ELSEVIERJournal of Statistical Planning and

Inference 63 (1997) 55 70

journal ofstatistical planningand inference

Sequential estimation of the mean of NEF-PVF distributions

Abstract

ILet .F : {F;!: II E 0) denote the class of natural exponential family of distributions having

power variance function, (NEF-PVF). WC consider the problem of’ sequentially estimating themean /L of 15 E .P, based on i.i.d. observations from 6,. WC propose an appropriate scclucntialestimation procedure under a combined loss of estimation error and sampling cost. We provideexpansion for the regret .R, and study its asymptotic properties. We show that A+‘,, : &( // ) T o( I )as (I - X. where c > 0 is a known constant and r(/l) denotes the coefficient of variation of I-Y,.0 1997 Elsevicr Science B.V.

.4 MS c~lrrssjfic~trtion: primary 62L 12; secondary 62F I2

k’o~~t~~rl.s: Sequential estimation: Stopping time; Regret expansion; Exponential families: Powervariance function; Compound Poisson: Stable distributions

1. Introduction

Let A’, .A’?. . > be a sequence of .i.d. random variables from a distribution F‘ havingan unknown mean /A and variance 0:. Consider the problem of estimating the mean /Jsubject to minimum risk under a combined loss of squared error and sampling cost

L,, = L,,(&) =A(&, - /I)2 + II. (1.1)

where ,i,, is an estimator of /I and A denotes the known weight of the estimation errorrelative to the sampling cost. Based on a sample of size II (fixed and nonrandom ). theestimator ,i,, =x,, s 11-l C X, has a risk with respect to ( 1.1) given by

(1.2)

This risk is minimized by taking a sample of size (an integer adjacent to) (I :: .4 ’ ‘CT.The corresponding minimum risk is E(L,,) = 2~. The problem however. is that often~___

* C’orrcsponding author. E-mail: boukaifir math.iupui.cdu.

(13%3758:97:$17.00 @ 1997 Elsevicr Science H.V. All rights rcscrvrdPI1 so37x-37~~(9~)ool97-x

56 A. Bose, B. Boukail Journul qf’ Statistical Planning and Injermcr 63 (1997) 55-70

cr* is an unknown (nuisance) parameter and no fixed sample size n minimizes (1.2).This then calls for a sequential procedure which involves a sample of random size N.The basic idea is to continue sampling the observations, one at a time, until a goodestimate of a or equivalently of cr2 is obtained. This leads to a sample of random sizeas determined by the stopping rule:

N = inf{n 3mo, sf <!,,n2/A}.

where of is a sample estimate of cr2 (usually the maximum likelihood estimate), mo(32) is some initial sample size and 4, = 1 + &o/rz +o( 1) is some converging sequenceof constants, introduced to reduce bias. Such sequential procedures are usually analyzedby studying the behavior of E(L,+r ) (the risk associated with xv) as compared to E(L,)as A-x.

Sequential procedures of this nature were initially developed by Stein (1945, 1949)and Robbins (1959) as two-stage procedures and fully sequential procedures for pointestimation and interval estimation of prescribed accuracy for the mean of normallydistributed random variables.

In the normal distribution case, si is taken to be the sample variance and the estimate2, and the event {N = n} are independent for every n. This independence property washeavily exploited by most researchers who subsequently worked on the normal problem.Starr (1966) showed that for the normal case the sequential estimation procedure isrisk ejficz’ent (i.e. lim,~3X E(L,\;)/E(L,) = 1). Starr and Woodroofe (1969) found thatthe difference .J?A, = E(L,+f ) - E(L,) (the so-called reyret) is bounded and eventuallyWoodroofe (I 977) used nonlinear renewal theory to provide an asymptotic expansionto .2(, as d, = t + o( 1).

Recently, Bose and Boukai (1993) considered the sequential estimation problem ina class of two-parameter exponential family of distributions. They presented a similarindependence result and utilized it to study the second-order asymptotic properties ofthe resulting regret.

