modified maximum likelihood predictors of the sth-order statistic based on this data, where r < s < n. We suggest four types of modifications to the predictive ...
COMPUTATIONAL STATISTICS & DATAANALYSIS ELSEVIER
Computational Statistics & Data Analysis 25 (1997) 91-106
Modified maximum likelihood predictors of future order statistics from normal samples M o h a m m a d Z. Raqab Department of Mathematics, University of Jordan, Amman 11942 Jordan Received 1 May 1995; revised 1 September 1996
Abstract Suppose we have a Type II censored sample consisting of the first r-order statistics of a random sample of size n from a normal population with unknown mean. In this paper, we look at some modified maximum likelihood predictors of the sth-order statistic based on this data, where r < s < n. We suggest four types of modifications to the predictive likelihood equations in order to find such predictors. We compute their mean square prediction errors by simulation and compare them with the best linear unbiased predictors and alternative linear unbiased predictors for n = 5 and 10 and for selected r and s values. A modification based on first-order Taylor series expansion applied in a two-step procedure appears to yield good predictors when s > r + 1. © 1997 Elsevier Science B.V.
Keywords." Prediction; Order statistics; Normal distribution; Maximum likelihood predictor; Modified maximum likelihood predictor
AMS classification." 62G30; 62F99; 62M20
1. Introduction
Let X I : . < X2:, < --- < X,:, denote the order statistics of a r a n d o m sample from an absolutely continuous cumulative distribution function (cdf) F(x; O)having probability density function (pdf)f, where 0 is possibly a vector-valued parameter. Let Fk:. andfk:, denote the cdf and pdf of Xk:., respectively, for k = 1 to n. Also, let f,~:, denote the pdf of Xi:, and Xj:,, for i 0), the best-known predictor is the best linear unbiased predictor (BLUP) (see, for example, David, 1981, p. 156). This is based on the work by Kaminsky and Nelson (1975), who applied Goldberger's (1962) results in the context of a linear model for the prediction of Xs:,. Let Zi:, = ( X ~ : , - p)/a (i = 1, 2, ..., r), Z = (ZI:,, ..., Zr:,), and let ~t and V denote the mean vector and the covariance matrix of Z, respectively. Then, the B L U P of Xs:. is given by 5L(1) = (ill + aS~L) + w ' V - l ( X - -
fill -- aL:t),
(1.1)
where fiL and aL are the least-squares estimators of # and a, respectively, as = E(Zs:,) and w' = (wl, . . . , wr) with wi = Cov(Zi:,, Zs:,) (i = 1, 2, . . . , r), and 1 is a vector whose elements are all unity. Gupta (1952) has proposed alternative linear estimators of p and a by replacing the variance-covariance matrix of Z, V, by an identity matrix I. Raqab (1996) has used Gupta's estimators to derive an alternative linear unbiased predictor (ALUP) of Xs:, and proposed another A L U P by replacing V by D, where D = diag(vl 1, ..., vr,). Let us denote these ALUP's by 5L(2) and 6L(3), respectively. When a is known, we use/~L = ( I ' V - 1 X ) / ( I ' V - 11) in (1.1) to obtain the BLUP, and replace V by I in this expression for/2L for 6L(2) and replace V by D for 6L(3). Kaminsky and Rhodin (1985) have extended the method of maximum likelihood to allow for the joint prediction of a future random variable and estimation of an unknown parameter. The resulting predictors is being called the maximum likelihood predictor (MLP). They gave some sufficient conditions for the existence and uniqueness of the M L P and illustrated the method with several examples. It is not possible to solve the likelihood equation to obtain a closed-form expression for the M L P in most cases. In the pure estimation problem, Mehrotra and Nanda (1974) obtained approximate maximum likelihood estimators (MLEs) for the normal and gamma distributions by replacing h(x) or xh(x) by their respective expected values, where h, the hazard rate function, is given by f(x) h(x) - 1 - F(x)"
(1.2)
Using numerical computations they showed this procedure produces estimators that are efficient when compared to the best linear unbiased estimators (BLUE's). Balakrishnan and Cohen (1991, Ch. 6) used the Taylor series expansion of h(x) and f ( x ) / F ( x ) around the p,th quantile where p, = r/(n + 1), to obtain modified MLEs of the parameters of the normal and Rayleigh distributions. The main point in their approach is that the likelihood equation involves messy terms and it is impossible to obtain an explicit form for the MLE. Similar is the case with prediction as well because of the nature of the predictive likelihood function (PLF).
