SEQUENTIAL ESTIMATION OF THE SQUARE OF THE RAYLEIGH PARAMETER

where, a is a known positive number, determined by the cost of estimation relative to the cost of a single  observation, β>1 is a given number and w n is an appropriate point estimate of θ (defined below). In practice, one might be interested in estimating the population variance  σ = 1⁄2(4-π)θ or the population second moment μ2 = 2θ. Since both of these parameters are linear functions of θ, it suffices to estimate θ. For observed values x1>0,..., xn>0, of X1,, ..., Xn, the log-likelihood function is: 2 2 1 1 1 ( ) ln 2 ln ln 2 n n


INTRODUCTION
Let X 1 ,,…,X n denote independent observations to be taken sequentially up to a predetermined stage n from the Rayleigh distribution with p.d.f: where, θ is an unknown positive number. It is desired to estimate θ 2 , subject to the loss function considered by (Chow and Yu, 1981;Martinsek, 1988) that is Eqution 1: where, a is a known positive number, determined by the cost of estimation relative to the cost of a single observation, β>1 is a given number and w n is an appropriate point estimate of θ 2 (defined below). In practice, one might be interested in estimating the population variance  σ 2 = ½(4-π)θ 2 or the population second moment µ 2 = 2θ 2 . Since both of these parameters are linear functions of θ 2 , it suffices to estimate θ 2 .
For observed values x 1 >0,…, x n >0, of X 1 ,, …, X n , the log-likelihood function is: , i = 1,…, n and where the random variables Y 1 ,…, Y n are independent with common distribution the Exponential distribution with mean µ Y = θ 2 and standard deviation σ Y = θ 2 .

JMSS
Since n a depends on the unknown value of θ, there is no fixed-sample-size procedure that attains the minimum risk * a R in practice. Therefore, we propose to use the sequential procedure ( ) T T,Y which stops the sampling process after observing Y 1 ,…,Y T and estimates θ 2 by with m a being a positive integer. Note that the standard deviation based on Y 1 ,…, Y n is used in (2) as the estimator of θ 2 , instead of n n W = Y , since θ 2 is also the standard deviation of Y 1 . (3) is such that δ√a ≤m a = o(a) as a→∞ for some δ>0, then Equation 4:

If m a in
As a θθ, by Martinsek (1988), since the skewness of Y 1 is equal to 2. This shows that T Y is biased for large values of a. Thus, consider the biased-corrected estimator Equation 5: where, * a R is as in (2). In this study we provide a secondorder asymptotic expansion, as as a→∞, for ( ) * T a r T,θ and show that this regret is asymptotically negative if we choose 6 0 < θ < (4β -4) (3.25β +1) . Starr and Woodroofe (1969) considered the case in which X 1 , X 2 , … are i.i.d. Normal random variables and showed that the regret of their procedure is O(1). Then, Woodroofe (1977) showed that the regret is 0.
Martinsek (1983) extended Woodroofe's result to the nonparametric case. Tahir (1989) proposed a class of bias-reduction estimators of the mean of the oneparameter exponential family and provided an asymptotic second-order lower bound for the regret. Kim and Han (2009) considered estimation of the scale parameter of the Rayleigh distribution under general progressive censoring. Mousa et al. (2005;Prakash, 2013) focused on Bayesian prediction and Bayesian estimation for Rayleigh models.

ASYMPTOTIC EXPANSION FOR THE REGRET OF THE SEQUENTIAL PROCEDURE
Rewrite the stopping time T in (3) as Equation 7: And let U a = t(V t /t) -1/2 -a denote the excess over the stopping boundary. Chang and Hsiung (1979) showed that the excess U a converges in distribution to a random variable U as a→∞.

Lemma 1
Let T be as in (3). Then is the asymptotic mean of the excess over the boundary.

Proof
The first assertion follows from Lemma 1 of Chow and Robbins (1985). For the second assertion: As a→∞, by Chang and Hsiung (1979), using the fact that the kurtosis of Y 1 is

Proposition 1
Let * n θ be defined by (5) and let T be defined by (3) with ma being such that δ√a≤m a = o(a)as a→∞ for some δ>0. Then, as a→∞.

Proof
For a>0 Equation 8: The proposition follows by taking the limit as a→∞ in (8) and using (4) and the fact that as a→∞ if β>1, by the first assertion of Lemma 1 and (2.2) of Martinsek (1983).

Lemma 2
Let T be defined by (3) with m a being such that δ√a≤ma = o(a) as a→∞ for some δ>0 and with β>1. Then: ( ) As α→∞.

Proof
First, observe that Equation 10: ( ) ( ) For a>0. Moreover Equation 11: As a→∞, by (4). Next, expand g(y) = y 1/θ 1 at : where, T * is a random variable such that Next, rewrite T in (3) as T = inf{n ≥ m a : n(V n /n) −β/2 >a}, where, V n is as in (7) and let: Denote the excess over the stopping boundary. Expanding h(y) = y −β/2 at 2 Y y σ = , substituting y = V T /T and multiplying by T yields: for a>0, where λ T is a random variable between V T /T and 2 Y σ . Furthermore, write: It follows easily that Equation 13: For a>0, where:
Theorem 1. Let T be defined by (3) with m a being such that δ√a ≤m a = o(a) as a→∞ for some δ>0 and β>1.
Let the regret of the biased-corrected procedure ( ) * T T,θ be as in (6) As a→∞ if δ>1, by Martinsek (1988). Next, take the limit, as a→∞, in (9) and use (19), Lemma 2 and the fact that: as a→∞ if δ>1, by the first assertion of Lemma 1 and (2.2) of Martinsek (1983), to complete the proof.

NEGATIVE ASYMPTOTIC REGRET
Theorem 1 shows that the biased-corrected procedure  This means that for the values of θ in the interval (0, θ β ) with β>1, the sequential procedure ( ) * T T,µ performs better, for large values of a, than the best fixed-samplesize procedure ( ), * a * n a n ,Y where * a n is the greatest integer less than or equal to n a = aθ 2β (see Table 1).

CONCLUSION
We have proposed a sequential procedure for estimating the square of the shape parameter of the Rayleigh distribution and provided a second-order asymptotic expansion for the incurred regret. It is seen that the proposed procedure performs better than the best fixed-sample-size procedure if the shape parameter lies in a specific subinterval of the positive real numbers.
For future research, it would be worth considering Bayesian sequential estimation of a function of the shape parameter of the Rayleigh distribution, in which the focus will be on finding a sequential procedure and approximating the Bayes regret, as well as comparing the proposed procedure with existing procedures.