It is evident from the available literature that even though such an independenceresult reduces much of the technical difficulties in the analyses, it is by no meansnecessary to establish regret expansion. Extensions of this procedure to non-normalcases were considered by several authors.

Starr and Woodroofe (1972) dealt with the one-parameter gamma distribution, inwhich case a2 = rp2 (with known x), and proved the boundedness of the regret. Theregret expansion in this case was provided by Woodroofe ( 1977) by some ingenious andclever analysis which overcomes the lack of independence; see also Vardi (1979) forthe Poisson distribution case. Aras (1989) provided second-order results for the case ofcensored data from the negative exponential distribution. Tahir (1989) provided regretexpansion for a class of the one-parameter exponential family which satisfies a certainsmoothness condition and is based on a stopping time which requires the initial samplesize to approach infinity at a suitable rate (see also Remark 4 below).

To estimate the mean with a ‘distribution free’ approach, Ghosh and Mukhopadhyay( 1979) and Chow and Yu ( 198 I ) used the sample variance as the estimate of G’ and

allowed the initial sample size rn() to be a function of il. which tends to x as A - X.They show the risk efficiency of the estimation procedure. Similar results were provedby Sen and Ghosh (1981) for estimation of symmetric parametric functions using II-statistics. In the distribution free case, Chow and Martinsek (1982) have shown that theregret is bounded. With stronger conditions, Martinsek ( 1983) obtained an asymptoticexpression for the regret for nonlattice variables and bounds on the regret when thevariables are lattice.

Our objective in this paper is to analyze sequential estimation procedures for themean of NEF-PVF distributions. This is the class of one-parameter natural exponen-tial F&mily of distributions having a /~~JHXJ~ ~L//.~LIIXY fi~7cfio~, so that nT2 = Z/I’ . t’orsome Y and ;I. The cases of ;’ = 0, 1.2 correspond to the normal, Poisson and gamma

distributions, respectively. This family also includes the class of compound Poissondistributions and a class of stable distributions. See Bar-Lev and Finis ( 1986) t’or ageneral discussion on this family and its statistical importance.

We propose an appropriate sequential procedure for this filmily of distributions. ‘Theexact procedure depends of course on the value of ;‘. In Theorem 2 below, wc showthat for the NEF-PVF distributions. the regret -#,, is of the form

.#0

= A;, + 4) .?- - - - - - (/1)+0(I).

4(1.3)

as A - X. where c’(/~) denotes the coelficient of variation of the underlying distribu-tion E‘. The limiting regret (1.3) is positive for all 11. However. we show below (seeRemark 6) that if we consider a weighted version of the loss (I, 1 ). then with an ap-propriate choice of the weight, the limiting regret is negative for all IL. The phenomenaof negative regret was first discovered by Martinsek ( 1983) in a nonparametric settingwhich allowed nz~ to approach infinity at a suitable rate. The possibility of a ncga-tive regret was discussed earlier in Starr and Woodroofe ( 1972, pp. I 153~~1 IS3 ). whostudied the negative exponential case. Woodroofe (1985) showed that (set Example 2there) for the negative exponential case, under a weighted version of the loss function( 1.1 ), negative values for the regret are possible.

The analysis involves some interesting interplay between the stopping time .*L’. thestlopped partial sum S.1’ and the overshoot R\ Fortunately. many of the results ot‘b’oodroofe ( 1976, 1977) are available in a sufficiently general framework to bc of USC‘tc us. This reduces some of our technical hurdles.

This paper is organized as follows. In Section 2. we introduce the NEF-PVF family.define our procedure and state the main result. Section 3 is dedicated to the proofswhereas in Section 4, we provide auxiliary results on uniform integrability.

