ABSTRACT
F(T) = G(~) = f g(z) dz. (9.3.4) In making the transformation from X to Z, ~ has become the parameter to be
estimated. After ~ has been calculated, it then follows that fF = Tl~. The maximum likelihood estimating equation a In L/aa = 0 follows from
(9.3.2) as
a In L = _ np + _!._ i x 2 _ c aF = 0 aa <T a 3 i=I I 1-F(T) aa , (9.3.5)
where
- g(~) (:2). (9.3.6) aF aG aG a~ -=-=--= We substitute (9.3.6) into (9.3.5) and at the same time replace a with T!~ and simplify to obtain
(9.3.7)
where as a result of the transformation from X to Z, ~ has now become the parameter to be estimated. We calculate~ as the solution of (9.3.7), and it follows as
(9.3.8)
In order to evaluate HP(~. c!n) as defined in (9.3. 7), we need to evaluate the integral
lz lz 2-(p-2)12 , G (z) = g (t)dt = tp-le-t-12 dt. p 0 p 0 f(p/2) (9.3.9)
This can be done by repeated integration by parts. For p = 2, and 3, we find
G3(z) = 2[1f>(z)- z<\>(z)] - 1, (9.3.10) where <j>(z) and lf>(z) are the pdf and the cdf, respectively, of the standard normal distribution (0,1). Withp = 2 and 3, respectively, in (9.3.2), the standardized pdfs become
and (9.3.11)
For p = 2, we substitute g() and GO from (9.3.11) and (9.3.10) into (9.3.7), and simplify to obtain
H (~ ~) = ~ - ~ = 2a2 - ~ 2 'n e n yz n· (9.3.12)
it= - L x~ + cT2 • 1 [ I! ] 2n i=l (9.3.13) This estimator might be derived in a more direct manner with F 2(x; a) - exp[ -x2/2a2] by returning to (9.3.2) and writing
n 1 n yz lnL = -2n In a+ L lnx;- - 2 L x~- ~ + const. (9.3.14)
Accordingly,
(9.3.15)
For p = 3, we substitute g3() and G3() from (9.3.II) and (9.3.IO) into (9.3.7) and simplify to obtain
H3 ( ~. ~) = ~ - ~ L -<l>~b(~ ~4>(~) l (9.3.I6) and the estimate for~ follows from (9.3.7) as
(9.3.I7)
The loglikelihood function in this case is
- 2n In a + 2: ln(x; - "{) - - 2 2: (x; - "{)2 i= 1 2a i= 1
c 2 - 2a 2 (T - "{) + const, (9.3.I8)
and the maximum likelihood estimating equations are II I II a lnL
(Y (Y i=t (Y
We eliminate a between these two equations and simplify to obtain the following equation in 'Y only:
(9.3.20)
This equation can be solved iteratively for the MLE, :Y. Then fr 2 follows from the second equation of (9. 3. 19) as
(r2 = - 2: (x; - )')2 + c(T - )'? . I [ 11 J 2n i=1 (9.3.2I) Modified maximum likelihood estimators that employ the second equation of
(9. 3 .I9) plus the equation E(X 1 :N) = x 1 N exhibit a smaller bias than the MLE
and may therefore be preferred over the MLE in most applications. For these estimators, in place of the first equation of (9.3.19), we have
We estimate O" between the two applicable estimating equations to obtain
'TT n i=I
in which 'Y is the only unknown. We solve this equation for :y, and fr 2 then follows from (9.3.21).
