An application of the given procedures to a problem of selecting the “most probable event” from a multinomial The problem of selecting from given Poisson processes for the discrete case is related to the problem of selecting from given The value ofN is determined by a lower bound placed on the value of the Pcs. The Pcs for this configuration is independent of λ for two of the given procedures, and monotone increasing in λ for Given (λ/λ)≧0>1, it is shown that the probability of a correct selection (Pcs) is minimized whenθλ=θλ=.=θλ=θλ=θλ, say. In the discrete case, the selection involves certain randomization. In the continuous case, the process for which theNth arrival time is shortest, is selected. The processes are observed until there areN arrivals from any of the given processes, when the processes are observed continuously, or until there are at leastN arrivals, when the processes are observed at successive intervals of time whereN is a pre-determined positive integer.
We consider three procedures for selecting the process corresponding to λ. Let λ≦λ≦.≦λ denote the ordered set of values λ1.,λ. There are givenk Poisson processes with mean arrival times 1/λ1.1/λ Finally, we consider (h) pathway analysis for high-dimensional data by constructing a multiple test of correlation coefficients. Further, we consider variable selection for classification and propose (g) a two-stage discriminant procedure after screening some variables. Following the variable selection procedure, we consider (f) variable selection for high-dimensional regression to compare favorably with the lasso in terms of the assurance of accuracy and the computational cost. Moreover, we propose (e) a two-stage variable selection procedure that provides screening of variables in the first stage and selects a significant set of associated variables from among a set of candidate variables in the second stage. We also give (d) a two-stage discriminant procedure that controls misclassification rates being no more than a prespecified value. In addition, we give (b) a two-sample test to assure prespecified size and power simultaneously together with (c) an equality-test procedure for two covariance matrices. By developing asymptotic normality when p→∞, we first give (a) a given-bandwidth confidence region for the square loss. To develop theory and methodologies, the most important and basic idea is the asymptotic normality when p→∞. We offer the sample size determination for inference problems by creating new types of multivariate two-stage procedures. This is the first attempt to apply sequential analysis to high-dimensional statistical inference ensuring prespecified accuracy. The purpose of this article is to suggest directions for future research and possible solutions about p≫n problems by using new types of two-stage estimation methodologies.
We consider a variety of inference problems for high-dimensional data.