Flexible modelling in statistics: past, present and future

In times where more and more data become available and where the data exhibit rather complex structures (significant departure from symmetry, heavy or light tails), flexible modelling has become an essential task for statisticians as well as researchers and practitioners from domains such as economics, finance or environmental sciences. This is reflected by the wealth of existing proposals for flexible distributions; well-known examples are Azzalini's skew-normal, Tukey's $g$-and-$h$, mixture and two-piece distributions, to cite but these. My aim in the present paper is to provide an introduction to this research field, intended to be useful both for novices and professionals of the domain. After a description of the research stream itself, I will narrate the gripping history of flexible modelling, starring emblematic heroes from the past such as Edgeworth and Pearson, then depict three of the most used flexible families of distributions, and finally provide an outlook on future flexible modelling research by posing challenging open questions.Comment: 27 pages, 4 figure

Skew-symmetric distributions and Fisher information -- a tale of two densities

Skew-symmetric densities recently received much attention in the literature, giving rise to increasingly general families of univariate and multivariate skewed densities. Most of those families, however, suffer from the inferential drawback of a potentially singular Fisher information in the vicinity of symmetry. All existing results indicate that Gaussian densities (possibly after restriction to some linear subspace) play a special and somewhat intriguing role in that context. We dispel that widespread opinion by providing a full characterization, in a general multivariate context, of the information singularity phenomenon, highlighting its relation to a possible link between symmetric kernels and skewing functions -- a link that can be interpreted as the mismatch of two densities.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ346 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A tractable, parsimonious and flexible model for cylindrical data, with applications

In this paper, we propose cylindrical distributions obtained by combining the sine-skewed von Mises distribution (circular part) with the Weibull distribution (linear part). This new model, the WeiSSVM, enjoys numerous advantages: simple normalizing constant and hence very tractable density, parameter-parsimony and interpretability, good circular-linear dependence structure, easy random number generation thanks to known marginal/conditional distributions, flexibility illustrated via excellent fitting abilities, and a straightforward extension to the case of directional-linear data. Inferential issues, such as independence testing, circular-linear respectively linear-circular regression, can easily be tackled with our model, which we apply on two real data sets. We conclude the paper by discussing future applications of our model.Comment: 17 pages, 5 figure

Analysis of biological and biomedical data with circular statistics

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Efficient inference about the tail weight in multivariate Student tt distributions

We propose a new testing procedure about the tail weight parameter of multivariate Student $t$ distributions by having recourse to the Le Cam methodology. Our test is asymptotically as efficient as the classical likelihood ratio test, but outperforms the latter by its flexibility and simplicity: indeed, our approach allows to estimate the location and scatter nuisance parameters by any root-$n$ consistent estimators, hereby avoiding numerically complex maximum likelihood estimation. The finite-sample properties of our test are analyzed in a Monte Carlo simulation study, and we apply our method on a financial data set. We conclude the paper by indicating how to use this framework for efficient point estimation.Comment: 23 page

Bounds for the asymptotic normality of the maximum likelihood estimator using the Delta method

The asymptotic normality of the Maximum Likelihood Estimator (MLE) is a cornerstone of statistical theory. In the present paper, we provide sharp explicit upper bounds on Zolotarev-type distances between the exact, unknown distribution of the MLE and its limiting normal distribution. Our approach to this fundamental issue is based on a sound combination of the Delta method, Stein's method, Taylor expansions and conditional expectations, for the classical situations where the MLE can be expressed as a function of a sum of independent and identically distributed terms. This encompasses in particular the broad exponential family of distributions.Comment: 15 pages, 1 tabl

High-dimensional tests for spherical location and spiked covariance

Rotationally symmetric distributions on the p-dimensional unit hypersphere, extremely popular in directional statistics, involve a location parameter theta that indicates the direction of the symmetry axis. The most classical way of addressing the spherical location problem H_0:theta=theta_0, with theta_0 a fixed location, is the so-called Watson test, which is based on the sample mean of the observations. This test enjoys many desirable properties, but its implementation requires the sample size n to be large compared to the dimension p. This is a severe limitation, since more and more problems nowadays involve high-dimensional directional data (e.g., in genetics or text mining). In this work, we therefore introduce a modified Watson statistic that can cope with high-dimensionality. We derive its asymptotic null distribution as both n and p go to infinity. This is achieved in a universal asymptotic framework that allows p to go to infinity arbitrarily fast (or slowly) as a function of n. We further show that our results also provide high-dimensional tests for a problem that has recently attracted much attention, namely that of testing that the covariance matrix of a multinormal distribution has a "theta_0-spiked" structure. Finally, a Monte Carlo simulation study corroborates our asymptotic results