# lec2 - Binomial Distribution Y 1 Y n ind ∼ Bin n i π i...

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Binomial Distribution: Y 1 , . . . , Y n ind ∼ Bin ( n i , π i ), μ i = nπ i , ⇒ ‘ ( π , y ) = X i ‰ log n i y i ¶ + y i log π i + ( n i- y i ) log(1- π i ) ⇒ ‘ ( μ ; y ) = X i ‰ log n i y i ¶ + y i log μ i n i ¶ + ( n i- y i ) log n i- μ i n i ¶ ⇒ D * ( y ; ˆ μ ) = 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 ˆ μ i n i ¶ + ( n i- y i ) log n i- ˆ μ i n i ¶‚ ⇒ D * ( y ; ˆ μ ) = 2 X i ‰ y i log y i ˆ μ i ¶ + ( n i- y i ) log n i- y i n i- ˆ μ i ¶ In general, for a sample from an E.D. family density D * ( y ; ˆ μ ) = X i 2 w i n y i ( ˜ θ i- ˆ θ i )- ‡ b ( ˜ θ i )- b ( ˆ θ i ) ·o /φ = D ( y ; ˆ μ ) /φ where D ( y ; ˆ μ ) is the (unscaled) deviance, which is a function of the data only (it is free of φ ), and ˜ θ i = θ i ( y i ), ˆ θ i = θ i (ˆ μ i ). 101 Asymptotic Distribution of the Deviance and X 2 Statistics: From the asymptotic chi-square-ness of 2 log λ its tempting to conclude D * ( y , ˆ μ ) a ∼ χ 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 Pearson’s generalized X 2 statistic: X 2 ( y ; ˆ μ ) = X i ( y i- ˆ μ i ) 2 v (ˆ μ 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 ; ˆ μ ) = 2 g X i =1 { ‘ i ( y i ; y i )- ‘ i (ˆ μ i ; y i ) } X 2 ( y ; ˆ μ ) = g X i =1 ( y i- ˆ μ i ) 2 v (ˆ μ 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 → λ i where λ 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 ∼ χ 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)....
View Full Document

{[ snackBarMessage ]}

### Page1 / 100

lec2 - Binomial Distribution Y 1 Y n ind ∼ Bin n i π i...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online