A fast robust method for fitting gamma distributions

The art of fitting gamma distributions robustly is described. In particular we compare methods of fitting via minimizing a Cramér Von Mises distance, an L ₂ minimum distance estimator, and fitting a B-optimal M-estimator. After a brief prelude on robust estimation explaining the merits in terms of weak continuity and Fréchet differentiability of all the aforesaid estimators from an asymptotic point of view, a comparison is drawn with classical estimation and fitting. In summary, we give a practical example where minimizing a Cramér Von Mises distance is both efficacious in terms of efficiency and robustness as well as being easily implemented. Here gamma distributions arise naturally for “in control” representation indicators from measurements of spectra when using fourier transform infrared (FTIR) spectroscopy. However, estimating the in-control parameters for these distributions is often difficult, due to the occasional occurrence of outliers.     

On the performance of L2E estimation in modelling heterogeneous count responses with extreme values

In healthcare studies, count data sets measured with covariates often exhibit heterogeneity and contain extreme values. To analyse such count data sets, we use a finite mixture of regression model framework and investigate a robust estimation approach, called the L₂E [D.W. Scott, On fitting and adapting of density estimates, Comput. Sci. Stat. 30 (1998), pp. 124–133], to estimate the parameters. The L₂E is based on an integrated L₂ distance between parametric conditional and true conditional mass functions. In addition to studying the theoretical properties of the L₂E estimator, we compare the performance of L₂E with the maximum likelihood (ML) estimator and a minimum Hellinger distance (MHD) estimator via Monte Carlo simulations for correctly specified and gross-error contaminated mixture of Poisson regression models. These show that the L₂E is a viable robust alternative to the ML and MHD estimators. More importantly, we use the L₂E to perform a comprehensive analysis of a Western Australia hospital inpatient obstetrical length of stay (LOS) (in days) data that contains extreme values. It is shown that the L₂E provides a two-component Poisson mixture regression fit to the LOS data which is better than those based on the ML and MHD estimators. The L₂E fit identifies admission type as a significant covariate that profiles the predominant subpopulation of normal-stayers as planned patients and the small subpopulation of long-stayers as emergency patients.     

On least absolute values estimation

The resistance of least absolute values (L₁) estimators to outliers and their robustness to heavy-tailed distributions make these estimators useful alternatives to the usual least squares estimators. The recent development of efficient algorithms for L₁ estimation in linear models has permitted their use in practical data analysis. Although in general the L₁ estimators are not unique, there are a number of properties they all share. The set of all L₁ estimators for a given model and data set can be characterized as the convex hull of some extreme estimators. Properties of the extreme estimators and of the L₁-estimate set are considered.     

Choosing a robustness tuning parameter

A novel method is proposed for choosing the tuning parameter associated with a family of robust estimators. It consists of minimising estimated mean squared error, an approach that requires pilot estimation of model parameters. The method is explored for the family of minimum distance estimators proposed by [Basu, A., Harris, I.R., Hjort, N.L. and Jones, M.C., 1998, Robust and efficient estimation by minimising a density power divergence. Biometrika, 85, 549–559.] Our preference in that context is for a version of the method using the L ₂ distance estimator [Scott, D.W., 2001, Parametric statistical modeling by minimum integrated squared error. Technometrics, 43, 274–285.] as pilot estimator.     

Weighted L1-estimates for the First-order Bifurcating Autoregressive Model

We developed robust estimators that minimize a weighted L₁ norm for the first-order bifurcating autoregressive model. When all of the weights are fixed, our estimate is an L₁ estimate that is robust against outlying points in the response space and more efficient than the least squares estimate for heavy-tailed error distributions. When the weights are random and depend on the points in the factor space, the weighted L₁ estimate is robust against outlying points in the factor space. Simulated and artificial examples are presented. The behavior of the proposed estimate is modeled through a Monte Carlo study.     

$${\mathcal{L}}_p$$ loss functions: a robust bayesian approach

In bayesian inference, the Bayes estimator is the alternative with the minimum expected loss. In most cases, the loss function shows the distance between the alternative and the parameter. Therefore, any distance can lead to a loss function. Among the best known distance functions is L _p one, where the choice of value p may be difficult and arbitrary. This paper examines robust models where the loss function is modelled by family L _p . Our solution concept is the non-dominated alternative. We characterize the non-dominated set by having the posterior distribution function satisfy a particular asymmetry property. We also include an example to illustrate the methodology described.     

Minimum variance unbiased estimation of stress–strength reliability under bivariate normal and its comparisons

In many industrial and natural phenomena, we need the probability that a component is smaller than the other component. Under a stress–strength model, this is reliability of an item. Under independent setup, there are different approaches for the estimation of such reliability. Here, estimation is considered under the dependent case. Under bi-variate setup uniformly minimum variance unbiased estimator is obtained. Also comparison with available estimator based on Maximum Likelihood Estimate (MLE) is done through Mean Square Error (MSE) and bias. Also these are compared by computing L₁ distance between their distribution functions. From this idea and numerical computations, UMVUE appears to be good.     

Robust Coordinate Descent Algorithm Robust Solution Path for High-dimensional Sparse Regression Modeling

The L₁-type regularization provides a useful tool for variable selection in high-dimensional regression modeling. Various algorithms have been proposed to solve optimization problems for L₁-type regularization. Especially the coordinate descent algorithm has been shown to be effective in sparse regression modeling. Although the algorithm shows a remarkable performance to solve optimization problems for L₁-type regularization, it suffers from outliers, since the procedure is based on the inner product of predictor variables and partial residuals obtained from a non-robust manner. To overcome this drawback, we propose a robust coordinate descent algorithm, especially focusing on the high-dimensional regression modeling based on the principal components space. We show that the proposed robust algorithm converges to the minimum value of its objective function. Monte Carlo experiments and real data analysis are conducted to examine the efficiency of the proposed robust algorithm. We observe that our robust coordinate descent algorithm effectively performs for the high-dimensional regression modeling even in the presence of outliers.     