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Now, the P L F of Xs:, and 0 is given by Y
L ( X ~ : , , O; X ) = C . ( r , s) 1-[ f(Xj:,) [F(X~:. -- F ( X ~ : , ) ] ~ - r - I T(X~:,) j=l
x [1 - F(X~:,)] "-~,
(1.3)
where C , ( r , s ) = n!/{(s - r - 1)!(n - s)!}. The P L F of X~:, and 0 in (1.3) can be written as a product of two likelihood functions L1 and L2 where
- --- r)! j=~-5 ]q f(xj.,) - (n
L~ ( O ; X ) = f ( O ; X )
[1 - F(X,:,)] "-~
and L2(X~:.;O,X)
=
(n -- r)! [F(X~:.) - F(X,:.)] ~-'-1 (s -- r -- 1)!(n -- s) [1 -- F ( X r : , ) ] " - r x [1 -- F ( X , :,)]"-sf(Xs:,).
The predictive likelihood equations (PLEs) corresponding to L, L1, and L2 are in the following forms: OlogL _ 0 ~Xs:. ~3log L 1(0)
and
~?log____~L_ 0, ~0~
(1.4)
- 0,
(1.5)
where 0i is a component of 0, and 0 log L2 (X~:,) = 0.
(1.6)
So the PLEs involve h(Xi:,) (i = r, s), where h is defined in (1.2) and the extended hazard rate function h~(X,:,, X~:,), h2(Xr:n, Xs:n), defined by
fix) h~(x, y) - F ( y ) - F ( x ) '
f(y)
h z ( x , y) - F(y) - F ( x ) '
x < y.
(1.7)
The following section introduces some useful results for the expected values of these functions. In Section 3, we consider four types of modifications (Types I-IV) to the PLEs, to derive the modified m a x i m u m likelihood predictors (MMLPs). Two of these are based on replacing h, h~ and h2 by their expected values in the PLEs and the remaining two are based on Taylor series expansions. We apply these techniques in Section 4 to predict a future order statistic using a Type II censored sample from normal distribution with u n k n o w n mean /t and known standard deviation e. We compare the predictors so obtained on the basis of bias, MSPE, and the value of the likelihood function at the predicted value. Guided by these comparisons we select the one whose performance is the best most often and compare it with the B L U P s and ALUPs. All these numerical comparisons are discussed in Section 5.
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2. Some useful results We now introduce three lemmas which are important in our approximating h, hi and h2 appearing in the PLEs. We assume below that the derivative of the pdf, f'(x), exists almost everywhere and is absolutely integrable, so that E 7~(Xi:.) exists where
7J(x) = - {f'(x)/f (x)}.
(2.1)
Lemma 2.1. 1
Ef(Xi:,)
n+l
n+ 1 Z
k=i+l
E~(Xk:.+I),
i O. This means that (/~,(1), 6a(1)) corresponds to approximate m a x i m u m point of L. Note that E(Z~:,)s are known and already tabulated in Teichroew (1956, pp. 410-426). F r o m (4.12), we can notice that the modified P M L E of/~,/),(1), is unbiased estimator of #. While in the above derivation we have assumed r + 1 < s < n, it is easily seen that (4.13) holds even when s = n. We now consider the case where s = r + 1. Then the PLEs in (4.1) and (4.2) can be reduced into the following equations:
"
Zi:, + Z~+I:, +
'=1
i
E(Z~:n) = 0.
(4.14)
i=r+2
--Z~+x:n-- ~ E(Zi:n)=O.
(4.15)
i=r+2
The solutions of(4.14) and (4.15) (the modified P M L E o f # and M M L P of X,:.) turn out to be /~a(1)=X(,)
and
6,(1)=X(r)-a
~ ,=r+2
E(Z,:,).
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Since E(Zi:n) , S are monotonically increasing and Z T = l E(Zi:.) = 0, ZT=k E(Zi:,) > 0 for any k > 1. Thus 6a(1) < X(r) < Xr:.. The 6a(1) given in (4.13) need not exceed X,:. either. Thus, we take the Type I M M L P of Xs:, as ~'6,(1) i f g a ( 1 ) > X , : , and r + l < s < n , 6"(1) = ~(X.:. if ~Sa(1) ___Xr:n or s = r + 1, where 6a(1) is given by (4.13).
4.2. Type H modification In the first stage, we replace the hazard function appearing in (4.7) by its expected value. In view of (4.9), the resulting modified P L E will be r
?1
E Zi:n -~- Z i=1
E(Zi:n) =0"
i=r+l
On noting that All(L1)~- - ra -2 < 0, the modified P M L E of # can be given by o(21 =
+
e(z,:.).
r i=r+l
Using the facts that E(X~:.)=I~+aE(Z~:,) and Z~'=~E(Z~:.)=0, we get E(~,(2)) = #. Thus,/~,(2) is unbiased estimator of ~. This estimator was obtained by Mehrotra and N a n d a (1974) in the estimation set-up. In the second stage, we substitute ~a(2) given above, in (4.8), and replace h(Zs:,) and h2(Z,:., Zs:,) in that equation by their respective expected values in (4.9) and (4.10). Thus, the modified likelihood equation simplifies to E(Zs:,)-Zs:.
=0
(r+l
