Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring

In this paper, we consider the maximum likelihood estimator (MLE) of the scale parameter of the generalized exponential (GE) distribution based on a random censoring model. We assume the censoring distribution also follows a GE distribution. Since the estimator does not provide an explicit solution, we propose a simple method of deriving an explicit estimator by approximating the likelihood function. In order to compare the performance of the estimators, Monte Carlo simulation is conducted. The results show that the MLE and the approximate MLE are almost identical in terms of bias and variance.     

Point and Interval Estimation for a Generalized Logistic Distribution Under Progressive Type II Censoring

The likelihood equations based on a progressively Type II censored sample from a Type I generalized logistic distribution do not provide explicit solutions for the location and scale parameters. We present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We examine numerically the bias and variance of these estimators and show that these estimators are as efficient as the maximum likelihood estimators (MLEs). The probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. Therefore we suggest using unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. A wide range of sample sizes and progressive censoring schemes have been considered in this study. Finally, we present a numerical example to illustrate the methods of inference developed here.     

A note on comparison and improvement of estimators based on likelihood

The problem of estimation of a parameter of interest in the presence of a nuisance parameter, which is either location or scale, is considered. Three estimators are taken into account: usual maximum likelihood (ML) estimator, maximum integrated likelihood estimator and the bias-corrected ML estimator. General results on comparison of these estimators w.r.t. the second-order risk based on the mean-squared error are obtained. Possible improvements of basic estimators via the notion of admissibility and methodology given in Ghosh and Sinha [A necessary and sufficient condition for second order admissibility with applications to Berkson's bioassay problem. Ann Stat. 1981;9(6):1334–1338] are considered. In the recent paper by Tanaka et al. [On improved estimation of a gamma shape parameter. Statistics. 2014; doi:10.1080/02331888.2014.915842], this problem was considered for estimating the shape parameter of gamma distribution. Here, we perform more accurate comparison of estimators for this case as well as for some other cases.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

Approximate MLEs for the location and scale parameters of the skew logistic distribution

Azzalini (Scand J Stat 12:171–178, 1985) provided a methodology to introduce skewness in a normal distribution. Using the same method of Azzalini (1985), the skew logistic distribution can be easily obtained by introducing skewness to the logistic distribution. For the skew logistic distribution, the likelihood equations do not provide explicit solutions for the location and scale parameters. We present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We examine numerically the bias and variance of these estimators and show that these estimators are as efficient as the maximum likelihood estimators (MLEs). The coverage probabilities of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. To improve the coverage probabilities and for constructing confidence intervals, we suggest the use of simulated percentage points. Finally, we present a numerical example to illustrate the methods of inference developed here.     

On the calculations of the maximum likelihood estimates for the polynomial spline regression model with unknown knots and AR(1) errors

Asymptotic distributions of the maximum likelihood estimators of the regression coefficients and knot points for the polynomial spline regression models with unknown knots and AR(1) errors have been derived by Chan (1989). Chan showed that under some mild conditions the maximum likelihood estimators, after suitable standardization, asymptotically follow normal distributions as n diverges to infinity. For the calculations of the maximum likelihood estimators, iterative methods must be applied. But this is not easy to implement for the model considered. In this paper, we suggested an alternative method to compute the estimates of the regression parameters and knots. It is shown that the estimates obtained by this method are asymptotically equivalent to the maximum likelihood estimates considered by Chan.     

Robust estimation and testing in one-way ANOVA for Type II censored samples: skew normal error terms

Censoring can be occurred in many statistical analyses in the framework of experimental design. In this study, we estimate the model parameters in one-way ANOVA under Type II censoring. We assume that the distribution of the error terms is Azzalini's skew normal. We use Tiku's modified maximum likelihood (MML) methodology which is a modified version of the well-known maximum likelihood (ML) in the estimation procedure. Unlike ML methodology, MML methodology is non-iterative and gives explicit estimators of the model parameters. We also propose new test statistics based on the proposed estimators. The performances of the proposed estimators and the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. A real life data is analysed to show the implementation of the methodology presented in this paper at the end of the study.     

