Sequential Point Estimation of a Function of the Exponential Scale Parameter

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1 AUSTRIAN JOURNAL OF STATISTICS Volume 33 (2004), Number 3, Sequential Point Estimation of a Function of the Exponential Scale Parameter Chikara Uno Akita University, Japan Eiichi Isogai and Daisy Lou Lim Niigata University, Japan Abstract: We consider sequential point estimation of a function of the scale parameter of an exponential distribution subject to the loss function given as a sum of the squared error and a linear cost. For a fully sequential sampling scheme, we present a sufficient condition to get a second order approximation to the risk of the sequential procedure as the cost per observation tends to zero. In estimating the mean, our result coincides with that of Woodroofe (977). Further, in estimating the hazard rate for example, it is shown that our sequential procedure attains the minimum risk associated with the best fixed sample size procedure up to the order term. Zusammenfassung: Wir betrachten die sequentielle Punktschätzung einer Funktion des Skalenparameters einer Exponentialverteilung bezüglich einer Verlustfunktion, welche als Summe des quadratischen Fehlers und einer linearen Kostenfunktion gegeben ist. Für einen vollständigen sequentiellen Stichprobenplan presentieren wir eine hinreichende Bedingung, um für das Risiko der sequentiellen Prozedur eine Approximation zweiter Ordnung zu bekommen, wobei die Kosten je Beobachtung gegen Null streben. Bei der Schätzung des Erwartungswertes stimmt unser Ergebnis mit jenem in Woodroofe (977) überein. Schätzt man beispielsweise die Hazardrate, so wird gezeigt, dass unsere sequentielle Prozedur bis auf den Ordnungsterm das minimale Risiko erreicht, welches zur besten Prozedur bei festem Stichprobenumfang gehört. Keywords: Stopping Rule, Second Order Approximation, Regret, Uniform Integrability, Hazard Rate. Introduction Let X, X 2,... be independent and identically distributed random variables according to an exponential distribution having the probability density function f σ (x) = ( σ exp x ), x > 0, σ where the scale parameter σ (0, ) is unknown. It is interesting to estimate the mean σ and the variance σ 2. One may like to estimate the hazard rate σ and the reliability parameter, that is, P (X > b) = exp( b/σ) for some fixed b (> 0). For this reason, we consider the estimation of a function of the scale parameter.

2 282 Austrian Journal of Statistics, Vol. 33 (2004), No. 3, Suppose that θ(x) is a positive-valued and three times continuously differentiable function on x > 0 and that θ (x) 0 for x > 0, where θ stands for the first derivative of θ. Let θ and θ (3) denote the second and third derivatives of θ, respectively. Given a sample X,..., X n of size n, one wishes to estimate a function θ = θ(σ) by ˆθ n = θ(x n ), subject to the loss function L(ˆθ n ) = (ˆθn θ) 2 + cn, where X n = n n i= X i and c > 0 is the known cost per unit sample. The risk is given by R n = EL(ˆθ n ) = E(ˆθ n θ) 2 + cn. We want to find an appropriate sample size that will minimize the risk. By Taylor s expansion and the Hölder inequality we can show that under a certain condition, R n σ 2 θ (σ) 2 n + cn for sufficiently large n. Thus, R n is approximately minimized at n 0 σ θ (σ) c = n (say) () with R n0 2cn. Since σ is unknown, however, we can not use the best fixed sample size procedure n 0. Further, there is no fixed sample size procedure that will attain the minimum risk R n0 (see Takada, 986). Thus, it is necessary to find a sequential sampling rule. For the estimation of the mean θ = σ, Woodroofe (977) proposed a fully sequential procedure and gave a second order approximation to the risk. Mukhopadhyay et al. (997) considered the sequential estimation of the reliability parameter θ = exp( b/σ) for some fixed b (> 0). For the normal case, Takada (997) constructed sequential confidence intervals for a function of normal parameters and Uno and Isogai (2002) considered the sequential estimation of the powers of a normal scale parameter. In this paper, motivated by (), we propose the following stopping rule: N = N c = inf n m : n X n θ (X n ), (2) c where m ( ) is the pilot sample size. By the strong law of large numbers we have P (N < + ) =. In estimating θ = θ(σ) by ˆθ N = θ(x N ), the risk is given by R N = E(ˆθ N θ) 2 + ce(n). The performance of the procedure is measured by the regret R N 2cn. The purpose of this paper is to derive second order approximations to the expected sample size E(N) and the risk of the above sequential procedure R N as c 0. In Section 2, we present a sufficient condition to get an asymptotic expansion of the risk. In Section 3, as an example of the function θ(x) we consider the estimation of the hazard rate θ(σ) = σ with simulation experiments and show that our sequential procedure attains the minimum risk 2cn up to the order term.

