A Nonparametric Shewhart-Type Signed-Rank Control Chart Based on Runs

Measures of distributional symmetry based on quantiles, L-moments, and trimmed L-moments are briefly reviewed, and (asymptotic) sampling properties of commonly used estimators considered. Standard errors are estimated using both analytical and computer-intensive methods. Simulation is used to assess results when sampling from some known distributions; bootstrapping is used on sample data to estimate standard errors, construct confidence intervals, and test a hypothesis of distributional symmetry. Symmetry measures based on 2- or 3-trimmed L-moments have some advantages over other measures in terms of their existence. Their estimators are generally well behaved, even in relatively small samples.     

Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics

We find the distribution that has maximum entropy conditional on having specified values of its first r  L$L$-moments. This condition is equivalent to specifying the expected values of the order statistics of a sample of size r. The maximum-entropy distribution has a density-quantile function, the reciprocal of the derivative of the quantile function, that is a polynomial of degree r; the quantile function of the distribution can then be found by integration. This class of maximum-entropy distributions includes the uniform, exponential and logistic, and two new generalizations of the logistic distribution. It provides a new method of nonparametric fitting of a distribution to a data sample. We also derive maximum-entropy distributions subject to constraints on expected values of linear combinations of order statistics.     

Moments of L-Statistics: A Divided Differences Approach

The derivation of the distributions of linear combinations of order statistics or L-statistics and the computation of their moments has been approached in the literature several ways. In this paper we use the properties of divided differences to obtain expressions for moments of some order statistics that arise as special cases of L-statistics. Expectations of some well-known L-statistics such as the trimmed mean and the winsorised mean for the pareto distribution are computed. The study also undertakes the computation of L-moments that are expectations of certain linear combinations of order statistics. The algorithms have been implemented using some well-known continuous distributions as examples.     

Bias-reduced estimates for skewness, kurtosis, L-skewness and L-kurtosis

Estimates based on L-moments are less non-robust than estimates based on ordinary moments because the former are linear combinations of order statistics for all orders, whereas the later take increasing powers of deviations from the mean as the order increases. Estimates based on L-moments can also be more efficient than maximum likelihood estimates. Similarly, L-skewness and L-kurtosis are less non-robust and more informative than the traditional measures of skewness and kurtosis. Here, we give nonparametric bias-reduced estimates of both types of skewness and kurtosis. Their asymptotic computational efficiency is infinitely better than that of corresponding bootstrapped estimates.     

Trimmed and Winsorized means based on a scaled deviation

Trimmed (and Winsorized) means based on a scaled deviation are introduced and studied. The influence functions of the estimators are derived and their limiting distributions are established via asymptotic representations. As a main focus of the paper, the performance of the estimators with respect to various robustness and efficiency criteria is evaluated and compared with leading competitors including the ordinary Tukey trimmed (and Winsorized) means. Unlike the Tukey trimming which always trims a fixed fraction of sample points at each end of data, the trimming scheme here only trims points at one or both ends that have a scaled deviation beyond some threshold. The resulting trimmed (and Winsorized) means are much more robust than their predecessors. Indeed they can share the best breakdown point robustness of the sample median for any common trimming thresholds. Furthermore, for appropriate trimming thresholds they are highly efficient at light-tailed symmetric models and more efficient than their predecessors at heavy-tailed or contaminated symmetric models. Detailed comparisons with leading competitors on various robustness and efficiency aspects reveal that the scaled deviation trimmed (Winsorized) means behave very well overall and consequently represent very favorable alternatives to the ordinary trimmed (Winsorized) means.     

Bayesian Analysis of a Queueing System with a Long-Tailed Arrival Process

Internet traffic data is characterized by some unusual statistical properties, in particular, the presence of heavy-tailed variables. A typical model for heavy-tailed distributions is the Pareto distribution although this is not adequate in many cases. In this article, we consider a mixture of two-parameter Pareto distributions as a model for heavy-tailed data and use a Bayesian approach based on the birth-death Markov chain Monte Carlo algorithm to fit this model. We estimate some measures of interest related to the queueing system k-Par/M/1 where k-Par denotes a mixture of k Pareto distributions. Heavy-tailed variables are difficult to model in such queueing systems because of the lack of a simple expression for the Laplace Transform (LT). We use a procedure based on recent LT approximating results for the Pareto/M/1 system. We illustrate our approach with both simulated and real data.     

