Limiting distributions of MLE and UMVUE in the biparametric uniform distribution

In this paper, we study the asymptotic distributions of MLE and UMVUE of a parametric functionh(θ₁, θ₂) when sampling from a biparametric uniform distributionU(θ₁, θ₂). We obtain both limiting distributions as a convolution of exponential distributions, and we observe that the limiting distribution of UMVUE is a shift of the limiting distribution of MLE.     

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.     

Best unbiased estimation of fixed effects in mixed ANOVA models

In a linear model with an arbitrary variance–covariance matrix, Zyskind (Ann. Math. Statist. 38 (1967) 1092) provided necessary and sufficient conditions for when a given linear function of the fixed-effect parameters has a best linear unbiased estimator (BLUE). If these conditions hold uniformly for all possible variance–covariance parameters (i.e., there is a UBLUE) and if the data are assumed to be normally distributed, these conditions are also necessary and sufficient for the parametric function to have a uniformly minimum variance unbiased estimator (UMVUE). For mixed-effects ANOVA models, we show how these conditions can be translated in terms of the incidence array, which facilitates verification of the UBLUE and UMVUE properties and facilitates construction of designs having such properties.     

Statistical aspects of linkage analysis in interlaboratory studies

This paper investigates statistical issues that arise in interlaboratory studies known as Key Comparisons when one has to link several comparisons to or through existing studies. An approach to the analysis of such a data is proposed using Gaussian distributions with heterogeneous variances. We develop conditions for the set of sufficient statistics to be complete and for the uniqueness of uniformly minimum variance unbiased estimators (UMVUE) of the contrast parametric functions. New procedures are derived for estimating these functions with estimates of their uncertainty. These estimates lead to associated confidence intervals for the laboratories (or studies) contrasts. Several examples demonstrate statistical inference for contrasts based on linkage through the pilot laboratories. Monte Carlo simulation results on performance of approximate confidence intervals are also reported.     

Guided Censored Regression

Parametrically guided non‐parametric regression is an appealing method that can reduce the bias of a non‐parametric regression function estimator without increasing the variance. In this paper, we adapt this method to the censored data case using an unbiased transformation of the data and a local linear fit. The asymptotic properties of the proposed estimator are established, and its performance is evaluated via finite sample simulations.     

Improvement of goodness-of-fit test for normal distribution based on entropy and power comparison

Vasicek's entropy test for normality is based on sample entropy and a parametric entropy estimator. These estimators are known to have bias in small samples. The use of Vasicek's test could affect the capability of detecting non-normality to some extent. This paper presents an improved entropy test, which uses bias-corrected entropy estimators. A Monte Carlo simulation study is performed to compare the power of the proposed test under several alternative distributions with some other tests. The results report that as anticipated, the improved entropy test has consistently higher power than the ordinary entropy test in nearly all sample sizes and alternatives considered, and compares favorably with other tests.     

An extension of parametric ROC analysis for calculating diagnostic accuracy when underlying distributions are mixture of Gaussian

The semiparametric LABROC approach of fitting binormal model for estimating AUC as a global index of accuracy has been justified (except for bimodal forms), while for estimating a local index of accuracy such as TPF, it may lead to a bias in severe departure of data from binormality. We extended parametric ROC analysis for quantitative data when one or both pair members are mixture of Gaussian (MG) in particular for bimodal forms. We analytically showed that AUC and TPF are a mixture of weighting parameters of different components of AUCs and TPFs of a mixture of underlying distributions. In a simulation study of six configurations of MG distributions:{bimodal, normal} and {bimodal, bimodal} pairs, the parameters of MG distributions were estimated using the EM algorithm. The results showed that the estimated AUC from our proposed model was essentially unbiased, and that the bias in the estimated TPF at a clinically relevant range of FPF was roughly 0.01 for a sample size of n=100/100. In practice, with severe departures from binormality, we recommend an extension of the LABROC and software development for future research to allow for each member of the pair of distributions to be a mixture of Gaussian that is a more flexible parametric form.     

