This is the first book on an evaluation of (weak) consistency of an
information criterion for variable selection in high-dimensional
multivariate linear regression models by using the high-dimensional
asymptotic framework. It is an asymptotic framework such that the
sample size n and the dimension of response variables vector p are
approaching â simultaneously under a condition that p/n goes to a
constant included in [0,1).Most statistical textbooks evaluate
consistency of an information criterion by using the large-sample
asymptotic framework such that n goes to â under the fixed p. The
evaluation of consistency of an information criterion from the
high-dimensional asymptotic framework provides new knowledge to us,
e.g., Akaike's information criterion (AIC) sometimes becomes
consistent under the high-dimensional asymptotic framework although
it never has a consistency under the large-sample asymptotic
framework; and Bayesian information criterion (BIC) sometimes
becomes inconsistent under the high-dimensional asymptotic
framework although it is always consistent under the large-sample
asymptotic framework. The knowledge may help to choose an
information criterion to be used for high-dimensional data
analysis, which has been attracting the attention of many
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