HIGHER THAN SECOND-ORDER APPROXIMATIONS VIA TWO-STAGE SAMPLING

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1 Sankhyā : The Indian Journal of Statistics 1999, Volume 61, Series A, Pt. 2, pp HIGHER THAN SECOND-ORDER APPROXIMATIONS VIA TWO-STAGE SAMPING By NITIS MUKHOPADHYAY University of Connecticut, Storrs SUMMARY. We consider the classical fixed-width (= 2d) confidence interval estimation problem for the mean µ of a normal population whose variance σ 2 is unknown, but it is assumed that σ > σ where σ (> 0) is known. Under these circumstances, the seminal two-stage procedure of Stein (1945, 1949) has been recently modified by Mukhopadhyay and Duggan (1997), and that modified methodology was shown to enjoy asymptotic second-order characteristics, similar to those found in Woodroofe (1977) and Ghosh and Mukhopadhyay (1981) in the case of the purely sequential estimation strategies, that is, expanding E(N) and the coverage probability respectively up to the orders o(1) and o(d 2 ) as d 0. In Theorem 1.1, we first obtain expansions of both lower and upper bounds of E(N) up to the order O(d 6 ). In Theorem 1.2, we then provide expansions of the lower and upper bounds for the coverage probability associated with the two-stage procedure of Mukhopadhyay and Duggan (1997) up to the order o(d 4 ), whereas Theorem 1.3 further sharpens this order of approximation to O(d 6 ). These results amount to what may be referred as the third-order approximations and beyond via double sampling. Such results are not available in the case of any existing purely sequential and other multistage estimation strategies. 1. Introduction We consider a sequence of independent and identically distributed random variables X 1, X 2,... from a N(µ, σ 2 ) population where µ R and σɛr + are assumed unknown parameters. Given two preassigned numbers d(> 0) and 0 < α < 1, let us address the classical problem of constructing a confidence interval I for µ such that P {µ I} 1 α for all µ and σ 2, and that the length of I is fixed at 2d. Based on X 1,..., X n where n is predetermined, this problem cannot be solved (Dantzig (1940)). However, Stein (1945, 1949) came up with an ingenious technique involving a two-stage sampling strategy in order to tackle this problem. Paper received. April AMS (1991) subject classification. Primary 6212; secondary Key words and phrases. Fixed-width interval, consistency, third-order expansions, higher-order expansions.

2 third-order approximations and beyond 255 One starts with X 1,..., X m where m( 2) is a fixed initial sample size, and obtains X m = m 1 Σ m i=1 X i, Sm 2 = (m 1) 1 Σ m i=1 (X i X m ) 2. et b m be the upper 50α% point of the Student s t distribution with (m 1) degrees of freedom. Define N = max {m, [ b 2 msm/d 2 2] } (1.1) where [u] = largest integer < u. If N = m, no sampling is needed in the second stage. If N > m, then one samples the difference (N m) in the second stage. Now, based on the combined observations X 1,..., X N we propose the fixed-width confidence interval I N = [ X N ± d ] for µ. Under this scheme, Stein (1945) showed that P {µ I N } 1 α for all fixed µ, σ 2, d and α. This property is known as consistency. If one, however, pretends that σ is known, then for fixed n, P {µ I n } = 2Φ(n 1 2 d/σ) 1, which would be at least (1 α) if n is the smallest integer a 2 σ 2 /d 2 = C, say, where a is the upper 50α% point of the standard normal distribution. Ghosh and Mukhopadhyay (1981) showed that the modified twostage procedure of Mukhopadhyay (1980), where m is allowed to go to infinity at a certain rate as d 0, was first-order efficient, but not second-order efficient, in the asymptotic case. In their terminology, the purely sequential procedure of Ray (1957) and Chow and Robbins (1965) was asymptotically second-order efficient (Woodroofe (1977)), but the consistency property was lost. Even though σ is assumed unknown, throughout this paper, we suppose that σ > σ where σ (> 0) is known. In this situation, Mukhopadhyay and Duggan (1997) appropriately modified the original two-stage procedure (1.1) and this modified methodology was shown to enjoy the consistency property together with the asymptotic second-order characteristics, that is expanding E(N) and the coverage probability respectively up to o(1) and o(d 2 ), which were similar to those enjoyed by the fully sequential estimation strategies under the frequentist paradigm. For a Bayesian overview, one may look at Cohen and Sackrowitz (1984). For small and moderate values of C, the modified methodology worked exceptionally well when, for all practical purposes, σ is not within twenty percent of the magnitude of σ. In this paper, we set out to derive the third- and higher order approximations, that is up to o(d 4 ) and O(d 6 ), for the average sample size as well as the coverage probability associated with the double sampling technique of Mukhopadhyay and Duggan (1997). Since σ > σ where σ (> 0) is known, let us define m = max {m 0, [ a 2 σ/d 2 2] } (1.2) where m 0 ( 2) is a fixed integer. et X 1,..., X m be the pilot observations which provide X m and Sm. 2 Then, let N = max {m, [ b 2 msm/d 2 2] } (1.3)

