A Parametric Test of Perfect Ranking in Balanced Ranked Set Sampling

Many techniques based on data which are drawn by Ranked Set Sampling (RSS) scheme assume that the ranking of observations is perfect. Therefore it is essential to develop some methods for testing this assumption. In this article, we propose a parametric location-scale free test for assessing the assumption of perfect ranking. The results of a simulation study in two special cases of normal and exponential distributions indicate that the proposed test performs well in comparison with its leading competitors.     

MLE of parameters of location-scale distribution for complete and partially grouped data

This article studies the MLEs of parameters of location-scale distribution functions. It gives the necessary and sufficient conditions under which the MLEs of the location and scale parameters uniquely exist with completely grouped data. The results for the completely grouped data further imply that the Pearson–Fisher test is applicable to location-scale families. It also gives sufficient conditions under which the MLEs of the two parameters uniquely exist with partially grouped data. Here, the partially grouped data include complete data, Type-I censored data and others as special cases.     

Ordered partially ordered judgment subset sampling with applications to parametric inference

ABSTRACT

In this paper, we use the idea of order statistics from independent and non-identically distributed random variables to propose ordered partially ordered judgment subset sampling (OPOJSS) and then develop optimal linear parametric inferences. The best linear unbiased and invariant estimators of the location and scale parameters of a location-scale family are developed based on OPOJSS. It is shown that, despite the presence or absence of ranking errors, the proposed estimators with OPOJSS are uniformly better than the existing estimators with simple random sampling (SRS), ranked set sampling (RSS), ordered RSS (ORSS) and partially ordered judgment subset sampling (POJSS). Moreover, we also derive the best linear unbiased estimators (BLUEs) of the unknown parameters of the simple linear regression model with replicated observations using POJSS and OPOJSS. It is found that the BLUEs with OPOJSS are more precise than the BLUEs based on SRS, RSS, ORSS and POJSS.     

Location-scale mixed models and goodness-of-fit assessment applied to insect ecology

Survival models have been extensively used to analyse time-until-event data. There is a range of extended models that incorporate different aspects, such as overdispersion/frailty, mixtures, and flexible response functions through semi-parametric models. In this work, we show how a useful tool to assess goodness-of-fit, the half-normal plot of residuals with a simulated envelope, implemented in the hnp package in R, can be used on a location-scale modelling context. We fitted a range of survival models to time-until-event data, where the event was an insect predator attacking a larva in a biological control experiment. We started with the Weibull model and then fitted the exponentiated-Weibull location-scale model with regressors both for the location and scale parameters. We performed variable selection for each model and, by producing half-normal plots with simulated envelopes for the deviance residuals of the model fits, we found that the exponentiated-Weibull fitted the data better. We then included a random effect in the exponentiated-Weibull model to accommodate correlated observations. Finally, we discuss possible implications of the results found in the case study.     

ℒ1-Limit of Trimmed Sums of Order Statistics from Location-Scale Distributions with Applications to Type II Censored Data Analysis

We derive the ?¹-limit of trimmed sums of order statistics from location-scale distributions satisfying certain assumptions. Based on this limit, an approximation to the asymptotic variance of a Best-Asymptotic-Normal (BAN) estimator for the location parameter is developed. Associated formulae are derived for four location-scale distributions commonly used in lifetime data analysis. The approximation is analyzed via the properties of the approximating function and by comparison to the exact values for a special case. Applications are illustrated by applying the approximation to comparing location parameters and to selecting the population with the largest location parameter, using censored samples from location-scale populations.     

Some remarks on conditional and unconditional inference for location-scale models

Conditional and unconditional confidence intervals have been compared by Grice, Bain, and Engelhardt (Commun. Statist. B7 (1978), 515–524) in terms of the location-scale model with double-exponential distribution form. Preference was found for the conditional intervals based on mean length and coverage probability for untrue parameters values. These two criteria for a location-scale system are shown to be inappropriate criteria for assessing the conditional versus unconditional approaches to inference. The usual ancillarity concept is also noted to be inappropriate. Support for many conditional analyses, however, is found in a more careful formulation of the statistical model.     

