the maximum-likelihood procedure to estimate model parameters, the survival distribution, the mean residual life function, and the hazard rate as well. The proportional odds assumption is also justified by the newly proposed bootstrap Komogorov–Smirnov type goodness-of-fit test. A simulation study on the MLE of model parameters and the median survival time is carried out to assess the finite sample performance of the model. Finally, we implement the proposed model on two real-life data sets.     

A new measure of association for bivariate survival data

Time dependent association measures between variables are of interest in bivariate survival data. Several such measures have been proposed in literature for the modelling and analysis of survival data. In this paper, we introduce a new measure of association for bivariate survival data using product moment residual life function and mean residual life function. Various properties of the proposed measure and its relationship with existing measures are discussed. We also develop a non-parametric estimator of the measure and study its asymptotic properties. The application of the result is illustrated using a real life data. Finally, a stimulation study is carried out to assess the performance of the estimator.     

A Simple Approach to Inference in Random Coefficient Models

Random coefficient regression models have been used to analyze cross-sectional and longitudinal data in economics and growth-curve data from biological and agricultural experiments. In the literature several estimators, including the ordinary least squares and the estimated generalized least squares (EGLS), have been considered for estimating the parameters of the mean model. Based on the asymptotic properties of the EGLS estimators, test statistics have been proposed for testing linear hypotheses involving the parameters of the mean model. An alternative estimator, the simple mean of the individual regression coefficients, provides estimation and hypothesis-testing procedures that are simple to compute and teach. The large sample properties of this simple estimator are shown to be similar to that of the EGLS estimator. The performance of the proposed estimator is compared with that of the existing estimators by Monte Carlo simulation.     

Estimation of average causal effect using the restricted mean residual lifetime as effect measure

Although mean residual lifetime is often of interest in biomedical studies, restricted mean residual lifetime must be considered in order to accommodate censoring. Differences in the restricted mean residual lifetime can be used as an appropriate quantity for comparing different treatment groups with respect to their survival times. In observational studies where the factor of interest is not randomized, covariate adjustment is needed to take into account imbalances in confounding factors. In this article, we develop an estimator for the average causal treatment difference using the restricted mean residual lifetime as target parameter. We account for confounding factors using the Aalen additive hazards model. Large sample property of the proposed estimator is established and simulation studies are conducted in order to assess small sample performance of the resulting estimator. The method is also applied to an observational data set of patients after an acute myocardial infarction event.     

Testing discontinuities in nonparametric regression

In nonparametric regression, it is often needed to detect whether there are jump discontinuities in the mean function. In this paper, we revisit the difference-based method in [13] and propose to further improve it. To achieve the goal, we first reveal that their method is less efficient due to the inappropriate choice of the response variable in their linear regression model. We then propose a new regression model for estimating the residual variance and the total amount of discontinuities simultaneously. In both theory and simulation, we show that the proposed variance estimator has a smaller mean-squared error compared to the existing estimator, whereas the estimation efficiency for the total amount of discontinuities remains unchanged. Finally, we construct a new test procedure for detection of discontinuities using the proposed method; and via simulation studies, we demonstrate that our new test procedure outperforms the existing one in most settings.     

Estimation of variance in bivariate normal distribution after the preliminary test of homogeneity

A preliminary test estimator of variance in the bivariate normal distribution is proposed after the Pitman–Morgan test of homogeneity of two variances. The bias and mean square error of the estimator are derived. The relative efficiency (RE) of the preliminary test estimator is studied. Computations and 3D graphs of RE for different parameters are analyzed. In order to get the maximum RE, recommendations of the significance level for the preliminary test are given for various sample sizes by using the max–min criterion.     

Shrinkage estimation of reliability in the exponential distribution

In this paper we propose two shrinkage testimators for the reliability of the exponential distribution and study their properties. The optimum shrinkage coefficients for the shrinkage testimators are obtained based on a regret function and the minimax regret criterion. Shrinkage testimators are compared with a preliminary test estimator and with the usual estimator in terms of mean squared error. The proposed shrinkage testimators are shown to be preferable to the preliminary test estimator and the usual estimator when the prior value of mean life is close to the true mean life.     

