Inference and optimal censoring schemes for progressively censored Birnbaum–Saunders distribution

The aim of this paper is twofold. First we discuss the maximum likelihood estimators of the unknown parameters of a two-parameter Birnbaum–Saunders distribution when the data are progressively Type-II censored. The maximum likelihood estimators are obtained using the EM algorithm by exploiting the property that the Birnbaum–Saunders distribution can be expressed as an equal mixture of an inverse Gaussian distribution and its reciprocal. From the proposed EM algorithm, the observed information matrix can be obtained quite easily, which can be used to construct the asymptotic confidence intervals. We perform the analysis of two real and one simulated data sets for illustrative purposes, and the performances are quite satisfactory. We further propose the use of different criteria to compare two different sampling schemes, and then find the optimal sampling scheme for a given criterion. It is observed that finding the optimal censoring scheme is a discrete optimization problem, and it is quite a computer intensive process. We examine one sub-optimal censoring scheme by restricting the choice of censoring schemes to one-step censoring schemes as suggested by Balakrishnan (2007), which can be obtained quite easily. We compare the performances of the sub-optimal censoring schemes with the optimal ones, and observe that the loss of information is quite insignificant.     

Nonparametric (smoothed) likelihood and integral equations

We show that there is an intimate connection between the theory of nonparametric (smoothed) maximum likelihood estimators for certain inverse problems and integral equations. This is illustrated by estimators for interval censoring and deconvolution problems. We also discuss the asymptotic efficiency of the MLE for smooth functionals in these models.     

A bias-reduced estimator for the mean of a heavy-tailed distribution with an infinite second moment

We use bias-reduced estimators of high quantiles of heavy-tailed distributions, to introduce a new estimator for the mean in the case of infinite second moment. The asymptotic normality of the proposed estimator is established and checked in a simulation study, by four of the most popular goodness-of-fit tests. The accuracy of the resulting confidence intervals is evaluated as well. We also investigate the finite sample behavior and compare our estimator with some versions of Peng's estimator of the mean (namely those based on Hill, t-Hill and Huisman et al. extreme value index estimators). Moreover, we discuss the robustness of the tail index estimators used in this paper. Finally, our estimation procedure is applied to the well-known Danish fire insurance claims data set, to provide confidence bounds for the means of weekly and monthly maximum losses over a period of 10 years.     

Testing the Null Hypothesis of Nonstationary Long Memory Against the Alternative Hypothesis of a Nonlinear Ergodic Model

Interest in the interface of nonstationarity and nonlinearity has been increasing in the econometric literature. This paper provides a formal method of testing for nonstationary long memory against the alternative of a particular form of nonlinear ergodic processes; namely, exponential smooth transition autoregressive processes. In this regard, the current paper provides a significant generalization to existing unit root tests by allowing the null hypothesis to encompass a much larger class of nonstationary processes. The asymptotic theory associated with the proposed Wald statistic is derived, and Monte Carlo simulation results confirm that the Wald statistics have reasonably correct size and good power in small samples. In an application to real interest rates and the Yen real exchange rates, we find that the tests are able to distinguish between these competing processes in most cases, supporting the long-run Purchasing Power Parity (PPP) and Fisher hypotheses. But, there are a few cases in which long memory and nonlinear ergodic processes display similar characteristics and are thus confused with each other in small samples.     

Bayesian inference based on a jointly type-II censored sample from two exponential populations

In this paper, based on a jointly type-II censored sample from two exponential populations, the Bayesian inference for the two unknown parameters are developed with the use of squared-error, linear-exponential and general entropy loss functions. The problem of predicting the future failure times, both point and interval prediction, based on the observed joint type-II censored data, is also addressed from a Bayesian viewpoint. A Monte Carlo simulation study is conducted to compare the Bayesian estimators with the maximum likelihood estimator developed by Balakrishnan and Rasouli [Exact likelihood inference for two exponential populations under joint type-II censoring. Comput Stat Data Anal. 2008;52:2725–2738]. Finally, a numerical example is utilized for the purpose of illustration.     

A weighted quantile regression for left-truncated and right-censored data

Quantile regression methods have been used to estimate upper and lower quantile reference curves as the function of several covariates. In this article, it is demonstrated that the estimating equation of Zhou [A weighted quantile regression for randomly truncated data, Comput. Stat. Data Anal. 55 (2011), pp. 554–566.] can be extended to analyse left-truncated and right-censored data. We evaluate the finite sample performance of the proposed estimators through simulation studies. The proposed estimator β?(q) is applied to the Veteran's Administration lung cancer data reported by Prentice [Exponential survival with censoring and explanatory variables, Biometrika 60 (1973), pp. 279–288].     

Approximate maximum likelihood estimation of the mean and standard deviation of the normal distribution based on type ii censored samples

It is known that the maximum likelihood methods does not provide explicit estimators for the mean and standard deviation of the normal distribution based on Type II censored samples. In this paper we present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We obtain the variances and covariance of these estimators. We also show that these estimators are almost as eficient as the maximum likelihood (ML) estimators and just as eficient as the best linear unbiased (BLU), and the modified maximum likelihood (MML) estimators. Finally, we illustrate this method of estimation by applying it to Gupta's and Darwin's data.     

Exact inference on multiple exponential populations under a joint type-II progressive censoring scheme

ABSTRACT

Recently Mondal and Kundu [Mondal S, Kundu D. A new two sample type-II progressive censoring scheme. Commun Stat Theory Methods. 2018. doi:10.1080/03610926.2018.1472781] introduced a Type-II progressive censoring scheme for two populations. In this article, we extend the above scheme for more than two populations. The aim of this paper is to study the statistical inference under the multi-sample Type-II progressive censoring scheme, when the underlying distributions are exponential. We derive the maximum likelihood estimators (MLEs) of the unknown parameters when they exist and find out their exact distributions. The stochastic monotonicity of the MLEs has been established and this property can be used to construct exact confidence intervals of the parameters via pivoting the cumulative distribution functions of the MLEs. The distributional properties of the ordered failure times are also obtained. The Bayesian analysis of the unknown model parameters has been provided. The performances of the different methods have been examined by extensive Monte Carlo simulations. We analyse two data sets for illustrative purposes.     

Estimation of δ=P(X<Y) for Burr XII distribution based on the progressively first failure-censored samples

Let X and Y have two-parameter Burr XII distributions. The maximum-likelihood estimator of δ=P(X<Y) is studied under the progressively first failure-censored samples. Three confidence intervals of δ are constructed by using an asymptotic distribution of the maximum-likelihood estimator of δ and two bootstrapping procedures, respectively. Some computational results from intensive simulations are presented. An illustrative example is provided to demonstrate the application of the proposed method.