A generalization of the log-series distribution

A new generalized logarithmic series distribution (GLSD) with two parameters is proposed.The proposed model is flexible enough to describe short-tailed as well as long-tailed data.Some recurence relations for its probabilities and the factorial moments are presente.These recurrence relations are utilized to obtain the minimum chi-square estimators for the parmaters.Maximum likelihood estimators and some other estimators based on first few moments and probabilities are also suggested.Asymptotic relative efficiency of some of these estimators is also obtained and compared.Two test statistics based on the minimum chi-square estimators fo testing some hypotheses regarding the GLSD are proposed.The fit of the model and the application of the test statistics are exemplified by some data sets.Finally, a graphical method is suggested for differentiating between the ordinary logarithmic series distribution and the GLSD.     

Weighted quantile regression and testing for varying-coefficient models with randomly truncated data

This paper develops a varying-coefficient approach to the estimation and testing of regression quantiles under randomly truncated data. In order to handle the truncated data, the random weights are introduced and the weighted quantile regression (WQR) estimators for nonparametric functions are proposed. To achieve nice efficiency properties, we further develop a weighted composite quantile regression (WCQR) estimation method for nonparametric functions in varying-coefficient models. The asymptotic properties both for the proposed WQR and WCQR estimators are established. In addition, we propose a novel bootstrap-based test procedure to test whether the nonparametric functions in varying-coefficient quantile models can be specified by some function forms. The performance of the proposed estimators and test procedure are investigated through simulation studies and a real data example.     

Testing and Merging Information for Effect Size Estimation

A large-sample test for testing the equality of two effect sizes is presented. The null and non-null distributions of the proposed test statistic are derived. Further, the problem of estimating the effect size is considered when it is a priori suspected that two effect sizes may be close to each other. The combined data from all the samples leads to more efficient estimator of the effect size. We propose a basis for optimally combining estimation problems when there is uncertainty concerning the appropriate statistical model-estimator to use in representing the sampling process. The objective here is to produce natural adaptive estimators with some good statistical properties. In the context of two bivariate statistical models, the expressions for the asymptotic mean squared error of the proposed estimators are derived and compared with the parallel expressions for the benchmark estimators. We demonstrate that the suggested preliminary test estimator has superior asymptotic mean squared error performance relative to the benchmark and pooled estimators. A simulation study and application of the methodology to real data are presented.     

Estimation of the area under ROC curve with censored data

The area under the receiver operating characteristic curve is the most commonly used summary measure of diagnostic accuracy for a continuous-scale diagnostic test. In this paper, we develop methods to estimate the area under the curve (AUC) with censored data. Based on two different integration representations of this parameter, two nonparametric estimators are defined by the “plug in” method. Both the proposed estimators are shown to be asymptotically normal based on counting process and martingale theory. A simulation study is conducted to evaluate the performances of the proposed estimators.     

A new extended generalized Gompertz distribution with statistical properties and simulations

Abstract

Statistical distributions are very useful in describing and predicting real world phenomena. In many applied areas there is a clear need for the extended forms of the well-known distributions. Generally, the new distributions are more flexible to model real data that present a high degree of skewness and kurtosis. The choice of the best-suited statistical distribution for modeling data is very important.

In this article, we proposed an extended generalized Gompertz (EGGo) family of EGGo. Certain statistical properties of EGGo family including distribution shapes, hazard function, skewness, limit behavior, moments and order statistics are discussed. The flexibility of this family is assessed by its application to real data sets and comparison with other competing distributions. The maximum likelihood equations for estimating the parameters based on real data are given. The performances of the estimators such as maximum likelihood estimators, least squares estimators, weighted least squares estimators, Cramer-von-Mises estimators, Anderson-Darling estimators and right tailed Anderson-Darling estimators are discussed. The likelihood ratio test is derived to illustrate that the EGGo distribution is better than other nested models in fitting data set or not. We use R software for simulation in order to perform applications and test the validity of this model.     

Comparison of correlated proportions based on paired binary data from clustered samples

In this article, we explore hypothesis testing problems related to correlated proportions from clustered matched-pair binary data. Null hypotheses of equality in proportions, homogeneity, and non-inferiority of one to another are similar testing problems of linear contrasts of correlated proportions with suitable transformation. The covariance estimators of the test statistics are based on moment estimation under the null hypotheses. We present a general framework for testing linear contrasts of the correlated proportions from clustered matched-pair data based upon a class of unbiased estimators of the proportions. The corresponding testing procedures do not impose structure assumptions on the correlation matrix and are easy to use. Simulation results suggest that the proposed method is more likely to maintain the proper significance level and to improve power than the test proposed by Obuchowski.     

