Robust quasi-likelihood inference in generalized linear mixed models with outliers

It is well known that in a traditional outlier-free situation, the generalized quasi-likelihood (GQL) approach [B.C. Sutradhar, On exact quasilikelihood inference in generalized linear mixed models, Sankhya: Indian J. Statist. 66 (2004), pp. 261–289] performs very well to obtain the consistent as well as the efficient estimates for the parameters involved in the generalized linear mixed models (GLMMs). In this paper, we first examine the effect of the presence of one or more outliers on the GQL estimation for the parameters in such GLMMs, especially in two important models such as count and binary mixed models. The outliers appear to cause serious biases and hence inconsistency in the estimation. As a remedy, we then propose a robust GQL (RGQL) approach in order to obtain the consistent estimates for the parameters in the GLMMs in the presence of one or more outliers. An extensive simulation study is conducted to examine the consistency performance of the proposed RGQL approach.     

Robust regression for estimating the Burr XII parameters with outliers

The Burr XII distribution offers a more flexible alternative to the lognormal, log-logistic and Weibull distributions. Outliers can occur during reliability life testing. Thus, we need an efficient method to estimate the parameters of the Burr XII distribution for censored data with outliers. The objective of this paper is to present a robust regression (RR) method called M-estimator to estimate the parameters of a two-parameter Burr XII distribution based on the probability plotting procedure for both the complete and multiply-censored data with outliers. The simulation results show that the RR method outperforms the unweighted least squares and maximum likelihood methods in most cases in terms of bias and errors in the root mean square.     

Robust estimation in long-memory processes under additive outliers

In this paper, we introduce an alternative semiparametric estimator of the fractional differencing parameter in ARFIMA models which is robust against additive outliers. The proposed estimator is a variant of the GPH estimator [Geweke, J., Porter-Hudak, S., 1983. The estimation and application of long memory time series model. Journal of Time Series Analysis 4, 221–238]. In particular, we use the robust sample autocorrelations of Ma, Y. and Genton, M. [2000. Highly robust estimation of the autocovariance function. Journal of Time Series Analysis 21, 663–684] to obtain an estimator for the spectral density of the process. Numerical results show that the estimator we propose for the differencing parameter is robust when the data contain additive outliers.     

Robust non-parametric smoothing of non-stationary time series

Motivated by the need of extracting local trends and low frequency components in non-stationary time series, this paper discusses methods of robust non-parametric smoothing. Basic approach is the combination of the parametric M-estimation with kernel and local polynomial regression methods. The result is an iterative estimator that retains a linear structure, but has kernel weights also in the direction of the prediction errors. The design of smoothing coefficients is carried out with robust cross-validation criteria and rules of thumb. The method works well both to remove the influence of patches of outliers and to detect the local breaks and persistent structural change in time series.     

Monte Carlo Kalman filter and smoothing for multivariate discrete state space models

The author studies state space models for multivariate binomial time series, focussing on the development of the Kalman filter and smoothing for state variables. He proposes a Monte Carlo approach employing the latent variable representation which transplants the classical Kalman filter and smoothing developed for Gaussian state space models to discrete models and leads to a conceptually simple and computationally convenient approach. The method is illustrated through simulations and concrete examples.     

Robust estimation of AR coefficients under simultaneously influencing outliers and missing values

A family of robust estimators for coefficients of Gaussian AR(p) time series under simultaneously influencing distortions of two types: outliers and missing values, is proposed. The estimators are based on special properties of the Cauchy probability distribution; consistency and the asymptotic normality of these estimators are proven. An approximate solution of the problem of minimization of the asymptotic variance within the proposed family of estimators is found. Performance of the proposed estimators is illustrated for simulated time series and for real data sets.     

Local smoothing with given marginals

In models using categorical data, one may use adjacency relations to justify smoothing to improve upon simple histogram approximations of the probabilities. This is particularly convenient for sparsely observed or rather peaked distributions. Moreover, in a few models, prior knowledge of a marginal distribution is available. We adapt local polynomial estimators to include this partial information about the underlying distribution and give explicit representations for the proposed estimators. An application to a set of anthropological data is included.     