3 C. Uno et al Main Results In this section, we shall investigate second order asymptotic properties of the sequential procedure. Let h(x) = x for x > 0. θ (x) 2 The stopping rule N defined by (2) becomes N = infn m : Z n n, where Z n = n h(x n) h(σ). Let Y i = (X i /σ), for i =, 2,..., S n = n i= Y i and Y n = n S n. By Taylor s theorem, h(x n ) = h(σ) + h (σ)(x n σ) + 2 (X n σ) 2 h (η n ), where η n is a random variable between σ and X n. Then we have Z n = n + αs n + ξ n with ( ) α = + σθ (σ), ξ θ n = n(x n σ) 2 h (η n ) (3) (σ) 2h(σ) and Let h (x) = θ (x) 2 θ (x) + xθ (x) 2 2θ (x) + xθ (3) (x). θ (x) x 3 θ (x) 3 x 2 θ (x) 2 T = infn : n + αs n > 0 and ρ = E(T + αs T ) 2 2E(T + αs T ). (4) Consider the following assumptions: [ ( (A) Z n n ) ] + 3, n m is uniformly integrable for some 0 < ε 0 <, ε 0 (A2) where x + = max(x, 0). np ξ n < ε n < for some 0 < ε <. n=m Then we obtain the following approximation to the expected sample size for all σ (0, ) but not uniformly in σ. Theorem If (A) and (A2) hold, then E(N) = n + ρ l + o() as c 0, where l = + σθ (σ) θ (σ) + σ2 θ (σ) 2 σ2 θ (3) (σ). θ (σ) 2 2θ (σ)

4 284 Austrian Journal of Statistics, Vol. 33 (2004), No. 3, Proof. Let W be distributed according to a standard normal distribution N(0, ). Then, from (3) d ξ l W 2 as n, ξ n where d denotes convergence in distribution. We shall check conditions (C) (C6) of Aras and Woodroofe (993). Clearly, (C) holds. (C2) with p = 3 and (C3) are identical with (A) and (A2), respectively. Letting g(y) = h(σy + σ)/h(σ), (C4), (C5) and (C6) follow from Proposition 4 of Aras and Woodroofe (993). Hence, from Theorem of Aras and Woodroofe (993), E(N) = n + ρ E(ξ) + o() = n + ρ l + o() as c 0, which concludes the theorem. The proposition below gives sufficient conditions for (A2) which are useful in actual estimation problems. Proposition (i) (ii) If h (η n ) 0 for all n m, then (A2) holds. If sup n m E h (η n ) s < for some s > 2, then (A2) holds. Proof. From (3), (i) implies (A2). Suppose that (ii) holds. For 0 < ε < and q > 2, P (ξ n < εn) = P h (η n )(X n σ) 2 > 2h(σ)ε (2h(σ)ε) q E (X n σ) 2 h (η n ) q (2h(σ)ε) q E X n σ 2qu /u E h (η n ) qv /v, where u + v = and u >. Choose (u, v) and 2 < q < s such that s = qv. By the Marcinkiewicz-Zygmund inequality (see Chow and Teicher, 988) we get E X n σ 2qu = O(n qu ). Thus we have np (ξ n < εn) = O(n q ) as n, which implies (A2). We shall now assess the regret R N 2cn. By Taylor s theorem, θ(x N ) θ(σ) = θ (σ)(x N σ) + 2 θ (σ)(x N σ) (X N σ) 3 θ (3) (φ c ), (5) where φ c is a random variable between σ and X N. We impose the following assumption: (A3) For some a >, u > and c 0 > 0, sup c au E X N σ 4au < and sup E θ (3) (ζ c ) 2au/(u ) <, 0<c c 0 0<c c 0 where ζ c is any random variable between σ and X N. Remark If θ (3) (x) is bounded, then the second part of (A3) is satisfied. If θ (3) (x) is convex on (0, ) and sup 0<c c0 E θ (3) (X N ) 2au/(u ) <, then the second part of (A3) is satisfied. Let θ(x) = x r for x > 0 with any fixed r, for instance. Then θ (3) (x) is convex unless 3 < r < 4.