Robust Estimation for Zero‐Inflated Poisson Regression

Abstract. The zero‐inflated Poisson regression model is a special case of finite mixture models that is useful for count data containing many zeros. Typically, maximum likelihood (ML) estimation is used for fitting such models. However, it is well known that the ML estimator is highly sensitive to the presence of outliers and can become unstable when mixture components are poorly separated. In this paper, we propose an alternative robust estimation approach, robust expectation‐solution (RES) estimation. We compare the RES approach with an existing robust approach, minimum Hellinger distance (MHD) estimation. Simulation results indicate that both methods improve on ML when outliers are present and/or when the mixture components are poorly separated. However, the RES approach is more efficient in all the scenarios we considered. In addition, the RES method is shown to yield consistent and asymptotically normal estimators and, in contrast to MHD, can be applied quite generally.     

Some theory and practical uses of trimmed L-moments

Trimmed L-moments, defined by Elamir and Seheult [2003. Trimmed L-moments. Comput. Statist. Data Anal. 43, 299–314], summarize the shape of probability distributions or data samples in a way that remains viable for heavy-tailed distributions, even those for which the mean may not exist. We derive some further theoretical results concerning trimmed L-moments: a relation with the expansion of the quantile function as a weighted sum of Jacobi polynomials; the bounds that must be satisfied by trimmed L-moments; recurrences between trimmed L-moments with different degrees of trimming; and the asymptotic distributions of sample estimators of trimmed L-moments. We also give examples of how trimmed L-moments can be used, analogously to L-moments, in the analysis of heavy-tailed data. Examples include identification of distributions using a trimmed L-moment ratio diagram, shape parameter estimation for the generalized Pareto distribution, and fitting generalized Pareto distributions to a heavy-tailed data sample of computer network traffic.     

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.     

Robust ridge and robust Liu estimator for regression based on the LTS estimator

In the multiple linear regression analysis, the ridge regression estimator and the Liu estimator are often used to address multicollinearity. Besides multicollinearity, outliers are also a problem in the multiple linear regression analysis. We propose new biased estimators based on the least trimmed squares (LTS) ridge estimator and the LTS Liu estimator in the case of the presence of both outliers and multicollinearity. For this purpose, a simulation study is conducted in order to see the difference between the robust ridge estimator and the robust Liu estimator in terms of their effectiveness; the mean square error. In our simulations, the behavior of the new biased estimators is examined for types of outliers: X-space outlier, Y-space outlier, and X-and Y-space outlier. The results for a number of different illustrative cases are presented. This paper also provides the results for the robust ridge regression and robust Liu estimators based on a real-life data set combining the problem of multicollinearity and outliers.     

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.     

Tukey’s M-estimator of the Poisson parameter with a special focus on small means

We treat robust M-estimators for independent and identically distributed Poisson data. We introduce modified Tukey M-estimators with bias correction and compare them to M-estimators based on the Huber function as well as to weighted likelihood and other estimators by simulation in case of clean data and data with outliers. In particular, we investigate the problem of combining robustness and high efficiencies at small Poisson means caused by the strong asymmetry of such Poisson distributions and propose a further estimator based on adaptive trimming. The advantages of the constructed estimators are illustrated by an application to smoothing count data with a time varying mean and level shifts.     

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.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.     

The optimal Lp norm estimator in linear regression models

The least squares estimator is usually applied when estimating the parameters in linear regression models. As this estimator is sensitive to departures from normality in the residual distribution, several alternatives have been proposed. The L_p norm estimators is one class of such alternatives. It has been proposed that the kurtosis of the residual distribution be taken into account when a choice of estimator in the L_p norm class is made (i.e. the choice of p). In this paper, the asymtotic variance of the estimators is used as the criterion in the choice of p. It is shown that when this criterion is applied, other characteristics of the residual distribution than the kurtosis (namely moments of order p-2 and 2p-2) are important.     

New robust estimators for detecting non-random patterns in multivariate control charts: a simulation approach

In the past decade, different robust estimators have been proposed by several researchers to improve the ability to detect non-random patterns such as trend, process mean shift, and outliers in multivariate control charts. However, the use of the sample mean vector and the mean square successive difference matrix in the T ² control chart is sensitive in detecting process mean shift or trend but less sensitive in detecting outliers. On the other hand, the minimum volume ellipsoid (MVE) estimators in the T ² control chart are sensitive in detecting multiple outliers but less sensitive in detecting trend or process mean shift. Therefore, new robust estimators using both merits of the mean square successive difference matrix and the MVE estimators are developed to modify Hotelling's T ² control chart. To compare the detection performance among various control charts, a simulation approach for establishing control limits and calculating signal probabilities is provided as well. Our simulation results show that a multivariate control chart using the new robust estimators can achieve a well-balanced sensitivity in detecting the above-mentioned non-random patterns. Finally, three numerical examples further demonstrate the usefulness of our new robust estimators.     

A comparison of robust alternatives to Hotelling’s T 2 control chart

Control charts are one of the widest used techniques in statistical process control. In Phase I, historical observations are analysed in order to construct a control chart. Because of the existence of multiple outliers that are undetected by control charts such as Hotelling’s T ² due to the masking effect, robust alternatives to Hotelling’s T ² have been developed based on minimum volume ellipsoid (MVE) estimators, minimum covariance determinant (MCD) estimators, reweighted MCD estimators or trimmed estimators. In this paper, we use a simulation study to analyse the performance of each alternative in various situations and offer guidance for the correct use of each estimator.