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Second-order asymptotics in a class of purely sequential minimum risk point estimation (MRPE) methodologies

  • Jun Hu
  • Nitis Mukhopadhyay
Original Paper
  • 23 Downloads

Abstract

Under the squared error loss plus linear cost of sampling, we revisit the minimum risk point estimation (MRPE) problem for an unknown normal mean \(\mu\) when the variance \(\sigma ^{2}\) also remains unknown. We begin by defining a new class of purely sequential MRPE methodologies based on a general estimator \(W_{n}\) for \(\sigma\) satisfying a set of conditions in proposing the requisite stopping boundary. Under such appropriate set of sufficient conditions on \(W_{n}\) and a properly constructed associated stopping variable, we show that (i) the normalized stopping time converges in law to a normal distribution (Theorem 3.3), and (ii) the square of such a normalized stopping time is uniformly integrable (Theorem 3.4). These results subsequently lead to an asymptotic second-order expansion of the associated regret function in general (Theorem 4.1). After such general considerations, we include a number of substantial illustrations where we respectively substitute appropriate multiples of Gini’s mean difference and the mean absolute deviation in the place of the general estimator \(W_{n}\). These illustrations show a number of desirable asymptotic first-order and second-order properties under the resulting purely sequential MRPE strategies. We end this discourse by highlighting selected summaries obtained via simulations.

Keywords

Asymptotic first-order properties Asymptotic second-order properties Linear cost Regret Risk efficiency Sequential strategy Simulations Squared error loss 

Mathematics Subject Classification

62L10 62L12 62G05 62G20 

1 Introduction and a brief review

Purely sequential estimation methodologies date back to path-breaking papers of Anscombe (1950, 1952, 1953), Ray (1957), and Chow and Robbins (1965). Anscombe, Ray, and Chow and Robbins gave a solid foundation to establish purely sequential fixed-width confidence interval estimation methodologies for an unknown normal mean \(\mu\) when the population variance \(\sigma ^{2}\) remained unknown. Indeed, Chow and Robbins (1965) brought forward the fundamental nature of the theory of purely sequential nonparametric fixed-width confidence interval estimation methodologies.

The far reaching purely sequential minimum risk point estimation (MRPE) methodology was originally developed by Robbins (1959) for an unknown normal mean \(\mu\) when the population variance \(\sigma ^{2}\) remained unknown. Under the squared error loss (SEL) plus linear cost of sampling, the sequential estimation strategy of Robbins (1959) was subsequently broadened by Starr (1966) and Starr and Woodroofe (1969) where desirable asymptotic properties associated with efficiency, risk efficiency, and regret were proved. Second-order properties of the associated regret were further developed by Lai and Siegmund (1977, 1979), Woodroofe (1977), and Mukhopadhyay (1988). It is appropriate to mention that Ghosh and Mukhopadhyay (1981) first introduced the notion of asymptotic second-order efficiency.

In a distribution-free situation, Mukhopadhyay (1978) and Ghosh and Mukhopadhyay (1979) first developed MRPE problems for the estimation of a population mean under the squared error loss plus linear cost of sampling and proved asymptotic risk efficiency. Chow and Yu (1981), Chow and Martinsek (1982), and a series of follow-up broadened this area significantly. Sen and Ghosh (1981) developed nonparametric sequential point estimation of the mean of a U-statistic. They concluded the asymptotic first-order efficiency, risk efficiency, as well as other elegant asymptotics.

Mukhopadhyay (1982) suggested using a broader class of (nonparametric) estimators of \(\sigma ^{2}\) instead of using the customary sample variance (or sample standard deviation) as an estimator of the unknown parameter \(\sigma ^{2}\) (or \(\sigma\)) in the stopping rules. Chattopadhyay and Mukhopadhyay (2013) and Mukhopadhyay and Hu (2017, 2018) have recently looked into purely sequential estimation strategies using appropriate multiples of functions of Gini’s mean difference (GMD) or mean absolute deviation (MAD) as possible substitutes of the traditional sample variance (or sample standard deviation).

In this paper, we primarily focus on purely sequential sampling strategies rather than two-stage and other multi-stage sampling methods. We also deliberately keep the literature on numerous multivariate and regression problems out of this present discourse. To obtain the status of wide-ranging two-stage inference methods, one may additionally refer to Ghosh and Mukhopadhyay (1976, 1981), Aoshima and Mukhopadhyay (2002), Chattopadhyay and Mukhopadhyay (2013), and other more recent sources. One may also achieve a comprehensive review of this field by looking at the monographs by Sen (1981, 1985), Siegmund (1985), Ghosh and Sen (1991), Mukhopadhyay and Solanky (1994), Jurečkovā and Sen (1996), Ghosh et al. (1997), Mukhopadhyay et al. (2004), Mukhopadhyay and de Silva (2009), Zacks (2009, 2017), and other relevant sources.

Under the squared error loss plus linear cost of sampling, we revisit in this paper the minimum risk point estimation (MRPE) problem for an unknown normal mean \(\mu\) when the variance \(\sigma ^{2}\) also remains unknown. We begin with the formulation of a new and general class of purely sequential sampling methodologies based on an arbitrary estimator
$$\begin{aligned} W_{n}\equiv W_{n}(X_{1},\ldots ,X_{n}) \end{aligned}$$
for \(\sigma\) with \(n\ge 2\) in proposing the stopping boundary in Sect. 2. Under an appropriate set of sufficient conditions (C1–C7, Sect. 3) on \(W_{n}\) and the properly constructed associated stopping variable (2.4), we show that (i) the normalized stopping time converges in law to a normal distribution (Theorem 3.3), (ii) the square of such a normalized stopping time is uniformly integrable (Theorem 3.4). These, along with other additional properties (Theorem 3.1, Lemma 3.1), subsequently lead to asymptotic risk efficiency (Theorem 3.2) and asymptotic second-order expansion of the associated regret function (Theorem 3.4) in general.

After these general considerations, we include substantial illustrations where we respectively substitute Gini’s mean difference (GMD) and the mean absolute deviation (MAD) in the place of the general estimator \(W_{n}\). These illustrations are followed by a number of desirable asymptotic first-order and second-order properties under the resulting purely sequential sampling strategies. We end this discourse by highlighting selected summaries obtained via simulations.

The objective of this paper is to revisit in depth the purely sequential minimum risk point estimation methodologies involving GMD or MAD established in Mukhopadhyay and Hu (2017). Having proposed a new purely sequential methodology based on nonparametric estimators with some certain conditions satisfied, we develop asymptotic second-order results which are considerably stronger than those reported. The formulations of the newly proposed methodology are presented in Sect. 2. The main theorems are laid down in Sects. 3 and 4 under generality along with some of the substantial proofs. In Sect. 5, some illustrations are provided followed by summaries from simulations presented in Sect. 6. We end with some concluding thoughts.

2 A general formulation

We begin with the assumption that we have a sequence of independent observations available from a \(N\left( \mu ,\sigma ^{2}\right)\) population with two unknown parameters \(-\infty<\mu <\infty\) and \(0<\sigma <\infty\). Having recorded \(X_{1},\ldots ,X_{n},n\ge 2\), we denote the customarily used unbiased estimators for \(\mu\) and \(\sigma ^{2}\) by
$$\begin{aligned} \begin{array}{rl} \hbox {Sample mean:} &{} \overline{X}_{n}=n^{-1}\Sigma _{i=1}^{n}X_{i}, \\ \hbox {Sample variance:} &{} \overset{}{S_{n}^{2}=(n-1)^{-1}\Sigma _{i=1}^{n}(X_{i}- \overline{X}_{n})^{2},n\ge 2,} \end{array} \end{aligned}$$
respectively.

