Parametric Distribution (parametric + distribution)

Distribution by Scientific Domains


Selected Abstracts


A unified approach to estimation of nonlinear mixed effects and Berkson measurement error models

THE CANADIAN JOURNAL OF STATISTICS, Issue 2 2007
Liqun Wang
Abstract Mixed effects models and Berkson measurement error models are widely used. They share features which the author uses to develop a unified estimation framework. He deals with models in which the random effects (or measurement errors) have a general parametric distribution, whereas the random regression coefficients (or unobserved predictor variables) and error terms have nonparametric distributions. He proposes a second-order least squares estimator and a simulation-based estimator based on the first two moments of the conditional response variable given the observed covariates. He shows that both estimators are consistent and asymptotically normally distributed under fairly general conditions. The author also reports Monte Carlo simulation studies showing that the proposed estimators perform satisfactorily for relatively small sample sizes. Compared to the likelihood approach, the proposed methods are computationally feasible and do not rely on the normality assumption for random effects or other variables in the model. Une stratégie d'estimation commune pour les modèles non linéaires à effets mixtes et les modèles d'erreur de mesure de Berkson Les modèles à effets mixtes et les modèles d'erreur de mesure de Berkson sont très usités. Ils par-tagent certaines caractéristiques que l'auteur met à profit pour élaborer une stratégie d'estimation commune. II considère des modèles dans lesquels la loi des effets aléatoires (ou des erreurs de mesure) est paramé-trique tandis que celles des coefficients de régression aléatoires (ou de variables exogènes non observées) et des termes d'erreur ne le sont pas. II propose une estimation des moindres carrés au second ordre et une approche par simulation fondées sur les deux premiers moments conditionnels de la variable endogène, sachant les variables exogènes observées. Les deux estimateurs s'avèrent convergents et asymptotiquement gaussiens sous des conditions assez générales. L'auteur fait aussi état d'études de Monte-Carlo attestant du bon comportement des deux estimations dans des échantillons relativement petits. Les méthodes proposées ne posent aucune difficulté particulière au plan numérique et au contraire de l'approche par vraisemblance, ne supposent ni la normalité des effets aléatoires, ni celle des autres variables du modèle. [source]


Efficiency measure, modelling and estimation in combined array designs

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 4 2003
Tak Mak
Abstract In off-line quality control, the settings that minimize the variance of a quality characteristic are unknown and must be determined based on an estimated dual response model of mean and variance. The present paper proposes a direct measure of the efficiency of any given design-estimation procedure for variance minimization. This not only facilitates the comparison of different design-estimation procedures, but may also provide a guideline for choosing a better solution when the estimated dual response model suggests multiple solutions. Motivated by the analysis of an industrial experiment on spray painting, the present paper also applies a class of link functions to model process variances in off-line quality control. For model fitting, a parametric distribution is employed in updating the variance estimates used in an iteratively weighted least squares procedure for mean estimation. In analysing combined array experiments, Engel and Huele (Technometrics, 1996; 39:365) used log-link to model process variances and considered an iteratively weighted least squares leading to the pseudo-likelihood estimates of variances as discussed in Carroll and Ruppert (Transformation and Weighting in Regression, Chapman & Hall: New York). Their method is a special case of the approach considered in this paper. It is seen for the spray paint data that the log-link may not be satisfactory and the class of link functions considered here improves substantially the fit to process variances. This conclusion is reached with a suggested method of comparing ,empirical variances' with the ,theoretical variances' based on the assumed model. Copyright © 2003 John Wiley & Sons, Ltd. [source]


Finite Mixture Models for Mapping Spatially Dependent Disease Counts

BIOMETRICAL JOURNAL, Issue 1 2009
Marco Alfó
Abstract A vast literature has recently been concerned with the analysis of variation in disease counts recorded across geographical areas with the aim of detecting clusters of regions with homogeneous behavior. Most of the proposed modeling approaches have been discussed for the univariate case and only very recently spatial models have been extended to predict more than one outcome simultaneously. In this paper we extend the standard finite mixture models to the analysis of multiple, spatially correlated, counts. Dependence among outcomes is modeled using a set of correlated random effects and estimation is carried out by numerical integration through an EM algorithm without assuming any specific parametric distribution for the random effects. The spatial structure is captured by the use of a Gibbs representation for the prior probabilities of component membership through a Strauss-like model. The proposed model is illustrated using real data (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


