OMB Meeting Book - January 8, 2015

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GAMMA DISTRIBUTION The gamma distribution with shape = 2 and scale = 1 (rate = 1) has mean 2 and standard deviation of 1.4142. It is an example of a asymmetric distribution skewed to the right, but less so than the lognormal distribution. Based on 100,000 realizations of samples of size 24, the estimated mean standard deviation (‘s’) is 1.396, the equivalent estimate based on MAD is 1.187, and the equivalent estimate based on the IQR is 1.229. The biases in the MAD- and IQR-based estimates again are substantial. 0.3075745 0.3059277 0.3226098 The standard errors of the statistics (i.e., standard deviations of the sampling distributions) are 0.3076 for s, 0.3059 for the MAD-based estimate and 0.3226 for the IQR-based estimate. All estimates are comparable in precision, but the MAD- and IQR-based estimates are biased. Use of ‘robust’ estimators for the standard deviation when the underlying distribution is skewed results in estimates which are too small by 20%.

Gamma (Shape=2, Rate=1) Distribution

Gamma MAD IQR

0.0 0.1 0.2 0.3

Density

0

2

4

6

8

10

z

7

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