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Register a new userA pairwise likelihood approach for the empirical estimation of the underlyingvariograms in the plurigaussian models
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Added by
Nicolas Desassis
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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The plurigaussian model is particularly suited to describe categorical regionalized variables. Starting from a simple principle, the thresh-olding of one or several Gaussian random fields (GRFs) to obtain categories, the plurigaussian model is well adapted for a wide range ofsituations. By acting on the form of the thresholding rule and/or the threshold values (which can vary along space) and the variograms ofthe underlying GRFs, one can generate many spatial configurations for the categorical variables. One difficulty is to choose variogrammodel for the underlying GRFs. Indeed, these latter are hidden by the truncation and we only observe the simple and cross-variogramsof the category indicators. In this paper, we propose a semiparametric method based on the pairwise likelihood to estimate the empiricalvariogram of the GRFs. It provides an exploratory tool in order to choose a suitable model for each GRF and later to estimate its param-eters. We illustrate the efficiency of the method with a Monte-Carlo simulation study .The method presented in this paper is implemented in the R packageRGeostats.
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The saddlepoint approximation gives an approximation to the density of a random variable in terms of its moment generating function. When the underlying random variable is itself the sum of $n$ unobserved i.i.d. terms, the basic classical result is that the relative error in the density is of order $1/n$. If instead the approximation is interpreted as a likelihood and maximised as a function of model parameters, the result is an approximation to the maximum likelihood estimate (MLE) that can be much faster to compute than the true MLE. This paper proves the analogous basic result for the approximation error between the saddlepoint MLE and the true MLE: subject to certain explicit identifiability conditions, the error has asymptotic size $O(1/n^2)$ for some parameters, and $O(1/n^{3/2})$ or $O(1/n)$ for others. In all three cases, the approximation errors are asymptotically negligible compared to the inferential uncertainty. The proof is based on a factorisation of the saddlepoint likelihood into an exact and approximate term, along with an analysis of the approximation error in the gradient of the log-likelihood. This factorisation also gives insight into alternatives to the saddlepoint approximation, including a new and simpler saddlepoint approximation, for which we derive analogous error bounds. As a corollary of our results, we also obtain the asymptotic size of the MLE error approximation when the saddlepoint approximation is replaced by the normal approximation.
The James-Stein estimator is an estimator of the multivariate normal mean and dominates the maximum likelihood estimator (MLE) under squared error loss. The original work inspired great interest in developing shrinkage estimators for a variety of problems. Nonetheless, research on shrinkage estimation for manifold-valued data is scarce. In this paper, we propose shrinkage estimators for the parameters of the Log-Normal distribution defined on the manifold of $N times N$ symmetric positive-definite matrices. For this manifold, we choose the Log-Euclidean metric as its Riemannian metric since it is easy to compute and is widely used in applications. By using the Log-Euclidean distance in the loss function, we derive a shrinkage estimator in an analytic form and show that it is asymptotically optimal within a large class of estimators including the MLE, which is the sample Frechet mean of the data. We demonstrate the performance of the proposed shrinkage estimator via several simulated data experiments. Furthermore, we apply the shrinkage estimator to perform statistical inference in diffusion magnetic resonance imaging problems.
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We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributions by deriving upper and lower bounds for the contraction rates of the corresponding empirical and hierarchical Bayes posterior distributions.
We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=expvarphi_0$ where $varphi_0$ is a concave function on $mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the lower invelope of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $varphi_0=log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.
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