On shrinkage estimation of the exponential location parameter

Under the assumption that the exponential distribution is a reasonable model for a given population, some shrinkage estimators for the location parameter based on type 1 and type II censored samples have been derived. It is shown that these estimators dominate maximum likelihood estimators (MLE's) asymptotically under the mean squared error (MSE) criterion. A Monte Carlo study shows a significant improvement of our estimators over MLE's in terms of MSE for small samples.     

Performance of nonparametric maximum likelihood estimator in a

When the probability of selecting an individual in a population is propor­tional to its lifelength, it is called length biased sampling. A nonparametric maximum likelihood estimator (NPMLE) of survival in a length biased sam­ple is given in Vardi (1982). In this study, we examine the performance of Vardi's NPMLE in estimating the true survival curve when observations are from a length biased sample. We also compute estimators based on a linear combination (LCE) of empirical distribution function (EDF) estimators and weighted estimators. In our simulations, we consider observations from a mix­ture of two different distributions, one from F and the other from G which is a length biased distribution of F. Through a series of simulations with vari­ous proportions of length biasing in a sample, we show that the NPMLE and the LCE closely approximate the true survival curve. Throughout the sur­vival curve, the EDF estimators overestimate the survival. We also consider a case where the observations are from three different weighted distributions, Again, both the NPMLE and the LCE closely approximate the true distribu­tion, indicating that the length biasedness is properly adjusted for. Finally, an efficiency study shows that Vardi's estimators are more efficient than the EDF estimators in the lower percentiles of the survival curves.     

Linear approximate ML estimation in scaled Type I generalized logistic distribution based on Type-II censored samples

The scaled (two-parameter) Type I generalized logistic distribution (GLD) is considered with the known shape parameter. The ML method does not yield an explicit estimator for the scale parameter even in complete samples. In this article, we therefore construct a new linear estimator for scale parameter, based on complete and doubly Type-II censored samples, by making linear approximations to the intractable terms of the likelihood equation using least-squares (LS) method, a new approach of linearization. We call this as linear approximate maximum likelihood estimator (LAMLE). We also construct LAMLE based on Taylor series method of linear approximation and found that this estimator is slightly biased than that based on the LS method. A Monte Carlo simulation is used to investigate the performance of LAMLE and found that it is almost as efficient as MLE, though biased than MLE. We also compare unbiased LAMLE with BLUE based on the exact variances of the estimators and interestingly this new unbiased LAMLE is found just as efficient as the BLUE in both complete and Type-II censored samples. Since MLE is known as asymptotically unbiased, in large samples we compare unbiased LAMLE with MLE and found that this estimator is almost as efficient as MLE. We have also discussed interval estimation of the scale parameter from complete and Type-II censored samples. Finally, we present some numerical examples to illustrate the construction of the new estimators developed here.     

Usefulness of robust estimators in sample survey

It Is illustrated that the adaptation of robust estimators, Tiku's (1967, 1978, 1980) modified maximum likelihood estimators for example, in sample survey can lead to considerable improvements In efficiencies in estimating the mean of a finite population (a random sample of Independent observations from a super-popularlon). In this paper, however, we confine our attention to symmetric super-populations     

Inference for the extreme value distribution under progressive Type-II censoring

The extreme value distribution has been extensively used to model natural phenomena such as rainfall and floods, and also in modeling lifetimes and material strengths. Maximum likelihood estimation (MLE) for the parameters of the extreme value distribution leads to likelihood equations that have to be solved numerically, even when the complete sample is available. In this paper, we discuss point and interval estimation based on progressively Type-II censored samples. Through an approximation in the likelihood equations, we obtain explicit estimators which are approximations to the MLEs. Using these approximate estimators as starting values, we obtain the MLEs using an iterative method and examine numerically their bias and mean squared error. The approximate estimators compare quite favorably to the MLEs in terms of both bias and efficiency. Results of the simulation study, however, show that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory for both these estimators and particularly so when the effective sample size is small. We, therefore, suggest the use of unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. The results are presented for a wide range of sample sizes and different progressive censoring schemes. We conclude with an illustrative example.     

