Class Of Symmetric Distributions Research Articles

Robust Minimax Variance Estimation of Location under Bounded Distribution Interquantile Ranges

The performance of the minimax variance (in the Huber sense) robust M-estimates of a location parameter designed in the class of symmetric distributions with bounded interquantile ranges is studied both on large and small samples. Although the general structure of these minimax M-estimates has been described by Huber (1981) long ago, they were not studied as main attention was focused on the estimates designed for the various neighborhoods of a Gaussian. In this work, the proposed estimates are compared to the sample mean, sample median, and the classical robust Huber and Hampel M-estimates under the Gaussian, contaminated Gaussian, Laplace, and Cauchy distributions. The performance of an estimate is measured by its efficiency, bias, and mean squared error. The following conclusions are made: (i) under the Gaussian, Laplace, Cauchy, and moderately contaminated Gaussian distributions, the proposed minimax M-estimates outperform the robust Huber and Hampel M-estimates with respect to asymptotic efficiency; (ii) under heavily contaminated Gaussian distributions, the Huber and Hampel M-estimates are slightly better; (iii) on small samples, these classical robust estimates also slightly outperform the proposed minimax estimates in mean squared error.

Linear estimation for the extended exponential power distribution

Tiao and Lund [The use of OLUMV estimators in inference robustness studies of the location parameter of a class of symmetric distributions. J Amer Statist Assoc. 1970;65(329):370–386] tabulated the coefficients of the best linear unbiased estimators (BLUEs) of location and scale for a particular family of symmetric distributions. This family was a reparameterization of the extended exponential power distribution (EEPD) with the shape parameter restricted to be greater than or equal to one. In this work, we consider the BLU estimation of the location and scale parameters of the EEPD when the shape parameter is one-third and one-half. We obtain closed-form expressions for the single and product moments of the order statistics when the shape parameter is in general in the form of a reciprocal of an integer. These expressions are then used to determine the BLUEs and the corresponding variances for complete samples of size 20 and less. We consider some other linear estimators of the location and scale parameters and then compare them with the BLUEs. Finally, we present a numerical example to illustrate the developed results.

On a variance stabilizing model and its application to genomic data

In this paper, we propose a model based on a class of symmetric distributions, which avoids the transformation of data, stabilizes the variance of the observations, and provides robust estimation of parameters and high flexibility for modeling different types of data. Probabilistic and statistical aspects of this new model are developed throughout the article, which include mathematical properties, estimation of parameters and inference. The obtained results are illustrated by means of real genomic data.

Influence diagnostics in generalized symmetric linear models

The aim of this paper is to introduce generalized symmetric linear models (GSLMs) in the same sense of generalized linear models (GLMs), in which a link function is defined to establish a relationship between the mean values of symmetric distributions and linear predictors. The class of symmetric distributions contains various distributions with lighter and heavier tails than normal and hence offers a more flexible basis for analyzing symmetric data. An iteratively reweighed least squares (IRLS) algorithm is derived to obtain maximum likelihood estimates. The local influence methodology is applied to study the sensitivity of the maximum likelihood estimates under some usual perturbation schemes, such as case-weight, response variable, continuous explanatory variable and scale parameter perturbations. We also discuss generalized leverage and residual analysis. Finally, an illustration is given in which the methodology developed in this paper is applied to a real data set.

Optimal ranked set sampling estimation based on medians from multiple set sizes

Ranked set sampling (RSS) is a sample selection technique that makes use of expert knowledge to rank sample units before measuring them. Even though rankings are not always perfect, RSS is useful in situations where obtaining measurements is costly, difficult, or destructive. Research in this area has tended to focus on the case where all set sizes are equal. This article represents a departure from that setting because we encounter different set sizes within a single sample. More specifically, we propose an alternative estimator for the median of a symmetric distribution using medians of ranked set samples of various set sizes from such a distribution. This estimator is seen to be robust over a wide class of symmetric distributions.

Dependence Structure of a Class of Symmetric Distributions

In this article, dependence structure of a class of symmetric distributions is considered. Let X and Y be two n-dimensional random vectors having such distributions. We investigate conditions on the generators of densities of X and Y such that X is MTP2, and X and Y can be compared in the multivariate likelihood ratio order. Nonnegativity of the covariance between functions of two adjacent order statistics of X is also given.

The Karcher mean of a class of symmetric distributions on the circle

We derive unique Karcher means for symmetric probability distributions on the unit circle, under the conditions that: (i) the distributions have differentiable density functions and (ii) the distributions are concentrated at the symmetry points.

Inference for mixtures of symmetric distributions

This article discusses the problem of estimation of parameters in finite mixtures when the mixture components are assumed to be symmetric and to come from the same location family. We refer to these mixtures as semi-parametric because no additional assumptions other than symmetry are made regarding the parametric form of the component distributions. Because the class of symmetric distributions is so broad, identifiability of parameters is a major issue in these mixtures. We develop a notion of identifiability of finite mixture models, which we call k-identifiability, where k denotes the number of components in the mixture. We give sufficient conditions for k-identifiability of location mixtures of symmetric components when k=2 or 3. We propose a novel distance-based method for estimating the (location and mixing) parameters from a k-identifiable model and establish the strong consistency and asymptotic normality of the estimator. In the specific case of L2-distance, we show that our estimator generalizes the Hodges–Lehmann estimator. We discuss the numerical implementation of these procedures, along with an empirical estimate of the component distribution, in the two-component case. In comparisons with maximum likelihood estimation assuming normal components, our method produces somewhat higher standard error estimates in the case where the components are truly normal, but dramatically outperforms the normal method when the components are heavy-tailed.

