STAT4290 April 25 2011

# STAT4290 April 25 2011 - Shapiro-Wilk test for Normality...

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Shapiro-Wilk test for Normality Motivation – look ( from regress/ANOVA perspective) at relationship ordered values in sample and expected values of order statistics (basically-quantiles) of the null distribution – N(0,1) Can’t directly fir a regression, becauses: (i) Ordered values in the sample are not independent (ii) Order statistics are not identically distributed Their approach – look at the squared slope of the regression line. Under the Normality hypothesis is an estimate of the population variance multiplied by a constant. Compare this to the residual mean square about the regression line, another estimate of the variance. Following the basic idea leads, eventually to W = = ( - + - , = W 1Di 1kai Xn i 1 Xi where k n2 , ai is the expected values of the order statistic from N0 1 = = ( - ) D i 1n Xi X 2 W is essentially a Pearson correlation (squared) between the ordered values from the sample and the expected quantiles of N(0,1). So, W is between 0 and 1. The closer it is to 1, the higher the correlation => better match

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STAT4290 April 25 2011 - Shapiro-Wilk test for Normality...

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