Post on 02-Jun-2018
8/11/2019 GLMs - List of Standard Pearson Residual Results
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The Actuarial Education Company
Pearson residuals
var( )
i i
i
y m
m
-
where var( )i is the variance of the distribution with any im s replaced by the
fitted results.
Useful for 2~ ( , )i iY N m s only as the Pearson residuals are (0,1)N .
non-normal distributions produce skewed (non-normal) Pearson residuals, which
makes them hard to interpret.
Poisson distribution ~ ( )i iY Poi
Pearson residual
var( )i i i i
i i
y y mm
- -
= = since var( )i iY =
Exponential distribution ~ ( ) ~ (1 )i i iY Exp Expl m
Pearson residual
var( )
i i i i
ii
y y m
mm
- -= = since
2
21var( )i
i iY lm= =
Gamma distribution ~ ( , ) ~ ( , )i i iY Ga Gaa l a a m
Pearson residual
var( )
i i i i
ii
y y m
am
- -= = since
2
2var( ) i
iiY
aal
= =
Normal distribution2~ ( , )i iY N m s
Pearson residual
var( )
i i i i
i
y y m
sm
- -= = since 2var( )iY s=
Binomial distribution
iZ
i nY =
where ~ ( , )i iZ Bin n m
Pearson residual ( )
var( ) (1 )
i i i i
i i i
y y
n
m
m m m
- -= =
-
since ( ) 2 2(1 ) (1 )1var( ) var var( )i i i i iZ ni in nn nY Z m m m- -
= = = =
8/11/2019 GLMs - List of Standard Pearson Residual Results
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The Actuarial Education Company
Poisson distribution ~ ( )i iY Poi with linear predictor, ih a= .
Pearson residual
var( )
i i i i i
i i
y y y e
e
a
a
m m
m m
- - -= = =
since var( )i iY m= and, using inv. link function, ii e eh am = =
Exponential distribution ~ ( ) ~ (1 )i i iY Exp Expl m with linear predictor, ih a= .
Pearson residual
1var( )
i i i ii
ii
y yy
ma
mm
- -= = = -
since2
21var( )i
i iY l= = and, using inv. link function, 1 1
ii ah
m = =
Gamma distribution ~ ( , ) ~ ( , )i i iY Ga Gaa l a a m with linear predictor, ih a=
.
Pearson residual
( 1)var( )
i i i ii
ii
y ya y
a
m ma
mm
- -= = = -
since2
2var( ) i
i
ai a
Y m
l= = and, using inv. link function, 1 1
ii ah
m = =
Normal distribution2~ ( , )i iY N m s with linear predictor, ih a= .
Pearson residual
var( )
i i i i i
i
y y ym a
s sm
- - -= = =
since var( )iY s= , and, using inv. link function, i ih a= =
Binomial distribution iZ
i nY = where ~ ( , )i iZ Bin n m with linear predictor,
ih a= .
Pearson residual
( ) (1 )
var( ) (1 )
i i i i i
i i i
y y y e e
n e n
a a
a
m m
m m m
- - + -= = =
-
since ( ) 2 2(1 ) (1 )1var( ) var var( )i i i i iZ ni in nn nY Z m m m- -
= = = =
and, using inv. link function,
11
ii
e ei
ee
h a
ahm
++
= =