5 C. Uno et al. 285 The main theorem of this paper is as follows. Theorem 2 If (A), (A2) and (A3) hold, then R N 2cn = σθ (σ) θ (σ) + 7σ2 θ (σ) 2 σ2 θ (3) (σ) c + o(c) as c 0. 4θ (σ) 2 θ (σ) Remark 2 Theorem 2 shows that in estimating the mean θ = σ, the regret becomes 3c + o(c), which coincides with the result of Woodroofe (977). Proof of Theorem 2. From (5), R N 2cn = E(ˆθ N θ) 2 + ce(n) 2cn = θ (σ) 2 E(X N σ) 2 + ce(n) 2cn +θ (σ)θ (σ)e(x N σ) θ (σ) 2 E(X N σ) θ (σ)e(x N σ) 4 θ (3) (φ c ) + 6 θ (σ)e(x N σ) 5 θ (3) (φ c ) + 36 E[(X N σ) 6 θ (3) (φ c ) 2 ]. (6) From Theorems 2 and 3 of Aras and Woodroofe (993), we obtain (7) (9) below, as c 0. θ (σ) 2 E(X N σ) 2 + ce(n) 2cn = c(n ) 2 E(Y N ) 2 + E(N) 2n = 2E(ξW 2 ) 2E(ξ) + 3α 2 + 2αE(Y ) 3 c + o(c) = 4l + 3α 2 + 4αc + o(c) = σθ (σ) θ (σ) + θ (σ) 2 7σ2 2 σ2 θ (3) (σ) c + o(c). (7) θ (σ) 2 θ (σ) θ (σ)θ (σ)e(x N σ) 3 = σθ (σ) θ (σ) c(n ) 2 E(Y N ) 3 = σθ (σ) θ (σ) 6α + E(Y ) 3 c + o(c) = 4 σθ (σ) θ (σ) + θ (σ) 2 6σ2 c + o(c). (8) θ (σ) 2 4 θ (σ) 2 E(X N σ) 4 = σ2 θ (σ) 2 4θ (σ) 2 c(n ) 2 E(Y N ) 4 = 3 4 σ2 θ (σ) 2 θ (σ) 2 c + o(c). (9) For b >, p >, q = p/(p ) and v = u/(u ), we have E (n ) /2 4 Y N θ (3) (φ c ) 2 (Y N ) 2 b E (n ) /2 Y N 4bpu /pu E θ (3) (φ c ) 2bpv /pv E(Y N ) 2bq /q

6 286 Austrian Journal of Statistics, Vol. 33 (2004), No. 3, and by Doob s maximal inequality, ( 2bq E(Y N ) 2bq E sup(y n ) 2bq 2bq ) E(Y ) 2bq <. n Choosing b and p such that a = bp, (A3) yields the uniform integrability of (n ) /2 Y N 4 θ (3) (φ c ) 2 (Y N ) 2, 0 < c c 0. Since (n ) /2 Y N 4 θ (3) (φ c ) 2 get E[(X 36 N σ) 6 θ (3) (φ c ) 2 ] = d θ (3) (σ) 2 W 4 and (Y N ) 2 0 a.s. as c 0, we σ 4 [ (n 36θ (σ) c E ) /2 4 ] Y 2 N θ (3) (φ c ) 2 (Y N ) 2 = o(c). (0) By arguments similar to (0), we obtain and 3 θ (σ)e (X N σ) 4 θ (3) (φ c ) [ = σ2 (n 3θ (σ) c E ) /2 4 ] Y N θ (3) (φ c ) = σ2 3θ (σ) c 3θ (3) (σ) + o() = σ2 θ (3) (σ) c + o(c) () θ (σ) 6 θ (σ)e(x N σ) 5 θ (3) (φ c ) = σ3 θ (σ) [ (n 6θ (σ) c E ) /2 4 ] Y 2 N θ (3) (φ c )Y N = o(c). (2) Substituting (7) (2) into (6), we get R N 2cn = 3 + (6 4) σθ (σ) θ (σ) + ( θ (σ) 2 4 )σ2 + ( 2 + ) σ2 θ (3) (σ) c + o(c) θ (σ) 2 θ (σ) = σθ (σ) θ (σ) + 7σ2 θ (σ) 2 σ2 θ (3) (σ) c + o(c) as c 0, 4θ (σ) 2 θ (σ) which concludes the theorem. 3 Example As an example of the function θ(x), we consider the estimation of the hazard rate θ = θ(σ) = σ. Ali and Isogai (2003) considered the bounded risk point estimation problem for the power of scale parameter σ r of a negative exponential distribution. In this case θ(x) = x r. In estimating θ = σ by ˆθ n = X n, the risk is given by R n = EL(ˆθ n ) = E(ˆθ n θ) 2 + cn = E(X n σ ) 2 + cn,

