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Modern Applied Statistics With S

Probability and Measure

A bivariate distribution function H(x, y) with marginals F(x) and G(y) is said to be generated by an Archimedean copula if it can be expressed in the form H(x, y) = ϕ–1[ϕ{F(x)} + ϕ{G(y)}] for some convex, decreasing function ϕ defined on [0, 1] in such a way that ϕ(1) = 0. Many well-known systems of bivariate distributions belong to this class, including those of Gumbel, Ali-Mikhail-Haq-Thelot, Clayton, Frank, and Hougaard. Frailty models also fall under that general prescription. This article examines the problem of selecting an Archimedean copula providing a suitable representation of the dependence structure between two variates X and Y in the light of a random sample (X 1, Y 1), …, (X n , Y n ). The key to the estimation procedure is a one-dimensional empirical distribution function that can be constructed whether the uniform representation of X and Y is Archimedean or not, and independently of their marginals. This semiparametric estimator, based on a decomposition of Kendall's tau statistic...

Statistical Inference Procedures for Bivariate Archimedean Copulas

This article introduces actuaries to the concept of “copulas,” a tool for understanding relationships among multivariate outcomes. A copula is a function that links univariate marginals to their full multivariate distribution. Copulas were introduced in 1959 in the context of probabilistic metric spaces. The literature on the statistical properties and applications of copulas has been developing rapidly in recent years. This article explores some of these practical applications, including estimation of joint life mortality and multidecrement models. In addition, we describe basic properties of copulas, their relationships to measures of dependence, and several families of copulas that have appeared in the literature. An annotated bibliography provides a resource for researchers and practitioners who wish to continue their study of copulas. For those who wish to use copulas for statistical inference, we illustrate statistical inference procedures by using insurance company data on losses and expen...

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Understanding Relationships Using Copulas

We propose an independence criterion based on the eigen-spectrum of covariance operators in reproducing kernel Hilbert spaces (RKHSs), consisting of an empirical estimate of the Hilbert-Schmidt norm of the cross-covariance operator (we term this a Hilbert-Schmidt Independence Criterion, or HSIC). This approach has several advantages, compared with previous kernel-based independence criteria. First, the empirical estimate is simpler than any other kernel dependence test, and requires no user-defined regularisation. Second, there is a clearly defined population quantity which the empirical estimate approaches in the large sample limit, with exponential convergence guaranteed between the two: this ensures that independence tests based on HSIC do not suffer from slow learning rates. Finally, we show in the context of independent component analysis (ICA) that the performance of HSIC is competitive with that of previously published kernel-based criteria, and of other recently published ICA methods.

Measuring statistical dependence with Hilbert-Schmidt norms

Linear algebra.- Random vectors.- Gamma, Dirichlet, and F distributions.- Invariance.- Multivariate normal.- Multivariate sampling.- Wishart distributions.- Tests on mean and variance.- Multivariate regression.- Principal components.- Canonical correlations.- Asymptotic expansions.- Robustness.- Bootstrap confidence regions and tests.

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Theory of multivariate statistics

Suppose the observations (ti,yi), i = 1,… n, follow the model

where gj are unknown functions. The estimation of the additive components can be done by approximating gj, with a function made up of the sum of a linear fit and a truncated Fourier series of cosines and minimizing a penalized least-squares loss function over the coefficients. This finite-dimensional basis approximation, when fitting an additive model with r predictors, has the advantage of reducing the computations drastically, since it does not require the use of the backfitting algorithm. The cross-validation (CV) [or generalized cross-validation (GCV)] for the additive fit is calculated in a further 0(n) operations. A search path in the r-dimensional space of degrees of freedom is proposed along which the CV (GCV) continuously decreases. The path ends when an increase in the degrees of freedom of any of the predictors yields an increase in CV (GCV). This procedure is illustrated on a meteorological data set.

Fourier smoother and additive models

Minimax estimators in the normal MANOVA model

https://www.dms.umontreal.ca/~bilodeau/ecfgiven.pdf

Martin Bilodeau

Papers

Theory of multivariate statistics

Fourier smoother and additive models

Minimax estimators in the normal MANOVA model

A multivariate empirical characteristic function test of independence with normal marginals

Stein estimation under elliptical distributions