Statistics for Spatial Data.

A survey of recent developments in the theory and application of com- posite likelihood is provided, building on the review paper of Varin(2008). A range of application areas, including geostatistics, spatial extremes, and space-time mod- els, as well as clustered and longitudinal data and time series are considered. The important area of applications to statistical genetics is omitted, in light ofLarribe and Fearnhead(2011). Emphasis is given to the development of the theory, and the current state of knowledge on e!ciency and robustness of composite likelihood inference.

/pdf/an-overview-of-composite-likelihood-methods-2lgbmk7ee2.pdf

An overview of composite likelihood methods

Composite fading (i.e., multipath fading and shadowing together) has increasingly been analyzed by means of the K channel and related models. Nevertheless, these models do have computational and analytical difficulties. Motivated by this context, we propose a mixture gamma (MG) distribution for the signal-to-noise ratio (SNR) of wireless channels. Not only is it a more accurate model for composite fading, but is also a versatile approximation for any fading SNR. As this distribution consists of N (≥ 1) component gamma distributions, we show how its parameters can be determined by using probability density function (PDF) or moment generating function (MGF) matching. We demonstrate the accuracy of the MG model by computing the mean square error (MSE) or the Kullback-Leibler (KL) divergence or by comparing the moments. With this model, performance metrics such as the average channel capacity, the outage probability, the symbol error rate (SER), and the detection capability of an energy detector are readily derived.

http://www.ece.ualberta.ca/~chintha/resources/papers/2011/6059452.pdf

A Mixture Gamma Distribution to Model the SNR of Wireless Channels

Constraint-based causal discovery (CCD) algorithms require fast and accurate conditional independence (CI) testing. The Kernel Conditional Independence Test (KCIT) is currently one of the most popular CI tests in the non-parametric setting, but many investigators cannot use KCIT with large datasets because the test scales at least quadratically with sample size. We therefore devise two relaxations called the Randomized Conditional Independence Test (RCIT) and the Randomized conditional Correlation Test (RCoT) which both approximate KCIT by utilizing random Fourier features. In practice, both of the proposed tests scale linearly with sample size and return accurate p-values much faster than KCIT in the large sample size context. CCD algorithms run with RCIT or RCoT also return graphs at least as accurate as the same algorithms run with KCIT but with large reductions in run time.

https://www.degruyter.com/document/doi/10.1515/jci-2018-0017/pdf

Approximate Kernel-Based Conditional Independence Tests for Fast Non-Parametric Causal Discovery

Composite likelihood may be useful for approximating likelihood based inference when the full likelihood is too complex to deal with. Stemming from a misspecified model, inference based on composite likelihood requires suitable corrections. Here we focus on the composite likelihood ratio statistic for a mul- tidimensional parameter of interest, and we propose a parameterization invariant adjustment that allows reference to the usual asymptotic chi-square distribution. Two examples dealing with pairwise likelihood are analysed through simulation.

/pdf/adjusting-composite-likelihood-ratio-statistics-2g333z1v5f.pdf

Adjusting composite likelihood ratio statistics

There are a number of cases where the moments of a distribution are easily obtained, but theoretical distributions are not available in closed form. This paper shows how to use moment methods to approximate a theoretical univariate distribution with mixtures of known distributions. The methods are illustrated with gamma mixtures. It is shown that for a certain class of mixture distributions, which include the normal and gamma mixture families, one can solve for a p-point mixing distribution such that the corresponding mixture has exactly the same first 2p moments as the targeted univariate distribution. The gamma mixture approximation to the distribution of a positive weighted sums of independent central χ2 variables is demonstrated and compared with a number of existing approximations. The numerical results show that the new approximation is generally superior to these alternatives.