Asymptotic Properties of Conditional Quantiles for a Class of Symmetric Distributions

This paper is devoted to studying the asymptotic properties of conditional distributions for one class of symmetric measures on space ${\bf R}^{\infty}$. Explicit formulae of infinite-dimensional conditional quantiles are obtained for distributions of this class.

Optimal Allocation for Symmetric Distributions in Ranked Set Sampling

Ranked set sampling (RSS) is a cost efficient method of sampling that provides a more precise estimator of population mean than simple random sampling. The benefits due to ranked set sampling further increase when appropriate allocation of sampling units is made. For highly skew distributions, allocation based on the Neyman criterion achieves a substantial precision gain over equal allocation. But the same is not true for symmetric distributions; in fact, the gains due to using the Neyman allocation are typically very marginal for symmetric distributions. This paper, determines optimal RSS allocations for two classes of symmetric distributions. Depending upon the class, the optimal allocation assigns all measurements either to the extreme ranks or to the middle rank(s). This allocation outperforms both equal and Neyman allocations in terms of the precision of the estimator which remains unbiased. The two classes of distributions are distinguished by different growth patterns in the variance of their order statistics regarded as a function of the rank order. For one class, the variance peaks for middle rank orders and tapers off in the tails; for the other class, the variance peaks for the two extreme rank orders and tapers off toward the middle. Kurtosis appears to effectively discriminate between the two classes of symmetic distributions. The Neyman allocation is required to quantify all rank orders at least once (to ensure general unbiasedness) but then quantifies most frequently the more variable rank orders. Under symmetry, unbiasedness can be obtained without quantifying all rank orders and the optimal allocation quantifies the least variable rank order(s), resulting in a high precision estimator.

Adaptive Jonckheere-type tests for ordered alternatives

Testing against ordered alternatives in the c -sample location problem plays an important role in statistical practice. The parametric test proposed by Barlow et al .-in the following, called the 'B-test'-is an appropriate test under the model of normality. For non-normal data, however, there are rank tests which have higher power than the B-test, such as the Jonckheere test or so-called Jonckheere-type tests introduced and studied by Buning and Kossler. However, we usually have no information about the underlying distribution. Thus, an adaptive test should be applied which takes into account the given data set. Two versions of such an adaptive test are proposed, which are based on the concept introduced by Hogg in 1974. These adaptive tests are compared with each of the single Jonckheere-type tests in the adaptive scheme and also with the B-test. It is shown via Monte Carlo simulation that the adaptive tests behave well over a broad class of symmetric distributions with short, medium and long tails, as well as for asymmetric distributions.

On the mean squared error, the mean absolute error and the like

The problem of finding the minimizer of the rth -mean error, is revisited, via a unified approach. The approach is discussed for arbitrary r and is illustrated for r = 1 (mean absolute error)r = 2 (mean squared error), and r = 4. This approach is also discussed in the context of maximum likelihood estimation in a class of symmetric distributions which includes, among others, the Laplace and the normal distributions.

On the handling of fuzziness for continuous-valued attributes in decision tree generation

In this paper, fuzziness existing in the process of generating decision trees by discretizing continuous-valued attributes is considered. In a sense a better way to express this fuzziness via fuzzy numbers is presented using possibility theory. The fact that selection of membership functions in a class of symmetric distributions does not influence the decision tree generation is proved. The validity of using the tree to classify future examples is explained. On the basis of likelihood possibility maximization, the existing algorithm is revised. The revised algorithm leads to more reasonable and more natural decision trees.

A probabilistic inequality for sums of bounded symmetric independent random variables

An inequality dt is proved which describes an extremal property of a two-point distribution within the class of symmetric distributions with bounded support.

Acceptance sampling plans by variables for a class of symmetric distributions

The estimation of percentage defectives using a normal sampling plan will not be appropriate when the assumption of normality is violated. In this paper, we propose a sampling plan based on a more general symmetric family of distributions with the parameters estimated using the modified maximum likelihood (MML) procedures introduced by Tiku and Suresh . This sampling plan works well for most of the symmetric non-normal distributions. Some numerical study has also been carried out to show the superiority of the proposed plan.

Asymptotically robust detection for known signals in contaminated multiplicative noise

Abstract Robust detection of known weak signals of unknown amplitude is considered for a class of combined additive and nonadditive noise models and an asymptotically large set of independent observations. Cases where the additive noise distribution is known only to be a member of the Tukey-Huber contamination class of symmetric distributions and cases where the observation model is also known only to be a member of a particular class of multiplicative noise models are considered. The finite-sample-size performance of our schemes is found to be relatively insensitive to variations in the additive noise distribution and the observation model in a specific example.