Showing papers in "Sankhya in 2013"
TL;DR: In this paper, a high-dimensional modification of Akaike information criterion (AIC) for selection of variables in discriminant analysis has been proposed, denoted by HAIC, which is an asymptotically unbiased estimator of the risk function.
Abstract: This paper is concerned with high-dimensional modifications of Akaike information criterion (AIC) for selection of variables in discriminant analysis. The AIC has been proposed as an asymptotically unbiased estimator of the risk function of the candidate model when the dimension is fixed and the sample size tends to infinity. On the other hand, Fujikoshi (2002) attempted to modify the AIC in two-group discriminant analysis when the dimension and the sample size tend to infinity. Such an estimator is called high-dimensional AIC, which is denoted by HAIC. However, its modification was obtained under a restrictive assumption, and furthermore, it was difficult to extend the method to multiple-group case. In this paper, by a new approach we propose HAIC which is an asymptotically unbiased estimator of the risk function in multiple-group discriminant analysis when both the dimension and the sample size tend to infinity, for a general class of candidate models. By simulation experiments it is shown that HAIC is more useful than other AIC type of criteria.
11 citations
TL;DR: A fundamental theorem is provided that can be used in conjunction with Kolmogorov asymptotic conditions to derive the first moments of well-known estimators of the actual error rate in linear discriminant analysis of a multivariate Gaussian model under the assumption of a common known covariance matrix.
Abstract: We provide a fundamental theorem that can be used in conjunction with Kolmogorov asymptotic conditions to derive the first moments of well-known estimators of the actual error rate in linear discriminant analysis of a multivariate Gaussian model under the assumption of a common known covariance matrix. The estimators studied in this paper are plug-in and smoothed resubstitution error estimators, both of which have not been studied before under Kolmogorov asymptotic conditions. As a result of this work, we present an optimal smoothing parameter that makes the smoothed resubstitution an unbiased estimator of the true error. For the sake of completeness, we further show how to utilize the presented fundamental theorem to achieve several previously reported results, namely the first moment of the resubstitution estimator and the actual error rate. We provide numerical examples to show the accuracy of the succeeding finite sample approximations in situations where the number of dimensions is comparable or even larger than the sample size.
6 citations
TL;DR: In this paper, the authors focus on characterizing and computing the probabilities of ruin in three mathematical models arising in economics: credit systems, exchange economies, and the problem of sustaining a constant consumption of a resource augmented by a random input.
Abstract: In this paper the focus is on characterizing and computing the probabilities of ruin in three mathematical models arising in economics. First, we examine a credit system in which small loans without collaterals are extended to a large number of costumers, and study the probability of collapse due to defaults. Next, we consider a Walrasian model of an exchange economy in which the endowments are random, and analyze the probability that at equilibrium prices an agent does not have the minimum income needed for survival. Finally, the problem of sustaining a constant consumption of a resource the stock of which is augmented by a random input is considered. The steady state of the resulting Markov process, the speed at which it is approached, and the possibility of exhaustion of the stock are examined.
3 citations