3 256 nitis mukhopadhyay and we implement this two-stage procedure as we did in the case of (1.1). Finally, based on X 1,..., X N, we consider the fixed-width confidence interval I N = [ X N ± d ] for µ. This methodology was proposed by Mukhopadhyay and Duggan (1997). Obviously, the random variables I(N = n) and X n are independent for all n m, where I( ) stands for the indicator function of ( ), and hence [ ( ) ] P (µɛi N ) = E 2Φ N 1 2 d/σ 1... (1.4) where = E [g (N/C)] ( ) g(x) = 2Φ ax 1 2 1, x > (1.5) The main results are now summarized as follows, while their proofs are deferred to Section 3. We write g (k) (x) to denote d k g(x)/dx k for k = 1,..., 6. Theorem 1.1. For the two-stage procedure (1.2) - (1.3), we have as d 0 : η 0 +η 1 C 1 +η 2 C 2 +O(C 3 ) E(N C) (1+η 0 )+η 1 C 1 +η 2 C 2 +O(C 3 ) where η i 1 = a i (σ 2 /σ 2 )i, i = 1, 2, 3, with a 1, a 2, a 3 defined in (2.3). A property such as η 0 +o(1) E(N C) (1+η 0 )+o(1) would be referred to as second-order approximation or second-order efficiency according to Woodroofe (1977) and Ghosh and Mukhopadhyay (1981), respectively. This result was first pointed out in Mukhopadhyay and Duggan (1997) for the particular procedure on hand. One may also look at Ghosh et al. (1997) for an overview. From Theorem 1.1, we can immediately claim that η 0 + η 1 C 1 + o(d 2 ) E(N C) (1 + η 0 ) + η 1 C 1 + o(d 2 ) which can then quite legitimately be viewed as thirdorder approximation. Theorem 1.1 actually provides significantly higher than third-order approximation for the ASN function. From the proof of this result given in Section 3, it becomes clear that bounds for E(N C) can be derived literally up to any desired order of approximation. For more brevity, we continue up to a remainder term which is O(C 3 ). Theorem 1.2. For the two-stage procedure (1.2) - (1.3), we have as d 0 : (1 α) + C 1 { η 0 g (1) (1) + ( σ 2 σ 2 1) g (2) (1) g (3) (1) } + C 2 { η 1 g (1) (1) B 2g (2) (1) B 3g (3) (1) σ4 σ 4 g(4) (1) } + o(d 4 ) P {µɛi N } (1 α) + C 1 { (η 0 + 1) g (1) (1) + ( σ 2 σ 2 1) g (2) (1) + g (3) (1) } +C 2 { η 1 g (1) (1) B 1g (2) (1) B 4g (3) (1) σ4 σ 4 g(4) (1) } + o(d 4 ) with η 0, η 1 defined in Theorem 1.1, B 1, B 2 defined in emma 2.3, and B 3, B 4 defined in emma 2.4.

4 third-order approximations and beyond 257 The bounds for the coverage probability given in Theorem 1.2 can be referred to as expansions up to third-order approximation. Theorem 1.3. For the two-stage procedure (1.2) - (1.3), we have as d 0 : (1 α) + A 1 C 1 + A 2 C 2 + O(d 6 ) P {µ I N } (1 α) + A 1U C 1 + A 2U C 2 + O(d 6 ) where A 1, A 2 are respectively the same coefficients of C 1, C 2 in the lower bound in Theorem 1.2, and A 1U, A 2U are respectively the same coefficients of C 1, C 2 in the upper bound in Theorem 1.2. The major difference between Theorems 1.2 and 1.3 amounts to significantly sharpening the rate of approximation by replacing the o(d 4 ) term in Theorem 1.2 with the O(d 6 ) term in Theorem 1.3. From Section 3, it will be clear that in order to verify Theorem 1.3 we need two additional emmas beyond what we need in providing a proof of Theorem 1.2. The types of higher order approximations developed in this paper are not available for any problem in the sequential estimation literature. Also, these investigations may provide the crucial impetus in the future to explore possibilities of achieving higher than second-order approximations via purely sequential and other sampling strategies. 2. Auxiliary emmas and Proofs Recall that b m is the upper 50α% point of the Student s t distribution with (m 1) degrees of freedom. From Johnson and Kotz (1970, page 102), one finds the following Cornish-Fisher expansion of b m in terms of a: b m = a a(a2 + 1)(m 1) a(5a4 + 16a 2 + 3)(m 1) a(3a6 + 19a a 2 15)(m 1) 3 + O(m 4 ).... (2.1) From (2.1), the following expansions can be immediately verified: b 2 ma 2 = 1 + a 1 m 1 + a 2 m 2 + a 3 m 3 + O(m 4 ), b 4 ma 4 = 1 + 2a 1 m 1 + (a a 2 )m 2 + 2(a 3 + a 1 a 2 )m 3 + O(m 4 ), b 6 ma 6 = 1 + 3a 1 m 1 + 3(a a 2 )m 2 + (3a 3 + 6a 1 a 2 + a 3 1)m 3 + O(m 4 ),

