Mathematics

A perennial objection against Bayes factor point-null hypothesis tests is that the point-null hypothesis is known to be false from the outset. Following Morey and Rouder (2011) we examine the consequences of approximating the sharp point-null hypothesis by a hazy `peri-null' hypothesis instantiated as a narrow prior distribution centered on the point of interest. The peri-null Bayes factor then equals the point-null Bayes factor multiplied by a correction term which is itself a Bayes factor. For moderate sample sizes, the correction term is relatively inconsequential; however, for large sample sizes the correction term becomes influential and causes the peri-null Bayes factor to be inconsistent and approach a limit that depends on the ratio of prior ordinates evaluated at the maximum likelihood estimate. We characterize the asymptotic behavior of the peri-null Bayes factor and discuss how to construct peri-null Bayes factor hypothesis tests that are also consistent.

We consider a sparse linear regression model with unknown symmetric error under the high-dimensional setting. The true error distribution is assumed to belong to the locally β -Hölder class with an exponentially decreasing tail, which does not need to be sub-Gaussian. We obtain posterior convergence rates of the regression coefficient and the error density, which are nearly optimal and adaptive to the unknown sparsity level. Furthermore, we derive the semi-parametric Bernstein-von Mises (BvM) theorem to characterize asymptotic shape of the marginal posterior for regression coefficients. Under the sub-Gaussianity assumption on the true score function, strong model selection consistency for regression coefficients are also obtained, which eventually asserts the frequentist's validity of credible sets.

We discuss Bayesian inference for parameters selected using the data. First, we provide a critical analysis of the existing positions in the literature regarding the correct Bayesian approach under selection. Second, we propose two types of non-informative priors for selection models. These priors may be employed to produce a posterior distribution in the absence of prior information as well as to provide well-calibrated frequentist inference for the selected parameter. We test the proposed priors empirically in several scenarios.

In this paper we investigate the estimation of the unknown parameters of a competing risk model based on a Weibull distributed decreasing failure rate and an exponentially distributed constant failure rate, under right censored data.likelihood estimators.

Conditional heteroscedastic (CH) models are routinely used to analyze financial datasets. The classical models such as ARCH-GARCH with time-invariant coefficients are often inadequate to describe frequent changes over time due to market variability. However we can achieve significantly better insight by considering the time-varying analogues of these models. In this paper, we propose a Bayesian approach to the estimation of such models and develop computationally efficient MCMC algorithm based on Hamiltonian Monte Carlo (HMC) sampling. We also established posterior contraction rates with increasing sample size in terms of the average Hellinger metric. The performance of our method is compared with frequentist estimates and estimates from the time constant analogues. To conclude the paper we obtain time-varying parameter estimates for some popular Forex (currency conversion rate) and stock market datasets.

In this paper, we propose Bayesian non-parametric tests for one-sample and two-sample multivariate location problems. We model the underlying distributions using a Dirichlet process prior. For the one-sample problem, we compute a Bayesian credible set of the multivariate spatial median and accept the null hypothesis if the credible set contains the null value. For the two-sample problem, we form a credible set for the difference of the spatial medians of the two samples and we accept the null hypothesis of equality if the credible set contains zero. We derive the local asymptotic power of the tests under shrinking alternatives, and also present a simulation study to compare the finite-sample performance of our testing procedures with existing parametric and non-parametric tests.

Classical learning theory suggests that strong regularization is needed to learn a class with large complexity. This intuition is in contrast with the modern practice of machine learning, in particular learning neural networks, where the number of parameters often exceeds the number of data points. It has been observed empirically that such overparametrized models can show good generalization performance even if trained with vanishing or negative regularization. The aim of this work is to understand theoretically how this effect can occur, by studying the setting of ridge regression. We provide non-asymptotic generalization bounds for overparametrized ridge regression that depend on the arbitrary covariance structure of the data, and show that those bounds are tight for a range of regularization parameter values. To our knowledge this is the first work that studies overparametrized ridge regression in such a general setting. We identify when small or negative regularization is sufficient for obtaining small generalization error. On the technical side, our bounds only require the data vectors to be i.i.d. sub-gaussian, while most previous work assumes independence of the components of those vectors.

We obtain a sufficient condition for benign overfitting of linear regression problem. Our result does not rely on concentration argument but on small-ball assumption and thus can holds in heavy-tailed case. The basic idea is to establish a coordinate small-ball estimate in terms of effective rank so that we can calibrate the balance of epsilon-Net and exponential probability. Our result indicates that benign overfitting is not depending on concentration property of the input vector. Finally, we discuss potential difficulties for benign overfitting beyond linear model and a benign overfitting result without truncated effective rank.

The linear regression model can be used even when the true regression function is not linear. The resulting estimated linear function is the best linear approximation to the regression function and the vector β of the coefficients of this linear approximation are the projection parameter. We provide finite sample bounds on the Normal approximation to the law of the least squares estimator of the projection parameters normalized by the sandwich-based standard error. Our results hold in the increasing dimension setting and under minimal assumptions on the distribution of the response variable. Furthermore, we construct confidence sets for β in the form of hyper-rectangles and establish rates on their coverage accuracy. We provide analogous results for partial correlations among the entries of sub-Gaussian vectors.

Let Z:={ Z t ,t??} be a stationary Gaussian process. We study two estimators of E[ Z 2 0 ] , namely f ? T (Z):= 1 T ??T 0 Z 2 t dt , and f ? n (Z):= 1 n ??n i=1 Z 2 t i , where t i =i ? n , i=0,1,??n , ? n ?? and T n :=n ? n ?��? . We prove that the two estimators are strongly consistent and establish Berry-Esseen bounds for a central limit theorem involving f ? T (Z) and f ? n (Z) . We apply these results to asymptotically stationary Gaussian processes and estimate the drift parameter for Gaussian Ornstein-Uhlenbeck processes.