Selection of error probability laws by generalized modified profile likelihood

Although error probability law selection of models of location-scale forms is of importance in some sense, the commonly used model selection procedures, such as AIC and BIC, do not apply to it. By treating error probability law as a “parameter” of interest, location and scale as nuisance parameters, this paper proposes that generalized modified profile likelihood (GMPL), considered as a quasi-likelihood function of error probability law, be used to select the error probability laws. The GMPL method achieves minimax rate optimality and proves to be consistent. Simulations show its good performance for finite and even small samples. Note that it is straightforward to generalize the GMPL of location-scale models to various models of location-scale forms particularly including the various linear regression models and their variations, to select their error probability laws. The author believes that GMPL and its variations would be quite promising for various model selection problems.     

“位置—尺度”分布族的近似构建——以收入分布函数序列为例

“位置-尺度”分布族具有数学表述的便利性和实证分析的易解释性,但从数据出发,验证“位置-尺度”分布族是否成立存在困难,因此提出“位置-尺度”分布族的近似构建,在估计分布函数统一形式的基础上,利用最小二乘法,得到位置和尺度参数的估计;以收入分布函数为例,将该构建方法用于2001-2010年中国城镇居民收入分布函数序列分布族的估计和预测,以验证中国城镇居民收入分布近似构成“位置-尺度”分布族.     

Random number generation and estimation with the bimodal asymmetric power-normal distribution

Recently, Bolfarine et al. [Bimodal symmetric-asymmetric power-normal families. Commun Statist Theory Methods. Forthcoming. doi:10.1080/03610926.2013.765475] introduced a bimodal asymmetric model having the normal and skew normal as special cases. Here, we prove a stochastic representation for their bimodal asymmetric model and use it to generate random numbers from that model. It is shown how the resulting algorithm can be seen as an improvement over the rejection method. We also discuss practical and numerical aspects regarding the estimation of the model parameters by maximum likelihood under simple random sampling. We show that a unique stationary point of the likelihood equations exists except when all observations have the same sign. However, the location-scale extension of the model usually presents two or more roots and this fact is illustrated here. The standard maximization routines available in the R system (Broyden–Fletcher–Goldfarb–Shanno (BFGS), Trust, Nelder–Mead) were considered in our implementations but exhibited similar performance. We show the usefulness of inspecting profile loglikelihoods as a method to obtain starting values for maximization and illustrate data analysis with the location-scale model in the presence of multiple roots. A simple Bayesian model is discussed in the context of a data set which presents a flat likelihood in the direction of the skewness parameter.     

Order restricted statistical inference for scale parameters based on sequential order statistics

In the situation of a multi-sample experiment consisting of differently equipped sequential k-out-of-n systems, scale parameters of underlying distributions from a general location-scale family of distributions are estimated under an order restriction. In each sample, the case of missing the smallest observations is included. Moreover, based on a profile likelihood a homogeneity test against an ordered alternative is proposed and analyzed. This work extends an approach of Bhattacharya [2007. Testing for ordered failure rates under general progressive censoring. J. Statist. Plann. Inference 137, 1775–1786] in the progressive Type-II censoring framework.     

On estimation and diagnostics analysis in log-generalized gamma regression model for interval-censored data

The interval-censored survival data appear very frequently, where the event of interest is not observed exactly but it is only known to occur within some time interval. In this paper, we propose a location-scale regression model based on the log-generalized gamma distribution for modelling interval-censored data. We shall be concerned only with parametric forms. The proposed model for interval-censored data represents a parametric family of models that has, as special submodels, other regression models which are broadly used in lifetime data analysis. Assuming interval-censored data, we consider a frequentist analysis, a Jackknife estimator and a non-parametric bootstrap for the model parameters. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some techniques to perform global influence.     

Using auxiliary data for parameter estimation with non-ignorably missing outcomes

We propose a method for estimating parameters in generalized linear models when the outcome variable is missing for some subjects and the missing data mechanism is non-ignorable. We assume throughout that the covariates are fully observed. One possible method for estimating the parameters is maximum likelihood with a non-ignorable missing data model. However, caution must be used when fitting non-ignorable missing data models because certain parameters may be inestimable for some models. Instead of fitting a non-ignorable model, we propose the use of auxiliary information in a likelihood approach to reduce the bias, without having to specify a non-ignorable model. The method is applied to a mental health study.     