Many zeros does not mean zero inflation: comparing the goodness-of-fit of parametric models to multivariate abundance data

ENVIRONMETRICS, Issue 3 2005
David I. Warton
Abstract An important step in studying the ecology of a species is choosing a statistical model of abundance; however, there has been little general consideration of which statistical model to use. In particular, abundance data have many zeros (often 50,80 per cent of all values), and zero-inflated count distributions are often used to specifically model the high frequency of zeros in abundance data. However, in such cases it is often taken for granted that a zero-inflated model is required, and the goodness-of-fit to count distributions with and without zero inflation is not often compared for abundance data. In this article, the goodness-of-fit was compared for several marginal models of abundance in 20 multivariate datasets (a total of 1672 variables across all datasets) from different sources. Multivariate abundance data are quite commonly collected in applied ecology, and the properties of these data may differ from abundances collected in autecological studies. Goodness-of-fit was assessed using AIC values, graphs of observed vs expected proportion of zeros in a dataset, and graphs of the sample mean,variance relationship. The negative binomial model was the best fitting of the count distributions, without zero-inflation. The high frequency of zeros was well described by the systematic component of the model (i.e. at some places predicted abundance was high, while at others it was zero) and so it was rarely necessary to modify the random component of the model (i.e. fitting a zero-inflated distribution). A Gaussian model based on transformed abundances fitted data surprisingly well, and rescaled per cent cover was usually poorly fitted by a count distribution. In conclusion, results suggest that the high frequency of zeros commonly seen in multivariate abundance data is best considered to come from distributions where mean abundance is often very low (hence there are many zeros), as opposed to claiming that there are an unusually high number of zeros compared to common parametric distributions. Copyright © 2005 John Wiley & Sons, Ltd. [source]


Memoirs of an indifferent trader: Estimating forecast distributions from prediction markets

QUANTITATIVE ECONOMICS, Issue 1 2010
Joyce E. Berg
C11; C93; D8; G1 Prediction markets for future events are increasingly common and they often trade several contracts for the same event. This paper considers the distribution of a normative risk-neutral trader who, given any portfolio of contracts traded on the event, would choose not to reallocate that portfolio of contracts even if transactions costs were zero. Because common parametric distributions can conflict with observed prediction market prices, the distribution is given a nonparametric representation together with a prior distribution favoring smooth and concentrated distributions. Posterior modal distributions are found for popular vote shares of the U.S. presidential candidates in the 100 days leading up to the elections of 1992, 1996, 2000, and 2004, using bid and ask prices on multiple contracts from the Iowa Electronic Markets. On some days, the distributions are multimodal or substantially asymmetric. The derived distributions are more concentrated than the historical distribution of popular vote shares in presidential elections, but do not tend to become more concentrated as time to elections diminishes. [source]


Test of Marginal Compatibility and Smoothing Methods for Exchangeable Binary Data with Unequal Cluster Sizes

BIOMETRICS, Issue 1 2007
Zhen Pang
Summary Exchangeable binary data are often collected in developmental toxicity and other studies, and a whole host of parametric distributions for fitting this kind of data have been proposed in the literature. While these distributions can be matched to have the same marginal probability and intra-cluster correlation, they can be quite different in terms of shape and higher-order quantities of interest such as the litter-level risk of having at least one malformed fetus. A sensible alternative is to fit a saturated model (Bowman and George, 1995, Journal of the American Statistical Association90, 871,879) using the expectation-maximization (EM) algorithm proposed by Stefanescu and Turnbull (2003, Biometrics59, 18,24). The assumption of compatibility of marginal distributions is often made to link up the distributions for different cluster sizes so that estimation can be based on the combined data. Stefanescu and Turnbull proposed a modified trend test to test this assumption. Their test, however, fails to take into account the variability of an estimated null expectation and as a result leads to inaccurate p -values. This drawback is rectified in this article. When the data are sparse, the probability function estimated using a saturated model can be very jagged and some kind of smoothing is needed. We extend the penalized likelihood method (Simonoff, 1983, Annals of Statistics11, 208,218) to the present case of unequal cluster sizes and implement the method using an EM-type algorithm. In the presence of covariate, we propose a penalized kernel method that performs smoothing in both the covariate and response space. The proposed methods are illustrated using several data sets and the sampling and robustness properties of the resulting estimators are evaluated by simulations. [source]