Dantzig (1940) proved the non-existence of fixed sample size procedures to construct the confidence interval of preassigned width and coverage probability for a normal mean in the ignorance of any knowledge about the variance. In order to deal with this estimation problem, Stein (1945) proposed a two-stage procedure. Starr (1966) and Woodroofe (1977) adopted purely sequential procedures for the same estimation problem. Two-stage and purely sequential procedures for estimating the parameters involved with different probabilistic models have been developed and studied by various authors. For a brief review, one may refer to the monographs of Ghosh
Hall (1983) deduced that both the two-stage and purely sequential estimation procedures have some drawbacks. On one side, two-stage procedure is easy to operate as it requires only two stages and achieves the exact coverage probability but the difference between the average sample number and the ‘optimal’ fixed-sample size does not remain asymptotically bounded. As a result, two-stage procedure leads us to considerable oversampling. On the other side, purely sequential procedure is complicated in nature to apply and achieves the target value of coverage probability only asymptotically. Hall (1983) developed a sampling scheme in order to construct fixed-width confidence interval for a normal mean, which could combine the advantages of two-stage and sequential procedures. He named it as an “accelerated” sequential procedure. In his procedure, sampling stages can be reduced by a predetermined factor at the cost of finite number of observations with very nearly the desired coverage and difference between the average sample number and ‘optimal’ fixed sample size remains bounded.
Mukhopadhyay and Solanky (1991) developed the second-order asymptotic theory for accelerated sequential stopping rules and provided several interesting applications of their general setup. They have also provided solutions to various interesting ranking and selection problems by using the accelerated sequential sampling design (Mukhopadhyay and Solanky, 1992a,b, 1993). Mukhopadhyay (1996) provided an alternative formulation of accelerated sequential procedures having applications to various parametric and non parametric estimation problems. Datta and Mukhopadhyay (1998) proposed the accelerated sequential procedures in partitioning a set of normal populations. Chattopadhyay (2000) and Chattopadhyay and Sengupta (2006) have developed accelerated sequential procedures for the parameters of exponential and normal populations under asymmetric loss structures. Recently, Hu (2021) provided an improved accelerated sequential procedure to tackle the problem of fixed-width confidence interval estimation of an unknown normal mean in the absence of any knowledge about the variance. Similar procedures to deal with some other estimation problems have been developed and studied by numerous authors. To cite a few, one may refer to Son and Hamdy (1990), Hamdy and Son (1991), Chaturvedi and Tomer (2003), Chaturvedi
Taking into consideration the relationship among the ‘optimal’ fixed sample size solutions of estimation and ranking and selection problems related to normal populations, Chaturvedi and Gupta (1994) developed a class of sequential procedures. The confidence interval and ranking and selection problems were linked with the bounded risk point estimation problems under zero-one loss function exploiting the common functional form of the risks associated with these problems and second-order approximations were obtained for the ‘regret’ (difference between the risks of optimal and fixed sample size procedures). It was shown that the results obtained under this general set-up provided second-order approximations to estimation and ranking and selection problems and, as such, no separate dealing was required.
In the present paper, motivated by the work of Chaturvedi and Gupta (1994), a general class of accelerated sequential procedures is developed. Second-order approximations are obtained for the expected sample size and ‘regret’ respectively. By means of examples, it is established that the estimation problems based on various probability distributions can be handled with the help of the proposed class of accelerated sequential procedures. Note that the work of Chaturvedi and Gupta (1994) covers various inferential problems only for normal distribution whereas our present paper provides a more generalized approach covering the wide range of problems (the problems based on normal distribution are one among them). Further, in the present work, we propose a general class of accelerated sequential procedures which reduces the number of sampling operations substantially, whereas Chaturvedi and Gupta (1994) dealt with a class of sequential procedures under the normal distribution case.
The rest of the paper is organized as follows. In Section 2, we give the set-up of the estimation problem and establish the failure of fixed sample size procedure to deal with ‘it’. In Section 3, we propose the accelerated sequential procedure to handle this estimation problem. Section 4 contains some important results associated with the proposed accelerated sequential procedure. In Section 5, we derive the associated second-order approximations. Section 6 deals with applications of the proposed methodology to tackle the inferential problems related to various distributions useful in reliability theory. Section 7 presents extensive simulation and real data analyses, to complement our proposed procedures. Finally, in Section 8, we provide brief set of conclusions and future scope of the proposed work.