Throughout this presentation, we propose to estimate the population mean \(\mu\) by the sample mean, \(\overline{X}_{n}\), based on n observations \(X_{1},\ldots ,X_{n}\). The sample variance \(S_{n}^{2}\) or the sample standard deviation \(S_{n}\) is regarded as a customary estimator of \(\sigma ^{2}\) or \(\sigma\). One should realize that sometimes one may use an appropriate multiple of \(S_{n}^{2}\) or \(S_{n}\) as needed and note that they are respectively consistent estimators of \(\sigma ^{2}\) or \(\sigma\). We first review the purely sequential MRPE methodology of Robbins (1959) very briefly in Sect. 2.1 followed by our new general theory.

2.1 The purely sequential MRPE methodology of Robbins (1959)

The purely sequential MRPE methodology for \(\mu\) due to Robbins (1959) would begin with a loss function given by
$$\begin{aligned} L_{n}\equiv L_{n}\left( \mu ,\overline{X}_{n}\right) =A\left( \overline{X} _{n}-\mu \right) ^{2}+cn,\text { where }A\text { and }c\text { are both known.} \end{aligned}$$
(2.1)
Here, \(A\left( >0\right)\) is an appropriate weight function, c is the unit cost of each observation, and n is the sample size. The loss function (2.1) would attempt to balance between the risk due to estimation error for using \(\overline{X}_{n}\) to estimate \(\mu\) and the sampling cost of n observations.
Associated with the loss function in (2.1), we can write the risk function as follows:
$$\begin{aligned} R_{n}\left( c\right) \equiv E_{\mu ,\sigma }\left[ L_{n}\left( \mu , \overline{X}_{n}\right) \right] =A\sigma ^{2}n^{-1}+cn, \end{aligned}$$
(2.2)
which we may minimize to obtain the requisite optimal fixed sample size \(n^{*}\) given by:
$$\begin{aligned} n^{*}\equiv n^{*}\left( c\right) =\sigma \sqrt{A/c}, \end{aligned}$$
(2.3)
had \(\sigma\) been known. We tacitly disregard the fact that \(n^{*}\) may not be an integer.

The minimum risk turns out to be \(R_{n^{*}}\left( c\right)\) which simplifies to \(2cn^{*}\). Our goal is to achieve this minimum risk approximately and we will remain mindful to point out whether this holds reasonably well in the sense of first-order or second-order of asymptotic approximation.

Since the magnitude of \(n^{*}\) remains unknown, one may begin with the pilot data \(X_{1},\ldots ,X_{m}\) of size \(m\left( \ge 2\right)\), and then sample one additional observation at-a-time sequentially as needed until we terminate according to the following stopping rule:
$$\begin{aligned} \hbox {Methodology } \mathcal {P}_{0}: \quad N_{\mathcal {P}_{0}}\equiv N_{\mathcal {P} _{0}}\left( c\right) =\inf \left\{ n\ge m:n\ge \sqrt{A/c}\left( S_{n}+n^{-\lambda }\right) \right\} , \end{aligned}$$
(2.4)
where \(\lambda \left( >1/2\right)\) is held fixed. This is nearly Robbins’s (1959) stopping rule except that neither Robbins (1959) nor Starr (1966) included the “\(n^{-\lambda }\)” term in defining the stopping boundary in (2.4). It is obvious that \(P_{\mu ,\sigma }\left( N_{\mathcal {P}_{0}}<\infty \right) =1\) and \(N_{\mathcal {P} _{0}}\uparrow \infty\) w.p.1 as \(c\downarrow 0\). Upon termination of sampling according to (2.4), one would finally estimate \(\mu\) with the corresponding sample mean \(\overline{X}_{N_{\mathcal {P}_{0}}}\equiv N_{ \mathcal {P}_{0}}^{-1}\Sigma _{i=1}^{N_{\mathcal {P}_{0}}}X_{i}\) based upon \(\left\{ N_{\mathcal {P}_{0}},X_{1},\ldots ,X_{m},\ldots ,X_{N_{\mathcal {P} _{0}}}\right\}\).

2.2 A new general purely sequential MRPE methodology

Next, we keep our option open by building a broad structure having considered an appropriate consistent nonparametric estimator in general for \(\sigma\), denoted by \(W_{n}\), assumed positive w.p.1. That is, \(W_{n}\) may not necessarily be a multiple of the sample standard deviation \(S_{n}\).

Since the magnitude of \(n^{*}\) from (2.3) remains unknown, one may begin with the pilot data \(X_{1},\ldots ,X_{m}\) of size \(m\left( \ge 2\right)\), and then sample one additional observation at-a-time sequentially as needed until we terminate according to the following stopping rule stated generally as:
$$\begin{aligned} \hbox {Methodology } \mathcal {P}: \quad N_{\mathcal {P}}\equiv N_{\mathcal {P}}\left( c\right) =\inf \left\{ n\ge m:n\ge \sqrt{A/c}\left( W_{n}+n^{-\lambda }\right) \right\} , \end{aligned}$$
(2.5)
where \(\lambda \left( >1/2\right)\) is held fixed. For the methodology (2.5), it is obvious that \(P_{\mu ,\sigma }\left( N_{\mathcal {P}}<\infty \right) =1\) and \(N_{\mathcal {P}}\uparrow \infty\) w.p.1 as \(c\downarrow 0\). Upon termination of sampling according to (2.5), one would finally estimate \(\mu\) with the corresponding sample mean \(\overline{X}_{N_{\mathcal {P} }}\equiv N_{\mathcal {P}}^{-1}\Sigma _{i=1}^{N_{\mathcal {P}}}X_{i}\) based upon \(\left\{ N_{\mathcal {P}},X_{1},\ldots ,X_{m},\ldots ,X_{N_{\mathcal {P} }}\right\}\).

Eventually, our aim is to prove uniform integrability of \(N_{\mathcal {P} }^{*2}\) where \(N_{\mathcal {P}}^{*}\) stands for the standardized stopping time, namely \(n^{*-1/2}(N_{\mathcal {P}}-n^{*}),\) associated with the methodology \(\mathcal {P}\) from (2.5). Under the methodology \(\mathcal {P}_{0}\) from (2.4), the uniform integrability of \(N_{\mathcal {P} _{0}}^{*2}\) was proved by Lai and Siegmund (1977, 1979) and Woodroofe (1977, 1982) by exploiting the nonlinear renewal theory in sequential analysis.

On the other hand, Ghosh and Mukhopadhyay (1980) verified an analogous result with the help of a directly constructed proof. See also Mukhopadhyay (1988), Mukhopadhyay and Solanky (1994, pp. 48–52), Ghosh et al. (1997, pp. 58–65), and Mukhopadhyay and de Silva (2009, pp. 155–160), and other relevant sources.

