Mathematics

We describe a general class of ie-merging functions and pose the problem of finding ie-merging functions outside this class.

The significance of Marshall-Olkin distribution in reliability theory has motivated us to introduce a generalized exponentiated Marshall-Olkin (GEMO), a family of distributions.

Covariance matrix testing for high dimensional data is a fundamental problem. A large class of covariance test statistics based on certain averaged spectral statistics of the sample covariance matrix are known to obey central limit theorems under the null. However, precise understanding for the power behavior of the corresponding tests under general alternatives remains largely unknown. This paper develops a general method for analyzing the power behavior of covariance test statistics via accurate non-asymptotic power expansions. We specialize our general method to two prototypical settings of testing identity and sphericity, and derive sharp power expansion for a number of widely used tests, including the likelihood ratio tests, Ledoit-Nagao-Wolf's test, Cai-Ma's test and John's test. The power expansion for each of those tests holds uniformly over all possible alternatives under mild growth conditions on the dimension-to-sample ratio. Interestingly, although some of those tests are previously known to share the same limiting power behavior under spiked covariance alternatives with a fixed number of spikes, our new power characterizations indicate that such equivalence fails when many spikes exist. The proofs of our results combine techniques from Poincaré-type inequalities, random matrices and zonal polynomials.

This article develops a periodic version of a time varying parameter fractional process in the stationary region. It is a partial extension of Hosking (1981)'s article which dealt with the case where the coefficients are invariant in time. We will describe the probabilistic theories of this periodic model. The results are followed by a graphical representation of the autocovariances functions.

We propose a theoretical framework for the problem of learning a real-valued function which meets fairness requirements. This framework is built upon the notion of α -relative (fairness) improvement of the regression function which we introduce using the theory of optimal transport. Setting α=0 corresponds to the regression problem under the Demographic Parity constraint, while α=1 corresponds to the classical regression problem without any constraints. For α∈(0,1) the proposed framework allows to continuously interpolate between these two extreme cases and to study partially fair predictors. Within this framework we precisely quantify the cost in risk induced by the introduction of the fairness constraint. We put forward a statistical minimax setup and derive a general problem-dependent lower bound on the risk of any estimator satisfying α -relative improvement constraint. We illustrate our framework on a model of linear regression with Gaussian design and systematic group-dependent bias, deriving matching (up to absolute constants) upper and lower bounds on the minimax risk under the introduced constraint. Finally, we perform a simulation study of the latter setup.

A novel approach towards construction of absolutely continuous distributions over the unit interval is proposed. Considering two absolutely continuous random variables with positive support, this method conditions on their convolution to generate a new random variable in the unit interval. This approach is demonstrated using some popular choices of the positive random variables such as the exponential, Lindley, gamma. Some existing distributions like the uniform and the beta are formulated with this method. Several new structures of density functions having potential for future application in real life problems are also provided. One of the new distributions having one parameter is considered for parameter estimation and real life modelling application and shown to provide better fit than the popular one parameter Topp-Leone model.

Robust estimators and Wald-type tests are developed for the multinomial logistic regression based on ? -divergence measures. The robustness of the proposed estimators and tests is proved through the study of their influence functions and it is also illustrated with two numerical examples and an extensive simulation study.

In this paper, we investigate estimators for symmetric α -stable CARMA processes sampled equidistantly. Simulation studies suggest that the Whittle estimator and the estimator presented in Garc\'ıa et al. (2011) are consistent estimators for the parameters of stable CARMA processes. For CARMA processes with finite second moments it is well-known that the Whittle estimator is consistent and asymptotically normally distributed. Therefore, in the light-tailed setting the properties of the Whittle estimator for CARMA processes are similar to those of the Whittle estimator for ARMA processes. However, in the present paper we prove that, in general, the Whittle estimator for symmetric α -stable CARMA processes sampled at low frequencies is not consistent and highlight why simulation studies suggest something else. Thus, in contrast to the light-tailed setting the properties of the Whittle estimator for heavy-tailed ARMA processes can not be transferred to heavy-tailed CARMA processes. We elaborate as well that the estimator presented in Garc\'ıa et al. (2011) faces the same problems. However, the Whittle estimator for stable CAR(1) processes is consistent.

In this note we consider the optimal design problem for estimating the slope of a polynomial regression with no intercept at a given point, say z. In contrast to previous work, which considers symmetric design spaces we investigate the model on the interval [0,a] and characterize those values of z , where an explicit solution of the optimal design is possible.

In this paper, we investigate the hypothesis testing problem that checks whether part of covariates / confounders significantly affect the heterogeneous treatment effect given all covariates. This model checking is particularly useful in the case where there are many collected covariates such that we can possibly alleviate the typical curse of dimensionality. In the test construction procedure, we use a projection-based idea and a nonparametric estimation-based test procedure to construct an aggregated version over all projection directions. The resulting test statistic is then interestingly with no effect from slow convergence rate the nonparametric estimation usually suffers from. This feature makes the test behave like a global smoothing test to have ability to detect a broad class of local alternatives converging to the null at the fastest possible rate in hypothesis testing. Also, the test can inherit the merit of lobal smoothing tests to be sensitive to oscillating alternative models. The performance of the test is examined by numerical studies and the analysis for a real data example for illustration.