6,938 research outputs found
Objective prior for the number of degrees of freedom of a t distribution
In this paper, we construct an objective prior for the degrees of freedom of a t distribution, when the parameter is taken to be discrete. This parameter is typically problematic to estimate and a problem in objective Bayesian inference since improper priors lead to improper posteriors, whilst proper priors may dom- inate the data likelihood. We find an objective criterion, based on loss functions, instead of trying to define objective probabilities directly. Truncating the prior on the degrees of freedom is necessary, as the t distribution, above a certain number of degrees of freedom, becomes the normal distribution. The defined prior is tested in simulation scenarios, including linear regression with t-distributed errors, and on real data: the daily returns of the closing Dow Jones index over a period of 98 days
Inference for Extremal Conditional Quantile Models, with an Application to Market and Birthweight Risks
Quantile regression is an increasingly important empirical tool in economics
and other sciences for analyzing the impact of a set of regressors on the
conditional distribution of an outcome. Extremal quantile regression, or
quantile regression applied to the tails, is of interest in many economic and
financial applications, such as conditional value-at-risk, production
efficiency, and adjustment bands in (S,s) models. In this paper we provide
feasible inference tools for extremal conditional quantile models that rely
upon extreme value approximations to the distribution of self-normalized
quantile regression statistics. The methods are simple to implement and can be
of independent interest even in the non-regression case. We illustrate the
results with two empirical examples analyzing extreme fluctuations of a stock
return and extremely low percentiles of live infants' birthweights in the range
between 250 and 1500 grams.Comment: 41 pages, 9 figure
User-Friendly Covariance Estimation for Heavy-Tailed Distributions
We offer a survey of recent results on covariance estimation for heavy-tailed
distributions. By unifying ideas scattered in the literature, we propose
user-friendly methods that facilitate practical implementation. Specifically,
we introduce element-wise and spectrum-wise truncation operators, as well as
their -estimator counterparts, to robustify the sample covariance matrix.
Different from the classical notion of robustness that is characterized by the
breakdown property, we focus on the tail robustness which is evidenced by the
connection between nonasymptotic deviation and confidence level. The key
observation is that the estimators needs to adapt to the sample size,
dimensionality of the data and the noise level to achieve optimal tradeoff
between bias and robustness. Furthermore, to facilitate their practical use, we
propose data-driven procedures that automatically calibrate the tuning
parameters. We demonstrate their applications to a series of structured models
in high dimensions, including the bandable and low-rank covariance matrices and
sparse precision matrices. Numerical studies lend strong support to the
proposed methods.Comment: 56 pages, 2 figure
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