lec2 - Binomial Distribution: Y 1 , . . . , Y n ind Bin ( n...

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Unformatted text preview: Binomial Distribution: Y 1 , . . . , Y n ind Bin ( n i , i ), i = n i , ( , y ) = X i log n i y i + y i log i + ( n i- y i ) log(1- i ) ( ; y ) = X i log n i y i + y i log i n i + ( n i- y i ) log n i- i n i D * ( y ; ) = 2 X i log n i y i + y i log y i n i + ( n i- y i ) log n i- y i n i - log n i y i + y i log i n i + ( n i- y i ) log n i- i n i D * ( y ; ) = 2 X i y i log y i i + ( n i- y i ) log n i- y i n i- i In general, for a sample from an E.D. family density D * ( y ; ) = X i 2 w i n y i ( i- i )- b ( i )- b ( i ) o / = D ( y ; ) / where D ( y ; ) is the (unscaled) deviance, which is a function of the data only (it is free of ), and i = i ( y i ), i = i ( i ). 101 Asymptotic Distribution of the Deviance and X 2 Statistics: From the asymptotic chi-square-ness of 2 log its tempting to conclude D * ( y , ) a 2 ( n- p ) This is not necessarily true! Wilks result is based on asymptotics under which n while t 1 , t 2 stay fixed. However, in some cases the saturated model requires estimation of an increasing number of parameters as n , so that standard theory does not apply. Therefore, we need to be careful in using the chi-square approximation to the distribution of the deviance, and in using the deviance as a true goodness-of-fit statistic. Similar comments apply to Pearsons generalized X 2 statistic: X 2 ( y ; ) = X i ( y i- i ) 2 v ( i ) /w i Under some grouped data situations, both X 2 and D * are asymp- totically chi-square. We assume here that data have been grouped as far as possible and that X 2 and D * are computed as sums of independent contributions over g groups: D * ( y ; ) = 2 g X i =1 { i ( y i ; y i )- i ( i ; y i ) } X 2 ( y ; ) = g X i =1 ( y i- i ) 2 v ( i ) /w i 102 Fixed-cells Asymptotics: Classical assumptions in asymptotic theory for grouped data imply i. a fixed number of groups ii. increasing sample sizes n i , i = 1 , . . . , g , such that n i /n i where i > 0, i = 1 , . . . , g , are fixed proportions iii. a fixed number of cells k in each group (think of a discrete outcome variable in each group with k possible values) iv. a fixed number of parameters being estimated in the current model Under these assumptions and certain regularity conditions, then both X 2 and D * are asymptotically chi-square under the null hypothesis that the current model holds with X 2 , D * a 2 ( t 1- p ) where t 1 = g ( k- 1) = the number of independent parameters estimated under the full model This result holds for a more general class of goodness-of-fit statis- tic, the power-divergence family , within which X 2 and D * are special cases (see F&T, sec. 3.4)....
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This note was uploaded on 11/13/2011 for the course STAT 8620 taught by Professor Hall during the Fall '11 term at University of Florida.

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lec2 - Binomial Distribution: Y 1 , . . . , Y n ind Bin ( n...

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