For the MRPE methodology \((\mathcal {P},\overline{X}_{N_{\mathcal {P}}})\) from (2.5), the achieved purely sequential risk would be given by:
$$\begin{aligned} R_{N_{\mathcal {P}}}\left( c\right) \equiv E_{\mu ,\sigma }\left[ L_{N_{ \mathcal {P}}}\left( \mu ,\overline{X}_{N_{\mathcal {P}}}\right) \right] =AE_{\mu ,\sigma }\left[ \left( \overline{X}_{N_{\mathcal {P}}}-\mu \right) ^{2}\right] +cE_{\mu ,\sigma }\left[ N_{\mathcal {P}}\right] . \end{aligned}$$
(2.6)
To quantify the closeness between the achieved risk \(R_{N_{\mathcal { P}}}\left( c\right)\) and the minimum risk \(R_{n^{*}}\left( c\right)\), Robbins (1959) defined two classic measures:
$$\begin{aligned} \begin{array}{rl} \hbox {Risk Efficiency:} &{} \xi _{\mathcal {P}}(c)\equiv R_{N_{\mathcal {P}}}\left( c\right) /R_{n^{*}}\left( c\right) \\ \hbox {Regret:} &{} \overset{}{\omega _{\mathcal {P}}(c)\equiv R_{N_{\mathcal {P} }}\left( c\right) -R_{n^{*}}\left( c\right) .} \end{array} \end{aligned}$$
(2.7)
In our subsequent analyses, we will study the behaviors of both the risk efficiency \(\xi _{\mathcal {P}}(c)\) and the regret \(\omega _{\mathcal {P}}(c)\) as \(c\rightarrow 0\).
Starr (1966) proved:
$$\begin{aligned} \xi _{\mathcal {P}_{0}}(c)\rightarrow 1 \quad \hbox { if } m\ge 3, \end{aligned}$$
(2.8)
whereas Woodroofe (1977) proved:
$$\begin{aligned} \omega _{\mathcal {P}_{0}}(c)=\frac{1}{2}c+o(c) \quad \hbox { if } m\ge 4, \end{aligned}$$
(2.9)
as \(c\rightarrow 0\). That is, the MRPE methodology \((\mathcal {P}_{0}, \overline{X}_{N_{\mathcal {P}_{0}}})\) from (2.4), is asymptotically (first-order) risk efficient when \(m\ge 3\), whereas the regret amounts to the cost of one-half of a single observation asymptotically when \(m\ge 4\).

3 Main results under the general methodology \(\mathcal {P}\) from (2.5)

In this section, we devote our attention to the purely sequential MRPE methodology \(\mathcal {P}\) alone. Also, we will generally use [u] to denote the largest integer that is smaller than \(u(>0)\). It should be clear from the context when this notation is used in the proofs. Also, I(A) would stand for the indicator function of an event A

Next, we itemize a set of conditions involving \(W_{n},\) assumed positive w.p. 1, as follows:
  1. (C1)
    Independence:
    $$\begin{aligned} \overline{X}_{n} \hbox { and } \{W_{\underset{}{k}}; 2\le k\le n\} \hbox { are distributed independently for all } n\ge 2. \end{aligned}$$
     
  2. (C2)
    Convergence in probability:
    $$\begin{aligned} W_{\underset{}{n}}\overset{P_{\mu ,\sigma }}{ \rightarrow }\sigma \hbox { as } n\rightarrow \infty . \end{aligned}$$
     
  3. (C3)
    Aysmptotic normality:
    $$\begin{aligned} \sqrt{n}\left( \sigma ^{-1}W_{\underset{}{n}}-1\right) \overset{\pounds }{\rightarrow }N\left( 0,\delta ^{2}\right) \hbox { for some } \delta (>0) \hbox { as } n\rightarrow \infty . \end{aligned}$$
     
  4. (C4)

    Uniform continuity in probability (u.c.i.p):

    For every \(\varepsilon >0\), there exists a large \(\nu \equiv \nu \left( \varepsilon \right)\) and small \(\gamma >0\) for which
    $$\begin{aligned} P_{\mu ,\sigma }\left( \max \limits _{0\le k\le n\gamma }\left| W_{n+k}-W_{n}\right| \ge \varepsilon \right) <\varepsilon \hbox { holds for any } n\ge \nu . \end{aligned}$$
     
  5. (C5)

    Kolmogorov’s inequality:

    For every \(\varepsilon >0\), and some \(2\le n_{1}\le n_{2}\),
    $$\begin{aligned} P_{\mu ,\sigma }\left( \max \limits _{n_{1}\le n\le n_{2}}\left| W_{n}-\sigma \right| \ge \varepsilon \right) \le \varepsilon ^{-r}E_{\mu ,\sigma }\left[ \left| W_{n_{1}}-\sigma \right| ^{r}\right] , \hbox { with } r\ge 2. \end{aligned}$$
     
  6. (C6)
    Order of central absolute moments:
    $$\begin{aligned} \hbox {For } \underset{}{n\ge 2} \hbox { and } r\ge 2, E_{\mu ,\sigma }\left[ \left| W_{n}-\sigma \right| ^{r}\right] =O\left( n^{-r/2}\right) . \end{aligned}$$
     
  7. (C7)
    Wiener’s condition:
    $$\begin{aligned} E_{\mu ,\sigma }\left[ \sup \nolimits _{n\ge 2}W_{n}\right] <\infty . \end{aligned}$$
     
Now, we proceed to lay down a number of lemmas and theorems associated with the purely sequential MRPE methodology \(\mathcal {P}\). In doing so, we collect a number of precisely stated conditions from the itemized list and then show a road-map to prove the claims under some of the selected conditions from (C1) to (C7) as needed.

Lemma 3.1

For the purely sequential MRPE methodology \((\mathcal {P},\overline{X}_{N_{\mathcal {P}}})\) given by (2.5), under the condition (C1), the expressions of the associated risk efficiency and regret are respectively given by:
$$\begin{aligned} \xi _{\mathcal {P}}(c)\equiv & {} \frac{1}{2}E_{\mu ,\sigma }\left[ n^{*}/N_{\mathcal {P}}\right] +\frac{1}{2}E_{\mu ,\sigma }\left[ N_{\mathcal {P} }/n^{*}\right] ; \\ \omega _{\mathcal {P}}(c)\equiv & {} cE_{\mu ,\sigma }\left[ N_{ \mathcal {P}}^{-1}\left( N_{\mathcal {P}}-n^{*}\right) ^{2}\right] ; \end{aligned}$$
with \(n^{*}\) defined by (2.3).

Proof

Under the condition (C1), from (2.2) we can claim that \(E_{\mu ,\sigma }\left[ L_{N_{\mathcal {P}}}\left( \mu ,\overline{X}_{N_{ \mathcal {P}}}\right) \right] =A\sigma ^{2}E_{\mu ,\sigma }\left[ N_{\mathcal { P}}^{-1}\right] +cE_{\mu ,\sigma }\left[ N_{\mathcal {P}}\right]\). Then, the results follow from (2.7) since \(R_{n^{*}}\left( c\right) =2cn^{*}\). \(\square\)

Lemma 3.2

The condition (C4) follows from the two conditions (C5) and (C6) combined.

Proof

We note that for any fixed \(n_{0}\ge \nu\), we may express:
$$\begin{aligned}&P_{\mu ,\sigma }\left( \max \limits _{0\le k\le n_{0}\gamma }\left| W_{n_{0}+k}-W_{n_{0}}\right| \ge \varepsilon \right) \nonumber \\&\quad \le P_{\mu ,\sigma }\left( \max \limits _{0\le k\le n_{0}\gamma }\left\{ \left| W_{n_{0}+k}-\sigma \right| +\left| W_{n_{0}}-\sigma \right| \right\} \ge \varepsilon \right) \nonumber \\&\quad \le P_{\mu ,\sigma }\left( \max \limits _{\nu \le n\le n_{0}\left( 1+\gamma \right) }\left| W_{n}-\sigma \right| \ge \varepsilon /2\right) +P_{\mu ,\sigma }\left( \max \limits _{\nu \le n\le n_{0}}\left| W_{n}-\sigma \right| \ge \varepsilon /2\right) \nonumber \\&\quad \le 2\left( \varepsilon /2\right) ^{-r}E_{\mu ,\sigma }\left[ \left| W_{\nu }-\sigma \right| ^{r}\right] , \hbox { by } \hbox {(C5)--(C6)} \nonumber \\&\quad =O\left( \left( \nu \varepsilon ^{2}\right) ^{-r/2}\right) . \end{aligned}$$
(3.1)
Then, by choosing large enough \(\nu\) and appropriate r such that \(O\left( \left( \nu \varepsilon ^{2}\right) ^{-r/2}\right) <\varepsilon\), it should be clear that the Anscombe’s (1952) u.c.i.p. property for the sequence \(\left\{ W_{n};\text { }n\ge 2\right\}\) holds. \(\square\)

Theorem 3.1

For the purely sequential MRPE methodology \((\mathcal {P},\overline{X}_{N_{\mathcal {P}}})\) given by (2.5), under the conditions (C2) and (C7), we have:
$$\begin{aligned} \lim \limits _{c\rightarrow 0}E_{\mu ,\sigma }[N_{\mathcal {P}}/n^{*}]=1 [ \textit{Asymptotic First-Order Efficiency}], \end{aligned}$$
with \(n^{*}\) defined by (2.3).