Binomial AR(1) processes with innovational outliers

Abstract

Binomial integer-valued AR processes have been well studied in the literature, but there is little progress in modeling bounded integer-valued time series with outliers. In this paper, we first review some basic properties of the binomial integer-valued AR(1) process and then we introduce binomial integer-valued AR(1) processes with two classes of innovational outliers. We focus on the joint conditional least squares (CLS) and the joint conditional maximum likelihood (CML) estimates of models’ parameters and the probability of occurrence of the outlier. Their large-sample properties are illustrated by simulation studies. Artificial and real data examples are used to demonstrate good performances of the proposed models.     

Nonlinear regression models for heterogeneous data with massive outliers

The income or expenditure-related data sets are often nonlinear, heteroscedastic, skewed even after the transformation, and contain numerous outliers. We propose a class of robust nonlinear models that treat outlying observations effectively without removing them. For this purpose, case-specific parameters and a related penalty are employed to detect and modify the outliers systematically. We show how the existing nonlinear models such as smoothing splines and generalized additive models can be robustified by the case-specific parameters. Next, we extend the proposed methods to the heterogeneous models by incorporating unequal weights. The details of estimating the weights are provided. Two real data sets and simulated data sets show the potential of the proposed methods when the nature of the data is nonlinear with outlying observations.     

Spuriosity and outliers in directional data

This paper discusses inference regarding the mean direction and the concentration parameters based on data from the von Mises distribution from a Bayesian point of view, when k(k < n/2) of the n observations are spurious, that is, are from a von Mises population with a shifted mean direction. The Bayesian analysis for this spuriosity case provides both detection, identification, and estimation for the mean direction and the concentration parameter when indeed spurious observations are present, possibly giving rise to outliers.     

A note on contamination models and outliers

In order to describe or generate so-called outliers in univariate statistical data, contamination models are often used. These models assume that k out of n independent random variables are shifted or multiplicated by some constant, whereas the other observations still come i.i.d. from some common target distribution. Of course, these contaminants do not necessarily stick out as the extremes in the sample. Moreover, it is the amount and magnitude of ‘contamination” which determines the number of obvious outliers. Using the concept of Davies and Gather (1993) to formalize the outlier notion we quantify the amount of contamination needed to produce a prespecified expected number of ‘genuine’ outliers. In particular, we demonstrate that for sample of moderate size from a normal target distribution a rather large shift of the contaminants is necessary to yield a certain expected number of outliers. Such an insight is of interest when designing simulation studies where outliers shoulod occur as well as in theoretical investigations on outliers.     

Estimating curves and derivatives with parametric penalized spline smoothing

Accurate estimation of an underlying function and its derivatives is one of the central problems in statistics. Parametric forms are often proposed based on the expert opinion or prior knowledge of the underlying function. However, these strict parametric assumptions may result in biased estimates when they are not completely accurate. Meanwhile, nonparametric smoothing methods, which do not impose any parametric form, are quite flexible. We propose a parametric penalized spline smoothing method, which has the same flexibility as the nonparametric smoothing methods. It also uses the prior knowledge of the underlying function by defining an additional penalty term using the distance of the fitted function to the assumed parametric function. Our simulation studies show that the parametric penalized spline smoothing method can obtain more accurate estimates of the function and its derivatives than the penalized spline smoothing method. The parametric penalized spline smoothing method is also demonstrated by estimating the human height function and its derivatives from the real data.     

