1. Introduction
Rayleigh distribution was presented by Rayleigh [
1] in 1880 and primarily proposed in the context of a problem in acoustics and optics. As a useful reference for the history of the distribution, see Johnson et al. [
2]. The distribution is extensively used in communication theory to describe the hourly median and immediate peak power of received radio signals. It plays a crucial role in survival analysis, reliability analysis, physical sciences, engineering, medical imaging science, applied statistics, and clinical studies. For more details related to its application, see Palovko [
3], Gross and Clark [
4], Lee and Wang [
5], Rosen et al. [
6], and Siddiqui [
7]; for more information about the distribution and its statistical parameter inference, see Siddiqui [
7], Hirano [
8], Dyer and Whisenand [
9], Howlader and Hossian [
10], and Johnson et al. [
2].
There have been many forms for the Rayleigh distribution to provide flexibility for modeling data. Vod [
11,
12] proposed a generalized form of the Rayleigh distribution and discussed its statistical and inferential properties. The probability density function of the generalized form with scale parameter
and shape parameter
is given by:
where
is the gamma function.
At
we obtain the standard Rayleigh distribution. The probability density function of the standard Rayleigh distribution with scale parameter
is:
Now, let
be independent and identically distributed random variables following a standard Rayleigh distribution with unknown scale parameter
. It can be shown from Johnson et al. [
2] that the population mean and population variance of the distribution are, respectively,
and
.
Recently, Yousef et al. [
13] discussed the Rayleigh distribution scale parameter’s multistage estimation using Hall’s [
14] three-stage procedure. They tackled two estimation problems, point and confidence interval estimation, under a unified optimal stopping rule. They obtained the three-stage regret while they discussed the coverage probability through Monte Carlo simulation. They proved that the procedure attains asymptotic second-order efficiency and asymptotic consistency in the sense of Chow and Robbins [
15] and Ghosh and Mukhopadhyay [
16]. Tahir [
17] proposed a purely sequential procedure to tackle the point estimation problem for the square of the scale parameter of the Rayleigh distribution, using a weighted squared-error loss function plus the cost of sampling. He found a second-order asymptotic expansion for the incurred regret and proved that the asymptotic regret is negative for a range of parameter values.
In this paper, the aim is to estimate the population variance
or the population second moment
of the Rayleigh distribution through estimating the scale parameter’s square of the Rayleigh distribution. We do so because both the variance and the second moment are linear functions of
. We use Hall’s [
14] procedure to carry out the study. Using sequential estimation goes back to the non-existence of a fixed-sample-size procedure that solves the problem analytically. For more details, see Mukhopadhyay and de Silva [
18] (chapters 6–13 and 16) and Ghosh et al. [
19] (Theorem 3.7.1). Since our focus is on the sequential estimation of the scale parameter’s square, we use the following transformation to ease the subsequent sections’ calculations. Let
, and
. Then the Jacobian transformation yields that:
which is the probability density function of the exponential distribution with mean
. It is readily known that
is distributed according to the chi-squared distribution with two degrees of freedom
. Let
be a sequence of independent and identically distributed random variables following the exponential distribution in (1), then the
raw moment is given by:
Hence, for any
and
we have:
Moreover, let
be the sample average of a random sample of size
, then:
3. Three-Stage Sequential Procedure for Inference
Hall [
14] introduced the three-stage sampling procedure. The objective was to obtain a fixed-width confidence interval for the mean of a normal distribution when the variance is finite but unknown. It was designed to overcome several technical problems in both one-by-one purely sequential schemes that were introduced by Anscombe [
24], Robbins [
25], and Chow and Robbins [
15] and the two-stage bulk sample that was introduced by Stein [
26,
27] and Cox [
28]. The procedure showed asymptotic second-order efficiency and asymptotic consistency in the sense of Chow and Robbins [
15]. Mukhopadhyay [
29] developed a unified framework for the three-stage procedure and laid out the theory associated with asymptotic second-order properties. As suggested by the name, the procedure is carried out in three consecutive sampling phases, the pilot phase, the main-study phase, and the fine-tuning phase.
The pilot-study phase: We start the process by observing a random pilot sample of size to initiate the sampling procedure and calculate the sample estimate .
The main-study phase: In this phase, only a portion
of the optimal sample size
is estimated to avoid the early stopping and the possibility of oversampling. Let
be the integer-valued function. Then the procedure is terminated in this stage according to the following stopping rule:
If , we terminate the process at this phase; otherwise, we continue to observe an additional sample of size say augment the two samples and update the estimate .
The fine-tuning phase: We define the fine-tuning stopping rule as:
If , sampling is terminated, else we continue to sample a sample of size , say , then we terminate the sampling course. Hence, we propose the point estimate and the confidence interval for the unknown parameter . As a result, the three-stage point estimate for the variance of the Rayleigh distribution is .
The asymptotic characteristics of each phase are given in the following section.
The following asymptotic results were developed under the general regularity assumptions set forward by Hall [
14] to develop a theory for the three-stage sampling procedure, which states:
Assumption A.
Let and , be constants, with , and ,.
3.1. Asymptotic Characteristics for the Main Study Phase
Theorem 1. Under assumption (A), for the stopping rules (6) and (7), we have as
(i)
(ii)
(iii)
Proof.
To prove we consider .
Conditioning on the generated by , ,, …, we get
.
, by Wald’s [
30] first equation. Therefore,
. We then expand
in a Taylor series expansion around
to obtain:
where
is a random variable that lies between
and
.
where we
in a Taylor series to the second-order around
and the fact that
at its derivative is bounded gives:
as
, by the regularity condition in Assumption (A) and
is a constant independent of
.
