Harmonic mean approach to uunbalanced random effects models under heteroscedasticity

In this paper we consider unbalanced random effects models under heteroscedastic variances. By using' the harmonic mean approach, it is shown that the problems are analogous to those from balanced random effects models under horaoscedastic variances. Thus, by using the harmonic mean approach, statistical inferences about variance components are derived by using procedures from balanced models under homoscedastic variances. Laguerre polynomial expansion is used to approximate the sampling distributions of relevant statistics.     

On quadratic estimation of heteroscedastic variances

Beginning with a brief introduction to the general theory the concept of Bayes invariant quadratic unbiased estimators (BAIQUEs) founded by Kleffe and Pingus(1974)is applied to combined samples with a common mean and different variances.Explicite formulas for Baique under these special assumptions are derived.Finally,some numerical comparisons of the variance function of Baiques under different prior distributions are given.     

Robust replicated heteroscedastic measurement error model using heavy-tailed distribution

Heteroscedastic measurement error models are widely used in epidemiological, analytical chemistry, and other research areas. In this article, we propose a heteroscedastic measurement error model for replicated data under scale mixtures of normal distributions with/without equation error, which covers unpair and/or unequal replication cases. We obtain iterative formulas of maximum likelihood estimations via EM algorithm, and provide closed forms of asymptotic variances of the estimators. Simulation studies and a real data application are reported to investigate the effective and robust performances of the model and estimates.     

Multivariate spatial interpolation and exposure to air pollutants

We develop and apply an approach to the spatial interpolation of a vector-valued random response field. The Bayesian approach we adopt enables uncertainty about the underlying models to be représentés in expressing the accuracy of the resulting interpolants. The methodology is particularly relevant in environmetrics, where vector-valued responses are only observed at designated sites at successive time points. The theory allows space-time modelling at the second level of the hierarchical prior model so that uncertainty about the model parameters has been fully expressed at the first level. In this way, we avoid unduly optimistic estimates of inferential accuracy. Moreover, the prior model can be upgraded with any available new data, while past data can be used in a systematic way to fit model parameters. The theory is based on the multivariate normal and related joint distributions. Our hierarchical prior models lead to posterior distributions which are robust with respect to the choice of the prior (hyperparameters). We illustrate our theory with an example involving monitoring stations in southern Ontario, where monthly average levels of ozone, sulphate, and nitrate are available and between-station response triplets are interpolated. In this example we use a recently developed method for interpolating spatial correlation fields.     

Use of prior information for estimating the variance components

The MINQUE and its modifications are considered for estimating the variances of the balanced one-way random effects model. The effects of the a priori values on the estimators of the variances are examined in detail. The Mean Square Errors of the estimators are compared for variations in the prior values of the unknown variances.     

Saddlepoint approximations to tail expectations under non-Gaussian base distributions: option pricing applications

The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of the saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We also demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatility model.     

Impact of Model Choice on LR Assessment in Case of Rare Haplotype Match (Frequentist Approach)

The likelihood ratio (LR) measures the relative weight of forensic data regarding two hypotheses. Several levels of uncertainty arise if frequentist methods are chosen for its assessment: the assumed population model only approximates the true one, and its parameters are estimated through a limited database. Moreover, it may be wise to discard part of data, especially that only indirectly related to the hypotheses. Different reductions define different LRs. Therefore, it is more sensible to talk about ‘a’ LR instead of ‘the’ LR, and the error involved in the estimation should be quantified. Two frequentist methods are proposed in the light of these points for the ‘rare type match problem’, that is, when a match between the perpetrator's and the suspect's DNA profile, never observed before in the database of reference, is to be evaluated.     

Bayesian estimates in a one-way ANOVA random effects model

Bayesian estimators of variance components are developed, based on posterior mean and posterior mode, respectively, in a one-way ANOVA random effects model with independent prior distributions. The formulas for the proposed estimators are simple. The estimators give sensible results for 'badly-behaved' datasets, where the standard unbiased estimates are negative. They are markedly robust as compared to the existing estimators such as the maximum likelihood estimators and the maximum posterior density estimators.     

LR cointegration tests when some cointegrating relations are known

This paper considers the asymptotic analysis of the likelihood ratio (LR), cointegration (CI) rank test in vector autoregressive models (VAR) when some CI vectors are known and fixed. It is shown that the limit law is free of nuisance parameters. In the case of LR tests against the alternative of completely unrestricted CI space, the limit law can be expressed as the convolution of known distributions. This deconvolution is employed to approximate the quantiles of the distribution, without resorting to new simulations.     

A Comparison of Bayesian Models of Heteroscedasticity in Nested Normal Data

We consider the fitting of a Bayesian model to grouped data in which observations are assumed normally distributed around group means that are themselves normally distributed, and consider several alternatives for accommodating the possibility of heteroscedasticity within the data. We consider the case where the underlying distribution of the variances is unknown, and investigate several candidate prior distributions for those variances. In each case, the parameters of the candidate priors (the hyperparameters) are themselves given uninformative priors (hyperpriors). The most mathematically convenient model for the group variances is to assign them inverse gamma distributed priors, the inverse gamma distribution being the conjugate prior distribution for the unknown variance of a normal population. We demonstrate that for a wide class of underlying distributions of the group variances, a model that assigns the variances an inverse gamma-distributed prior displays favorable goodness-of-fit properties relative to other candidate priors, and hence may be used as standard for modeling such data. This allows us to take advantage of the elegant mathematical property of prior conjugacy in a wide variety of contexts without compromising model fitness. We test our findings on nine real world publicly available datasets from different domains, and on a wide range of artificially generated datasets.     

Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

Testing for departures from nominal dispersion in generalized nonlinear models with varying dispersion and/or additive random effects

This paper discusses the tests for departures from nominal dispersion in the framework of generalized nonlinear models with varying dispersion and/or additive random effects. We consider two classes of exponential family distributions. The first is discrete exponential family distributions, such as Poisson, binomial, and negative binomial distributions. The second is continuous exponential family distributions, such as normal, gamma, and inverse Gaussian distributions. Correspondingly, we develop a unifying approach and propose several tests for testing for departures from nominal dispersion in two classes of generalized nonlinear models. The score test statistics are constructed and expressed in simple, easy to use, matrix formulas, so that the tests can easily be implemented using existing statistical software. The properties of test statistics are investigated through Monte Carlo simulations.     

Estimation of parameters in DNA mixture analysis

In [7], a Bayesian network for analysis of mixed traces of DNA was presented using gamma distributions for modelling peak sizes in the electropherogram. It was demonstrated that the analysis was sensitive to the choice of a variance factor and hence this should be adapted to any new trace analysed. In this paper, we discuss how the variance parameter can be estimated by maximum likelihood to achieve this. The unknown proportions of DNA from each contributor can similarly be estimated by maximum likelihood jointly with the variance parameter. Furthermore, we discuss how to incorporate prior knowledge about the parameters in a Bayesian analysis. The proposed estimation methods are illustrated through a few examples of applications for calculating evidential value in casework and for mixture deconvolution.     

Sensitivity of a bayesian inference to prior assumptions

The sensitivity of-a Bayesian inference to prior assumptions is examined by Monte Carlo simulation for the beta-binomial conjugate family of distributions. Results for the effect on a Bayesian probability interval of the binomial parameter indicate that the Bayesian inference is for the most part quite sensitive to misspecification of the prior distribution. The magnitude of the sensitivity depends primarily on the difference of assigned means and variances from the respective means and variances of the actually-sampled prior distributions. The effect of a disparity in form between the assigned prior and actually-sampled distributions was less important for the cases tested.     

Likelihood-based estimation for Gaussian MRFs

Markov random fields (MRFs) express spatial dependence through conditional distributions, although their stochastic behavior is defined by their joint distribution. These joint distributions are typically difficult to obtain in closed form, the problem being a normalizing constant that is a function of unknown parameters. The Gaussian MRF (or conditional autoregressive model) is one case where the normalizing constant is available in closed form; however, when sample sizes are moderate to large (thousands to tens of thousands), and beyond, its computation can be problematic. Because the conditional autoregressive (CAR) model is often used for spatial-data modeling, we develop likelihood-inference methodology for this model in situations where the sample size is too large for its normalizing constant to be computed directly. In particular, we use simulation methodology to obtain maximum likelihood estimators of mean, variance, and spatial-depencence parameters (including their asymptotic variances and covariances) of CAR models.     

ROBUST COMPARISON OF DISPERSIONS FOR SAMPLES OF DIRECTIONAL DATA

Robust methods are proposed for testing whether several directional distributions on the unit p-sphere have comparable dispersions. The families of distributions considered are the Langevin for random vectors, and the Generalised Scheidegger-Watson for random axes, with specific interest in the Fisher and Watson distributions on the sphere. The methods are analogues of Levene's procedure for comparing variances of normal distributions.     

On the distributions of components sums of squares in one-way unbalanced random model under heter06ene0us error variances

Exact sampling distributions of sums of squares in the unbalanced one-way random model are obtained under heterogeneous error variances. These distributions are used to investigate the effect of heteroscedasticity and unbalancedness on the probability of negative estimate of the group variance component. The computed results reveal that heteroscedasticity affects the probability of negative estimate in all situations of group sizes. Further, the probability decreases with heterogeneity of error variances for balanced situations and increases with variability among group size for equal error variances case.     

Bayesian estimation methods for contingency tables

Summary A method of inputting prior opinion in contingency tables is described. The method can be used to incorporate beliefs of independence or symmetry but extensions are straightforward. Logistic normal distributions that express such beliefs are used as priors of the cell probabilities and posterior estimates are derived. Empirical Bayes methods are also discussed and approximate posterior variances are provided. The methods are illustrated by a numerical example.     

Bayes estimates of mixing proportions in finite mixture distributions

With reference to the problem of estimating the mixing proportions in a finite mixture distribution with known components, employing Dirichlet prior, closed form expressions for the posterior means and variances are obtained. To avoid the difficulties in computing the estimates, an approximation procedure is introduced. Numerical studies carried out for normal mixtures indicate the closeness of the approximations and their superiority over the maximum likelihood estimates at least in the case of small samples.     

Size distribution stereology for quasiellipsoids in r

Weakly stationary fields of random quasiellipsoids (rigid or flat ellipsoids) in Rⁿ are intersected with a fixed hyperplane H. The stereological problem consists in determining the size and shape distribution of a “typoical” quasiellipsoid of the sample by selectional data. The size is assumed to be independent of shape and directins. In general the problem cannot be solved uniquely (s.[1]). In the present paper the question is answered for which shape–direction distributions the stereological formulas for all size distributions are the same as in the well–known spherical case