Abstract
We consider high-dimension low-sample-size data taken from the standard multivariate normal distribution under assumption that dimension is a random variable. The second order Chebyshev–Edgeworth expansions for distributions of an angle between two sample observations and corresponding sample correlation coefficient are constructed with error bounds. Depending on the type of normalization, we get three different limit distributions: Normal, Student’s t-, or Laplace distributions. The paper continues studies of the authors on approximation of statistics for random size samples.
Highlights
Let ~X1 = ( X11, ..., X1m ) T, . . . , ~Xk = ( Xk1, ..., Xkm ) T be a random sample from m-dimensional population
It became the basis of research in mathematical statistics for the analysis of high-dimensional data, see, e.g., Fujikoshi et al [2], which are an important part of the current data analysis fashionable area called Big data
Prediction intervals for the future observations for generalized order statistics and confidence intervals for quantiles based on samples of random sizes are studied in Barakat et al [10] and Al-Mutairi and Raqab [11], respectively
Summary
The aim of the present paper is to study approximation for the third statistic ang(~X1 , ~X2 ) under generalized assumption that m is a realization of a random variable, say Nn , which represents the sample dimension and is independent of ~X1 and ~X2. Prediction intervals for the future observations for generalized order statistics and confidence intervals for quantiles based on samples of random sizes are studied in Barakat et al [10] and Al-Mutairi and Raqab [11], respectively They illustrated their results with real biometric data set, the duration of remission of leukemia patients treated by one drug.
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