Asymmetric generalizations of symmetric univariate probability distributions obtained through quantile splicing

Abstract

Balakrishnan et al. proposed a two-piece skew logistic distribution by making use of the cumulative distribution function (CDF) of half distributions as the building block, to give rise to an asymmetric family of two-piece distributions, through the inclusion of a single shape parameter. This paper proposes the construction of asymmetric families of two-piece distributions by making use of quantile functions of symmetric distributions as building blocks. This proposition will enable the derivation of a general formula for the L-moments of two-piece distributions. Examples will be presented, where the logistic, normal, Student’s t(2) and hyperbolic secant distributions are considered.     

A Sieve model for extreme values

From the class of extreme value distributions, we focus on the set of heavy-tailed distributions which produce low-frequency, high-cost events. The regular Pareto distribution is the basic model of choice, being the simplest heavy-tailed distribution. Real data suggest that modifications of the Pareto distribution may be a better fit; an alternative model is the truncated Pareto distribution (TPD). For further study, this paper proposed a TPD Sieve class of distributions. The properties and estimation on the Sieve class are also discussed. We fit the models to the largest Black Sea bass caught in Buzzard's Bay, MA, USA and the costliest Atlantic hurricanes from 1900 to 2005. Using measures of model adequacy, the TPD Sieve model is generally found to be the best-fitting model.     

Symmetric regression quantile and its application to robust estimation for the nonlinear regression model

Populational conditional quantiles in terms of percentage α are useful as indices for identifying outliers. We propose a class of symmetric quantiles for estimating unknown nonlinear regression conditional quantiles. In large samples, symmetric quantiles are more efficient than regression quantiles considered by Koenker and Bassett (Econometrica 46 (1978) 33) for small or large values of α, when the underlying distribution is symmetric, in the sense that they have smaller asymptotic variances. Symmetric quantiles play a useful role in identifying outliers. In estimating nonlinear regression parameters by symmetric trimmed means constructed by symmetric quantiles, we show that their asymptotic variances can be very close to (or can even attain) the Cramer–Rao lower bound under symmetric heavy-tailed error distributions, whereas the usual robust and nonrobust estimators cannot.     

The minimum L2 distance estimator for Poisson mixture models

A robust estimator is developed for Poisson mixture models with a known number of components. The proposed estimator minimizes the L₂ distance between a sample of data and the model. When the component distributions are completely known, the estimators for the mixing proportions are in closed form. When the parameters for the component Poisson distributions are unknown, numerical methods are needed to calculate the estimators. Compared to the minimum Hellinger distance estimator, the minimum L₂ estimator can be less robust to extreme outliers, and often more robust to moderate outliers.     

Berry–Esseen type bounds for trimmed U-statistics

Trimmed U  -statistics can be constructed in two different ways: by basing the statistic on a trimmed sample or by averaging the trimmed set of kernel values. Mild conditions are given to ensure the rate of convergence to normality is O(n^-1/2)$\mathrm{O}(n^{-1/2})$ in both cases.     

Local Linear Estimation for Spatiotemporal Models Based on Least Absolute Deviation

When the data contain outliers or come from population with heavy-tailed distributions, which appear very often in spatiotemporal data, the estimation methods based on least-squares (L₂) method will not perform well. More robust estimation methods are required. In this article, we propose the local linear estimation for spatiotemporal models based on least absolute deviation (L₁) and drive the asymptotic distributions of the L₁-estimators under some mild conditions imposed on the spatiotemporal process. The simulation results for two examples, with outliers and heavy-tailed distribution, respectively, show that the L₁-estimators perform better than the L₂-estimators.     

L 1-estimation for varying coefficient models

The varying coefficient model is a useful extension of linear models and has many advantages in practical use. To estimate the unknown functions in the model, the kernel type with local linear least-squares (L ₂) estimation methods has been proposed by several authors. When the data contain outliers or come from population with heavy-tailed distributions, L ₁-estimation should yield better estimators. In this article, we present the local linear L ₁-estimation method and derive the asymptotic distributions of the L ₁-estimators. The simulation results for two examples, with outliers and heavy-tailed distribution, respectively, show that the L ₁-estimators outperform the L ₂-estimators.     

