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Register a new userFitting phase--type scale mixtures to heavy--tailed data and distributions
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Leonardo Rojas Nandayapa
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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We consider the fitting of heavy tailed data and distribution with a special attention to distributions with a non--standard shape in the body of the distribution. To this end we consider a dense class of heavy tailed distributions introduced recently, employing an EM algorithm for the the maximum likelihood estimates of its parameters. We present methods for fitting to observed data, histograms, censored data, as well as to theoretical distributions. Numerical examples are provided with simulated data and a benchmark reinsurance dataset. We empirically demonstrate that our model can provide excellent fits to heavy--tailed data/distributions with minimal assumptions
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We propose and analyze a new estimator of the covariance matrix that admits strong theoretical guarantees under weak assumptions on the underlying distribution, such as existence of moments of only low order. While estimation of covariance matrices corresponding to sub-Gaussian distributions is well-understood, much less in known in the case of heavy-tailed data. As K. Balasubramanian and M. Yuan write, data from real-world experiments oftentimes tend to be corrupted with outliers and/or exhibit heavy tails. In such cases, it is not clear that those covariance matrix estimators .. remain optimal and ..what are the other possible strategies to deal with heavy tailed distributions warrant further studies. We make a step towards answering this question and prove tight deviation inequalities for the proposed estimator that depend only on the parameters controlling the intrinsic dimension associated to the covariance matrix (as opposed to the dimension of the ambient space); in particular, our results are applicable in the case of high-dimensional observations.
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This paper considers Bayesian multiple testing under sparsity for polynomial-tailed distributions satisfying a monotone likelihood ratio property. Included in this class of distributions are the Students t, the Pareto, and many other distributions. We prove some general asymptotic optimality results under fixed and random thresholding. As examples of these general results, we establish the Bayesian asymptotic optimality of several multiple testing procedures in the literature for appropriately chosen false discovery rate levels. We also show by simulation that the Benjamini-Hochberg procedure with a false discovery rate level different from the asymptotically optimal one can lead to high Bayes risk.
We study the performance of the Least Squares Estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a $p$-th moment ($pgeq 1$). In such a heavy-tailed regression setting, we show that if the model satisfies a standard `entropy condition with exponent $alpha in (0,2)$, then the $L_2$ loss of the LSE converges at a rate begin{align*} mathcal{O}_{mathbf{P}}big(n^{-frac{1}{2+alpha}} vee n^{-frac{1}{2}+frac{1}{2p}}big). end{align*} Such a rate cannot be improved under the entropy condition alone. This rate quantifies both some positive and negative aspects of the LSE in a heavy-tailed regression setting. On the positive side, as long as the errors have $pgeq 1+2/alpha$ moments, the $L_2$ loss of the LSE converges at the same rate as if the errors are Gaussian. On the negative side, if $p<1+2/alpha$, there are (many) hard models at any entropy level $alpha$ for which the $L_2$ loss of the LSE converges at a strictly slower rate than other robust estimators. The validity of the above rate relies crucially on the independence of the covariates and the errors. In fact, the $L_2$ loss of the LSE can converge arbitrarily slowly when the independence fails. The key technical ingredient is a new multiplier inequality that gives sharp bounds for the `multiplier empirical process associated with the LSE. We further give an application to the sparse linear regression model with heavy-tailed covariates and errors to demonstrate the scope of this new inequality.
We consider X 1 ,. .. , X n a sample of data on the circle S 1 , whose distribution is a twocomponent mixture. Denoting R and Q two rotations on S 1 , the density of the X i s is assumed to be g(x) = pf (R --1 x) + (1 -- p)f (Q --1 x), where p $in$ (0, 1) and f is an unknown density on the circle. In this paper we estimate both the parametric part $theta$ = (p, R, Q) and the nonparametric part f. The specific problems of identifiability on the circle are studied. A consistent estimator of $theta$ is introduced and its asymptotic normality is proved. We propose a Fourier-based estimator of f with a penalized criterion to choose the resolution level. We show that our adaptive estimator is optimal from the oracle and minimax points of view when the density belongs to a Sobolev ball. Our method is illustrated by numerical simulations.
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