A class of mean residual life regression models with censored survival data

When describing a failure time distribution, the mean residual life is sometimes preferred to the survival or hazard rate. Regression analysis making use of the mean residual life function has recently drawn a great deal of attention. In this paper, a class of mean residual life regression models are proposed for censored data, and estimation procedures and a goodness-of-fit test are developed. Both asymptotic and finite sample properties of the proposed estimators are established, and the proposed methods are applied to a cancer data set from a clinic trial.     

Asymptotic relative efficiency of wald tests in measurement error models

In this paper, asymptotic relative efficiency (ARE) of Wald tests for the Tweedie class of models with log-linear mean, is considered when the aux¬iliary variable is measured with error. Wald test statistics based on the naive maximum likelihood estimator and on a consistent estimator which is obtained by using Nakarnura's (1990) corrected score function approach are defined. As shown analytically, the Wald statistics based on the naive and corrected score function estimators are asymptotically equivalents in terms of ARE. On the other hand, the asymptotic relative efficiency of the naive and corrected Wald statistic with respect to the Wald statistic based on the true covariate equals to the square of the correlation between the unobserved and the observed co-variate. A small scale numerical Monte Carlo study and an example illustrate the small sample size situation.     

A family of test statistics for trend change in mean residual life with unknown turning point using censored data

In this paper, we develop a family of test statistics for testing exponentiality against increasing then decreasing mean residual life alternatives to accommodate the randomly censored data when neither the change point nor the proportion is known. We establish the asymptotic null distribution of test statistics. Monte Carlo simulations are conducted to investigate the speed of convergence of test statistics to the asymptotic null distribution and study the performance of test statistics by the power of tests.     

Tests of fit for the Laplace distribution based on correcting moments of entropy estimators

ABSTRACT

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing test statistics against various alternatives. Performance of the proposed new test statistics is illustrated in real cases.     

A generalized quantile estimator

An alternative to the conventional sample quantlle Is proposed as a nonparametrlc estimator of a continuous population quantlle.The alternative estimator Is a "generalized sample quantlle" obtained by averaging an appropriate subsample quantlle over all subsamples of .a fixed size.Since the resulting statistic is a U-statistic with representation also as a linear combination of order statistics, known results are employed then to establish asymptotic normality.The alternative estimator is shown to be asymptotically efficient in the class of nonparametrlc models specified by Pfanzagl (1975).Analytic results and Monte Carlo studies with a moderate sample size for a variety of distributions Indicate that the proposed estimator usually provides mean square error of estimation less than that of the conventional sample quantile.     

Semiparametric regression for restricted mean residual life under right censoring

A mean residual life function (MRLF) is the remaining life expectancy of a subject who has survived to a certain time point. In the presence of covariates, regression models are needed to study the association between the MRLFs and covariates. If the survival time tends to be too long or the tail is not observed, the restricted mean residual life must be considered. In this paper, we propose the proportional restricted mean residual life model for fitting survival data under right censoring. For inference on the model parameters, martingale estimating equations are developed, and the asymptotic properties of the proposed estimators are established. In addition, a class of goodness-of-fit test is presented to assess the adequacy of the model. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and the approach is applied to a set of real life data collected from a randomized clinical trial.     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.     

Pooling reliability functions

In this article large sample pooling procedures for reliability functions of an exponential life testing model is considered. Asymptotic properties of shrinkage estimation procedure subsequent to preliminary tests are developed. It is shown that the proposed estimator possesses substantially snakker asymptotic mean squared error than the usual estimator in a region of the lparameter space. Relative efficiencies of the purposed estimators to the usual estimators are obtained and recommendations of the level of the preliminary tests are provided. Relative dominance picture of the estimators is presented. It is shown that the proposed estimator provides a wider dominance range over usual estimator than the usual preliminary test estimator. More importantly, the size of the preliminary test is meaningful. Simulation studies is also carried out to appraise the performance of the estimators when samples are small.     

Improved estimation of population mean using known median of auxiliary variable

In this article, we consider the problem of estimation of population mean using the known median of auxiliary variable. We proposed an estimator and its efficiency is studied analytically as well as empirically for different conditions. The proposed estimator is found to be more efficient than traditional estimators such as sample mean and linear regression estimator.