2. Estimating the mean of NEF-PVF distributions

Let -F = {fi,: 0 ( 0) be minimal and steep natural exponential family (NEF) 01‘distributions whose members are of the form

5 8 A. Bose, B. Boukuil Journd of Stutisticml Planning and Injhwice 63 (1997J 55-70

/+(dx) = exp{& + c(o)}/l(dx), 0 E 0, (2.1)

where n is a sigma-finite measure on the Bore1 sets of iw and the parameter set 0consists of all U E R for which s exp{tlx}n(dx) < cc. It is well known (see Bamdorff-Nielsen, 1978) that on int 0, 41 has moments of all order. Let p = ,u(O) = -dc(8)/d0and Sz = p(int 0) denote the mean value of fit and the mean parameter space, respec-tively. In addition, we denote by Y(U) the variance function corresponding to (1.1). Wewill assume in the following that the members of the NEF ,p have a power variancefunction (PVF) so that

V(,U) = R/L;‘, /l E 52 (2.2)

for some constants z # 0 and y. In the sequel we will denote by

(2.3)

the square of the coefficient of variation of fij.The class of NEF distributions with the PVF condition is known to be a broad

family which possesses many interesting properties. Bar-Lev and Enis (1986) haveshown, aside from a reproducibility property, that all NEF-PVFs are infinitely divisiblewith self-generating property. It is convenient to classify the members of .5 by their 7values. The characterization of NEF by means of their variance function was initiatedby Morris (1982). He considered a particular class of NEF having quadratic variancefunction (QVF) and showed that the NEF-QVF class consists of only six members,some of which are also members of the NEF-PVF class. As already mentioned in theintroduction, the 7 values of 0, 1 and 2 correspond to the normal, Poisson-type and thegamma distributions, respectively. Bar-Lev and Enis (1986) have shown that there existno steep NEF-PVF with ;’ values in (--x, 1) - (0). For 1 < j1 < 2 the correspondingNEF-PVF consists of compound Poisson distributions generated by gamma distributionand for 5’ > 2, the NEF-PVF class is induced by a class of stable distributions (seeBar-Lev and Enis, 1986 for further details).

It is clear from (2.2) that with a given value of CI, the case of 1’ = 0 corresponds to thenormal distribution. Moreover, Bar-Lev and Enis (1986) have shown that with 7 > 1,either Sz = [w- or n = Iw+ (according to as int 0 = [w+ or int 0 = iw-, respectively)and that these two cases are symmetrical in the sense that one can be considered asa reflection of the other about the origin. We will assume without loss of generalitythat 1: > 1 and that int 0 = lR_, so that the mean parameter space is s2 = lRf and thesupport of n in (2.1) is [0, co) (in which case SI > 0 trivially). We will maintain theseassumptions throughout the paper. The case of y = 1 (the Poisson case), being a lattice,will not be discussed in the present paper.

Let X1,X2.. . .,A!;, . , be a sequence of independent and identically distributed ran-dom variables from the NEF-PVF fin as defined by (2.1) and (2.2) with p E C? (orequivalently 0 t int 0) being an unknown parameter. For IZ > 1 we let S, =X1 +X2 +. . + A, and let x,, =S,,/n denote the usual average. For a fixed n, ,LI, = 2, is an

unbiased estimate of the mean AL. In view of the variance structure (2.2) and in or-der to achieve minimum risk with respect to the loss function (I. I ), we consider thesequential procedure based on the following stopping rule:

I = inf{n>mo: !,,n’ > A$;; }.

where 1~ > I is some initial sample size, and /,, is some converging sequence ofconstants, /,, = I + /o/n + o( l/n) as II -* X. Since II = [.4 V(/r )I’ ’ and I’( /I ) = zp . WCmay write the stopping rule t above as

where WC have put, for convenience, ii = 2;‘;, and h,, = 1 + A,,!77 with A,, --. ho 3 cVo ‘2as 17 - 3~. The stopping rule t in (2.4) is of the same form as the stopping rule definedin ( 1.1) of Woodroofe ( 1977). Alternatively, t can also be written in the form of ( I ) inLai and Siegmund (1977, 1979) as /,, = inf{n >,m(): S,: + <,( an}, where S(T is a partialsum of some suitable i.i.d random variables and &, is an appropriate slowly changingsequence of random variables. (see also discussion in Section 5 in Lai and Siegmund,1979).