The exponential–Weibull lifetime distribution

In this paper, we propose a new three-parameter model called the exponential–Weibull distribution, which includes as special models some widely known lifetime distributions. Some mathematical properties of the proposed distribution are investigated. We derive four explicit expressions for the generalized ordinary moments and a general formula for the incomplete moments based on infinite sums of Meijer's G functions. We also obtain explicit expressions for the generating function and mean deviations. We estimate the model parameters by maximum likelihood and determine the observed information matrix. Some simulations are run to assess the performance of the maximum likelihood estimators. The flexibility of the new distribution is illustrated by means of an application to real data.     

Maximum likelihood estimation of Laplace parameters based on general type-II censored examples

In this paper, we derive the maximum likelihood estimators of the parameters of a Laplace distribution based on general Type-II censored samples. The resulting explicit MLE's turn out to be simple linear functions of the order statistics. We then examine the asymptotic variance of the estimates by calculating the elements of the Fisher information matrix.     

Aalen's linear model for doubly censored data

Double censoring often occurs in registry studies when left censoring is present in addition to right censoring. In this work, we examine estimation of Aalen's nonparametric regression coefficients based on doubly censored data. We propose two estimation techniques. The first type of estimators, including ordinary least squared (OLS) estimator and weighted least squared (WLS) estimators, are obtained using martingale arguments. The second type of estimator, the maximum likelihood estimator (MLE), is obtained via expectation-maximization (EM) algorithms that treat the survival times of left censored observations as missing. Asymptotic properties, including the uniform consistency and weak convergence, are established for the MLE. Simulation results demonstrate that the MLE is more efficient than the OLS and WLS estimators.     

ON ESTIMATION OF LONG-MEMORY TIME SERIES MODELS

This paper discusses estimation associated with the long-memory time series models proposed by Granger & Joyeux (1980) and Hosking (1981). We consider the maximum likelihood estimator and the least squares estimator. Certain regularity conditions introduced by several authors to develop the asymptotic theory of these estimators do not hold in this model. However we can show that these estimators are strongly consistent, and we derive the limiting distribution and the rate of convergence.     

Improved estimation methods for unrelated question randomized response techniques

The randomized response technique (RRT) is an important tool, commonly used to avoid biased answers in survey on sensitive issues by preserving the respondents’ privacy. In this paper, we introduce a data collection method for survey on sensitive issues combining both the unrelated-question RRT and the direct question design. The direct questioning method is utilized to obtain responses to a non sensitive question that is related to the innocuous question from the unrelated-question RRT. These responses serve as additional information that can be used to improve the estimation of the prevalence of the sensitive behavior. Furthermore, we propose two new methods for the estimation of the proportion of respondents possessing the sensitive attribute under a missing data setup. More specifically, we develop the weighted estimator and the weighted conditional likelihood estimator. The performances of our estimators are studied numerically and compared with that of an existing one. Both proposed estimators are more efficient than the Greenberg's estimator. We illustrate our methods using real data from a survey study on illegal use of cable TV service in Taiwan.     

Inadmissibility of the uniformly minimum variance unbiased estimator of the inverse gaussian variance

The uniformly minimum variance unbiased estimator (UMVUE) of the variance of the inverse Gaussian distribution is shown to be inadmissible in terms of the mean squared error, and a dominating estimator is given. A dominating estimator to the maximum likelihood estimator (MLE) of the variance and estimators dominating the MLE's and the UMVUE's of other parameters are also given.     

Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes with jumps

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck processes with jumps based on continuous observations. We derive likelihood functions by using semimartingale theory. From this we get explicit formulas for estimators. The strong consistence and asymptotic normality of estimators are proved by using the method of stochastic integration.