Inequality for standardized normal samples
Source: Schweitzer 2009
November 13, 2009
inequalitiesvectorratioprobability and stats
Problem Statement
Let be -dimensional independent random (column) vectors with standard normal distribution, n \minus{} 1 > d. Furthermore let
\overline Z \equal{} \frac {1}{n}\sum_{i \equal{} 1}^n Z_i, S_n \equal{} \frac {1}{n \minus{} 1}\sum_{i \equal{} 1}^n(Z_i \minus{} \overline Z)(Z_i \minus{} \overline Z)^\top
be the sample mean and corrected empirical covariance matrix. Consider the standardized samples Y_i \equal{} S_n^{ \minus{} 1/2}(Z_i \minus{} \overline Z), i \equal{} 1,2,\dots,n. Show that
\frac {E|Y_1 \minus{} Y_2|}{E|Z_1 \minus{} Z_2|} > 1,
and that the ratio does not depend on , only on .