5 258 nitis mukhopadhyay b 8 ma 8 = 1 + 4a 1 m 1 + 2(3a a 2 )m 2 + 4(a 3 + 3a 1 a 2 + a 3 1)m 3 + O(m 4 ), b 10 m a 10 = 1 + 5a 1 m 1 + 5(2a a 2 )m 2 + 5(a 3 + 4a 1 a 2 + 2a 3 1)m 3 + O(m 4 ), b 12 m a 12 = 1 + 6a 1 m 1 + 3(5a a 2 )m 2 + 2(3a a 1 a a 3 1)m 3 where +O(m 4 ),... (2.2) a 1 = 1 2 (a2 + 1), a 2 = 1 24 (4a4 + 23a ), a 3 = 1 48 (2a6 + 26a a ).... (2.3) The first two lemmas are included here for completeness. These were proved in Mukhopadhyay and Duggan (1997). emma 2.1. For the stopping variable N from (1.3) we have as d 0 : P (N = m) = O (h 1 2 (m 1)) where h = (σ 2 /σ2 ) exp{1 (σ 2 /σ2 )} is a positive proper fraction. emma 2.2. For the stopping variable N from (1.3) we have: (i) C 1 2 (N C) N ( 0, 2σ 2 σ 2 ) as d 0; (ii) C 1 (N C) 2 is uniformly integrable for 0 < d < d 0, with sufficiently small d 0. We now set out to obtain both the lower and upper bounds for some of the positive moments of C 1 2 (N C) in the forms of expansions up to the appropriate order, to be made precise in the form of several lemmas. In the sequel, however, we repeatedly use expansions of some of the positive moments of Sm 2 which are given below. E ( ) Sm 4 = σ { m 1 + 2m 2 + 2m 3 + O(m 4 ) }, E ( ) Sm 6 E ( ) Sm 8 E ( ) Sm 10 E ( ) Sm 12 et us define and one notes that = σ 6 { 1 + 6m m m 3 + O(m 4 ) }, = σ 8 { m m m 3 + O(m 4 ) }, = σ 10 { m m m 3 + O(m 4 ) }, = σ 12 { m m m 3 + O(m 4 ) }.... (2.4) T = max { m, b 2 ms 2 md 2},... (2.5) T N T (2.6)