Many of the so-called first-order properties of t are well known in the literature (seeWaoodroofe, 1977) and are presented in the following proposition.

The nonlinear renewal theory developed by Lai and Siegmund ( 1977. 1979) and byWoodroofe (1976, 1977) provide a more refined approximation than that of Propositionl(b) to E(t). However, as is well known (see, for example, Woodroofe 1977. 1982;Bose and Boukai, 1993,1996), the initial sample size and the left tail behavior of theunderlying c.d.f. play a crucial role in such second order analysis. In Lemma 1, WCapproximate the left tail of the distribution of I.

(2.5)

60 A. Bose, B. BoukailJournal qf’ Statistical Planning and lflkrence 63 (1997) 55-70

Remark 1. Lemma 1 demonstrates the intimate ties between the PVF condition, theinitial sample size mo and the asymptotic left tail behavior of the stopping time t asdefined in (2.4).

(a) For ?: > 2, .p is the NEF family generated by stable distributions with support[O,oo) and characteristic exponent q = (2 - y)/( 1 - y). Special known cases are theinverse Gaussian family (q = $) and the family of modified Bessel-type distributions(q = f ) (see Bar-Lev and Enis, 1986). As it can be seen from Lemma 2 below, theleft tail behavior of these distribution is rather ‘nice’. In fact, it follows from Lemma 2that for x < ,u, F~(x)<Q..? for all ci > 0 (see conditions (2.5) in Woodroofe, 1977).Hence, (2.5) above holds for all mo 3 1.

(b) The case of 7 = 2, which corresponds to the gamma distribution, the conditionon mo and s is a restatement of conditions (2.5) and (2.6) in Woodroofe (1977).

(c) For 1 < y < 2, 9 is a family of compound Poisson generated by random sum ofgamma variates (see Bar-Lev and Enis, 1986 for details). These distributions have prob-ability mass at zero. In particular, condition (c) of Lemma 1 assures that P(&,, = 0) +0 as a --) w. See also Remark 2 below.

In the next two theorems we provide the main results of this paper. In Theorem 1,we present the second order approximation to E(t), whereas in Theorem 2, we providethe asymptotic expansion of the regret associated with t. In these expansions, theasymptotic behavior of the overshoot

6R,=b,pt t -s,

0a (2.6)

also plays a crucial role. By Theorem 2.1 of Woodroofe (1977) R, and t,* are asymp-

totically independent and R, 3 H, where H is a random variable whose mean is given

by

Remark 2. The developments in Woodroofe (1976, 1977), require the underlying dis-tribution to be sufficiently smooth (see Condition C in Woodroofe, 1976), so as toestablish the limiting density of the overshoot. However, it is evident from the ar-guments given in Lai and Siegmund (1979) (as well as in Woodroofe, 1976) thata sufficient condition to establish the limiting distribution of the overshoot and theasymptotic independence is that the underlying distribution is nonlattice. Observe thatthe NEF-PVF distributions with y 3 2 satisfy Woodroofe’s smoothness condition, andfor 1 < y < 2, these distributions are merely nonlattice.

The results of Lemma 3 in Section 4 (on uniform integrability), in conjunction withthe conditions of Lemma 1, are needed to establish the second-order approximationpresented in the two theorems below.

A. Bosr, B. Boukail Journal qf Strrtistical Planning anal Inferm? 63 i 1997) 55- 70 6 I

Theorem 1. Under the conditions q/’ Lrnma 1 \t,ith s = 1, MY Izaae us N + x

As in Section 1, we let .gu denote the regret, .?A, = E(L,) - 2~.