Exact Bounds for the Bias of Trimmed Means

Trimmed means remove outliers but affect the estimation efficiency in uncontaminated samples. This paper provides and analyses sharp upper and lower bounds on the bias in the non–parametric estimation of population means by various trimmed means in different scale units. It also presents sharp evaluations of expected trimmed means by the expectations of properly truncated parent distributions.     

Chernoff distance for doubly truncated distributions

In a recent paper, Nair et al. [Stat Pap 52:893–909, 2011] proposed Chernoff distance measure for left/right-truncated random variables and studied their properties in the context of reliability analysis. Here we extend the definition of Chernoff distance for doubly truncated distributions. This measure may help the information theorists and reliability analysts to study the various characteristics of a system/component when it fails between two time points. We study some properties of this measure and obtain its upper and lower bounds. We also study the interval Chernoff distance between the original and weighted distributions. These results generalize and enhance the related existing results that are developed based on Chernoff distance for one-sided truncated random variables.     

Power series estimator for binary data

There are several commonly used measures of association between treatment and control event rates in the population, including odds ratios, relative risk and number needed to treat. Conventionally those parameters are estimated by the sample proportion estimators. In this paper, we show that the sample proportional estimators tend to overestimate. Fortunately, those measurements are estimable by the power series estimators and they converge to UMVUE with a speed of convergency depending on big-O. For instance, it converges slowly for the number needed to treat if the difference between two sample proportions is close to zero.     

Finite Sample Comparison of Parametric,Semiparametric, and Wavelet Estimators of Fractional Integration

ABSTRACT

In this paper we compare through Monte Carlo simulations the finite sample properties of estimators of the fractional differencing parameter, d. This involves frequency domain, time domain, and wavelet based approaches, and we consider both parametric and semiparametric estimation methods. The estimators are briefly introduced and compared, and the criteria adopted for measuring finite sample performance are bias and root mean squared error. Most importantly, the simulations reveal that (1) the frequency domain maximum likelihood procedure is superior to the time domain parametric methods, (2) all the estimators are fairly robust to conditionally heteroscedastic errors, (3) the local polynomial Whittle and bias-reduced log-periodogram regression estimators are shown to be more robust to short-run dynamics than other semiparametric (frequency domain and wavelet) estimators and in some cases even outperform the time domain parametric methods, and (4) without sufficient trimming of scales the wavelet-based estimators are heavily biased.     

Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration

In this paper we compare through Monte Carlo simulations the finite sample properties of estimators of the fractional differencing parameter, d. This involves frequency domain, time domain, and wavelet based approaches, and we consider both parametric and semiparametric estimation methods. The estimators are briefly introduced and compared, and the criteria adopted for measuring finite sample performance are bias and root mean squared error. Most importantly, the simulations reveal that (1) the frequency domain maximum likelihood procedure is superior to the time domain parametric methods, (2) all the estimators are fairly robust to conditionally heteroscedastic errors, (3) the local polynomial Whittle and bias-reduced log-periodogram regression estimators are shown to be more robust to short-run dynamics than other semiparametric (frequency domain and wavelet) estimators and in some cases even outperform the time domain parametric methods, and (4) without sufficient trimming of scales the wavelet-based estimators are heavily biased.     

Sampling Some Truncated Distributions Via Rejection Algorithms

In this article, we develop rejection sampling algorithms to sample from some truncated and tail distributions. Such samplers are needed in many Markov chain Monte Carlo methods, often in connection with Bayesian inference. In addition to univariate normal, gamma, and beta distributions, we consider multivariate normal distributions truncated to certain sets.     

A Measure of Discrimination Between two Double Truncated Distributions

On the basis of Kullback-Leibler discrimination information, and of discrimination measures introduced by Ebrahimi and Kirmani (1996a) and by Di Crescenzo and Longobardi (2004), we propose a measure of discrepancy between double truncated distributions. Some properties of this measure are studied and some mistakes in the preceding literature are corrected.     