Proof

From the stopping rule defined in (2.5), we have the following two inequalities (w.p.1):
$$\begin{aligned} N_{\mathcal {P}}\ge \sqrt{A/c}\left( W_{N_{\mathcal {P}}}+N_{\mathcal {P} }^{-\lambda }\right) ,\text { as well as }N_{\mathcal {P}}<m+\sqrt{A/c}\left( W_{N_{\mathcal {P}}-1}+\left( N_{\mathcal {P}}-1\right) ^{-\lambda }\right) , \end{aligned}$$
(3.2)
from which we conclude:
$$\begin{aligned} \frac{W_{N_{\mathcal {P}}}+N_{\mathcal {P}}^{-\lambda }}{\sigma ^{{}}}\le \frac{N_{\mathcal {P}}}{n^{*}}<\frac{W_{N_{\mathcal {P}}-1}+\left( N_{ \mathcal {P}}-1\right) ^{-\lambda }}{\sigma ^{{}}}+\frac{m}{n^{*}}. \end{aligned}$$
(3.3)
Next, it is clear that \(N_{\mathcal {P}}/n^{*}\overset{P_{\mu ,\sigma }}{ \rightarrow }1\) as \(c\rightarrow 0\) under (C2). Also, we note that for some sufficiently small c, the right-hand side of (3.3) can be bounded as follows:
$$\begin{aligned} (0\le )\frac{N_{\mathcal {P}}}{n^{*}}<\sigma ^{-1}\left( \sup \nolimits _{n\ge 2}\left\{ W_{n}\right\} +1\right) +1. \end{aligned}$$
(3.4)
Thus, under (C7) and by the dominated convergence theorem, \(\lim \limits _{c\rightarrow 0}E_{\mu ,\sigma }[N_{\mathcal {P}}/n^{*}]=1\) holds immediately from (3.4). \(\square\)

Remark 3.1

We may use a technique similar to the one which led to (3.4) to claim that \(E_{\mu ,\sigma }[N_{\mathcal {P} }^{k}]<\infty\) for any fixed \(k>0\) after combining with (3.3). Indeed, one may verify: \(\lim \limits _{c\rightarrow 0}E_{\mu ,\sigma }[\left( N_{\mathcal { P}}/n^{*}\right) ^{k}]=1\) for any fixed \(k>0,\) under (C2) and (C7).

Lemma 3.3

For the purely sequential MRPE methodology \((\mathcal {P},\overline{X}_{N_{\mathcal {P}}})\) given by (2.5), under the conditions (C5)–(C6), for any arbitrary \(0<\eta <1\) with \(r\ge 2,\) we have:
$$\begin{aligned} P_{\mu ,\sigma }\left( N_{\mathcal {P}}\le \eta n^{*}\right) =O\left( n^{*^{-\frac{r}{2\left( 1+\lambda \right) }}}\right) , \end{aligned}$$
with \(n^{*}\) defined by (2.3).

Proof

Recall that [u] stand for the largest integer that is smaller than \(u(>0)\) and we define:
$$\begin{aligned} n_{1c}=\left[ (A/c)^{\frac{1}{2(1+\lambda )}}\right] =O(c^{-\frac{1}{ 2(1+\lambda )}}) \hbox { and } n_{2c}=\eta n^{*}=\eta \sigma \sqrt{A/c}. \end{aligned}$$
(3.5)
It should be obvious from the definition of \(N_{\mathcal {P}}\) in (2.5) that \(N_{\mathcal {P}}\ge n_{1c}\) w.p.1. Next, we set out to obtain the rate at which \(P_{\mu ,\sigma }(N_{\mathcal {P}}\le \eta n^{*})\) may converge to zero for small c. We may express:
$$\begin{aligned}&P_{\mu ,\sigma }(N_{\mathcal {P}}\le \eta n^{*})\nonumber \\&\quad \le P_{\mu ,\sigma }\left\{ W_{n}\le \eta \sigma \text { for some }n\text { such that }n_{1c}\le n\le n_{2c}\right\} \nonumber \\&\quad \le P_{\mu ,\sigma }\left\{ \max \limits _{n_{1c}\le n\le n_{2c}}\left| W_{n}-\sigma \right| \ge \left( 1-\eta \right) \sigma \right\} \nonumber \\&\quad \le \{\left( 1-\eta \right) \sigma \}^{-r}E_{\mu ,\sigma }\left[ \left| W_{n_{1c}}-\sigma \right| ^{r}\right] , \hbox {by (C5)} \nonumber \\&\quad =O\left( n_{1c}^{-r/2}\right) ,\text { by (C6)}, \end{aligned}$$
(3.6)
which leads to the desired order. \(\square\)

Theorem 3.2

For the purely sequential MRPE methodology \((\mathcal {P},\overline{X}_{N_{\mathcal {P}}})\) given by (2.5), under the conditions (C1)–(C2) and (C5)–(C7), we have:
$$\begin{aligned} \lim \limits _{c\rightarrow 0}\xi _{\mathcal {P}}(c)=1 [\textit{Asymptotic First-Order Risk Efficiency}], \end{aligned}$$
with the risk efficiency term \(\xi _{\mathcal {P}}(c)\) coming from Lemma 3.1.

Proof

From Lemma 3.1, we recall that
$$\begin{aligned} \xi _{\mathcal {P}}(c)=\frac{1}{2}E_{\mu ,\sigma }\left[ n^{*}/N_{ \mathcal {P}}\right] +\frac{1}{2}E_{\mu ,\sigma }\left[ N_{\mathcal {P} }/n^{*}\right] , \end{aligned}$$
(3.7)
under (C1). Also, under (C2) and (C7), we know from Theorem 3.1 that
$$\begin{aligned} \lim _{c\rightarrow 0}E_{\mu ,\sigma }[N_{\mathcal {P}}/n^{*}]=1. \end{aligned}$$
(3.8)
Thus, in view of (3.7), it will suffice to verify that \(\lim _{c\rightarrow 0}E_{\mu ,\sigma }[n^{*}/N_{\mathcal {P}}]=1\). Now, we express:
$$\begin{aligned} E_{\mu ,\sigma }[n^{*}/N_{\mathcal {P}}]= & {} E_{\mu ,\sigma }\left[ \frac{ n^{*}}{N_{\mathcal {P}}}I\left( N_{\mathcal {P}}>\frac{1}{2}n^{*}\right) \right] +E_{\mu ,\sigma }\left[ \frac{n^{*}}{N_{\mathcal {P}}} I\left( N_{\mathcal {P}}\le \frac{1}{2}n^{*}\right) \right] \nonumber \\= & {} E_{\mu ,\sigma }[J_{1}]+E_{\mu ,\sigma }[J_{2}], \hbox { say}, \end{aligned}$$
(3.9)
where I(A) stands for the indicator function of an event A.

We observe that \((0<)J_{1}<2\) and a bounded random variable is uniformly integrable. Also, \(J_{1}\overset{P_{\mu ,\sigma }}{\rightarrow }1\) as \(c\rightarrow 0\). Hence, \(E_{\mu ,\sigma }\left[ J_{1}\right] =1+o(1)\) as \(c\rightarrow 0\).