6 third-order approximations and beyond 259 Also, one has N C T C (2.7) From the proofs given in Mukhopadhyay and Duggan (1997), it is clear that emmas hold when N is replaced with T. Observe also from emma 2.1 that m s P (T = m) converges to zero at an arbitrarily fast rate, whatever be s(> 0) fixed. In the proofs of our emmas that follow, we combine terms such as m s P (T = m) for different values of s, and replace these with one generic expression, namely, O(m r ) or O(C r ) where r(> 0) can be chosen sufficiently large. Another point is to be noted here. A proof of Theorem 1.1 is given in Section 3, but it does not exploit any of the following lemmas, directly or otherwise, and hence in the proofs of such lemmas we use Theorem 1.1 whenever needed as N is replaced with T, that is, a result such as the following: E(T C) = η 0 + η 1 C 1 + η 2 C 2 + O(C 3 ).... (2.8) After looking at the expression in (3.7), it will become clear that in the proof of Theorem 1.2, we would need expansions or bounds for (i) E(N C) up to o(c 1 ), (ii) E{C 1 (N C) 2 } up to o(c 1 ), (iii) E{C 3 (N C) 3 } up to o(c 2 ), and (iv) E{C 2 (N C) 4 } up to o(1). The Theorem 1.1 and emmas provide results which are considerably stronger than the requirements in (i) - (iv) respectively. Next, from (3.8) it will also be clear that in the proof of Theorem 1.3, we would need expansions or bounds for (i) E(N C) up to O(C 2 ), (ii) E{C 1 (N C) 2 } up to O(C 2 ), (iii) E{C 3 (N C) 3 } up to O(C 3 ), (iv) E{C 2 (N C) 4 } up to O(C 1 ), (v) E{C 5 (N C) 5 } up to O(C 3 ), and (vi) E{C 3 (N C) 6 } up to O(1). The Theorem 1.1 provides a result which is stronger than what is needed in (i). The emmas respectively provide exactly what is needed in (ii) - (vi). emma 2.3. For the stopping variable N from (1.3), we have as d 0 : 2σ 2 σ B 1C 1 + O(C 2 ) E { C 1 (N C) 2} 2σ 2 σ B 2C 1 + O(C 2 ) where B 1 = (a a 1 + 2)σ 4 σ 4, B 2 = B 1 + 2η 0 + 1, with η 0 and a 1 respectively defined in Theorem 1.1 and (2.3). Proof. First, we will show that E { C 1 (T C) 2} = 2σ 2 σ 2 + B 1C 1 + O(C 2 ).... (2.9) From (2.5), we can write b 2 ms 2 md 2 T mi(t = m) + b 2 ms 2 md 2,... (2.10)

7 260 nitis mukhopadhyay which implies that b 4 ms 4 md 4 T 2 3m 2 I(T = m) + b 4 ms 4 md (2.11) Now, we utilize the expansions of b 4 m, b 2 m and E(S 4 m) from (2.2), (2.4) and the inequalities from (2.10) - (2.11) to write E { (T C) 2 /C } C 1 [ b 4 md 4 E(S 4 m) + C 2 2mCP (T = m) 2Cb 2 mσ 2 d 2] = C [{ 1 + 2a 1 m 1 + ( a a 2 ) m 2 + O(m 3 ) } {1 + 2m 1 + 2m 2 + O(m 3 )} + 1 2{1 + a 1 m 1 +a 2 m 2 + O(m 3 )} ] + O(m r ) = C [ 2m 1 + (2 + 4a 1 + a 2 1)m 2 + O(m 3 ) ] + O(m r ) Similarly, we obtain = 2σ 2 σ 2 + B 1C 1 + O(C 2 ).... (2.12) E { (T C) 2 /C } C 1 [ b 4 md 4 E(S 4 m) + 3m 2 P (T = m) + C 2 2Cb 2 mσ 2 d 2], and this can be simplified along the lines of (2.12). Thus, (2.9) follows. Next, using (2.6), one obtains E { (T C) 2 /C } 2 E { (N C) 2 /C } E { (T C) 2 /C } + 2E(T/C) + C 1,... (2.13) and then the result follows by combining (2.8), (2.9) and (2.13). emma 2.4. For the stopping variable N from (1.3), we have as d 0 : 6C 1 + B 3 C 2 + O(C 3 ) E { C 3 (N C) 3} 6C 1 + B 4 C 2 + O(C 3 ) where B 3 = q 6η 0 3, B 4 = q + 6σ 2 σ 2 + 6η with q = (8 + 6a 1 )σ 4 σ 4 and η 0, a 1 respectively defined in Theorem 1.1 and (2.3). Proof. First, we will show that From (2.5), observe that E { C 3 (T C) 3} = qc 2 + O(C 3 ).... (2.14) b 6 ms 6 md 6 T 3 7m 3 I(T = m) + b 6 ms 6 md 6,... (2.15)