Next,
consider the following two cases:
If
, we have:
We have used the assumption that and its derivatives are bounded and the fact that .
Therefore,
This proves of Theorem 1. □
To prove , we write and by of Theorem 1, we get . Arguments similar to those used above provide . Combining terms, the statement of the part of Theorem 1 is immediate. The proof of (ii) is complete. □
Part () of Theorem 1 is the direct application of parts and of Theorem 1. □
The following Theorem 2 provides second-order asymptotic expansion of a real-valued, continuously differentiable and bounded function of .
Theorem 2.
Under assumption (A), for the three-stage stopping rule (6) and (7) and aswe have:
Proof.
The proof is prompt if we consider the second-order Taylor expansion of around and make use of and of Theorem 1, and the assumption that the real-valued continuously differentiable function and its derivatives are bounded. The proof of (iii) is complete. □
In the following section, we present the asymptotic theory for the stopping variable
3.2. Asymptotic Characteristics for the Fine-Tuning Phase
Theorem 3.
Under assumption (A), for the stopping variable and as we have:
(i)
(ii)
(iii)
Proof.
First, write the random variable
as
,
(almost surely) except possibly on a set
, of measure zero such that
, and where the random variable
is distributed uniformly over
; see Hall [
14] for details. Therefore,
. By using Theorem 2 and
the proof is complete. □
Part (
.
. By taking the expectation we have
It can be shown that as , and are asymptotically uncorrelated.
Hence,
. By using Taylor expansion for
and utilizing Theorem 2, we have:
By using part (i), we obtain the proof. The proof is complete. □
Part follows directly from and of Theorem 3. □
The first part of Theorem 3 shows that
(first-order asymptotic efficiency) and
is bounded by a finite number that is unrelated to
. Such a property is called second-order asymptotic efficiency in the sense of Chow and Robbins [
15]. Part (
iii) shows that the variance increases as
increases.
Theorem 4 below provides a second-order asymptotic expansion of the moments of a real-valued function that is a continuously differentiable and bounded function of .
Theorem 4.
Let assumption (A) hold, and let (> 0) be a real-valued continuously differentiable function in a neighborhood around such that )|. Then as ,
Proof.
The proof is a direct substitution of and of Theorem 3 in the Taylor series expansion of the function . We omit any further details for brevity. The proof is complete. □
Lemma 1.
As , is an asymptotically standard normal distribution.
Proof.
According to Anscombe [
31], the central limit theorem
,
has an asymptotically standard normal distribution. By computing the moment generating function of
,
and using Theorem 4 we get the result. The proof is complete. □
Theorem 5.
Under assumption (A), for the stopping variable and as we have:
(i)
(ii)
(iii)
Proof.
To prove (i) write
Then, condition on the
generated by
,
to get:
Given
the term
is constant. Moreover, by Wald’s first equation [
30] we get:
Consider the second-order expansion of
in the Taylor series around
; we have:
where random variable
is between
and
. The assumption that
and its derivatives are bounded can be used to prove that
Consider that the first-order Taylor expansion of ) gives
. Again, we condition on the
generated by
. We get:
Arguments similar to those used above and the fact that yield statement of ( of Theorem 5. The proof is complete. □
Part of Theorem 5 is the direct use of of Theorem 5; we omit details. The proof is complete. □
Part (i) of Theorem 4 shows that is an asymptotically unbiased estimator of whereas the variance decreases as increases.
3.3. The Asymptotic Regret
The regret associated with the quadratic loss function with linear sampling cost given by (2) is:
where
is the loss associated with the three-stage sampling estimation procedure, and
is the optimal loss had the parameter
been known.
and by
of Theorem 5 and
of Theorem 3, we get:
and the optimal risk
. Therefore, the asymptotic regret is given as:
As shown above, negative regret is expected, since
. The issue of negative regret was addressed also by Martinsek [
22], Yousef [
32], and Hamdy [
33]. This phenomenon deserves an in-depth investigation shortly.
3.4. Three-Stage Asymptotic Coverage Probability for the Parameter
The three-stage coverage probability is defined as:
Meanwhile, the two events
and the event
are dependent for
. Therefore, we cannot obtain a mathematical expression of the coverage probability like those of Hall [
14], Hamdy et al. [
34], and Hamdy [
35]. Therefore, we conducted a Monte Carlo simulation using Microsoft Developer Studio software to study the performance of the three-stage fixed-width confidence interval for
when the optimal sample size varies from small to moderate and to large.
4. Simulation Study
We conducted a Monte Carlo simulation [
36] to study the performance of the fixed-width confidence interval for the parameter
. A series of 5
replications was generated from the exponential distribution with mean
using Microsoft Developer Studio software with the IMSL (International Mathematical and Statistical Library). The optimal sample sizes were chosen as recommended by Hall [
15]:
and 500. We took the design factor
and the pilot sample
. For brevity, we will consider
which gives
. Let
be the simulated estimate of the optimal sample size
with standard error
. Let
be the simulated estimate of the scale parameter’s square of the Rayleigh distribution with standard error
.
is the simulated estimate for the variance of the Rayleigh distribution and
is the simulated estimate of the coverage probability.
Table 1 below demonstrates the simulation results as the optimal sample size increases. We noticed that the simulation results agree with our findings while the coverage probability improves as the optimal sample size increases. It is evident from the simulation that the procedure provides coverage probabilities that are less than the prescribed nominal value, that is,
. At the same time, as
,
. Collectively, all estimates improve as the optimal sample size increases. For the simulation methodology, see Yousef [
37].