Remark 3. Theorem 2 demonstrates that the regret .Hu is asymptotically proportionalto the square of the coefficient of variation (2.3). For instance, in the case of the!g( 11, 0) distributions; y = 2, x = l/v c2(p) = l/v and .A, = 3jv + o( 1). This expansionagrees with that of Woodroofe (1977), who studied the gamma distribution case. More-over, it is easy to verify that for the NEF-PVF class (2.1), r(O) as defined in ( 16) ofWoodroofe ( 1985) is

By Theorem 2, R, --t r for all 0 t int 0, as a + x. Example 2 of Woodroofe ( 1985)shows that for the gamma distribution case (i.e. NEF-PVF with ;‘= 2) our procedurehali asymptotic local minima regret. We conjecture that this remains true for anyNEF-PVF with ?/ > 1.

Remark 4. Tahir (1989) provides a second-order expansion of the regret of his pro-cedure for an exponential family of distributions under the smoothness condition Cof Woodroofe (1976). However, Tahir also imposes a strong condition on the initialsample size 1120, required by his procedure. It appears (see (1.3), (1.4) and the proof ofLemma 2.2 in Tahir, 1989) that in order for his expansions to be valid, Tahir’s choiceof a stopping time, requires the initial sample size to approach infinity at a rate rn(j -& 4 30, as a + oo, for all distributions satisfying the smoothness condition. In con-trast, with only the PVF condition we impose, our Lemma 1 provides a careful examina-tion of the required initial sample size rn() and its relation to the underlying distribution.

Remark 5. Martinsek (1983) provided regret expansion in the nonparametric settingwith rno + ‘X at a particular rate and with an appropriate estimate of 0’. It is interestingto compare Theorem 2 to his regret expansion which is given by

.g; = y - $(_Zf ) + 2&z? ) + o( I ),

where Zi = (Xi - p)/cr. Using relations (2.2) and (2.3) for the Nef-PVF family (2.1 ),it follows that in our notation

_py = 1 + A3 ‘j 2y) &) + o( 1).

2

62 A. Bose, B. Boukail Journal of‘ Statistical Planning and Inference 63 (1997) 55-70

Hence, when compared to Theorem 2, the difference is

as A ---) M. Since 1/ > 1, this difference is positive for all ,U E 52. This is due to the factthat our estimation procedure is based on the parametric maximum likelihood estimateof #Ll.

Remark 6. When A in (1.1) is replaced by A s A*wi(p), the resulting loss is aweighted version of the loss function (1.1). The factor A*wi(p) can be interpreted asthe importance of the estimation error relative to the cost of one observation. When

w&J) = /, with p < y, the stopping rule (2.4) holds with 6 = 2/(y - /3). By astraightforward extension of Theorem 2 (see (3.13) below), we have

%I = $(Y - P)(4 + ‘/ - 5P)n2@) + o(l),

as A* + co. Here, negative values for the regret are possible (for all ,LL E Q, if (7+4)/5 <p < y, (compare with Example 2 in Woodroofe, 1985 with p E 2~).

Remark 7. Theorems 1 and 2 can be used to obtain an estimate for the mean p whichis corrected for bias. For further details in that direction, see Bose and Boukai (1995)who studied the (asymptotic) risk associated with such a bias-corrected estimates.

3. Proofs of main results

In this section we present the proofs of the results of Section 2. However, towardsthe proofs of Lemma 1, we first present some auxiliary results concerning the NEF-PVFdistributions.

Let F be the class of NEF-PVF distributions as defined by (2.1) and (2.2) above.For Q = Rf (thus 0 = iw-) and y > 1 (y # 2), relation (2.2) gives rise to explicitexpressions for the mean ,n(O) and the cumulants generating function c(O) (see Bar-Lev and Enis, 1986 for details). These are given by

/J z p(O) = [a(1 - y)O]‘/(‘-;‘) (3.1)

and

c(0) = &[a(l - y)q(2-7)1(‘--:~) (3.2)

for all 0 E 0. In Lemma 2 below we provide a bound on the left tail of the distributionof s,.

