Empirical likelihood intervals for the population mean and quantiles based on balanced ranked set samples

We consider nonparametric interval estimation for the population mean and quantiles based on a ranked set sample. The asymptotic distributions of the empirical log likelihood ratio statistic for the mean and quantiles are derived. Interval estimates of the population mean and quantiles are obtained by inverting the likelihood ratio statistic. Simulations are carried out to investigate and compare the performance of the empirical likelihood intervals with other known intervals.     

Kullback-Leibler divergence to evaluate posterior sensitivity to different priors for autoregressive time series models

In this paper we use the Kullback-Leibler divergence to measure the distance between the posteriors of the autoregressive (AR) model coefficients, aiming to evaluate mathematically the sensitivity of the coefficients posterior to different types of priors, i.e. Jeffreys’, g, and natural conjugate priors. In addition, we evaluate the impact of the posteriors distance in Bayesian estimates of mean and variance of the model coefficients by generating a large number of Monte Carlo simulations from the posteriors. Simulation study results show that the coefficients posterior is sensitive to prior distributions, and the posteriors distance has more influence on Bayesian estimates of variance than those of mean of the model coefficients. Same results are obtained from the application to real-world time series datasets.     

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.     

A study of the Gamma-Pareto (IV) distribution and its applications

ABSTRACT

Pareto distributions and their close relatives and generalizations provide very flexible families of heavy-tailed distributions that may be used to model income distributions as well as a wide variety of other social and economic distributions. On the other hand, gamma distribution has a wide application in various social and economic spheres such as survival analysis, to model aggregate insurance claims, and the amount of rainfall accumulated in a reservoir etc. Combining the above two heavy-tailed distributions, using the technique by Alzaatreh et al. (2012 Alzaatreh, A., Famoye, F., Lee, C. (2012). Gamma-Pareto distribution and its applications. J. Modern Appl. Stat. Methods. 11:78–94.[Crossref] , [Google Scholar]), we define a new distribution, namely Gamma-Pareto (IV) distribution, hereafter called as GPD^(IV) distribution. Various properties of the GPD^(IV) are investigated such as limiting behavior, moments, mode, and Shannon entropy. Also some characterizations of the GPD^(IV) distribution are mentioned in this paper. Maximum likelihood method is proposed for estimating the model parameters. For illustrative purposes, real data sets are considered as applications of the GPD^(IV) distribution.     

Asymptotic and Resampling-Based Confidence Intervals for P(X < Y)

ABSTRACT

We consider asymptotic and resampling-based interval estimation procedures for the stress-strength reliability P(X < Y). We developed and studied several types of intervals. Their performances are investigated using simulation techniques and compared in terms of attainment of the nominal confidence level, symmetry of lower and upper error rates, and expected length. Recommendations concerning their use are given.     

A CONTINUOUSLY ADAPTIVE RANK TEST FOR SHIFT IN LOCATION

This paper considers the problem of testing for shift in location when the symmetry of the underlying distribution is in doubt. Various adaptive test procedures have been suggested in the literature; they are mainly based on a preliminary test or measure of asymmetry, and then choosing between the sign or the Wilcoxon tests accordingly. However, as this paper demonstrates, there are some disadvantages with such procedures. This paper develops a test that does not suffer from such disadvantages. The proposed test is based on modifying the Wilcoxon scores according to the evidence of asymmetry of the distribution present in the data as indicated by the magnitude of the P‐value from a preliminary test of symmetry. A simulation study investigates and compares the performance of the proposed test and other known adaptive procedures in terms of power and attainment of the nominal size. The performance of a suitable bootstrap procedure for the situation under consideration is also studied. In most cases under consideration, the proposed test is found to be superior to the other tests.     

Bayesian identification of double seasonal autoregressive time series models

ABSTRACT

Seasonal autoregressive (SAR) models have been modified and extended to model high frequency time series characterized by exhibiting double seasonal patterns. Some researchers have introduced Bayesian inference for double seasonal autoregressive (DSAR) models; however, none has tackled the problem of Bayesian identification of DSAR models. Therefore, in order to fill this gap, we present a Bayesian methodology to identify the order of DSAR models. Assuming the model errors are normally distributed and using three priors, i.e. natural conjugate, g, and Jeffreys’ priors, on the model parameters, we derive the joint posterior mass function of the model order in a closed-form. Accordingly, the posterior mass function can be investigated and the best order of DSAR model is chosen as a value with the highest posterior probability for the time series being analyzed. We evaluate the proposed Bayesian methodology using simulation study, and we then apply it to real-world hourly internet amount of traffic dataset.     

Estimation of Pr(X < Y) Using Record Values in the One and Two Parameter Exponential Distributions

We consider the problem of estimating the stress-strength reliability when the available data is in the form of record values. The one parameter and two parameters exponential distribution are considered. In the case of two parameters exponential distributions we considered the case where the location parameter is common and the case where the scale parameter is common. The maximum likelihood estimators and the associated confidence intervals are derived.     

A new class of generalized logistic distribution

The logistic distribution and the S-shaped pattern of its cumulative distribution and quantile functions have been extensively used in many different spheres affecting human life. By far, the most well-known application of logistic distribution is in the logistic regression that is used for modeling categorical response variables. The exponentiated-exponential logistic distribution, a generalization of the logistic distribution, is obtained using the technique proposed by Alzaatreh et al. (2013 Alzaatreh, A., C. Lee, and F. Famoye. 2013. A new method for generating families of continuous distribution. Metron. 71:63–79.[Crossref] , [Google Scholar]) of mixing two distributions, hereafter called the EEL distribution. This distribution subsumes various types of logistic distribution. The structural analysis of the distribution in this paper includes limiting behavior, quantiles, moments, mode, skewness, kurtosis, order statistics, the large sample distributions of the sample maximum and the sample minimum, and the distribution of the sample median. For illustrative purposes, a real-life data set is considered as an application of the EEL distribution.     

Estimation of common location parameter of two exponential populations based on records

Consider the problem of estimating the common location parameter of two exponential populations using record data when the scale parameters are unknown. We derive the maximum likelihood estimator (MLE), the modified maximum likelihood estimator (MMLE) and the uniformly minimum variance unbiased estimator (UMVUE) of the common location parameter. Further, we derive a general result for inadmissibility of an equivariant estimator under the scaled-squared error loss function. Using this result, we conclude that the MLE and the UMVUE are inadmissible and better estimators are provided. A simulation study is conducted for comparing the performances of various competing estimators.