8 third-order approximations and beyond 261 and combine this with (2.10)- (2.11) and utilize (2.2), (2.4) to write E { C 3 (T C) 3} C [ 3 b 6 md 6 E ( { Sm) 6 3C 3m 2 P (T = m) + b 4 md 4 E ( )} Sm 4 +3C 2 b 2 mσ 2 d 2 C 3] = { 1 + 3a 1 m ( a a 2 ) m 2 + O(m 3 ) } { 1 + 6m m 2 +O(m 3 ) } 3 { 1 + 2a 1 m 1 + ( a a 2 ) m 2 + O(m 3 ) } {1 +2m 1 + 2m 2 + O(m 3 ) } + 3 { 1 + a 1 m 1 + a 2 m 2 +O(m 3 ) } 1 + O(m r ) = { 1 + 3(2 + a 1 )m 1 + ( a 1 + 3a a 2 )m 2 + O(m 3 ) } 3 { 1 + 2(1 + a 1 )m 1 + (2 + 4a 1 + a a 2 )m 2 +O(m 3 ) } + 3 { 1 + a 1 m 1 + a 2 m 2 + O(m 3 ) } 1 + O(m r ) = (8 + 6a 1 )m 2 + O(m 3 ). Similarly, one can obtain E { C 3 (T C) 3} C 3 [ b 6 md 6 E ( S 6 m) + 7m 3 P (T = m) 3Cb 4 md 4 E ( S 4 m) + 3C 2 b 2 mσ 2 d 2 + 3C 2 m 2 P (T = m) C 3],... (2.16) and this can be simplified along the lines of (2.16). Thus, (2.14) follows. Next, repeatedly using (2.6), we get C 3 (N C) 3 C 3 (T C) 3 + 3C 2 { (T C) 2 /C } and + 6C 2 T + C 3 (3T + 1),... (2.17) C 3 (N C) 3 C 3 (T C) 3 3C 2 (2T + 1).... (2.18) Now, the result follows by combining (2.8) - (2.9), (2.14), and (2.17) - (2.18). emma 2.5. For the stopping variable N from (1.3), we have as d 0 : E { C 2 (N C) 4} = 12σ 4 σ 4 + O(C 1 ).

9 262 nitis mukhopadhyay Proof. Following the lines of arguments as before, one obtains E { C 2 (T C) 4} C 2 [ b 8 ma 8 { m m 2 + O(m 3 ) } 4b 6 ma 6 { 1 + 6m m 2 + O(m 3 ) } +6b 4 ma 4 { 1 + 2m 1 + 2m 2 + O(m 3 ) } 4b 2 ma ] + O(m r ) = C 2 { 12m 2 + O(m 3 ) } = 12σ 4 σ 4 + O(C 1 ), and the same upper bound would similarly follow. Hence, we immediately conclude that E [ C 2 (T C) 4] = 12σ 4 σ 4 + O(C 1 ). et us denote U = C 1 2 (T C) and V = C 1 2 (N T ). From what we have shown so far, one can claim uniform integrability of U s for 0 < s 4. Next, let us write E { C 2 (N C) 4} = E(U 4 ) + 4E(U 3 V ) + 6E(U 2 V 2 ) + 4E(UV 3 ) + E(V 4 ).... (2.19) Since N T 1, we have E(V s ) = O(C s/2 ) for s = 1, 2, 3 and 4. Next, Holder s inequality leads to: E(U 3 V ) E 3 4 (U 4 )E 1 4 (V 4 ) = O(1)O(C 1 2 ) 0,... (2.20) and also E(U 3 )E(V ) = C 3 2 O(C 2 )O(C 1 2 ) 0, utilizing (2.14). Combining this with (2.20), one can claim that U 3 and V are asymptotically uncorrelated. Hence, E(U 3 V ) = O(C 1 ) asymptotically. Using similar arguments, one can see that E(U 2 V 2 ) E 2 3 ( U 3 )E 1 3 ( V 6 ) = O(1)O(C 1 ) 0, while E(U 2 )E(V 2 ) = O(1)O(C 1 ) 0, implying that U 2 and V 2 are then asymptotically uncorrelated. Thus, E(U 2 V 2 ) is asymptotically O(C 1 ). Similarly, one shows that E(UV 3 ) = O(C 1 ) as well. Hence, the result follows from (2.19), by taking expectations term by term. emma 2.6. For the stopping variable N from (1.3), we have as d 0 : E { C 5 (N C) 5} = O(C 3 ).

10 third-order approximations and beyond 263 Proof. As before, we start with E { C 5 (T C) 5} b 10 m a 10 { m m 2 + O(m 3 ) } 5b 8 ma 8 {1 + 12m 1 +56m 2 + O(m 3 ) } + 10b 6 ma 6 { 1 + 6m 1 14m 2 + O(m 3 ) } 10b 4 ma 4 { 1 + 2m 1 + 2m 2 + O(m 3 ) } + 5b 2 ma O(m r ) = { 1 + (20 + 5a 1 )m 1 + ( a a a 2 )m 2} 5 { 1 + (12 + 4a 1 )m 1 + ( a 1 + 6a a 2 )m 2} +10 { 1 + (6 + 3a 1 )m 1 + ( a 1 + 3a a 2 )m 2} 10 { 1 + (2 + 2a 1 )m 1 + (2 + 4a 1 + a a 2 )m 2} +5 { 1 + a 1 m 1 + a 2 m 2} 1 + O(m 3 ) + O(m r ) = O(C 3 ),... (2.21) and the upper bound can be tackled analogously as in previous lemmas. emma 2.7. For the stopping variable N from (1.3), we have as d 0 : E { C 3 (N C) 6} = 120σ 6 σ 6 + o(1). Proof. First, we will show that E { C 3 (T C) 6} = 120σ 6 σ 6 + O(C 1 ).... (2.22) Following the earlier lines of arguments, we get E { C 3 (T C) 6} C 3 [ b 12 m a 12 { m m m 3 + O(m 4 ) } 6b 10 m a 10 { m m m 3 + O(m 4 ) } +15b 8 ma 8 { m m m 3 + O(m 4 ) }