ANALYSIS OF VARIANCE IN EXPERIMENTAL DESIGN WITH NONNORMAL ERROR DISTRIBUTIONS

We consider a two-way classification model with interaction and assume that the errors have a location-scale nonnormal distribution. From an application of the modified likelihood estimation, we obtain efficient and robust estimators of the parameters. We define F statistics for testing main effects and interaction. We analyze the Box-Cox data and show that the method developed in this paper gives accurate results besides being easy theoretically and computationally.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

A bivariate regression model with cure fraction

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we introduce a location-scale model for bivariate survival times based on the copula to model the dependence of bivariate survival data with cure fraction. We create the correlation structure between the failure times using the Clayton family of copulas, which is assumed to have any distribution. It turns out that the model becomes very flexible with respect to the choice of the marginal distributions. For the proposed model, we consider inferential procedures based on constrained parameters under maximum likelihood. We derive the appropriate matrices for assessing local influence under different perturbation schemes and present some ways to perform global influence analysis. The relevance of the approach is illustrated using a real data set and a diagnostic analysis is performed to select an appropriate model.     

Estimation and prediction in the presence of an outlier under Type-II censoring

A single-outlier data set containing some independent random variables is considered such that all of observations expect one have the same distribution. To describe the model of interested, a location-scale family of distributions is used and the estimation problem of the parameters is studied when the data are collected under Type-II censoring scheme. Moreover, three different predictors are presented to predict the censored order statistics. They are also compared regarding both of mean squared prediction error and Pitman's measure of closeness criteria. The role of outlier parameter as well as censorship rate is studied on performance of proposed estimator and predictors. The results of the paper are illustrated via a real data set. Finally, some conclusions are stated.     

Empirical likelihood method for non-ignorable missing data problems

Missing response problem is ubiquitous in survey sampling, medical, social science and epidemiology studies. It is well known that non-ignorable missing is the most difficult missing data problem where the missing of a response depends on its own value. In statistical literature, unlike the ignorable missing data problem, not many papers on non-ignorable missing data are available except for the full parametric model based approach. In this paper we study a semiparametric model for non-ignorable missing data in which the missing probability is known up to some parameters, but the underlying distributions are not specified. By employing Owen (1988)’s empirical likelihood method we can obtain the constrained maximum empirical likelihood estimators of the parameters in the missing probability and the mean response which are shown to be asymptotically normal. Moreover the likelihood ratio statistic can be used to test whether the missing of the responses is non-ignorable or completely at random. The theoretical results are confirmed by a simulation study. As an illustration, the analysis of a real AIDS trial data shows that the missing of CD4 counts around two years are non-ignorable and the sample mean based on observed data only is biased.     

Best linear unbiased and invariant estimation in location-scale families based on double-ranked set sampling

Abstract

In this article, we propose the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the unknown parameters of location-scale family of distributions based on double-ranked set sampling (DRSS) using perfect and imperfect rankings. These estimators are then compared with the BLUEs and BLIEs based on ranked set sampling (RSS). It is shown that under perfect ranking, the proposed estimators are uniformly better than the BLUEs and BLIEs obtained via RSS. We also propose the best linear unbiased quantile (BLUQ) and the best linear invariant quantile (BLIQ) estimators for normal distribution under DRSS. It is observed that the proposed quantile estimators are more efficient than the BLUQ and BLIQ estimators based on RSS for both perfect and imperfect orderings.     

On prediction from the location-scale model with equi correlated response

The location-scale model with equi-correlated responses is discussed. The structure of the location-scale model is utilised to genera-te the prediction distribution of a future response and that of a set of future responses. The method avoids the integration procedures usually involved in derivation of prediction distributions and yields results same as those obtained by the Bayes method with the vague prior distribution* Finally the re-suits have been specialised to cover the case of the normal intra-class model.     

Bayesian Tobit quantile regression with single-index models

Based on the Bayesian framework of utilizing a Gaussian prior for the univariate nonparametric link function and an asymmetric Laplace distribution (ALD) for the residuals, we develop a Bayesian treatment for the Tobit quantile single-index regression model (TQSIM). With the location-scale mixture representation of the ALD, the posterior inferences of the latent variables and other parameters are achieved via the Markov Chain Monte Carlo computation method. TQSIM broadens the scope of applicability of the Tobit models by accommodating nonlinearity in the data. The proposed method is illustrated by two simulation examples and a labour supply dataset.