Discrete Truncated Power‐Law Distributions

Discrete power‐law distributions have significant consequences for understanding many phenomena in practice, and have attracted much attention in recent decades. However, in many practical applications, there exists a natural upper bound for the probability tail. In this paper, we develop maximum likelihood estimates for truncated discrete power‐law distributions based on the upper order statistics, and large sample properties are mentioned as well. Monte Carlo simulation is carried out to examine the finite sample performance of the estimates. Applications in real cyber attack data and peak gamma‐ray intensity of solar flares are highlighted.     

ON ESTIMATION FOLLOWING SELECTION FROM NONREGULAR DISTRIBUTIONS

Independent random samples are drawn from k (≥ 2) populations, having probability density functions belonging to a general truncation parameter family. The populations associated with the smallest and the largest truncation parameters are called the lower extreme population (LEP) and the upper extreme population (UEP), respectively. For the goal of selecting the LEP (UEP), we consider the natural selection rule, which selects the population corresponding to the smallest (largest) of k maximum likelihood estimates as the LEP (UEP), and study the problem of estimating the truncation parameter of the selected population. We unify some of the existing results, available in the literature for specific distributions, by deriving the uniformly minimum variance unbiased estimator (UMVUE) for the truncation parameter of the selected population. The conditional unbiasedness of the UMVUE is also checked. The cases of the left and the right truncation parameter families are dealt with separately. Finally, we consider an application to the Pareto probability model, where the performances of the UMVUE and three other natural estimators are compared with each other, under the mean squared error criterion.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

Pitfalls in using Weibull tailed distributions

By assuming that the underlying distribution belongs to the domain of attraction of an extreme value distribution, one can extrapolate the data to a far tail region so that a rare event can be predicted. However, when the distribution is in the domain of attraction of a Gumbel distribution, the extrapolation is quite limited generally in comparison with a heavy tailed distribution. In view of this drawback, a Weibull tailed distribution has been studied recently. Some methods for choosing the sample fraction in estimating the Weibull tail coefficient and some bias reduction estimators have been proposed in the literature. In this paper, we show that the theoretical optimal sample fraction does not exist and a bias reduction estimator does not always produce a smaller mean squared error than a biased estimator. These are different from using a heavy tailed distribution. Further we propose a refined class of Weibull tailed distributions which are more useful in estimating high quantiles and extreme tail probabilities.     

Selected singular-generalized-hyperbolic distributions with applications to order statistics and reliability

In this paper, the truncated version of the selected multivariate generalized-hyperbolic distributions is introduced. Considering special truncations, the joint distribution of the consecutive order statistics from the multivariate generalized-hyperbolic (GH) distribution is derived. It is shown that this joint distribution can be expressed as mixtures of the truncated selected-GH distributions. All of these truncated distributions are expressed as the selected singular-GH distributions. These results are used to obtain some expressions for the reliability measures such as mean residual life time, mean inactivity time and regression mean residual life for k-out-of-n systems.     

A Monte Carlo study of some limited and full information simultaneous equation estimators with normal and nonnormal autocorrelated disturbances

The purpose of this paper is to examine the small sample properties of various limited and full information estimators of the structural coefficients of a system of two equations. Specifically, we consider a first-order autoregressive error structure under normal and nonnormal disturbances — for four different covariance structures — and report on a Monte Carlo study of the small sample behavior of limited and full information estimators according to the criteria of bias and dispersion. The results show that the differences in performance of the estimators for the alternative forms of the disturbance distributions are large. Moreover, none of the examined estimators is superior relative to the others, in the sense that its bias and dispersion are the smallest for at least one form of the disturbance distribution. Finally, no combination of highly or lowly autocorrelated disturbances favors some specific limited or full information estimator.