Sequential Estimation Strategy for the MRPE Problem
Again, we begin with к-tuples (Х,д,..., Х,д), i = 1,2,..., шо(> 2), the pilot observation vectors. Then, we define the stopping time along the lines of Robbins (1959), Chow and Robbins (1965), Starr (1966b) and modify (11.26) as follows:
with S2rk coming from (11.23) and 7(> 0) is an appropriate number to be made more specific shortly. This Rkc estimates r'k from (11.20).
We continue to check the condition for boundary crossing in (11.41) successively with the updated pairs (r, Sr/k) at instances r = mo, mo + 1,... and then stop when r first crosses the corresponding boundary (A/c)^2(S,_t + г-7).The sampling strategy from (11.41) terminates w.p. 1, that is, Pi{Rk,c < oo} = 1. Having terminated sampling with the finally accrued data,
Asymptotic First-Order Results
As c -t 0, a number of asymptotic first-order properties associated with the estimation strategy (Ra-(1-, Gj?tjt) from (11.41)—(11.42) can be summarized as follows:
Again, the expression of rk comes from (11.20).
These properties will again customarily follow from Chow and Robbins (1965). One may also supply direct proofs of both parts from what is called the basic inequality. The second part in (11.43) may combine the first part, Fatou's Lemma, along with the result: Ер[Р*,с] < n* + 0(1). Else, one may follow along the steps we showed in (11.29) and improvise suitably as needed.
Under the loss function (11.18), the sequential risk function associated with the sequential MRPE strategy (R^, Gliiik) from (11.41)—(11.42) is described as:
In order to compare the performances of the MRPE strategy with the optimal fixed rk-strategy, even though this optimal fixed r£ -strategy is not implementable, Robbins (1959) defined the following metrics:
Robbins (1959) defined a sequential MRPE strategy asymptotically risk efficient if the following limiting result holds: liny.^o ip(c) = 1. Ghosh and Mukhopadhyay (1981) named this property as asymptotically first-order risk efficient.
Our proposed point estimator of Gf is clearly a sample mean of i.i.d. random variables Gijt, G2,k, ■ ■ • where we do not assume a specific common distribution which make this problem fall under the broad umbrella of sequential nonparametric point estimation scenarios. The sequential MRPE problem of estimating a population mean by a sample mean of i.i.d. random variables was first developed by Mukhopadhyay (1978). Ghosh and Mukhopadhyay (1979, Theorem 1) followed quickly and shaped the theory more fully by proving the following results as c —t 0:
where rk comes from (11.20).
Asymptotic Second-Order Results: A Brief Outline
In Ghosh and Mukhopadhyay's (1979) original distribution-free paper, they assumed high moment conditions which were later reduced substantially to a more acceptable level by Chow and Yu (1981) under weighted SEL loss plus linear cost while estimating a population mean. Sen and Ghosh (1981) had built an elegant theory for asymptotically risk efficient sequential estimation for the mean of a U-statistic under economical moment conditions. While estimating a population mean under SEL (or weighted SEL) plus linear cost, Chow and Martinsek (1982) and Martinsek (1983) came up with asymptotically sharp second-order results for the associated regret function from (11.45), co(c), under appropriate moment conditions.
We can immediately claim such asymptotic first-order and second- order results associated with our sequential MRPE strategy (Rk/C, Gr^) from (11.41)—(11.42), but we require no special moment condition because Ef[(Giiit)p] < oo for all fixed p{> 0) since G^ has a finite support, namely (0,1). We surely realize that Ef[Gi^-] = Gf + cf where the convergence of the remainder term cf to 0 will be fast enough so that our MRPE problem for Gf can be reduced to a MRPE problem for a population mean since к is large but held fixed.
More precisely, however, from Chow and Martinsek's (1982) and Mar- tinsek's (1983) results, the following conclusions will follow for a mildly modified version of the original stopping rule Rk/C from (11.41) which would be laid down as follows:
with the pilot size satisfying <5c-1/4 < rc = o(c-1/2) for some d > 0.
Theorem 11.1. For the sequential MRPE strategy (Rk,c,G^ k) associated with
(11.47), we have the following asymptotic second-order regret expansion as c —t 0:
with ар2, fp, w(c) coming from (11.11), (11.12) and (11.45) respectively where of2 « k~l£p for large but fixed k.
The next result follows from Chang and Hsiung (1979), also incorporated in Martinsek (1983).
Theorem 11.2. For the sequential MRPE strategy (Rk/C> k) associated with
(11.47) , we have the following asymptotic second-order efficiency result as c —> 0:
with r’k coming from (11.20) for large but fixed k. An expression ofd = dpy, free from c, comes from Martinsek (1983, p. 832).
These second-order results (Theorems 11.1-11.2) are very interesting, especially since we fully expect the stopping time Rk,c from (11.41) and Rk/C from
(11.47) to be asymptotically indistinguishable from one another. Now, in the spirits of Section 11.3.3 and assumption (11.33), we may presume (for large k) that the G,/s are practically N(Gp, <т/;2), and then clearly we would have: MGU - Gf)2] « 2dp4 and Ef[(Gu - Gf)3] « 0.
Under this scenario, from Theorem 11.1, we will conclude (as c —> 0):
which coincides with the regret expansion from Woodroofe (1977) in the normal case.
Now, one can readily see the marked difference between the technicalities developed by Chattopadhyay and De (2014, 2016), De and Chattopadhyay (2017) and those that we have developed in this paper. Our present construction of sequential MRPE strategies are very broad to deliver even the associated asymptotic second-order approximations.