Next, we handle the term \(E_{\mu ,\sigma }[J_{2}]\) and use Lemma 3.3 under (C5)–(C6) to express:
$$\begin{aligned} E_{\mu ,\sigma }[J_{2}]\le & {} E_{\mu ,\sigma }\left[ \frac{n^{*}}{n_{1c}} I\left( N_{\mathcal {P}}\le \frac{1}{2}n^{*}\right) \right] =O\left( n^{*}\right) O\left( n^{*-1/(1+\lambda )}\right) O\left( n^{*-r/(2(1+\lambda ))}\right) \nonumber \\= & {} O\left( n^{*1-\frac{1}{1+\lambda }-\frac{r}{ 2(1+\lambda )}}\right) =O\left( n^{*\frac{2\lambda -r}{2(1+\lambda )} }\right) \rightarrow 0,\text { as }c\rightarrow 0, \end{aligned}$$
(3.10)
as long as we pick some appropriately large \(r>\max \left\{ 2,2\lambda \right\}\). In view of (3.7)–(3.10), this proof is now complete. \(\square\)

Remark 3.2

We may use a technique similar to the one which led to (3.9)–(3.10) to claim that
$$\begin{aligned} \lim \limits _{c\rightarrow 0}E_{\mu ,\sigma }[\left( n^{*}/N_{\mathcal {P} }\right) ^{k}]=1 \end{aligned}$$
for any fixed \(k>0,\) under (C1)–(C2) and (C5)–(C7).

Remark 3.3

It is clear that our brief justifications of Theorems 3.1 and 3.2 under general conditions mildly overlap with the lines of proofs of Theorems 3.2 and 4.2 from Mukhopadhyay and Hu (2017). However, we keep these justifications for identifying which sufficient conditions from (C1) to (C7) are specifically used in validating the respective steps.

The next result follows from combining Anscombe’s (1952) random central limit theorem (random CLT) with a formal technique developed by Ghosh and Mukhopadhyay (1975) to transfer the asymptotic distributions of \(W_{N_{\mathcal {P}}},W_{N_{\mathcal {P}}-1}\) to conclude an asymptotic distribution of \(N_{\mathcal {P}}\). One may refer to Gut (2012) and Mukhopadhyay and Chattopadhyay (2012) to take into account some recent treatments of the random CLT.

The Ghosh-Mukhopadhyay theorem comes from Mukhopadhyay (1975, Chapter 2) which was fully utilized by Carroll (1977) right away to obtain asymptotic distributions of his stopping rules based on certain robust statistics. One may additionally review from Mukhopadhyay and Solanky (1994, Section 2.4), Ghosh et al. (1997, Exercise 2.7.4), among other sources.

Theorem 3.3

For the purely sequential MRPE methodology \((\mathcal {P},\overline{X}_{N_{\mathcal {P}}})\) given by (2.5), under the conditions (C2)–(C6) we have as \(c\rightarrow 0\):
$$\begin{aligned} \begin{array}{rl} (\textit{i}) &{} N_{\mathcal {P}}^{*}\equiv \left( N_{\mathcal {P}}-n^{*}\right) /n^{*1/2}\overset{\pounds }{\rightarrow }N\left( 0,\delta ^{2}\right) ; \\ (\textit{ii}) &{} \overset{}{\left( N_{\mathcal {P}}-n^{*}\right) /N_{ \mathcal {P}}^{1/2}\overset{\pounds }{\rightarrow }N\left( 0,\delta ^{2}\right) ;} \end{array} \end{aligned}$$
with \(n^{*}\) defined by (2.3) and \(\delta ^{2}(>0)\) coming from (C3).

Proof

To verify part (i), we recall that \(N_{\mathcal {P}}/n^{*}\overset{P_{\mu ,\sigma }}{\rightarrow }1\) as \(c\rightarrow 0\) under (C2). Having this settled, under (C3)–(C4), Anscombe’s (1952) random CLT would imply:
$$\begin{aligned} n^{*1/2}\left( \sigma ^{-1}W_{N_{\mathcal {P}}}-1\right) \overset{ \pounds }{\rightarrow }N\left( 0,\delta ^{2}\right) \hbox { and } n^{*1/2}\left( \sigma ^{-1}W_{N_{\mathcal {P}}-1}-1\right) \overset{\pounds }{ \rightarrow }N\left( 0,\delta ^{2}\right) \text {as }c\rightarrow 0, \end{aligned}$$
(3.11)
with \(\delta ^{2}(>0)\) coming from (C3). The Ghosh-Mukhopadhyay theorem (1975) immediately leads to part (i) from (3.11), that is,
$$\begin{aligned} n^{*-1/2}\left( N_{\mathcal {P}}-n^{*}\right) \overset{\pounds }{ \rightarrow }N\left( 0,\delta ^{2}\right) \text { as }c\rightarrow 0. \end{aligned}$$
Next, we obviously have \(\sqrt{n^{*}}/\sqrt{N_{\mathcal {P}}}\overset{ P_{\mu ,\sigma }}{\rightarrow }1\) as \(c\rightarrow 0\). Then, by Slutsky’s theorem, the result follows from part (i). \(\square\)

Theorem 3.4

For the purely sequential MRPE methodology \((\mathcal {P},\overline{X}_{N_{\mathcal {P}}})\) given by (2.5), under the conditions (C2)–(C6) we have as \(c\rightarrow 0\):
$$\begin{aligned} N_{\mathcal {P}}^{*2}\equiv \left( N_{\mathcal {P}}-n^{*}\right) ^{2}/n^{*} \hbox { is uniformly integrable}, \end{aligned}$$
for sufficiently small \(c\le c_{0}\) with some \(c_{0}(>0)\), \(n^{*}\) defined by (2.3) and \(\delta ^{2}(>0)\) coming from (C3).

Proof

Instead of appealing to nonlinear renewal theory, we proceed to prove this lemma in the spirit of the direct proofs from Ghosh and Mukhopadhyay (1980) and Ghosh et al. (1997, Lemma 7.2.3, pp. 217–219). Recall that [u] denotes the largest integer that is smaller than \(u(>0)\).