Testing hypotheses in truncated samples by means of divergence statistics

In this paper we consider the problem of testing hypotheses in parametric models, when only the first r (of n) ordered observations are known.Using divergence measures, a procedure to test statistical hypotheses is proposed, Replacing the parameters by suitable estimators in the expresion of the divergence measure, the test statistics are obtained.Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators for truncated samples are considered.Applications of these results in testing statistical hypotheses, on the basis of truncated data, are presented.The small sample behavior of the proposed test statistics is analyzed in particular cases.A comparative study of power values is carried out by computer simulation.     

Comparison of preliminary test estimators based on generalized order statistics from proportional hazard family using Pitman measure of closeness

In this article, based on generalized order statistics from a family of proportional hazard rate model, we use a statistical test to generate a class of preliminary test estimators and shrinkage preliminary test estimators for the proportionality parameter. These estimators are compared under Pitman measure of closeness (PMC) as well as MSE criteria. Although the PMC suffers from non transitivity, in the first class of estimators, it has the transitivity property and we obtain the Pitman-closest estimator. Analytical and graphical methods are used to show the range of parameter in which preliminary test and shrinkage preliminary test estimators perform better than their competitor estimators. Results reveal that when the prior information is not too far from its real value, the proposed estimators are superior based on both mentioned criteria.     

An Improved Estimation in Regression Parameter Matrix in Multivariate Regression Model

We consider shrinkage and preliminary test estimation strategies for the matrix of regression parameters in multivariate multiple regression model in the presence of a natural linear constraint. We suggest a shrinkage and preliminary test estimation strategies for the parameter matrix. The goal of this article is to critically examine the relative performances of these estimators in the direction of the subspace and candidate subspace restricted type estimators. Our analytical and numerical results show that the proposed shrinkage and preliminary test estimators perform better than the benchmark estimator under candidate subspace and beyond. The methods are also applied on a real data set for illustrative purposes.     

On the estimation of reliabilty function in a Weibull lifetime distribution

The estimation of the reliability function of the Weibull lifetime model is considered in the presence of uncertain prior information (not in the form of prior distribution) on the parameter of interest. This information is assumed to be available in some sort of a realistic conjecture. In this article, we focus on how to combine sample and non-sample information together in order to achieve improved estimation performance. Three classes of point estimatiors, namely, the unrestricted estimator, the shrinkage estimator and shrinkage preliminary test estimator (SPTE) are proposed. Their asymptotic biases and mean-squared errors are derived and compared. The relative dominance picture of the estimators is presented. Interestingly, the proposed SPTE dominates the unrestricted estimator in a range that is wider than that of the usual preliminary test estimator. A small-scale simulation experiment is used to examine the small sample properties of the proposed estimators. Our simulation investigations have provided strong evidence that corroborates with asymptotic theory. The suggested estimation methods are applied to a published data set to illustrate the performance of the estimators in a real-life situation.     

Extropy estimators with applications in testing uniformity

Two estimators for estimating the extropy of an absolutely continuous random variable with known support were introduced by using spacing. It is shown that the proposed estimators are consistent and their mean square errors are shift invariant. Their behaviours were also studied by means of real data and Monte Carlo simulation. The winner estimator of extropy in the Monte Carlo experiment was used to develop goodness-of-fit test for standard uniform distribution. It is shown that the extropy-based test that we proposed performs well by comparing its powers with that of other tests for uniformity.     

Estimating and testing effect size from an arbitrary population

In this article, we develop inference tools for an effect size parameter in a paired experiment. A class of estimators is defined that includes natural, shrinkage and shrinkage preliminary test estimators. The shrinkage and preliminary test methods incorporate uncertain prior information on the parameter. This information may be available in the form of a realistic guess on the basis of the experimenter’s knowledge and experience, which can be incorporated into the estimation process to increase the efficiency of the estimator. Asymptotic properties of the proposed estimators are investigated both analytically and computationally. A simulation study is also conducted to assess the performance of the estimators for moderate and large samples. For illustration purposes, the method is applied to a data set.     

Order restricted inference for multivariate binary data with application to toxicology

In many applications researchers collect multivariate binary response data under two or more, naturally ordered, experimental conditions. In such situations one is often interested in using all binary outcomes simultaneously to detect an ordering among the experimental conditions. To make such comparisons we develop a general methodology for testing for the multivariate stochastic order between K ≥ 2 multivariate binary distributions. The proposed test uses order restricted estimators which, according to our simulation study, are more efficient than the unrestricted estimators in terms of mean squared error. The power of the proposed test was compared with several alternative tests. These included procedures which combine individual univariate tests for order, such as union intersection tests and a Bonferroni based test. We also compared the proposed test with unrestricted Hotelling's T(2) type test. Our simulations suggest that the proposed method competes well with these alternatives. The gain in power is often substantial. The proposed methodology is illustrated by applying it to a two-year rodent cancer bioassay data obtained from the US National Toxicology Program (NTP). Supplemental materials are available online.     