11 264 nitis mukhopadhyay 20b 6 ma { m m m 3 + O(m 4 ) } +15b 4 ma { m 1 + 2m 2 + 2m 3 + O(m 4 ) } 6b 2 ma ] + O(m r ) = C [{ (30 + 6a 1 )m 1 + ( a 1 + 6a a 2 1)m 2 +( a a a 2 + 6a a 1 a 2 +20a 3 1)m 3} 6 { 1 + (20 + 5a 1 )m 1 + ( a a a 2 )m 2 + ( a a a 2 + 5a a 1 a a 3 1)m 3} + 15{1+ (12 + 4a 1 )m 1 + ( a 1 + 6a a 2 )m 2 + ( a a a 2 + 4a a 1 a 2 + 4a 3 1)m 3} 20 { 1 + (6 + 3a 1 )m 1 + ( a 1 + 3a a 2 )m 2 +( a a a 2 + 3a 3 + 6a 1 a 2 + a 3 1)m 3} +15 { 1 + (2 + 2a 1 )m 1 + (2 + 4a 1 + a a 2 )m 2 +(2 + 4a 1 + 2a a 2 + 2a 3 + 2a 1 a 2 )m 3} 6 { 1 + a 1 m 1 +a 2 m 2 + a 3 m 3} O(m 4 ) ] + O(m r ) = C 3 { 120m 3 + O(m 4 ) } = 120σ 4 σ 4 + O(C 1 ),... (2.23) and the upper bound can be treated in the same fashion. Thus, (2.22) follows. Hence, uniform integrability of C 3 (T C) 6 immediately follows, since C 1 2 (T C) N(0, 2σ 2 σ 2 ). From (2.7), one notes that C 1 2 N C C 1 2 T C + 1 for C > 1, and thus C 3 (N C) 6 is uniformly integrable. In view of emma 2.2, part (i), the result then follows from (2.22). Remark 2.1. One should note that the full expansions given in (2.2) and (2.4) up to the order O(m 4 ) have been utilized only in the proof of emma 2.7. In the proofs of emmas , we had used the very same expansions

12 third-order approximations and beyond 265 terminated at the O(m 3 ) term. 3. Proofs of Theorems Recall that g(x) = 2Φ(ax 1 2 ) 1 for x > 0, defined in (1.5). Observe that, for all x > 0, we have g (1) (x) = ax 1 2 φ(ax 1 2 ), g (2) (x) = 1 2 a {x a 2 x 1 2 } φ(ax 1 2 ), g (3) (x) = 1 4 a {3x a 2 x a 4 x 1 2 } φ(ax 1 2 ), g (4) (x) = 1 8 a {15x a 2 x a 4 x a 6 x 1 2 } φ(ax 1 2 ), { } g (5) (x) = 1 16 a 105x a 2 x a 4 x a 6 x a 8 x 1 2 φ(ax 1 2 ), g (6) (x) = 1 32 a { 945x a 2 x a 4 x a 6 x a 8 x a 10 x 1 2 } φ(ax 1 2 ), with g (k) (x) = d k g(x)/dx k, k = 1,..., 6, and hence... (3.1) g (1) (1) = aφ(a), g (2) (1) = 1 2 a(a2 + 1)φ(a), g (3) (1) = 1 4 a(a4 + 2a 2 + 3)φ(a), g (4) (1) = 1 8 a(a6 + 3a 4 + 9a )φ(a), g (5) (1) = 1 16 a(a8 + 4a a a )φ(a), g (6) (1) = 1 32 a(a10 + 5a a a a )φ(a).... (3.2) Now we provide the proofs of Theorems But, first we prove the following lemma. emma 3.1. d 0 : For the stopping variable N defined in (1.3), we have as (i) (ii) E [ C 2 (N C) 4 g (4) (W ) ] = 12σ 4 σ 4 g(4) (1) + o(1); E [ C 3 (N C) 6 g (6) (W ) ] = 120σ 6 σ 6 g(6) (1) + o(1);