We may write for any \(b>b_{0}+1\), \(b_{0}=\left( \sigma \sqrt{A/c_{1}}\right) ^{-1/2}\), where \(c_{1}\) is some appropriate positive constant such that \(c\le c_{1}\). We have:
$$\begin{aligned} E_{\mu ,\sigma }\left\{ \frac{\left( N_{\mathcal {P}}-n^{*}\right) ^{2}}{ n^{*}}I\left( \frac{\left( N_{\mathcal {P}}-n^{*}\right) ^{2}}{ n^{*}}>b^{2}\right) \right\} =2\int _{b}^{\infty }xP_{\mu ,\sigma }\left( \left| N_{\mathcal {P}}-n^{*}\right| >x\sqrt{n^{*}}\right) dx. \end{aligned}$$
(3.12)
We denote \(k_{1}=\left[ n^{*}+x\sqrt{n^{*}}\right] +1\) with \(x\ge b.\) Then, we obtain:
$$\begin{aligned} P_{\mu ,\sigma }\left( N_{\mathcal {P}}>n^{*}+x\sqrt{n^{*}}\right)\le & {} P_{\mu ,\sigma }\left( N_{\mathcal {P}}>k_{1}-1\right) \le P_{\mu ,\sigma }\left( k_{1}-1\le W_{k_{1}-1}\sqrt{A/c}\right) \\\le & {} P_{\mu ,\sigma }\left( \frac{W_{k_{1}-1}}{ \sigma }\ge \frac{n^{*}+x\sqrt{n^{*}}-1}{n^{*}}\right) , \end{aligned}$$
so that
$$\begin{aligned} P_{\mu ,\sigma }\left( N_{\mathcal {P}}>n^{*}+x\sqrt{n^{*}}\right)\le & {} P_{\mu ,\sigma }\left( \left| \frac{W_{k_{1}-1}}{\sigma } -1\right| \ge \frac{x\sqrt{n^{*}}-1}{n^{*}}\right) \\\le & {} \left( \frac{x}{\sqrt{n^{*}}}-\frac{1}{ n^{*}}\right) ^{-2r_{1}}E_{\mu ,\sigma }\left[ \left| \frac{ W_{k_{1}-1}}{\sigma }-1\right| ^{2r_{1}}\right] . \end{aligned}$$
Under the condition (C6), we claim that there exists \(\lambda _{1}\left( >0\right)\) depending only on \(r_{1}\) such that we can write:
$$\begin{aligned} P_{\mu ,\sigma }\left( N_{\mathcal {P}}>n^{*}+x\sqrt{n^{*}}\right) \le \lambda _{1}\left( k_{1}-1\right) ^{-r_{1}}n^{*^{r_{1}}}\left( x-1/ \sqrt{n^{*}}\right) ^{-2r_{1}}. \end{aligned}$$
(3.13)
But, note that \(\left( k_{1}-1\right) ^{-r_{1}}n^{*^{r_{1}}}<1\) for \(x\ge b>b_{0}+1=\left( \sigma \sqrt{A/c_{1}}\right) ^{-1/2}+1\) and \(n^{*}=\sigma \sqrt{A/c}\ge \sigma \sqrt{A/c_{1}}\). Thus, it follows that
$$\begin{aligned}&\int _{b}^{\infty }xP_{\mu ,\sigma }\left( \left| N_{\mathcal {P} }-n^{*}\right| >x\sqrt{n^{*}}\right) dx\le \lambda _{1}\int _{b}^{\infty }x\left( x-b_{0}\right) ^{-2r_{1}}dx \nonumber \\&\quad =\lambda _{1}\frac{\left( b-b_{0}\right) ^{1-2r_{1}}\left( \left( 3-2r_{1}\right) b-b_{0}\right) }{\left( 1-2r_{1}\right) \left( 2-2r_{1}\right) }\rightarrow 0 \end{aligned}$$
(3.14)
as \(b\rightarrow \infty\) by appropriately choosing \(r_{1}>1\).
Next, note that if \(\sqrt{n^{*}}\le b,\) we can claim:
$$\begin{aligned} \int _{b}^{\infty }xP_{\mu ,\sigma }\left( N_{\mathcal {P}}-n^{*}<-x\sqrt{ n^{*}}\right) dx=0. \end{aligned}$$
But, if \(\sqrt{n^{*}}>b\), there exists some \(0<\gamma <1\) such that \(\left( 1-\gamma \right) \sqrt{n^{*}}>b\) when \(c\le c_{2}\), for some suitably picked \(c_{2}(>0)\). Then,
$$\begin{aligned}&\int _{b}^{\infty }xP_{\mu ,\sigma }\left( N_{\mathcal {P}}-n^{*}<-x\sqrt{ n^{*}}\right) dx\le \int _{b}^{\sqrt{n^{*}}}xP_{\mu ,\sigma }\left( N_{\mathcal {P}}\le \gamma n^{*}\right) dx \nonumber \\&\quad \overset{\overset{}{}}{+\text { }\int _{b}^{\left( 1-\gamma \right) \sqrt{ n^{*}}}xP_{\mu ,\sigma }\left( \gamma n^{*}<N_{\mathcal {P}}<n^{*}-x\sqrt{n^{*}}\right) dx} \end{aligned}$$
(3.15)
Now, by Lemma 3.3, \(P_{\mu ,\sigma }\left( N_{\mathcal {P}}\le \gamma n^{*}\right) \le \lambda _{2}n^{*^{-r_{2}/\left( 2+2\lambda \right) }}\) with some appropriate \(r_{2}\left( >2+2\lambda \right)\) and \(\lambda _{2}\left( >0\right)\) depending on \(r_{2}\) alone. Hence, we have:
$$\begin{aligned} \int _{b}^{\sqrt{n^{*}}}xP_{\mu ,\sigma }\left( N_{\mathcal {P}}\le \gamma n^{*}\right) dx\le \lambda _{2}b^{2-\frac{r_{2}}{\left( 1+\lambda \right) }}\rightarrow 0\text { as }b\rightarrow \infty . \end{aligned}$$
(3.16)
Also, for \(b\le x\le\) \(\left( 1-\gamma \right) \sqrt{n^{*}}\), we write:
$$\begin{aligned} k_{2}=\left[ \gamma n^{*}\right] +1\text { and }k_{3}=\left[ n^{*}-x \sqrt{n^{*}}\right] . \end{aligned}$$
Thus, we obtain:
$$\begin{aligned}&P_{\mu ,\sigma }\left( \gamma n^{*}<N_{\mathcal {P}}<n^{*}-x\sqrt{ n^{*}}\right) =P_{\mu ,\sigma }\left( \cup _{n=k_{2}}^{k_{3}}\left\{ N_{ \mathcal {P}}=n\right\} \right) \nonumber \\&\quad \le P_{\mu ,\sigma }\left( \cup _{n=k_{2}}^{k_{3}}\left\{ \frac{W_{n}}{\sigma }<\frac{n}{n^{*}}\right\} \right) . \end{aligned}$$
(3.17)
But, for small c, say \(c\le c_{3}\) with some \(c_{3}(>0),\) we get:
$$\begin{aligned} \frac{n}{n^{*}}\le \frac{k_{3}}{n^{*}}\le \frac{n^{*}-x\sqrt{ n^{*}}}{n^{*}}=1-\frac{x}{\sqrt{n^{*}}}. \end{aligned}$$
Hence, it follows from (3.17) that
$$\begin{aligned}&P_{\mu ,\sigma }\left( \gamma n^{*}<N_{\mathcal {P}}<n^{*}-x\sqrt{ n^{*}}\right) \nonumber \\&\quad \le P_{\mu ,\sigma }\left( \frac{W_{n}}{\sigma }-1<- \frac{x}{\sqrt{n^{*}}},\text { for some }k_{2}\le n\le k_{3}\right) \\&\quad \le P_{\mu ,\sigma }\left( \left| \frac{W_{n}}{ \sigma }-1\right|>\frac{x}{\sqrt{n^{*}}},\text { for some }k_{2}\le n\le k_{3}\right) \\&\quad \le P_{\mu ,\sigma }\left( \max _{k_{2}\le n\le k_{3}}\left| \frac{W_{n}}{\sigma }-1\right| >\frac{x}{\sqrt{n^{*} }}\right) \\&\quad \le \left( x/\sqrt{n^{*}}\right) ^{-2r_{3}}E_{\mu ,\sigma }\left[ \left| \frac{W_{k_{2}}}{\sigma } -1\right| ^{2r_{3}}\right] , \end{aligned}$$
so that we obtain:
$$\begin{aligned} P_{\mu ,\sigma }\left( \gamma n^{*}<N_{\mathcal {P}}<n^{*}-x\sqrt{ n^{*}}\right) \le \lambda _{3}k_{2}^{-r_{3}}x^{-2r_{3}}n^{*^{r_{3}}}\le \lambda _{4}x^{-2r_{3}}, \end{aligned}$$
(3.18)
for some appropriate \(\lambda _{3}\left( >0\right)\) and \(\lambda _{4}\left( >0\right)\), both depending only on \(r_{3}\).
Choosing \(r_{3}>1\), we get from (3.18):
$$\begin{aligned}&\int _{b}^{\left( 1-\gamma \right) \sqrt{n^{*}}}xP_{\mu ,\sigma }\left( \gamma n^{*}<N_{\mathcal {P}}<n^{*}-x\sqrt{n^{*}}\right) dx\le \lambda _{4}\int _{b}^{\left( 1-\gamma \right) \sqrt{n^{*}} }x^{1-2^{r_{3}}}dx \nonumber \\&\quad \le \frac{\lambda _{4}}{2-2r_{3}} b^{2-2r_{3}}\rightarrow 0\text { as }b\rightarrow \infty . \end{aligned}$$
(3.19)
Now choosing \(c_{0}=\min \left\{ c_{1},c_{2},c_{3}\right\}\), with (3.13), (3.15) and (3.18) combined, we can quickly prove the uniform integrability of \(\left( N_{\mathcal {P}}-n^{*}\right) ^{2}/n^{*}\) in \(c\le c_{0}\).
To complete the proof of part (i), we observe that
$$\begin{aligned}&E_{\mu ,\sigma }\left\{ \frac{\left( N_{\mathcal {P}}-n^{*}\right) ^{2}}{ n^{*}}I\left( \frac{\left( N_{\mathcal {P}}-n^{*}\right) ^{2}}{N}>b^{2}\right) I\left( N_{\mathcal {P}}>\frac{1}{2}n^{*}\right) \right\} \nonumber \\&\quad \le 2E_{\mu ,\sigma }\left\{ \frac{\left( N_{ \mathcal {P}}-n^{*}\right) ^{2}}{n^{*}}I\left( \left( N_{\mathcal {P} }-n^{*}\right) ^{2}>\frac{1}{2}b^{2}n^{*}\right) \right\} \rightarrow 0, \end{aligned}$$
(3.20)
as \(b\rightarrow \infty\) uniformly in \(c\le c_{0}\).
Furthermore, choosing \(b>n^{*}\), it follows that for \(c\le c_{0},\) we can also express:
$$\begin{aligned}&E_{\mu ,\sigma }\left\{ \frac{\left( N_{\mathcal {P}}-n^{*}\right) ^{2}}{ n^{*}}I\left( \frac{\left( N_{\mathcal {P}}-n^{*}\right) ^{2}}{N} >b^{2}\right) I\left( N_{\mathcal {P}}\le \frac{1}{2}n^{*}\right) \right\} \nonumber \\&\quad \le m^{-1}n^{*^{2}}P_{\mu ,\sigma }\left( N_{ \mathcal {P}}\le \frac{1}{2}n^{*}\right) \le \lambda _{5}n^{*^{2-r_{4}}}\le \lambda _{5}b^{2-r_{4}}\rightarrow 0, \end{aligned}$$
(3.21)
as \(b\rightarrow \infty\), with \(r_{4}\left( >2\right)\) appropriately chosen and some \(\lambda _{5}\left( >0\right)\) depending only on \(r_{4}\). In view of (3.20)–(3.21), the intended result holds. \(\square\)

