Estimating the variance of a normal population by utilizing the information in a sample from a second related normal population

A class of estimators of the variance σ₁ ² of a normal population is introduced, by utilization the information in a sample from a second normal population with different mean and variance σ₂ ², under the restriction that σ₁ ²?≤?σ₂ ². Simulation results indicate that some members of this class are more efficient than the usual minimum variance unbiased estimator (MVUE) of σ₁ ², Stein estimator and Mehta and Gurland estimator. The case of known and unknown means are considered.     

Some small-sample properties of some recently proposed multivariate outlier detection techniques

Recently, several new robust multivariate estimators of location and scatter have been proposed that provide new and improved methods for detecting multivariate outliers. But for small sample sizes, there are no results on how these new multivariate outlier detection techniques compare in terms of p _n , their outside rate per observation (the expected proportion of points declared outliers) under normality. And there are no results comparing their ability to detect truly unusual points based on the model that generated the data. Moreover, there are no results comparing these methods to two fairly new techniques that do not rely on some robust covariance matrix. It is found that for an approach based on the orthogonal Gnanadesikan–Kettenring estimator, p _n can be very unsatisfactory with small sample sizes, but a simple modification gives much more satisfactory results. Similar problems were found when using the median ball algorithm, but a modification proved to be unsatisfactory. The translated-biweights (TBS) estimator generally performs well with a sample size of n≥20 and when dealing with p-variate data where p≤5. But with p=8 it can be unsatisfactory, even with n=200. A projection method as well the minimum generalized variance method generally perform best, but with p≤5 conditions where the TBS method is preferable are described. In terms of detecting truly unusual points, the methods can differ substantially depending on where the outliers happen to be, the number of outliers present, and the correlations among the variables.     

Maximum likelihood parameter estimation in the three-parameter gamma distribution with the use of Mathematica

The number of solutions of the system of the log-likelihood equations for the three-parameter case is still an open problem. Several methods have been developed for finding the solutions. In this article we present a program in Mathematica that can find all the solutions of the system of equations. Furthermore, we examine the case where the global maximum appears at the boundary of the domain of the log-likelihood function and we prove that any consistent estimators appear at the interior with probability tending to one.     

Complexity of penalized likelihood estimation

We show that for a class of penalty functions, finding the global optimizer in the penalized least-squares estimation is equivalent to the ‘exact cover by 3-sets’ problem, which belongs to a class of NP-hard problems. The NP-hardness result is then extended to the cases of penalized least absolute deviations regression and a special class of penalized support vector machines. We discuss its implication in statistics. To the best of our knowledge, this is the first formal documentation on the complexity of this type of problem.     

Variance estimation of the Buckley–James estimator under discrete assumptions

The Buckley–James estimator (BJE) is a widely recognized approach in dealing with right-censored linear regression models. There have been a lot of discussions in the literature on the estimation of the BJE as well as its asymptotic distribution. So far, no simulation has been done to directly estimate the asymptotic variance of the BJE. Kong and Yu [Asymptotic distributions of the Buckley–James estimator under nonstandard conditions, Statist. Sinica 17 (2007), pp. 341–360] studied the asymptotic distribution under discontinuous assumptions. Based on their methodology, we recalculate and correct some missing terms in the expression of the asymptotic variance in Theorem 2 of their work. We propose an estimator of the standard deviation of the BJE by using plug-in estimators. The estimator is shown to be consistent. The performance of the estimator is accessed through simulation studies under discrete underline distributions. We further extend our studies to several continuous underline distributions through simulation. The estimator is also applied to a real medical data set. The simulation results suggest that our estimation is a good approximation to the true standard deviation with reference to the empirical standard deviation.     

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.     

On estimation of R=P(Y<X) for exponential distribution under progressive type-II censoring

This paper deals with the estimation of the stress–strength parameter R=P(Y<X), when X and Y are independent exponential random variables, and the data obtained from both distributions are progressively type-II censored. The uniformly minimum variance unbiased estimator and the maximum-likelihood estimator (MLE) are obtained for the stress–strength parameter. Based on the exact distribution of the MLE of R, an exact confidence interval of R has been obtained. Bayes estimate of R and the associated credible interval are also obtained under the assumption of independent inverse gamma priors. An extensive computer simulation is used to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose.     

Bootstrapping probability-proportional-to-size samples via calibrated empirical population

A collection of six novel bootstrap algorithms, applied to probability-proportional-to-size samples, is explored for variance estimation, confidence interval and p-value production. Developed according to bootstrap fundamentals such as the mimicking principle and the plug-in rule, these algorithms make use of an empirical bootstrap population informed by sampled units each with assigned weight. Starting from the natural choice of Horvitz–Thompson (HT)-type weights, improvements based on calibration to known population features are fostered. Focusing on the population total as the parameter to be estimated and on the distribution of the HT estimator as the target of bootstrap estimation, simulation results are presented with the twofold objective of checking practical implementation and of investigating the statistical properties of the bootstrap estimates supplied by the algorithms explored.     

Confidence sets based on the positive part James–Stein estimator with the asymptotically constant coverage probability

The asymptotic expansions for the coverage probability of a confidence set centred at the James–Stein estimator presented in our previous publications show that this probability depends on the non-centrality parameter τ² (the sum of the squares of the means of normal distributions). In this paper we establish how these expansions can be used for a construction of confidence region with constant confidence level, which is asymptotically (the same formula for both case τ→0 and τ→∞) equal to some fixed value 1?α. We establish the shrinkage rate for the confidence region according to the growth of the dimension p and also the value of τ for which we observe quick decreasing of the coverage probability to the nominal level 1?α. When p→∞ this value of τ increases as O(p^1/4). The accuracy of the results obtained is shown by the Monte-Carlo statistical simulations.     

Tests of fit for the Laplace distribution based on correcting moments of entropy estimators

ABSTRACT

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing test statistics against various alternatives. Performance of the proposed new test statistics is illustrated in real cases.