Moment-Based Approximations of Distributions Using Mixtures: Theory and Applications

The focus of this article is on fitting regression models and testing of general linear hypotheses for correlated data using quasi-likelihood based techniques. The class of generalized method of moments or GMMs provides an elegant approach for estimating a vector of regression parameters from a set of score functions. Extending the principle of the GMMs, in the generalized estimating equation framework, leads to a quadratic inference function or QIF approach for the analysis of correlated data. We derive an iteratively reweighted generalized least squares or IRGLS algorithm for finding the QIF estimator and establish its convergence properties. A software library implementing the techniques is demonstrated through several datasets.

https://www.jstor.org/stable/info/27594282

Iteratively Reweighted Generalized Least Squares for Estimation and Testing With Correlated Data: An Inference Function Framework

In this research, inferential theory for hypothesis testing under general convex cone alternatives for correlated data is developed. While there exists extensive theory for hypothesis testing under smooth cone alternatives with independent observations, extension to correlated data under general convex cone alternatives remains an open problem. This long-pending problem is addressed by (1) establishing that a "generalized quasi-score" statistic is asymptotically equivalent to the squared length of the projection of the standard Gaussian vector onto the convex cone and (2) showing that the asymptotic null distribution of the test statistic is a weighted chi-squared distribution, where the weights are "mixed volumes" of the convex cone and its polar cone. Explicit expressions for these weights are derived using the volume-of-tube formula around a convex manifold in the unit sphere. Furthermore, an asymptotic lower bound is constructed for the power of the generalized quasi-score test under a sequence of local alternatives in the convex cone. Applications to testing under order restricted alternatives for correlated data are illustrated.

Inference Under Convex Cone Alternatives for Correlated Data

Hansen (1982) proposed a class of "generalized method of moments" (GMMs) for estimating a vector of regression parameters from a set of score functions. Hansen established that, under certain regularity conditions, the estimator based on the GMMs is consistent, asymptotically normal and asymptotically efficient. In the generalized estimating equation framework, extending the principle of the GMMs to implicitly estimate the underlying correlation structure leads to a "quadratic inference function" (QIF) for the analysis of correlated data. The main objectives of this research are to (1) formulate an appropriate estimated covariance matrix for the set of extended score functions defining the inference functions; (2) develop a unified large-sample theoretical framework for the QIF; (3) derive a generalization of the QIF test statistic for a general linear hypothesis problem involving correlated data while establishing the asymptotic distribution of the test statistic under the null and local alternative hypotheses; (4) propose an iteratively reweighted generalized least squares algorithm for inference in the QIF framework; and (5) investigate the effect of basis matrices, defining the set of extended score functions, on the size and power of the QIF test through Monte Carlo simulated experiments.

On large-sample estimation and testing via quadratic inference functions for correlated data

An important and yet difficult problem in fitting multivariate mixture models is determining the mixture complexity. We develop theory and a unified framework for finding the nonparametric maximum likelihood estimator of a multivariate mixing distribution and consequently estimating the mixture complexity. Multivariate mixtures provide a flexible approach to fitting high-dimensional data while offering data reduction through the number, location and shape of the component densities. The central principle of our method is to cast the mixture maximization problem in the concave optimization framework with finitely many linear inequality constraints and turn it into an unconstrained problem using a "penalty function". We establish the existence of parameter estimators and prove the convergence properties of the proposed algorithms. The role of a "sieve parameter'' in reducing the dimensionality of mixture models is demonstrated. We derive analytical machinery for building a collection of semiparametric mixture models, including the multivariate case, via the sieve parameter. The performance of the methods are shown with applications to several data sets including the cdc15 cell-cycle yeast microarray data.

Ramani S. Pilla

Papers

Moment-Based Approximations of Distributions Using Mixtures: Theory and Applications

Iteratively Reweighted Generalized Least Squares for Estimation and Testing With Correlated Data: An Inference Function Framework

Inference Under Convex Cone Alternatives for Correlated Data

On large-sample estimation and testing via quadratic inference functions for correlated data

Model Building for Semiparametric Mixtures