Bias Correction in the Type I Generalized Logistic Distribution

Four strategies for bias correction of the maximum likelihood estimator of the parameters in the Type I generalized logistic distribution are studied. First, we consider an analytic bias-corrected estimator, which is obtained by deriving an analytic expression for the bias to order n ^?1; second, a method based on modifying the likelihood equations; third, we consider the jackknife bias-corrected estimator; and fourth, we consider two bootstrap bias-corrected estimators. All bias correction estimators are compared by simulation. Finally, an example with a real data set is also presented.     

A Class of s–K Type Principal Components Estimators in the Linear Model

In this article, we introduce a new class of estimators called the s–K type principal components estimators to combat multicollinearity, which include the principal components regression (PCR) estimator, the r–k estimator and the s–K estimator as special cases. Necessary and sufficient conditions for the superiority of the new estimator over the PCR estimator, the r–k estimator and the s–K estimator are derived in the sense of the mean squared error matrix criterion. A Monte Carlo simulation study and a numerical example are given to illustrate the performance of the proposed estimator.     

Inference of R = P[X < Y] for skew normal distribution

This article studies the estimation of R = P[X < Y] when X and Y are two independent skew normal distribution with different parameters. When the scale parameter is unknown, the maximum likelihood estimator of R is proposed. The maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimation, and confidence interval of R are obtained when the common scale parameter is known. In the general case, the maximum likelihood estimator of R is also discussed. To compare the different proposed methods, Monte Carlo simulations are performed. At last, the analysis of a real dataset has been presented for illustrative purposes too.     

Umvu estimation for the inverse gaussian distribution I(μ,cμ2) with known c

The uniformly minimum variance unbiased estimator and the maximum likelihood estimator of μ for the inverse Gaussian distribution I(μc,μ ) with known c are constructed, and they are shown to be asymptoti- cally equivalent.     

Estimation of P(Y < X) for Weibull Distribution Under Progressive Type-II Censoring

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.     

Comparative Study of Blu and ML Estimators of Location and Scale Parameters of an Extreme Value Distribution for Small Samples and Type I Censorship

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

Pseudo‐R2 statistics under complex sampling

Model summaries based on the ratio of fitted and null likelihoods have been proposed for generalised linear models, reducing to the familiar R² coefficient of determination in the Gaussian model with identity link. In this note I show how to define the Cox–Snell and Nagelkerke summaries under arbitrary probability sampling designs, giving a design‐consistent estimator of the population model summary. It is also shown that for logistic regression models under case–control sampling the usual Cox–Snell and Nagelkerke R² are not design‐consistent, but are systematically larger than would be obtained with a cross‐sectional or cohort sample from the same population, even in settings where the weighted and unweighted logistic regression estimators are similar or identical. Implementation of the new estimators is straightforward and code is provided in R.     

Robust variance estimators for fixed-effect estimates with hierarchical-likelihood

In longitudinal studies, robust sandwich variance estimators are often used, and are especially useful when model assumptions are in doubt. However, the usual sandwich estimator does not allow for models with crossed random effects. The hierarchical likelihood extends the idea of the sandwich estimator to models not currently covered. By simulation studies, we show that the new sandwich estimator is robust against heteroscedastic errors and against misspecification of overdispersion in the y | v component.     

Pooled estimators for the shape parameter of the three parameter gamma distribution

The aim of the paper is to study the pooled estimator of the shape parameter of the three parameter gamma distribution when k independent samples are available. Sufficient conditions for the existence of the pooled estimator are given and the small as well as the large sample properties are studied. The harmonic mean of the k estimators of the independent samples is proposed in the place of the pooled estimator, in the case in which the latter does not exist.     

A hypothesis testing procedure for the evaluation on the lifetime performance index of products with Burr XII distribution under progressive type I interval censoring

ABSTRACT

It is a very important topic these days to assessing the lifetime performance of products in manufacturing or service industries. Lifetime performance indices C_L is used to measure the larger-the-better type quality characteristics to evaluate the process performance for the improvement of quality and productivity. The lifetimes of products are assumed to have Burr XII distribution. The maximum likelihood estimator is used to estimate the lifetime performance index based on the progressive type I interval censored sample. The asymptotic distribution of this estimator is also developed. We use this estimator to build the new hypothesis testing algorithmic procedure with respect to a lower specification limit. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.     

Comparisons of the <Emphasis Type="Italic">r</Emphasis> − <Emphasis Type="Italic">k</Emphasis> class estimator to the ordinary least squares estimator under the Pitman’s closeness criterion

In the presence of multicollinearity, the r − k class estimator is proposed as an alternative to the ordinary least squares (OLS) estimator which is a general estimator including the ordinary ridge regression (ORR), the principal components regression (PCR) and the OLS estimators. Comparison of competing estimators of a parameter in the sense of mean square error (MSE) criterion is of central interest. An alternative criterion to the MSE criterion is the Pitman’s (1937) closeness (PC) criterion. In this paper, we compare the r − k class estimator to the OLS estimator in terms of PC criterion so that we can get the comparison of the ORR estimator to the OLS estimator under the PC criterion which was done by Mason et al. (1990) and also the comparison of the PCR estimator to the OLS estimator by means of the PC criterion which was done by Lin and Wei (2002).     

Robust estimates of ordered means in normal models

In this paper we consider the problem of estimating the locations of several normal populations when an order relation between them is known to be true. We compare the maximum likelihood estimator, the M-estimators based on Huber’s ψ function, a robust weighted likelihood estimator, the Gastworth estimator and the trimmed mean estimator. A Monte-Carlo study illustrates the performance of the methods considered.     