13 266 nitis mukhopadhyay where W is a random variable between NC 1 and 1, with derivatives g (4) ( ), g (6) ( ) defined in (3.1) - (3.2). Proof. We sketch a proof of part (i) only, since the proof of part (ii) is very similar. From (3.1), observe that g (4) (x) 4 k i x bi... (3.3) where k i s are positive numbers and b 1 = 7 2, b 2 = 5 2, b 3 = 3 2 and b 4 = 1 2. On the set {N = m}, we have W > N/C = m/c. Hence, using emma 2.1 and (3.3), [ E C 2 (N C) 4 g (4) (W )I(N = m) ] 4 i=1 k bi (m C)4 i(c/m) C P (N = m) 2 = O(1)O(m 2 )O(h 1 2 (m 1) ) = o(1).... (3.4) Next, on the set {N > m}, we have W > σ 2 σ 2. Also, C 2 (N C) 4 is uniformly integrable, and hence utilizing (3.3) again, C 2 (N C) 4 g (4) (W )I(N > m) { 4 } C 2 (N C) 4 i=1 k i(σ 2 σ 2 ) bi, so that C 2 (N C) 4 g (4) (W )I(N > m) is then uniformly integrable. Thus, in view of emma 2.2 and the fact that g (4) (x) is continuous at x = 1, we obtain [ ] E C 2 (N C) 4 g (4) (W )I(N > m) = 12σ 4 σ 4 + o(1).... (3.5) Now, (3.4) - (3.5) together complete the proof of part (i). To prove part (ii), observe from (3.1) a bound like that in (3.3) for g (6) (x) and the fact that g (6) (x) is also continuous at x = 1. Further details are left out. We keep combining terms like m s P (N = m) and replace these by a generic term O(m r ) or O(C r ) for appropriately large r(> 0). Proof of theorem 1.1. From (1.3), we note the basic inequality, i=1 b 2 ms 2 md 2 N mi(n = m) + b 2 ms 2 md which implies ( b 2 m a 2) σ 2 d 2 E(N C) ( b 2 m a 2) σ 2 d O(m r ),... (3.6) with sufficiently large r(> 0). From the expansion of b 2 ma 2 given in (2.2), it follows immediately that ( b 2 m a 2) σ 2 d 2 = a 1 Cm 1 + a 2 Cm 2 + a 3 Cm 3 + O(C 3 ) = a 1 (σ 2 /σ 2 ) + a 2(σ 2 /σ 2 )2 C 1 + a 3 (σ 2 /σ 2 )3 C 2 +O(C 3 ),

14 third-order approximations and beyond 267 and hence the result follows from (3.6). Proof of theorem 1.2. From (1.5), recall that P (µɛi N ) = E[g(N/C)], and with some random variable W between 1 and NC 1, we obtain E[g(N/C)] = g(1) + C 1 (N C)g (1) (1) C 1 (N C) 2 C g (2) (1) + 1 (N C) 3 6 C g (3) (1) C 2 (N C) 4 C g (4) (W ) (3.7) Since g (2) (1) is negative, utilizing the upper and lower bounds from emmas as well as Theorem 1.1 and emma 3.1, part (i), we get: E[g(N/C)] g(1) + { η 0 C 1 + η 1 C 2 + o(c 2 ) } g (1) (1) and also C 1 { 2 + 2σ 2 σ 2 +B 2C 1 + o(c 1 ) } g (2) (1) { 6C 1 + B 3 C 2 +o(c 2 ) } g (3) (1) C 2 { 12σ 4 σ 4 g(4) (1) + o(1) } = (1 α) + C 1 { η 0 g (1) (1) + ( σ 2 σ 2 + 1) g (2) (1) g (3) (1) } +C 2 { η 1 g (1) (1) B 2g (2) (1) B 3g (3) (1) σ4 σ 4 g4 (1) } +o(c 2 ), E[g(N/C)] g(1) + { (η 0 + 1)C 1 + η 1 C 2 + o(c 2 ) } g (1) (1) since g(1) = 1 α C 1 { 2σ 2 σ 2 2 +B 1C 1 + o(c 1 ) } g (2) (1) { 6C 1 + B 4 C 2 + o(c 2 ) } g (3) (1) C 2 { 12σ 4 σ 4 g(4) (1) + o(1) } = (1 α) + C 1 { (η 0 + 1)g (1) (1) + ( σ 2 σ 2 1) g (2) (1) + g (3) (1) } +C 2 { η 1 g (1) (1) B 1g (2) (1) B 4g (3) (1) σ4 σ 4 g4 (1) } +o(c 2 ), Proof of theorem 1.3. With some random variable W between 1 and