A Marshall–Olkin family of heavy-tailed distributions which includes the lognormal one

Abstract

A new class of heavy-tailed distribution functions,, containing the lognormal distribution as a particular case is introduced. The class thus obtained depends on a set of three parameters, incorporating an additional distribution to the classical lognormal one. This new class of heavy-tailed distribution is presented as an alternative to other useful heavy-tailed distributions, such as the lognormal, Weibull, and Pareto distributions. The density and distribution functions of this new class are given by a closed expression which allows us to easily compute probabilities, quantiles, moments, and related measurements. Finally, some applications are shown as examples.     

A class of distributions with the quadratic mean residual quantile function

Abstract

The present paper introduces a new family of distributions with quadratic mean residual quantile function. Various distributional properties as well as reliability characteristics are discussed. Some characterizations of the class of distributions are presented. The estimation of parameters of the model using method of L-moments is studied. The practical application of the class of models is illustrated with a real life data set.     

On the Behrens–Fisher problem from the spatial median point of view

We present a non-parametric affine-invariant test for the multivariate Behrens–Fisher problem. The proposed method based on the spatial medians is asymptotic and does not require normality of the data. To improve its finite sample performance, we apply a correction of the type which was already used in a similar test based on trimmed means, however, our simulations show that in the case of heavy-tailed distributions our method performs better. Also in a simulation comparison with a recently published rank-based test our test yields satisfactory results.     

Predicting observables from a general class of distributions

A general class of distributions is proposed to be the underlying population model from which observables are to be predicted using the Bayesian approach. This class of distributions includes, among others, the Weibull, compound Weibull (or three-parameter Burr-type XII), Pareto, beta, Gompertz and compound Gompertz distributions. A proper general prior density function is suggested and the predictive density functions are obtained in the one- and two-sample cases. The informative sample is assumed to be a type II censored sample. Illustrative examples of Weibull (α,β), Burr-type XII (α,β), and Pareto (α,β) distributions are given and compared with the results obtained by previous researchers.     

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.     

Trimmed likelihood estimators for lifetime experiments and their influence functions

We study the behaviour of trimmed likelihood estimators (TLEs) for lifetime models with exponential or lognormal distributions possessing a linear or nonlinear link function. In particular, we investigate the difference between two possible definitions for the TLE, one called original trimmed likelihood estimator (OTLE) and one called modified trimmed likelihood estimator (MTLE) which is the finite sample version of a form for location and linear regression used by Bednarski and Clarke [Trimmed likelihood estimation of location and scale of the normal distribution. Aust J Statist. 1993;35:141–153, Asymptotics for an adaptive trimmed likelihood location estimator. Statistics. 2002;36:1–8] and Bednarski et al. [Adaptive trimmed likelihood estimation in regression. Discuss Math Probab Stat. 2010;30:203–219]. The OTLE is always an MTLE but the MTLE may not be unique even in cases where the OLTE is unique. We compare especially the functional forms of both types of estimators, characterize the difference with the implicit function theorem and indicate situations where they coincide and where they do not coincide. Since the functional form of the MTLE has a simpler form, we use it then for deriving the influence function, again with the help of the implicit function theorem. The derivation of the influence function for the functional form of the OTLE is similar but more complicated.     

Cross-validation Revisited

Data-based choice of the bandwidth is an important problem in kernel density estimation. The pseudo-likelihood and the least-squares cross-validation bandwidth selectors are well known, but widely criticized in the literature. For heavy-tailed distributions, the L₁ distance between the pseudo-likelihood-based estimator and the density does not seem to converge in probability to zero with increasing sample size. Even for normal-tailed densities, the rate of L₁ convergence is disappointingly slow. In this article, we report an interesting finding that with minor modifications both the cross-validation methods can be implemented effectively, even for heavy-tailed densities. For both these estimators, the L₁ distance (from the density) are shown to converge completely to zero irrespective of the tail of the density. The expected L₁ distance also goes to zero. These results hold even in the presence of a strongly mixing-type dependence. Monte Carlo simulations and analysis of the Old Faithful geyser data suggest that if implemented appropriately, contrary to the traditional belief, the cross-validation estimators compare well with the sophisticated plug-in and bootstrap-based estimators.