Lemma 2. Let Fo E g with ,u E [w+ and y > 1 (y # 2) then for all positive 8 < 1 wehave

(3.3)

and

(3.4)

Proof. By (2.1) we have for all x < 11

P(X, <x)< ~~ec((‘)-c((l+r)-r.~.

Set ,f’(p) = c(n)-c(Q+r)-rx; then f”(r) = p(H+r)-x and .f“‘(r) = V(p(H+r)) > 0.Hence, it follows by using (3.1) and (3.2) that ,f(r) has a unique minimum at r* =(.Y’-;’ - ,~‘-:‘)/r(l - y) at which point

,f’(r*) = Lx*-; - P2V _ x[xI -:, _ p’-;‘]

x(2 - y) X(1&j’)(3.5)

Let y(x) denote the right-hand side of (3.5). The proof of the first part is completedby taking x = EP and noting that .cl(r+) = &,I-:). By Theorem 2.2 of Bar-Lcv andEnis (1986), n(‘p:‘)‘(T-2).S,, has the same distribution as that of X*, where X” has anNEF-PVF distribution with mean p* = ,~n -‘V(;*’ Hence it follows immediately from. ,(3.3) that

P(S,, dr:np) = P(X* fF.jM ‘+(I -:‘) (Y-2)) = P(X* <f:/~*) < exp{$(pnP’ (;+‘)_J:)}.

which completes the proof of the second part. 0

Proof of Lemma 1. We will only prove parts (a) and (c), since the proof of part (b)(7 = 2; the gamma distribution case) is essentially given in Woodroofe (1977). Leti < v < 1 be fixed and let c denote a generic constant (positive). Then (2.5) may bewritten as

a’fi,(,(m()dtba/2) = us ‘2 P(t = k)k=mo

l ta"P(a" < t <a/2) z I, + 12.

Consider at first 12 and recall that moments of all order exist for (2.1). Then bydefinition (2.3) of t we have by a standard submartingale inequality

I2 = a”P(a” < t<a/2) <a”P(& <cpk’+“!a” for some kE(a”, a.i2])

<a”P( I.$ - kp( >c’k for some k E (a”, u/2])

64 A. Bose, B. Boukail Journal of’ Statistical Planning and Injtirence 63 (1997) 55-70

for sufficiently large Y. Now let &k = k/a and consider the first term 11. Clearly, by(2.4) and Lemma 2,

<as ‘3 exp{ $(pk-“(lk=mo

where 4(.) is given in (3.4). Using (3.4) it can be easily verified that for ‘/ > 2 andsufficiently large a,

II <a” E exp{&~kk”(Y-2), cf )}k=ma

<a$ k’z exp{ -ck(a/k)-“(2-7)}0

for all s > 0 and mo 3 1. This completes the proof of part (a). As for the proof of part(c), with 1 < y < 2, let CO = p2-7/a(y - 1) 5 [(y - I)u2(,u)]-’ and observe that (forsufficiently large a)

Ia’ I

-<a’rkgO exp{cok(l - ?)A2 - Y)>

<d+“c exp{como( 1 - y)/(2 - y)} -+O,

provided that mo > cl log a, where cl > (s + v)(2 - y)/co( 1 - y) or equivalently, ifct > (1 + s)2v2(p). This completes the proof of the third part of the lemma. 0

Proof of Theorem 1. By Lemma 1, with s = 1, we have Po(mo d t <a/2) = o( l/a) andby Lemma 3(b) (below) ]t,“l* is uniformly integrable. The proof now follows exactlyalong the same lines as those of Theorem 2.4 of Woodroofe (1977). q

Proof of Theorem 2. We will first show the formal expectation computation by usingthe relation