15 268 nitis mukhopadhyay NC 1, we can again write E[g(N/C)] = g(1) + C 1 (N C)g (1) (1) C 1 (N C) 2 C g (2) (1) + 1 (N C) 3 6 C g (3) (1) C 2 (N C) 4 C g (4) (1) 2... (3.8) + 1 (N C) C g (5) (1) C 3 (N C) 6 C g (6) (W ). 3 Recall that g (2) (1), g (4) (1) are both negative. Again utilizing the upper and lower bounds from emmas as well as Theorem 1.1 and emma 3.1, part (ii), we get: E[g(N/C)] g(1) + { η 0 C 1 + η 1 C 2 + O(C 3 ) } g (1) (1) C 1 { 2 + 2σ 2 σ 2 + B 2C 1 +O(C 2 ) } g (2) (1) { 6C 1 + B 3 C 2 + O(C 3 ) } g (3) (1) C 2 { 12σ 4 σ 4 + O(C 1 ) } g (4) (1) + O(C 3 ) +O(C 3 ) { 120σ 6 σ 6 + o(1)} = (1 α) + C 1 { η 0 g (1) (1) + ( σ 2 σ 2 + 1) g (2) (1) g (3) (1) } +C 2 { η 1 g (1) (1) B 2g (2) (1) B 4g (3) (1) σ4 σ 4 g(4) (1) } +O(C 3 ), and the upper bound is similar. Remark 3.1. In order to replace o(d 4 ) order for the remainder term in Theorem 1.2 by the significantly sharper rate O(d 6 ), the emmas made all the difference. Remark 3.2. Mukhopadhyay and Duggan (1997) had included some simulation results with µ = 5, σ = 5, σ = 2, 4, m 0 = 10 and C = 30, 40, 50, 100, 200, 300, 400, 500, 800, based on 5000 independent replications for each entry. Their second-order interval for E(N C) was taken to be (η 0, η 0 +1), and it was found that (n C) values were largely within this interval while, for the most part, staying closer to η 0 than (η 0 +1) for larger values of C(> 200). For C = 30, 40, 50 and 100, the corresponding (n C) values stayed within (η, η + 1) where η = η 0 +η 1 C 1 +η 2 C 2, given by Theorem 1.1, and usually staying much closer to η. The usefulness of the third- and higher order approximations, particularly for small and moderate values of C, is demonstrably clear.

16 third-order approximations and beyond 269 Acknowledgement. The author thanks Professor Bimal K. Sinha for his help in preparing a clearer presentation. References Chow, Y.S. and Robbins, H. (1965). On the asymptotic theory of fixed-width sequential confidence intervals for the mean. Ann. Math. Statist. 36, Cohen, A. and Sackrowitz, H.B. (1984). Bayes double sampling estimation procedures. Ann. Statist. 12, Dantzig, G.B. (1940). On the nonexistence of tests of Student s hypothesis having power functions independent of σ. Ann. Math. Statist. 11, Ghosh, M. and Mukhopadhyay, N. (1981). Consistency and asymptotic efficiency of twostage and sequential estimation procedures. Sankhya Ser. A 43, Ghosh, M., Mukhopadhyay, N. and Sen, P.K. (1997). Sequential Estimation. John Wiley & Sons., Inc., New York. Johnson, N.. and Kotz, S. (1970). Continuous Univariate Distributions 2. John Wiley & Sons., Inc., New York. Mukhopadhyay, N. (1980). A consistent and asymptotically efficient two-stage procedure to construct fixed-width confidence intervals for the mean. Metrika 27, Mukhopadhyay, N. and Duggan, W.T. (1997). Can a two-stage procedure enjoy secondorder properties? Sankhya Ser. A 59, Ray, W.D. (1957). Sequential confidence intervals for the mean of a normal population with unknown variance. J. Roy. Statist. Soc. Ser. B 19, Stein, C. (1945). A two sample test for a linear hypothesis whose power is independent of the variance. Ann. Math. Statist. 16, (1949). Some problems in sequential estimation (abstract). Econometrica 17, Woodroofe, M. (1977). Second-order approximations for sequential point and interval estimation. Ann. Statist. 5, N. Mukhopadhyay Department of Statistics University of Connecticut Storrs CT mukhop@uconnvm.uconn.edu