Estimation of generalized linear latent variable models

Summary.  Generalized linear latent variable models (GLLVMs), as defined by Bartholomew and Knott, enable modelling of relationships between manifest and latent variables. They extend structural equation modelling techniques, which are powerful tools in the social sciences. However, because of the complexity of the log-likelihood function of a GLLVM, an approximation such as numerical integration must be used for inference. This can limit drastically the number of variables in the model and can lead to biased estimators. We propose a new estimator for the parameters of a GLLVM, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The new estimator can be viewed as an M -estimator, leading to readily available asymptotic properties and correct inference. A simulation study shows its excellent finite sample properties, in particular when compared with a well-established approach such as LISREL. A real data example on the measurement of wealth for the computation of multidimensional inequality is analysed to highlight the importance of the methodology.     

Approximation of density functions by orthogonal series with grouped data

Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕ_j} be a complete orthonormal sequence of functions onL ₂(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the firstm of the ϑ_j’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. Research partially supported by: CNR grant, n. 93. 00837. CT10.     

L p –Estimators as Estimates of a Parameter of Location for a Sharp–pointed Symmetric Density

We study the asymptotics of L _p estimators, p > 0, over a sample having a symmetric density with a sharp–point at the centre of symmetry of the distribution. The rates of convergence of the L _p estimators in this situation depend on p and on the shape of the density. To obtain some of the limit distributions, we present new results in the asymptotics of M–estimators. We extend the delta method to the case when the Euclidean norm of the conveniently normalized M–estimators converge to a power of the Euclidean norm of a (possibly Gaussian) stable distribution.     

Robust m-estimators

This paper provides a summary of the influence function approach to robust estimation of parametric models. Hampel's optimality results for M-estimators with a bounded influence function is generalized to allow for arbitrary choices of the asymptotic efficiency criterion and the norm of the influence function. Further extensions to other cases of practical interest are also considered.     

De l'unicité des estimateurs robustes en régression lorsque le paramètre d'échelle et le paramètre de la régression sont estimés simultanément

Several methods have been suggested to calculate robust M- and G-M -estimators of the regression parameter β and of the error scale parameter σ in a linear model. This paper shows that, for some data sets well known in robust statistics, the nonlinear systems of equations for the simultaneous estimation of β, with an M-estimate with a redescending ψ-function, and σ, with the residual median absolute deviation (MAD), have many solutions. This multiplicity is not caused by the possible lack of uniqueness, for redescending ψ-functions, of the solutions of the system defining β with known σ; rather, the simultaneous estimation of β and σ together creates the problem. A way to avoid these multiple solutions is to proceed in two steps. First take σ as the median absolute deviation of the residuals for a uniquely defined robust M-estimate such as Huber's Proposal 2 or the L₁-estimate. Then solve the nonlinear system for the M-estimate with σ equal to the value obtained at the first step to get the estimate of β. Analytical conditions for the uniqueness of M and G-M-estimates are also given.     

ROBUST ESTIMATION UNDER TYPE II CENSORING

The properties of robust M-estimators with type II censored failure time data are considered. The optimal members within two classes of ψ-functions are characterized. The first optimality result is the censored data analogue of the optimality result described in Hampel et al. (1986); the estimators corresponding to the optimal members within this class are referred to as the optimal robust estimators. The second result pertains to a restricted class of ψ-functions which is the analogue of the class of ψ-functions considered in James (1986) for randomly censored data; the estimators corresponding to the optimal members within this restricted class are referred to as the optimal James-type estimators. We examine the usefulness of the two classes of ψ-functions and find that the breakdown point and efficiency of the optimal James-type estimators compare favourably with those of the corresponding optimal robust estimators. From the computational point of view, the optimal James-type ψ-functions are readily obtainable from the optimal ψ-functions in the uncensored case. The ψ-functions for the optimal robust estimators require a separate algorithm which is provided. A data set illustrates the optimal robust estimators for the parameters of the extreme value distribution.     

M-Estimators of Scale with Minimum Gross Errors Sensitivity

The median absolute deviation (MAD) is known to be the M-estimator of scale with minimum gross errors sensitivity (GES) when the error distribution is known to be symmetric and strongly unimodal. The problem considered here is to find the Fisher consistent M-estimator with minimum GES when the error distribution is symmetric but not necessarily unimodal. Under some general conditions, the score function χ corresponding to the minimizing M-estimator has the form χ(x) = ?1 when |x| < a; χ(x) = c when a < |x| < b; χ(x) = 1 when |x| > b. An example is given in which the M-estimator with minimum GES does not correspond to the MAD.     

A Robust Procedure in Nonlinear Models for Repeated Measurements

Nonlinear regression models arise when definite information is available about the form of the relationship between the response and predictor variables. Such information might involve direct knowledge of the actual form of the true model or might be represented by a set of differential equations that the model must satisfy. We develop M-procedures for estimating parameters and testing hypotheses of interest about these parameters in nonlinear regression models for repeated measurement data. Under regularity conditions, the asymptotic properties of the M-procedures are presented, including the uniform linearity, normality and consistency. The computation of the M-estimators of the model parameters is performed with iterative procedures, similar to Newton–Raphson and Fisher's scoring methods. The methodology is illustrated by using a multivariate logistic regression model with real data, along with a simulation study.     

Robust M- and L-Estimators of Scale Parameter

We consider first the class of M-estimators of scale that are location-scale equivariant and Fisher consistent at the error distribution of the shrinking contamination neighborhood and derive an expression for the maximal asymptotic mean-squared-error, for a suitably regular score function, followed by a lower bound on it. We next show that the minimax asymptotic mean-squzred-error is attained at an M-estimator of scale with the truncated MLE score function which, when specialized to the Standard Normal error distribution has the form of Huber's Proposal 2. The latter minimax property is also shown to hold for α-trimmed variance as an L-estimator of scale.