(St - Pt>~ = spt; + Op(ud2), (3.6)

which follows directly from (2.6) and a Taylor’s expansion of (t/u)& about 1. This re-lation will be use repeatedly throughout the proof. The rigorous justification of expec-tation computations follow from the uniform integrability of the relevant expressions.These are given separately in Section 4.

where A = a’!V(p). Also, by a simple algebra,

? (S, - ptyn-

t2 = (S, - PO’ + (S, - pq2 ; - 1[ 1

Since by Theorem 1 of Chow et al. (1965) E((S, - /it)‘) = V(/r)E(t) it follows that

E(L) = 2E(t) + &(Q,).

where

Q, = (S, ~ pty 1 - $ + (S, - PC)2 1 - t’ 7 2 = I + ff‘,,[ I [ 1.z $ - (2 say

We will treat I, and I& separately. We begin with II,, the easier of the two. To evaluatethe asymptotic distribution of ZIO, we use (3.6) to obtain

II,, = (S, - pry [I + :I2 ; [ 1 - ;1;

= 1 + f 2 ;t;;‘[psr: + op(a-‘/2)]2[ 1

= 4p262t,*4 + op(u-‘.2),

where the asymptotic distribution of t,* is given in Proposition 1. Thus, provided thatII,, is uniformly integrable (see Section 4 below), we have

lim E(&) = I~V’(~L)/~~/L~.U’X’ (3.7)

A,s for the first term, we write I, as 7I,,=(&/a)* 1-12+&& t- fi -I,+Z2.L 1a a [ 1a

The first term lr is treated similar to /I,. Again by using (3.6) it is easy to see that

I = t*2(St - WI2 =1 ‘I

a_ d21**t,T4 + Op(a-i ),

Hence,

provided that Ir is uniformly integrable (see Section 4).

(3.8)

66 A. Bose, B. BoukailJournal of’ Statistical Planning and Imjhrence 63 (1997) 55-70

The treatment of I1 is more complicated. By using the definition (2.6) of the over-shoot Rt we may write 12 as

12 = $(S, - /~t)~ + G(S, - pQ2(Rt - Y,) = Z21 + 122:cc

where

Clearly (see Proposition l), ~Ld~((t/a)~ - 1)+ 0 almost surely. Moreover, by a Taylorseries expansion of t’ about a6 we obtain that

@[(($l)-q-l)] =~lL(t)6(6_l)t~2(~)6-?;

where j(al/a) - 1) < I(t/a) -- 1 I. Hence, it follows immediately that

Yr = ;I&? - l)t,*’ $ ;I& + op( l),

Accordingly we have by (3.6) and (3.9) that

(3.9)

122 = s[dpti + 0p(a-‘12)]26P [

R, - ~,uij(h - l)ti2 - ;pc%) + q?(l) 1= $[a2p2t,‘2] R, - $6 - l)tz2 - ;y&I

+ q,(l).

Since by Theorem 2.1 of Woodroofe (1977) Rt and t,* are asymptotically independent,we find that

I22 5 -2@Z2r2H + d2(6 - 1),u2Z4r4 + 62~2r2Z2&

where r2 = V(IL)/~~~‘. Hence, with the uniform integrability as is established below,we obtain

lim E(Iz2) = -2p V(p)O’CX ,,_+3(h-l)$$) + fo V(h).

The last term to consider is 121. By Theorem 8 of Chow et al. (1965)

(3.10)

(3.11)

where 14 = E(& - ,u)~. As for the second term of (3.1 l), we find by an applicationof Wald’s Lemma that

A. Bose, B. Boukail Journal qf Sta/istic~crl Plannirq and It~fbmcc 63 (1997) 5.7 ~70 67

Again, by using (3.6), it is easy to see that

2 V(p)[(r - a)($ - ,~t)] = -6V(~()t,‘* + O&-I’*) z -6V(p)Z’z’

Hence, with the uniform integrability ofhave

t:(& - /it)/& as is established below, we

(3.12)

Now, by combining (3.7), (3.8), (3.10) and (3.12), we find that

(3.13)

where c’(p) is defined in (2.3). Moreover, since for the family (2.1), ~3 = ;xc~;‘-’ I’(/!)and 6 = 2/1,!, we find that

E(L,) = 2E(t) + $(2~ + 6)c*(/1) - y + tC, + o( 1).