Modelling and smoothing parameter estimation with multiple quadratic penalties

Penalized likelihood methods provide a range of practical modelling tools, including spline smoothing, generalized additive models and variants of ridge regression. Selecting the correct weights for penalties is a critical part of using these methods and in the single-penalty case the analyst has several well-founded techniques to choose from. However, many modelling problems suggest a formulation employing multiple penalties, and here general methodology is lacking. A wide family of models with multiple penalties can be fitted to data by iterative solution of the generalized ridge regression problem minimize || W ^1/2 ( Xp − y ) ||²ρ+Σ _{i =1} ^m  θ _i p ' S _i p ( p is a parameter vector, X a design matrix, S _i a non-negative definite coefficient matrix defining the i th penalty with associated smoothing parameter θ _i , W a diagonal weight matrix, y a vector of data or pseudodata and ρ an 'overall' smoothing parameter included for computational efficiency). This paper shows how smoothing parameter selection can be performed efficiently by applying generalized cross-validation to this problem and how this allows non-linear, generalized linear and linear models to be fitted using multiple penalties, substantially increasing the scope of penalized modelling methods. Examples of non-linear modelling, generalized additive modelling and anisotropic smoothing are given.     

A modified ridge m-estimator for linear regression model with multicollinearity and outliers

The ordinary least-square estimators for linear regression analysis with multicollinearity and outliers lead to unfavorable results. In this article, we propose a new robust modified ridge M-estimator (MRME) based on M-estimator (ME) to deal with the combined problem resulting from multicollinearity and outliers in the y-direction. MRME outperforms modified ridge estimator, robust ridge estimator and ME, according to mean squares error criterion. Furthermore, a numerical example and a Monte Carlo simulation experiment are given to illustrate some of the theoretical results.     

Multivariate outliers and decompositions of mahalanobis distance

Two decompositions of the Mahalanobis distance are considered. These decompositions help to explain some reasons for the outlyingness of multivari- ate observations. They also provide a graphical tool for identifying outliers including those that have a large influence on the multiple correlation coefficient, Illustrative examples are given.     

Nonparametric particle filtering and smoothing with quasi-Monte Carlo sampling

Sequential Monte Carlo methods (also known as particle filters and smoothers) are used for filtering and smoothing in general state-space models. These methods are based on importance sampling. In practice, it is often difficult to find a suitable proposal which allows effective importance sampling. This article develops an original particle filter and an original particle smoother which employ nonparametric importance sampling. The basic idea is to use a nonparametric estimate of the marginally optimal proposal. The proposed algorithms provide a better approximation of the filtering and smoothing distributions than standard methods. The methods’ advantage is most distinct in severely nonlinear situations. In contrast to most existing methods, they allow the use of quasi-Monte Carlo (QMC) sampling. In addition, they do not suffer from weight degeneration rendering a resampling step unnecessary. For the estimation of model parameters, an efficient on-line maximum-likelihood (ML) estimation technique is proposed which is also based on nonparametric approximations. All suggested algorithms have almost linear complexity for low-dimensional state-spaces. This is an advantage over standard smoothing and ML procedures. Particularly, all existing sequential Monte Carlo methods that incorporate QMC sampling have quadratic complexity. As an application, stochastic volatility estimation for high-frequency financial data is considered, which is of great importance in practice. The computer code is partly available as supplemental material.     

Online monitoring with local smoothing methods and adaptive ridging

We consider online monitoring of sequentially arising data as e.g. met in clinical information systems. The general focus thereby is to detect breakpoints, i.e. timepoints where the measurement series suddenly changes the general level. The method suggested is based on local estimation. In particular, local linear smoothing is combined by ridging with local constant smoothing. The procedure is demonstrated by examples and compared with other available online monitoring routines.     

Approximate bounded influence estimation for longitudinal data with outliers and measurement errors

Mixed effects models or random effects models are popular for the analysis of longitudinal data. In practice, longitudinal data are often complex since there may be outliers in both the response and the covariates and there may be measurement errors. The likelihood method is a common approach for these problems but it can be computationally very intensive and sometimes may even be computationally infeasible. In this article, we consider approximate robust methods for nonlinear mixed effects models to simultaneously address outliers and measurement errors. The approximate methods are computationally very efficient. We show the consistency and asymptotic normality of the approximate estimates. The methods can also be extended to missing data problems. An example is used to illustrate the methods and a simulation is conducted to evaluate the methods.