7 C. Uno et al. 287 which is finite for n > 2. In fact, R n = n + 2 (n )(n 2) σ 2 + cn = σ 2 n + cn + O(n 2 ) as n. Then, n = c /2 σ and the stopping rule N in (2) becomes N = inf n m : n c /2 X n. (3) A second order approximation to the expected sample size is given in the next theorem. Theorem 3 Suppose m. Then, E(N) = n + + o() as c 0. Proof. From (3), N = infn m : Z n n where Z n nx n /σ = ng(y n ) and g(x) = x +. Since g(x) is convex and E[g(Y ) + ] 3 = E(X /σ) 3 <, from Proposition 5 of Aras and Woodroofe (993), (A) and (A2) hold. Since θ = σ, we have ( ) α = + σθ (σ) = and l = + σθ (σ) θ (σ) θ (σ) + σ2 θ (σ) 2 σ2 θ (3) (σ) = 0. θ (σ) 2 2θ (σ) The stopping time T in (4) becomes T = infn : n + S n > 0 =, so that ρ = E(T + αs T ) 2 2E(T + αs T ) = E( + Y ) 2 2E( + Y ) =. Theorem with l = 0 and ρ = yields the theorem. Remark 3 The referee pointed out that for this example E(N) can be calculated by the following elementary method. Let Sn = n σ i= X i. Then E(N) = m + = m + k=m = m + n P (N > k) = m + k=m n 0 n 0 (k )! xk e x dx, m 2 k=0 m 2 = n + + e n k=0 P (Sk < n ) k=m x k k! e x dx, by interchanging summation and integral k (n ) i = n + + o() as c 0. i! i=0

8 288 Austrian Journal of Statistics, Vol. 33 (2004), No. 3, We cannot always calculate E(N) by elementary methods. Take for example, the function θ(x) = x r with r and r 0. In estimating θ = σ by ˆθ N = X N, the risk is given by R N = EL(ˆθ N ) = E(X N σ ) 2 + cn. The regret of the procedure (3) is given as follows. Theorem 4 If m > 2, then R N 2cn = o(c) as c 0. It follows from Theorem 4 that our procedure attains the minimum risk 2cn up to the o(c) term. We need the following lemmas to show Theorem 4. Let M stand for a generic positive constant not depending on c and let c 0 > 0 be a constant such that n. Lemma Let q. (i) (N/n ) q, c > 0 is uniformly integrable. (ii) If m > q, then (N/n ) q, 0 < c c 0 is uniformly integrable. Proof. From (3), (N/n ) q (X N /σ) q. Then, sup c>0 (X N ) q sup n (X n ) q and by Doob s maximal inequality, E sup n (X n ) q ( qe(x q q ) ) q <. Thus (i) holds. To show (ii), observe that (N )X N /σ < n on N > m, so that for 0 < c < c 0, N/n σx N + (/n )I N>m + (m/n )I N=m (σx N )I N>m + (m + ), where I denotes the indicator function. Therefore, for 0 < c c 0, (N/n ) q M (σx N ) q I N>m + (m + ) q. We have sup 0<c c0 (X N ) q I N>m sup n m (X n ) q and by Doob s maximal inequality, E sup n m (X n ) q ME(X m ) q < if m > q. Thus, the second assertion holds. From Theorem 2 of Chow et al. (979), we have the next lemma. Lemma 2 Let q. If (N/n ) q, 0 < c c 0 is uniformly integrable, then (n ) /2 N i= (X i σ) q, 0 < c c 0 is uniformly integrable.

