The FWCI Problem: Determination of the Optimal Number r
Having prefixed half-width d > 0 and recorded the к-tuples (X,,i,... ,Х,д.) at the ith instance, i =1,... ,r, we propose:
But we also require that the associated confidence coefficient should be asymptotically 1 - a where d(> 0) and 0 < a < 1 are prechosen both nearly zero!
In other words, the confidence interval Jrik from (11.16) will be very narrow, centered at Grk, but sitting almost on top of Gf with its associated confidence coefficient nearly 1 - a which will be close to 1. Next, we must have:
in view of (11.15) where
the standard normal d.f., and Ф(2а/г) = 1 - That is, za/2 is the upper 100(jo:)% point of the standard normal distribution. The expression Ck from (11.17) is referred to as the required optimal fixed number of к-tuples had been known.
Obviously, the magnitude of Ck remains unknown since F remains unknown, but we expect Ck to be very large. We will develop an easy-to- implement sequential estimation strategy in Section 11.3 along with some of its associated theoretical asymptotic characteristics.
The MRPE Problem: Determination of the Optimal Number r
In the spirits of Robbins (1959), Mukhopadhyay (1978), and Ghosh and Mukhopadhyay (1979), we propose to work under a loss function of the following form:
where A(> 0) is a known weight and c is a known positive constant but ck representing the cost for sampling one к-tuple observation vector where A-1, c are prechosen near zero!
The associated risk function for a fixed number r of /с-tuple vector observations may be approximately described as:
The risk function, Risk,.^), is convex in r and it is approximately minimized when we determine:
The expression r*k from (11.20) is referred to as the required optimal fixed number of к-tuples had been known.
Obviously, the magnitude of rk remains unknown since F remains unknown, but we expect rk to be very large. We will develop an easy-to- implement sequential estimation strategy in Section 11.4 along with some of the associated theoretical asymptotic characteristics.
A Suggested Guide for Choices of k
One point should be made clear. Determining a suitable dimension к for an observation vector should involve as many aspects of practical considerations as possible. The fact is: A choice such as к = 2,3,4 or 5 may work alright, even though we may face a big data scenario requiring possibly very large Q or rj*, but then sampling will continue over a very long period in real-time. That will be unacceptable and undesirable in many practical applications.
We should realize that the number of steps (equivalently, the cost of waiting for a decision) required by Chattopadhyay and De's (2016) or De and Chattopadhyay's (2017) strategies for termination of one observation at-a- time sampling will probably be unusually large compared with the number of steps (equivalently, cost) required by the methodologies proposed here to terminate drawing к-tuples at a time.
From our practical experience we may suggest a following breakdown:
a. к = 15 — 20 if prior belief suggests that Q or rf may be <500;
b. к = 30 - 50 if prior belief suggests that Ci or r may lie between [501, 1500];
c. к = 75 - 100 if prior belief suggests that C; or r may lie between [1501, 5000];
d. k = 125 - 200 if prior belief suggests that Ci or r may lie between [5001,10000];
e. к = 250 - 400 if prior belief suggests that Ci or r may lie between
[10001,20000];
f. к > 500 if prior belief suggests that Q or r may be >20001.
We emphasize that these are guidelines only. Surely, к should be much larger if prior belief suggests that Ci or rj may exceed 25000 or 50000 or 100000. Obviously no choice of к should come in "one size" in a hurry that "fits all" scenarios. It just cannot be that way. But, in the hands of an experienced statistical scientist, in serious consultations with his/her research collaborators in a subject-matter area of interest, suggesting an appropriate choice for к will not be the biggest hurdle to overcome satisfactorily.
Estimation of the Asymptotic Variance
Since we intend to carry out purely sequential sampling strategies, we must estimate the functional from (11.11); that is, as we continue to move ahead from one step to the next. Recall from (11.8) that we had already defined the i.i.d. estimators Gyt for Gf across i instances with i = 1,..., r(<= 2).
Thus, we may simply obtain the customary sample variance from the G,^/s, namely S2k defined as:
with fixed r. Indeed, S2k will provide a consistent estimator of the functional
ak,F ~ that is, S2rJ(^k~l£p as r -> oo by the weak law of large numbers (WLLN) as we hold к fixed.
We should emphasize that the versatility of S2k truly has no bearing on the exact expression of akp or that of shown in (11.12). Indeed, if we decide to use another estimator of Gf that is more involved or complicated in its appearance than Gr,f, we do not initially need to first derive an exact expression for the functional along the line of £j?. This is a very important point which will be briefly illustrated later with examples (Section 11.5).
Surely, we could propose another more esoteric nonparametric estimator of aft but we deliberately want to keep our methodologies simple and pretty for users to feel attracted by them and then implement them when applicable. The word "simplicity" used here is not to be equated to a sense of weakness. Indeed, we believe that "simplicity" is "golden" and very hard to come by.
We contemplate having to fix к large and then expect to arrive at some large r sequentially. So, a recursive formula for S2r k will be highly desirable in an environment of large-scale computing. We may undertake the following process of orthogonal (uncorrelated) decomposition of S2k:
so that we can express (for fixed r):
The sample statistics shown in (11.22) correspond to Helmert's orthogonal (uncorrelated) variables which are not independent unless F is normal (Mukhopadhyay, 2000, pp. 197-200). Thus, we put forward the following recursive relationships:
where the new Helmert variable, beyond (11.22), will look like:
At stage or step r, one will have stored the values of G,^, S2k. Then, at stage
r + 1, one will have a new к-tuple leading to new Gr+1A. so that (11.24) part (i) will update to Gr+i,t-. Then, one would determine U, using (11.25) so that (11.24) part (ii) will now conveniently update to S2r+l k. Such recursions will reduce computing complexities and storage issues significantly because we expect both г, к to be large.
