Purely Sequential Procedure

We will follow the same approach as Ghosh and Mukhopadhyay (1979) and Chattopadhyay and Mukhopadhyay (2013) to come up with the purely sequential methodology. Using (5.4), if 2 is known, the optimal fixed sample size would be

Неге, w denotes the ceiling function of w, since when a is known, J~^cг may not be an integer. Also, the expected cost of estimating the population mean // using the sample mean based on nc observations, in other words the minimized risk, is given by

As indicated earlier, the population variance may not be known in practice. So we use a purely sequential procedure in which the estimation of parameter proceeds in stages. In the first stage, a sample called the pilot sample is collected. The pilot sample is then used to estimate the population SD, which is then used to estimate the final sample size that is expected to be close to the optimal sample size. We note that we will use A,„ defined in (5.1), as an estimator of the population SD. Using Sen and Ghosh (1981), we note that A„ is a strong consistent estimator of a and hence, A„ + и-7 is also a strong consistent estimator of nc, required to minimize the expected cost of estimation of population mean, let us consider a pilot sample of size m. We will discuss the expression of m and choice of 7 momentarily. So we begin with the pilot sample X,„ and then in successive stages we draw an additional observation using a stopping rule defined as

If m > JA/c{A,„ + m-7), no more observations are observed beyond the pilot sample and we set N = m.Ifm < jA/c{ A„, + m-7), an additional observation is collected and the estimate for GMD is updated to A,„+i. The stopping rule is checked again for m + 1 > /Ajc{A„,+i + (m + l)-7). If the stopping rule is satisfied, no additional observation is made; otherwise, an additional observation is added to the existing sample and the estimate for GMD is updated accordingly. This process continues sequentially until the number of observations is n such that n > jA/c( A„ + n-7). Then the estimated optimal sample size needed to obtain the risk point estimator is N = n. Based on the combined data X,..., X„„..., Xy, we compute X,y. So, unlike the fixed sample size procedure, the final sample size in a purely sequential procedure is random as it depends on the estimator of the population SD. In what follows, we discuss how to obtain the expression of the pilot sample size.

Pilot Sample Size Computation

If the pilot sample is smaller, the sampling error in estimation of a is larger, which may make the estimate of a very unstable. On the other hand, if m is very large, we may end up using a larger number of observations than we actually need (i.e., N = m^> nc). We now proceed along the lines of Mukhopadhyay (1980) in order to find an estimate of the pilot sample size. Using the stopping rule proposed in Equation (5.7), we have

Hence, the final sample size,

So, we recommend the use of the pilot sample size

with mo = 2, because at least two observations must be needed to get an estimate of 0. We note that for any 7 > 0, this method can be used to get a pilot sample size.


Let us now look at the characteristics of the purely sequential procedure.

Lemma 5.1. Under the assumption that the second moment exists, for any c > 0, N is finite, i.e., P(N < oc) = 1.

Proof: We note that A„ is a strong consistent estimator of a. Therefore, for any fixed c > 0,

The last equality is obtained since A„ -t A almost surely as n —> oc. ■

Thus we can say that if the observations are collected using the stopping rule defined in (5.7), then with probability one the sampling will stop at some stage, in other words the computed sample size based on the GMD-based sequential procedure is finite.

We now prove a lemma which is required for the proof of Theorem 3.1.

Lemma 5.2. Suppose that X,...,Xn are i.i.d. random variables such that E(Xr) is finite for some r >2. For any e e (0,1) and 7 > 0,

Proof: Let us define

Using the definition of stopping rule N in (5.7), we have

The asymptotic order in Equation (5.10) is observed using Sen and Ghosh (1981) for r >2, provided E(X2r) < oc.

We now look at Theorem 5.1 which lists the characteristics of the purely sequential procedure in general. Since, both the sample mean and sample GMD are U-statistics, the theorem can be proved using Sen and Ghosh (1981) and De and Chattopadhyay (2017).

Theorem 5.1. For all fixed and finite p, о1, A and A, the purely sequential procedure has the following characteristics:

Proof: (a) From the expression of the final sample size, N defined in (5.7), we have,

Since N —> oc a.s. as с l 0, P(N = m) = 0, and A„ —> Д a.s. as n -t oo, by Theorem 2.1 of Gut (2009) Д^ -> Д a.s. Hence, dividing all sides of (5.11) by nc and letting c 0, we prove N/nc -t A/о a.s. as c i 0.

(b) Because N > m a.s. and nc > 1, dividing (5.11) by nc yields

Now, A„ is a U-statistic with a kernel of degree 2. Using Lemma 9.2.4 of Ghosh et al. (1997), E(supn>m A„) < oc if second moment exist. This implies E(supc.>0 A,v_i) < oo. Now, since, N/nc^>A/a a.s. as с|0, by the dominated convergence theorem and P(N = m) = 0, we conclude that lim^o E(N/nc) = A/cr.

(c) We have

We know, lim^o E(^) = ^ from (b). Let us consider lim1>,o(;^-)E(X!v — д)2. It is enough to show that limc-j,o jJ)-E(Xn _ д)2 = 1, i.e., lim^o ncE(X,v - д)2 = 2. Note that,

Thus it is sufficient to show that,


Using the Cauchy-Schwarz inequality,

using Lemma 9.2.4 of Ghosh et al. (1997)

Now, E(X„cnf = 0(n~2) provided E(Xj) exist.

Using the Cauchy-Schwarz inequality, so,

One can show that E[max„>„3r(X„ - /л)4] = 0(и£.2) provided £(X4) exist. Let Using the Cauchy-Schwarz inequality, we have

Following the same approach as in (5.14), we have Thus

Thus we have to prove that Suppose,

We note,

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