Sequential Point Estimation of a Function of the Exponential Scale Parameter Chikara Uno Akita

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 (1977). 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.


Introduction
Let X 1 , X 2 , . . .be independent and identically distributed random variables according to an exponential distribution having the probability density function 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 −1 and the reliability parameter, that is, P (X 1 > b) = exp(−b/σ) for some fixed b (> 0).For this reason, we consider the estimation of a function of the scale parameter.
Austrian Journal of Statistics, Vol. 33 (2004), No. 3, 281-291 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 1 , . . ., X n of size n, one wishes to estimate a function θ = θ(σ) by θn = θ(X n ), subject to the loss function where X n = n −1 n i=1 X i and c > 0 is the known cost per unit sample.The risk is given by R n = E{L( θn )} = E( θn − 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 −1 + cn for sufficiently large n.Thus, R n is approximately minimized at with R n 0 ≈ 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 n 0 (see Takada, 1986).Thus, it is necessary to find a sequential sampling rule.
For the estimation of the mean θ = σ, Woodroofe (1977) proposed a fully sequential procedure and gave a second order approximation to the risk.Mukhopadhyay et al. (1997) considered the sequential estimation of the reliability parameter θ = exp(−b/σ) for some fixed b (> 0).For the normal case, Takada (1997) 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 (1), we propose the following stopping rule: where m (≥ 1) is the pilot sample size.By the strong law of large numbers we have P (N < +∞) = 1.In estimating θ = θ(σ) by θN = θ(X N ), the risk is given by 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 −1 with simulation experiments and show that our sequential procedure attains the minimum risk 2cn * up to the order term.
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2 Main Results
In this section, we shall investigate second order asymptotic properties of the sequential procedure.Let h(x) = 1 x {θ (x)} 2 for x > 0 .
The stopping rule N defined by (2) becomes , where η n is a random variable between σ and X n .Then we have Consider the following assumptions: Then we obtain the following approximation to the expected sample size for all σ ∈ (0, ∞) but not uniformly in σ.
The proposition below gives sufficient conditions for (A2) which are useful in actual estimation problems.
We shall now assess the regret R N − 2cn * .By Taylor's theorem, where φ c is a random variable between σ and X N .We impose the following assumption: (A3) For some a > 1, u > 1 and c 0 > 0, sup where ζ c is any random variable between σ and X N .
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The main theorem of this paper is as follows.

Example
As an example of the function θ(x), we consider the estimation of the hazard rate θ = θ(σ) = σ −1 .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 −1 by θn = X −1 n , the risk is given by

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which is finite for n > 2. In fact, Then, n * = c −1/2 σ −1 and the stopping rule N in (2) becomes A second order approximation to the expected sample size is given in the next theorem.