A fair question one may ask is the following: What kinds of statistics \(W_{n}\) would certainly satisfy the condition (C1)? Next result points in that direction.

Lemma 3.4

For any fixed \(n(\ge 2)\), suppose that \(\overline{X}_{n}\) continues to stand for the customary sample mean and let \(W_{n}\) be a statistic which exclusively involves only \(\mathbf {Y}_{n}=(X_{1}-X_{n},X_{2}-X_{n},\ldots ,X_{n-1}-X_{n})\) and n. Then, \(\overline{X}_{n}\) and \((W_{2},W_{3},\ldots ,W_{n})\) are independent for all fixed \(n\ge 2\).

Proof

With n fixed, the distribution of \(\mathbf {Y} _{n}\) does not involve the unknown parameter \(\mu\). Now, fix \(\sigma =\sigma _{0}(>0)\) and consider the family of distributions, \(N(\mu ,\sigma _{0}^{2})\). In this family \(N(\mu ,\sigma _{0}^{2})\), \(\overline{X}_{n}\) is a complete and sufficient statistic for \(\mu ,\) but \(\mathbf {Y}_{n}\) is an ancillary statistic, that is, \(\mathbf {Y}_{n}\)’s distribution does not involve \(\mu\). Thus, by appealing to Basu’s (1955) theorem, we can conclude that \(\overline{X}_{n}\) and \(\mathbf {Y}_{n}\) are independently distributed statistics. This statement is true for every fixed \(\sigma =\sigma _{0}(>0)\).

Thus, \(\overline{X}_{n}\) and \(\mathbf {Y}_{n}\) are independently distributed statistics in the family \(N(\mu ,\sigma ^{2})\) where \(\mu ,\sigma ^{2}\) are both unknown parameters.

Next, since \(W_{n}\) involves only \({\mathbf {Y}}_{n}\), clearly \(\overline{X} _{n}\) and \(W_{n}\) are independently distributed statistics in the family \(N(\mu ,\sigma ^{2})\) where \(\mu ,\sigma ^{2}\) are both unknown parameters. \(\square\)

Remark 3.4

The reader will find a number of concrete examples of \(W_{n}\) in Sect. 5 satisfying the condition (C1). One may note that the result will hold even if \(\mathbf {W}_{n}\) is vector-valued. There are more applications of Basu’s (1955) theorem along these lines in Mukhopadhyay (2000, pp. 324–327).

4 The main result: asymptotic second-order expansion of the regret

The regret function, \(\omega _{\mathcal {P}}\left( c\right)\) from (2.7), associated with the purely sequential MRPE methodology \(\mathcal {P}\) from (2.5) was explicitly shown in Lemma 3.1. Now, we proceed with the second-order expansion of \(\omega _{\mathcal {P}}\left( c\right)\).

Theorem 4.1

Consider the regret function \(\omega _{\mathcal {P}}\left( c\right)\) from (2.7), associated with the purely sequential MRPE methodology \(\mathcal {P}\) from (2.5). Under the conditions (C1)–(C7), we have as \(c\rightarrow 0\):
$$\begin{aligned} \omega _{\mathcal {P}}\left( c\right) =\delta ^{2}c+o\left( c\right) [ \textit{Asymptotic Second-Order Regret}], \end{aligned}$$
with \(\delta ^{2}(>0)\) coming from (C3).

