OMB Meeting Book - January 8, 2015

56

STUDENT-t WITH 4 DEGREES OF FREEDOM The standard student-t distribution with 4 degrees of freedom has mean 0 and standard deviation of 1.4142. It is an example of a symmetric distribution with long tails. Based on 100,000 realizations of samples of size 24, the estimated mean standard deviation (‘s’) is 1.376, the equivalent estimate based on MAD is 1.090, and the equivalent estimate based on the IQR is 1.061. The biases in the MAD- and IQR-based estimates are substantial. The standard errors of the statistics (i.e., standard deviations of the sampling distributions) are 0.3895 for s, 0.2752 for the MAD-based estimate and 0.2638 for the IQR-based estimate. The sample standard deviation s is less precise, but unbiased. The MAD- and IQR-based estimates are precise, but biased. Use of ‘robust’ estimators for the standard deviation when the underlying distribution is platykurtic results in estimates which are too small by 30+%.

Student-t (4 d.f.) Distribution

Student-t (4 d.f.) MAD IQR

0.0 0.1 0.2 0.3

Density

-4

-2

0

2

4

z

6

25

Recommended to OMB by Committee on Statistics: 07-17-2013 Reviewed and approved by OMB: 07-18-2013