The proof is then completed by applying Theorem I. 0

4. Uniform integrability

ln Lemma 3 below, we restate (and therefore omit the proof), the results of Lemma2.1 and Theorem 2.3 of Woodroofe (1977). One should realize however, that condi-tions (2.5) and (2.6) in Woodroofe (1977) are now replaced by the conditions of ourLemma 1 as stated for the NEF-PVF class.

Lemma 3. For any s 3 1,(a) (t/a>” und Ri are unijormly integrable,(b) under the conditions of Lemma I, (tij2” is umfi)rmly intecqrable.

We now turn to establish the uniform integrability of II,, I,, 122 and 11, andto provide a full justification to the expectation computations given in the proof ofTheorem 2 (i.e. to justify the relations given in (3.7), (3.8), (3.10) and (3.12)). Themethod of proof we use follows largely that of Woodroofe (1977), however somewhatmore technical, in view of the current form of the overshoot R, (see (2.6)). WC givethe details only for II* and 122. The treatment of 11 and 121 is similar to ,I(,.

Let c > 0 denote a generic constant. We let % denote the event [t <a/2], ~9’ itscomplement and I(.@) the indicator of the set :‘A. On .B we have (S, ~ pt )’ < p’t*,hence it follows that

68 A. Bose, B. Boukail Journal of’ Statistical Planning and Inj&ence 63 (1997) 55-70

Accordingly, we have

J’II,dP<ca’P(t,<a/2)+0

.d

by Lemma I (with s = 2) as a -+ co. On the other hand,

On P, the first term Ji d&~t,“~[l +(t/a)12Rf/a. Hence, it is an immediate consequenceof Lemma 3 (with s = 2 + c) that J,Z(@) is uniformly integrable. Note that I(P)&(t/a)” - 1) = Z(.W)Gt,*(a~/a)‘-‘, where jai/al d 1 + t/a (on SF). Hence,

t 2+26

J2 <ctz4 1 + -L 1 GBC 1, (4.1)a

which in view of Lemma 3 is u.i provided (t, j* 4+1: is u.i. for some a > 0. We now focusattention on the second term 122. Recall that

where Rt is given in (2.6) and

(4.2)

It is easy to see that on 98, 1 Y, 1 <et <ca. Moreover, since Z(%9)(S, - ,ut)2 dct2 we have

J’ IZnl dPdc.?8 s

cRrdP+c.a a J’

t2 dP ,<caE(R,Z(~)) + ca2P($)I

<cE’!2(Rf)aP’i2(@) + ca2P(g).

It follows from Lemmas 3(a) and 1 (with s = 2) that E(I&(g)) + 0 as a ---f 00. Itremains to show that Z22Z(ac) is ui. Clearly,

By Lemma 3, J3 is u.i. In a similar manner to the bound in (4.1), we find that

which in view of Lemma 3 is u.i provided It,*\*+’ is u.i. for some c > 0. Moreover,using the definition (4.2) of Yl we find by a Taylor’s series expansion of I” about N”that

where lalial < 1 + (t/a) (on F). Accordingly,

which in view of Lemma 3 is ui. provided lt,*(‘i” is u.i. for some i: > 0. Similarly,I(B”).J4 is u.i. is provided It,j* ‘+I. is u.i. for some c > 0. Hence, 122 is u.i. providedIt:)“+” is u.i. for some E > 0 .

Acknowledgements

We thank Professor Michael Woodroofe for bringing to our attention the paper otTahir (1989). We also thank the referee for providing us with illuminating commentsand remarks. This work was done while Arup Bose was visiting in Department ofMathematical Sciences at IUPUI in 1993.

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