Inference of stress-strength for the Type-II generalized logistic distribution under progressively Type-II censored samples

This article studies the estimation of the reliability R = P[Y < X] when X and Y come from two independent generalized logistic distributions of Type-II with different parameters, based on progressively Type-II censored samples. When the common scale parameter is unknown, the maximum likelihood estimator and its asymptotic distribution are proposed. The asymptotic distribution is used to construct an asymptotic confidence interval of R. Bayes estimator of R and the corresponding credible interval using the Gibbs sampling technique have been proposed too. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimation, and confidence interval of R are extracted. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real dataset is given for illustrative purposes. Finally, methods are extended for proportional hazard rate models.     

Testing symmetry based on empirical likelihood

In this paper, we propose a general kth correlation coefficient between the density function and distribution function of a continuous variable as a measure of symmetry and asymmetry. We first propose a root-n moment-based estimator of the kth correlation coefficient and present its asymptotic results. Next, we consider statistical inference of the kth correlation coefficient by using the empirical likelihood (EL) method. The EL statistic is shown to be asymptotically a standard chi-squared distribution. Last, we propose a residual-based estimator of the kth correlation coefficient for a parametric regression model to test whether the density function of the true model error is symmetric or not. We present the asymptotic results of the residual-based kth correlation coefficient estimator and also construct its EL-based confidence intervals. Simulation studies are conducted to examine the performance of the proposed estimators, and we also use our proposed estimators to analyze the air quality dataset.     

EMPIRICAL LIKELIHOOD-BASED KERNEL DENSITY ESTIMATION

This paper considers the estimation of a probability density function when extra distributional information is available (e.g. the mean of the distribution is known or the variance is a known function of the mean). The standard kernel method cannot exploit such extra information systematically as it uses an equal probability weight n^-1 at each data point. The paper suggests using empirical likelihood to choose the probability weights under constraints formulated from the extra distributional information. An empirical likelihood-based kernel density estimator is given by replacing n^-1 by the empirical likelihood weights, and has these advantages: it makes systematic use of the extra information, it is able to reflect the extra characteristics of the density function, and its variance is smaller than that of the standard kernel density estimator.     

Estimation of non-occurrence probability in periodic model

Dimitrov and Khalil (1992) introduced a class of new probability distributions for modeling environmental evolution with periodic behavior. One of the key parameters in these distributions is α, the probability that the event being studied does not occur. In that article the authors derive an estimator for this parameter assuming a series of conditions. In this article it is shown that the estimator is valid under more general conditions, i.e. same of the assumptions are not necessary. It is shown that under the assumption that the elapsed time measured from the starting point of a period until the first occurrence time of the event given that the event occurred in this cycle is related to α, an approximate maximum likelihood estimator of a is proposed. The large sample properties of the estimator are discussed. Monte Carlo study is done for supporting the theoretical results.     

On the performance of some new Liu parameters for the gamma regression model

The maximum likelihood (ML) method is used to estimate the unknown Gamma regression (GR) coefficients. In the presence of multicollinearity, the variance of the ML method becomes overstated and the inference based on the ML method may not be trustworthy. To combat multicollinearity, the Liu estimator has been used. In this estimator, estimation of the Liu parameter d is an important problem. A few estimation methods are available in the literature for estimating such a parameter. This study has considered some of these methods and also proposed some new methods for estimation of the d. The Monte Carlo simulation study has been conducted to assess the performance of the proposed methods where the mean squared error (MSE) is considered as a performance criterion. Based on the Monte Carlo simulation and application results, it is shown that the Liu estimator is always superior to the ML and recommendation about which best Liu parameter should be used in the Liu estimator for the GR model is given.     

On the incidence–prevalence relation and length‐biased sampling

For many diseases, logistic constraints render large incidence studies difficult to carry out. This becomes a drawback, particularly when a new study is needed each time the incidence rate is investigated in a new population. By carrying out a prevalent cohort study with follow‐up it is possible to estimate the incidence rate if it is constant. The authors derive the maximum likelihood estimator (MLE) of the overall incidence rate, λ, as well as age‐specific incidence rates, by exploiting the epidemiologic relationship, (prevalence odds) = (incidence rate) × (mean duration) (P/[1 ? P] = λ × µ). The authors establish the asymptotic distributions of the MLEs and provide approximate confidence intervals for the parameters. Moreover, the MLE of λ is asymptotically most efficient and is the natural estimator obtained by substituting the marginal maximum likelihood estimators for P and µ into P/[1 ? P] = λ × µ. Following‐up the subjects allows the authors to develop these widely applicable procedures. The authors apply their methods to data collected as part of the Canadian Study of Health and Ageing to estimate the incidence rate of dementia amongst elderly Canadians. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Objective Bayesian Analysis of Skew‐t Distributions

Abstract. We study the Jeffreys prior and its properties for the shape parameter of univariate skew‐t distributions with linear and nonlinear Student's t skewing functions. In both cases, we show that the resulting priors for the shape parameter are symmetric around zero and proper. Moreover, we propose a Student's t approximation of the Jeffreys prior that makes an objective Bayesian analysis easy to perform. We carry out a Monte Carlo simulation study that demonstrates an overall better behaviour of the maximum a posteriori estimator compared with the maximum likelihood estimator. We also compare the frequentist coverage of the credible intervals based on the Jeffreys prior and its approximation and show that they are similar. We further discuss location‐scale models under scale mixtures of skew‐normal distributions and show some conditions for the existence of the posterior distribution and its moments. Finally, we present three numerical examples to illustrate the implications of our results on inference for skew‐t distributions.