9 C. Uno et al. 289 Proof of Theorem 4. We shall show (A3) with u = 3. Choose a > such that m > 2a. For p > and q = p/(p ), c 3a E X N σ 2a M E (n ) 2 N 2ap /p E(n (X i σ) /N) 2aq /q, i= which, together with Lemmas and 2, implies sup 0<c c0 c 3a E X N σ 2a <. Since θ (3) (x) = 6x 4, by Doob s maximal inequality and the condition that m > 2a we have sup E θ (3) (X N ) 3a M 0<c c 0 ME sup E(X N ) 2a 0<c c 0 sup(x n ) 2a ME(X m ) 2a <. n m So from Remark, (A3) holds. Thus, Theorem 2 with proves Theorem σθ (σ) θ (σ) + 7σ2 θ (σ) 2 σ2 θ (3) (σ) 4θ (σ) 2 θ (σ) Simulation. In order to justify the results of Theorems 3 and 4 we shall give brief simulation results. We are interested in the performance of our sequential procedure for various values of σ, and so we consider the cases when σ = 0.5, and 2, with corresponding θ = 2, and 0.5. Since the cost c is sufficiently small in our theorems, the values of c are chosen such that n = c /2 σ = 20, 30. The pilot sample size is set at m = 3 which is sufficient for Theorem 4. The simulation results in Tables and 2 are based on 00,000 repetitions by means of the stopping rule N defined by (3). From Theorems 3 and 4 we have that E(N) = n + + o() and (R N 2cn )/c = o() as c 0. Tables and 2 show that these results are justified. Further, it appears that the estimates E(ˆθ N ) for θ in both tables are very close to the true values. Therefore, our sequential procedure ˆθ N seems to be effective and useful. Table : n = 20 = 0 σ = 0.5 σ = σ = 2 m = 3 c = 0.0 c = c = θ = 2 θ = θ = 0.5 E(N) E(ˆθ N ) (R N 2cn )/c

10 290 Austrian Journal of Statistics, Vol. 33 (2004), No. 3, Table 2: n = 30 σ = 0.5 σ = σ = 2 m = 3 c = c = 0.00 c = θ = 2 θ = θ = 0.5 E(N) E(ˆθ N ) (R N 2cn )/c Acknowledgement We would like to thank an anonymous referee for the valuable comments. References Ali, M., and Isogai, E. (2003). Sequential point estimation of the powers of an exponential scale parameter. Scientiae Mathematicae Japonicae, 58, Aras, G., and Woodroofe, M. (993). Asymptotic expansions for the moments of a randomly stopped average. The Annals of Statistics, 2, Chow, Y. S., Hsiung, C. A., and Lai, T. L. (979). Extended renewal theory and moment convergence in anscombe s theorem. The Annals of Probability, 7, Chow, Y. S., and Teicher, H. (988). Probability theory (second ed.). New York: Springer- Verlag. Mukhopadhyay, N., Padmanabhan, A. R., and Solanky, T. K. S. (997). On estimating the reliability after sequentially estimating the mean: the exponential case. Metrika, 45, Takada, Y. (986). Non-existence of fixed sample size procedures for scale families. Sequential Analysis, 5, Takada, Y. (997). Fixed-width confidence intervals for a function of normal parameters. Sequential Analysis, 6, Uno, C., and Isogai, E. (2002). Sequential point estimation of the powers of a normal scale parameter. Metrika, 55, Woodroofe, M. (977). Second order approximations for sequential point and interval estimation. The Annals of Statistics, 5,

11 C. Uno et al. 29 Author s addresses: Dr. Chikara Uno Department of Mathematics Akita University - Tegata-gakuenmachi Akita Japan Tel Fax uno@math.akita-u.ac.jp Univ.-Prof. Dr. Eiichi Isogai Department of Mathematics Niigata University 8050 Ikarashi 2-No-Cho Niigata Japan Tel Fax isogai@scux.sc.niigata-u.ac.jp Daisy Lou Lim Graduate School of Science and Technology Niigata University 8050 Ikarashi 2-No-Cho Niigata Japan dlc lim@yahoo.com