We know that the ‘optimal’ fixed sample size required by a ‘given precision problem’ often takes the form,
where
where
Suppose we are interested in estimating the mean
where Φ(·) is the distribution function of a standard normal variate. Since, the associated confidence coefficient must be 1 −
where
It is obvious that the above optimal fixed sample size takes the form
where
In a similar manner, one can observe the assumptions
Coming back to
Taking
Based on these
The stopping rule
where
where
Obviously, the ‘optimal’ fixed sample size risk can be obtained by putting
and the ‘regret’ associated with the accelerated sequential procedure
In this section, we provide some important lemmas associated with the accelerated sequential procedure
To prove the result
We may consider ‘n’ sufficiently large, i.e., when (
Since from
we can rewrite
Thus, we have,
and
which can further be written as
where
We have,
Firstly, we focus on
We focus on
with
This is positive obviously when
We observe that for all
Since
The infinite sum
The lemma now follows on making substitutions from
On the event ‘
which on using
Moreover, on the event ‘
which on applying Lemma 2 gives that
Further, when
for all
In this section, we provide second-order approximations for the expected sample size (theorem 1) and ‘regret’ (theorem 2) associated with the proposed accelerated sequential procedure
Denoting by,
where,
It follows from Hall (1983) that, as
which clearly tends to 0 as
We evaluate
Let us define another stopping rule
Comparing
It now follows from this theorem 2.4 that, for all
where the value of
Since
Let us consider the difference,
It follows from Woodroofe (1977) that the mean of the asymptotic distribution of
Using (
and
Substituting
Expanding
It immediately follows from
as
and the theorem follows.
In this section, we discuss some inferential problems based on various distributions useful in reliability theory and provide their solutions by using the proposed class of accelerated sequential procedures given in
Let
where
The value
Since
Then, we move foreward ahead and take
After stopping, we estimate
Comparing
where
Comparing
Let
as the unbiased and consistent estimators of
where
The sample size
Since
After stopping, we estimate
Let
where
as the estimators of
where
The sample size
Since
After stopping, we estimate
Let
where
where
The value
and the corresponding minimum risk is
However, in the absence of any knowledge about
After stopping, we estimate
Let
where
Let
where
The corresponding risk is
The value
and the corresponding minimum risk is
However, in the absence of any knowledge about
After stopping, we estimate log
We will now showcase our proposed accelerated sequential procedures using extensive simulation and real data analyses. For brevity alone, we will only present results for estimating the scale parameter of a Pareto distribution, which precisely corresponds to Section 6.5.
The following simulation results are for the accelerated sequential procedure outlined in
Each block in Table 1 shows
As one can note, the values of
We now present analysis using a real data, by implementing our proposed accelerated sequential strategy. The dataset consists of breaking stress on carbon fibers (in Gba). This dataset has been studied and analyzed by many researchers, Fatima and Roohi (2015) who fitted a transmuted exponentiated Pareto distribution, Aljarrah
The full data consists of breaking stress of 100 carbon fibers. Treating these data as the universe, the maximum likelihood estimates were &
We have developed a general class of accelerated sequential procedures and obtained the associated second-order approximations for the expected sample size and ‘regret’ function. We have discussed the applications of the proposed class to estimate the parameters of various distributions such as normal, exponential, Pareto, inverse Gaussian, multivariate normal etc. Considering the special case of Pareto distribution, we have also studied the associated properties via simulations and presented a real data set on carbon fibers in support of the practical aspect of the proposed methodology. In application part, we deliberately focus on one particular topic (minimum risk point estimation) for brevity alone. One can easily tackle the problems of bounded risk point estimation and confidence interval (region) estimation respectively using the proposed class of accelerated sequential procedures. One may also consider these problems for some other models under various loss structures.
Moreover, there is a wide scope to extend the idea of this paper because several other problems fall under the proposed set-up. One such area is “ranking and selection” where the problems like selection of the largest of
One may also think about the development of other multi-stage sampling designs (two-stage, three-stage or
Simulation results from 10,000 replications of the accelerated sequential methodology equations (6.30)–(6.31) with
50 | 8e–4 | 3.0299, 0.0009 | 49.54, 0.1339 | 0.99 | 0.0011 | 0.0004 | −0.46 | −0.76 |
100 | 1e–4 | 3.0161, 0.0005 | 99.27, 0.1793 | 0.99 | 1.4e–4 | 5.3e–4 | −0.73 | −0.75 |
250 | 6.4e–6 | 3.0060, 0.0002 | 249.85, 0.6742 | 0.99 | 3.3e–5 | 1.8e–5 | −0.15 | −0.31 |
500 | 8e–7 | 3.0030, 0.0001 | 499.61, 1.2893 | 0.99 | 7.7e–6 | 4.1e–6 | −0.39 | −0.59 |
650 | 3.7e–7 | 3.0023, 7.3e–5 | 649.36, 0.5544 | 0.99 | 5.3e–7 | 3e–7 | −0.64 | −0.44 |
800 | 2e–7 | 3.0018, 6e–5 | 799.49, 0.6218 | 0.99 | 3.2e–7 | 1.4e–7 | −0.51 | −0.58 |
Analysis of breaking stress data using accelerated sequential procedure equations (6.30)–(6.31) with
30 | 0.0490 | 0.32 | 28 | 0.93 | 0.0107 | 0.1025 |
50 | 0.0105 | 0.31 | 53 | 1.06 | 0.0026 | 0.0221 |
60 | 0.0061 | 0.35 | 62 | 1.03 | 0.0050 | 0.0128 |
70 | 0.0038 | 0.34 | 75 | 1.07 | 0.0018 | 0.0080 |
80 | 0.0025 | 0.38 | 82 | 1.03 | 0.0008 | 0.0054 |