Consistency of modified Fisher prediction function

Journal of Statistical Planning and Inference 17 (1987) 401-403 North-Holland

401

CONSISTENCY OF M O D I F I E D FISHER PREDICTION FUNCTION W. LIANG Institute o f Statistics, Academia Sinica, Taipei, Taiwan, R.O.C. Received 22 July 1986 Recommended by M.L. Puri

Abstract: This short note gives an easy proof of weak and strong consistencies of the modified Fisher prediction function under simplified assumptions. AMS Subject Classification: 62F99. Key words: Modified Fisher prediction function; Consistency.

1. Introduction Let ~ l = {f(" ; 0): 0 e O} and ~2 = {g(" ; 0): 0 e 6)} be two families of probability density functions with common parameter space 6) having a topology. Let Xt, .... X,, be a random sample from the p.d.f, f(x; 00), describing an informative experiment, and Y be another random variable having p.d.f, go'; 00) describing a future experiment. The forms of the densities f and g are known but the common parameter 00 is unknown. The estimate of gO'; 00) based on X1, ... ,Xn is called a prediction function. Levy and Perng (1984, 1986) suggested the so-called modified Fisher prediction function as follows: L L , ( y i XI, ..., Xn) = s u p o Lp(O; Xl, ..., Xn)Lf(O; y) supo Lp(O; X 1, ... ,Xn) '

(1)

where Lp(O; Xl, ... ,Xn) and Ly(O; y) are the parametric likelihood functions for 0 based on the past and future respectively. In Levy and Perng (1986), the weak consistency of the modified Fisher prediction function was proved under a set of regularity conditions. This short note will give an easy proof of weak and strong consistencies of the modified Fisher prediction function under some simplified conditions. Centrumvoor Wisku;~de en In(ormatic~ Amsterdam

0378-3758/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

W. Liang / Fisherpredictionfunction

402

2. Main result

Let t~n and O* be the m a x i m i z e r o f Lp (0; X l , ... , X n) and Lp (0; X l , ... , X n)Lf(O, y) on 0 respectively. The regularity conditions (vii) and (ix) of Levy and Perng (1986) are

t~n p ' 00 as n ~ co,

(2)

n'~(O*-G,,) P,O

(3)

and

as n - - , ~ for some a > ½ .

Actually, (3) can be weakened as follows: O* p ' 00 as n--' oo.

(4)

Further, in the cited paper it is required that the gradient of g(O; y) is bounded. Here we only need that

g(O; y) is continuous in 0 when y is fixed.

(5)

In fact, these are enough for the weak consistency of modified Fisher prediction function. For the strong consistency, we need the strong version of (2) and (4), namely, On a.s., 00 as n o r a ,

(2')

and

0*

a.s.

' 00

as n ~ ~,.

(4')

Theorem. We have

LL*(y IX~, ...,X,,) p , g(y; 0o)

provided (2), (4) and (5) hold, and a°s°

LL*(Y IXl, ..., Xn) -----' g(Y; 0o)

provided (2'), (4') and (5) hold. Proof. Observe that

1LL*(YIXI,...,Xn)-g(Y; 00)1 ---max

I

Lp(O* ; X I , ... , X n ) L f ( O * ; Y) Lp(~n;Xl,...,Xn)

-g(Y; ~o),

g(y; Oo) _ Lp(On , XI, ..., X n)Lf(O* ; y) 1 Lp(On;Xl,...,Xn)

W. L&ng / Fisher prediction function

__

( L A o z , x~, ... , X , ) L f ( O , , , y)

< mn.x ~ (

~,

. . . . . . .

Lp(On,Xl,...,Xn)

403

gO'; 00),

Lp(On; XI,...,Xn)Lf(On; Y) 1

Oo)-

7p- .i

7 ,-x-5)

=max{gO'; O*)-gO'; Oo),g(y; 00)-gO'; On)}" If (2), (4) and (5) or (2'), (4') and (5) hold, then both gO,; O*)-g(y; 00) and g(y; 00)-g(y;/~,) converge correspondingly to 0 in probability or almost surely. This completes the proof. By the way, we notice that the independences among Xi's and between Xi's and Y are not directly needed.

References Levy, M.S. and S.K. Perng (1984). A maximum likelihood prediction function for the linear model with consistency results. Commun. Statist. - Theor. Meth. 13(10), 1257-1273. Levy, M.S. and S.K. Perng (1986). A note on the consistency of the modified Fisher prediction function. J. Statist. Plann. Inference 13, 263-267.