On preliminary test almost unbiased two-parameter estimator in linear regression model with student's t errors

In this paper, the preliminary test approach to the estimation of the linear regression model with student's t errors is considered. The preliminary test almost unbiased two-parameter estimator is proposed, when it is suspected that the regression parameter may be restricted to a constraint. The quadratic biases and quadratic risks of the proposed estimators are derived and compared under both null and alternative hypotheses. The conditions of superiority of the proposed estimators for departure parameter and biasing parameters k and d are derived, respectively. Furthermore, a real data example and a Monte Carlo simulation study are provided to illustrate some of the theoretical results.     

A james-stein type detour of U-statistics

For a vector of estimable parameters, a modified version of the James-Stein rule (incorporating the idea of preliminary test estimators) is utilized in formulating some estimators based on U-statistics and their jackknifed estimator of dispersion matrix. Asymptotic admissibility properties of the classical U-statistics, their preliminary test version and the proposed estimators are studied.     

Performance of the shrinkage preliminary test ridge regression estimators based on the conflicting of W,LR and LM tests

The shrinkage preliminary test ridge regression estimators (SPTRRE) based on the Wald (W), the likelihood ratio (LR) and the Lagrangian multiplier (LM) tests are considered in this paper. The bias and the risk functions of the proposed estimators are derived. The regions of optimality of the estimators are determined under the quadratic risk function. Under the null hypothesis, the SPTRRE based on LM test has the smallest risk, followed by the estimators based on LR and W tests. However, the SPTRRE based on W test performs the best followed by the LR and LM based estimators when the parameter moves away from the subspace of the restrictions. The conditions of superiority of the proposed estimator for both ridge and departure parameters are discussed. The optimum choice of the level of significance becomes the traditional choice by using the W test for all non-negative ridge parameters.     

An efficient and fast algorithm for estimating the parameters of two-dimensional sinusoidal signals

In this paper we propose a computationally efficient algorithm to estimate the parameters of a 2-D sinusoidal model in the presence of stationary noise. The estimators obtained by the proposed algorithm are consistent and asymptotically equivalent to the least squares estimators. Monte Carlo simulations are performed for different sample sizes and it is observed that the performances of the proposed method are quite satisfactory and they are equivalent to the least squares estimators. The main advantage of the proposed method is that the estimators can be obtained using only finite number of iterations. In fact it is shown that starting from the average of periodogram estimators, the proposed algorithm converges in three steps only. One synthesized texture data and one original texture data have been analyzed using the proposed algorithm for illustrative purpose.     

Assessing accuracy of a continuous screening test in the presence of verification bias

Summary.  In studies to assess the accuracy of a screening test, often definitive disease assessment is too invasive or expensive to be ascertained on all the study subjects. Although it may be more ethical or cost effective to ascertain the true disease status with a higher rate in study subjects where the screening test or additional information is suggestive of disease, estimates of accuracy can be biased in a study with such a design. This bias is known as verification bias. Verification bias correction methods that accommodate screening tests with binary or ordinal responses have been developed; however, no verification bias correction methods exist for tests with continuous results. We propose and compare imputation and reweighting bias-corrected estimators of true and false positive rates, receiver operating characteristic curves and area under the receiver operating characteristic curve for continuous tests. Distribution theory and simulation studies are used to compare the proposed estimators with respect to bias, relative efficiency and robustness to model misspecification. The bias correction estimators proposed are applied to data from a study of screening tests for neonatal hearing loss.     

Two-way ANOVA when the distribution of the error terms is skew t

In this article, we assume that the distribution of the error terms is skew t in two-way analysis of variance (ANOVA). Skew t distribution is very flexible for modeling the symmetric and the skew datasets, since it reduces to the well-known normal, skew normal, and Student's t distributions. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies. We also propose new test statistics based on these estimators for testing the equality of the treatment and the block means and also the interaction effect. The efficiencies of the ML and the MML estimators and the power values of the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. Simulation results show that the proposed methodologies are more preferable. We also show that the test statistics based on the ML estimators are more powerful than the test statistics based on the MML estimators as expected. However, power values of the test statistics based on the MML estimators are very close to the corresponding test statistics based on the ML estimators. At the end of the study, a real life example is given to show the implementation of the proposed methodologies.     

A New Estimation Method Based on Order Statistics in the Families of Symmetric Location-scale Distributions

In this study, new unbiased and nonlinear estimators based on order statistics are proposed for the family of symmetric location-scale distributions and these estimators can be computed from both uncensored and symmetric doubly Type II censored samples. In addition, other relevant unbiased estimators are proposed to estimate standard deviations of these new estimators. A simulation study has been performed to evaluate the performance of the new estimators compared to BLU estimators for small sample sizes. As a result of the simulation study, the new estimators proposed for the location-scale family in general performed nearly as good as BLU estimators. Furthermore, the computational advantage of the proposed estimators over BLU and ML estimators are worthy of notice. In addition, these new estimators have been applied to real data, and the estimation results obtained have been compatible with those of BLUE methods.