Proof

We recall from (2.7) that
$$\begin{aligned} \omega _{\mathcal {P}}\left( c\right) =cE_{\mu ,\sigma }\left\{ \frac{\left( N_{\mathcal {P}}-n^{*}\right) ^{2}}{N_{\mathcal {P}}}\right\} . \end{aligned}$$
(4.1)
First, we observe:
$$\begin{aligned} E_{\mu ,\sigma }\left[ \left( N_{\mathcal {P}}-n^{*}\right) ^{2}/N_{ \mathcal {P}}\right]= & {} E_{\mu ,\sigma }\left[ N_{\mathcal {P}}^{-1}\left( N_{ \mathcal {P}}-n^{*}\right) I\left( N_{\mathcal {P}}>\frac{1}{2}n^{*}\right) \right] \nonumber \\&+ E_{\mu ,\sigma }\left[ N_{\mathcal {P} }^{-1}\left( N_{\mathcal {P}}-n^{*}\right) ^{2}I\left( N_{\mathcal {P} }\le \frac{1}{2}n^{*}\right) \right] . \end{aligned}$$
(4.2)
Next, obviously, \(N_{\mathcal {P}}^{-1}\left( N_{\mathcal {P}}-n^{*}\right) ^{2}I\left( N_{\mathcal {P}}>\frac{1}{2}n^{*}\right) \le 2\left( N_{\mathcal {P}}-n^{*}\right) ^{2}/n^{*}\) w.p.1 which would clearly imply that the random variable \(N_{\mathcal {P}}^{-1}\left( N_{ \mathcal {P}}-n^{*}\right) ^{2}I\left( N_{\mathcal {P}}>\frac{1}{2}n^{*}\right)\) must be uniformly integrable in view of Theorem 3.4. We recall that \(N_{\mathcal {P}}/n^{*}\overset{P_{\mu ,\sigma }}{ \rightarrow }1\) as \(c\rightarrow 0\) so that \(N_{\mathcal {P}}^{-1}\left( N_{ \mathcal {P}}-n^{*}\right) ^{2}I\left( N_{\mathcal {P}}>\frac{1}{2}n^{*}\right) \overset{\pounds }{\rightarrow }\delta ^{2}\chi _{1}^{2}\) by Theorem 3.3, part (i). Hence, combining these features, we conclude:
$$\begin{aligned} E_{\mu ,\sigma }\left[ N_{\mathcal {P}}^{-1}\left( N_{\mathcal {P}}-n^{*}\right) ^{2}I\left( N_{\mathcal {P}}>\frac{1}{2}n^{*}\right) \right] =\delta ^{2}+o(1). \end{aligned}$$
(4.3)
Now, recall \(n_{1c}\) from (3.5) and write:
$$\begin{aligned}&E_{\mu ,\sigma }\left[ N_{\mathcal {P}}^{-1}\left( N_{\mathcal {P}}-n^{*}\right) ^{2}I\left( N_{\mathcal {P}}\le \frac{1}{2} n^{*}\right) \right] \nonumber \\&\quad \le E_{\mu ,\sigma }\left[ N_{\mathcal {P} }^{-1}\left( N_{\mathcal {P}}^{2}+n^{*2}\right) I\left( N_{\mathcal {P} }\le \frac{1}{2}n^{*}\right) \right] \nonumber \\&\quad \le E_{\mu ,\sigma }\left[ n_{1c}^{-1}\left( \frac{1}{4} n^{*2}+n^{*2}\right) I\left( N_{\mathcal {P}}\le \frac{1}{2}n^{*}\right) \right] \le \frac{5}{4}n^{*2}n_{1c}^{-1}P_{\mu ,\sigma }\left( N_{\mathcal {P}}\le \frac{1}{2}n^{*}\right) \nonumber \\&\quad =O\left( n^{*-\frac{r-4\lambda -2}{2+2\lambda }}\right) \rightarrow 0 \hbox { as } c\rightarrow 0, \hbox { in view of Lemma } 3.3, \hbox { if we pick } r>2+4\lambda \nonumber \\&\qquad \Rightarrow E_{\mu ,\sigma }\left[ N_{\mathcal {P} }^{-1}\left( N_{\mathcal {P}}-n^{*}\right) ^{2}I\left( N_{\mathcal {P} }\le \frac{1}{2}n^{*}\right) \right] =o(1). \end{aligned}$$
(4.4)
Combining (4.2)–(4.4), we conclude that
$$\begin{aligned} E_{\mu ,\sigma }\left[ \left( N_{\mathcal {P}}-n^{*}\right) ^{2}/N_{ \mathcal {P}}\right] =\delta ^{2}+o(1), \end{aligned}$$
which leads to the theorem in view of (4.1). \(\square\)

5 Illustrations

In this section, we substitute the statistic \(W_{n}\) with a number of practically useful (unbiased) and consistent estimators of the population standard deviation, \(\sigma\). For any fixed \(n(\ge 2)\), each such statistic \(W_{n}\) is a function of only
$$\begin{aligned} \mathbf {Y}_{n}=(X_{1}-X_{n},X_{2}-X_{n},\ldots ,X_{n-1}-X_{n})\textrm{ and } n. \end{aligned}$$
Thus, Lemma 3.4 will show that \(\overline{X} _{n}\) and \((W_{2},W_{3},\ldots ,W_{n})\) are independent for every fixed \(n\ge 2\).

5.1 Illustration 0: \(\mathcal {P}\equiv \mathcal {P}_{0}\)

Recall the MRPE methodology \(\mathcal {P}_{0}\) from (2.4) due to Robbins (1959) where \(S_{n}\) was used in the stopping boundary. Woodroofe (1977) proved that
$$\begin{aligned} \omega _{\mathcal {P}_{0}}\left( c\right) =\frac{1}{2}c+o(c), \end{aligned}$$
(5.1)
under (C1)–(C7). Indeed, a direct proof of (5.1) follows from our main result, Theorem 4.1. One may check that \(\delta ^{2}=\frac{1}{2}\).
For every fixed \(n(\ge 2),\) since \(\{S_{k};\) \(2\le k\le n\}\) is location invariant, (C1) is satisfied via Lemma 3.4. (C2)–(C3) should be straightforward. (C4)–(C6) can be justified by combining the inequality (w.p.1)
$$\begin{aligned} \left| S_{n}-\sigma \right| =\frac{\left| S_{n}^{2}-\sigma ^{2}\right| }{\left| S_{n}+\sigma \right| }\le \sigma ^{-1}\left| S_{n}^{2}-\sigma ^{2}\right| , \end{aligned}$$
with the facts that \(S_{n}^{2}\) satisfies (C4)–(C6) because \(S_{n}^{2}\) is indeed a U-statistic. See Hoeffding (1948, 1961), Sen and Ghosh (1981), Lee (1990), Mukhopadhyay and Hu (2017, 2018) and other sources for more details.
Next, observe that
$$\begin{aligned} \sup \nolimits _{n\ge 2}\left\{ S_{n}^{2}\right\} \le 2\sup \nolimits _{n\ge 2}\left\{ n^{-1}\Sigma _{i=1}^{n}(X_{i}-\mu )^{2}\right\} =2Z, \hbox { say}. \end{aligned}$$
In view of Wiener’s (1939) ergodic theorem, Z has all positive moments finite since the X’s have all positive moments finite. In other words, \(W_{n}\) substituted by \(S_{n}\) will satisfy all the stated conditions (C1)–(C7).

5.2 Illustration 1: \(\mathcal {P}\equiv \mathcal {P}_{1}\)

In this section, we provide an application with \(W_{n}\) in the stopping rule (2.5) substituted by a multiple of \(S_{n}\) defined as follows. Under the normality assumption, we can accordingly construct the unbiased and consistent estimator for \(\sigma\) based on \(S_{n}\) denoted by \(Q_{n}\), where
$$\begin{aligned} Q_{n}=a_{n}S_{n}\text { with }a_{n}=\left( \frac{n-1}{2}\right) ^{1/2}\Gamma \left( \frac{n-1}{2}\right) /\Gamma \left( \frac{n}{2}\right) . \end{aligned}$$
(5.2)
We may consider the following stopping rule:
$$\begin{aligned} N_{\mathcal {P}_{1}}\equiv N_{\mathcal {P}_{1}}\left( c\right) =\inf \left\{ n\ge m:n\ge \sqrt{A/c}\left( Q_{n}+n^{-\lambda }\right) \right\} . \end{aligned}$$
(5.3)
We denote this estimation strategy \(\mathcal {P}_{1}\). Now, we know that the following result holds:
$$\begin{aligned} \sqrt{n}\left( \sigma ^{-1}Q_{n}-1\right) \overset{\pounds }{\rightarrow } N\left( 0,\frac{1}{2}\right) \text { as }n\rightarrow \infty , \end{aligned}$$
so that \(\delta ^{2}=\frac{1}{2}\). Hence, we can immediately write:
$$\begin{aligned} \omega _{\mathcal {P}_{1}}\left( c\right) =\frac{1}{2}c+o(c), \end{aligned}$$
(5.4)
under (C1)–(C7).
This regret expansion is no different from that in (5.1) up to o(c). Its direct proof will follow right away from our main result, Theorem 4.1. One additional comment may be in order: Observe that \(\lim _{n\rightarrow \infty }a_{n}=1\) and hence for large enough \(n(\ge n_{0})\), we may claim that \(\left| a_{n}\right| \le 2\) and \(\left| a_{n}-1\right| \le 1\). Thus, we can express (w.p.1)
$$\begin{aligned} \left| a_{n}S_{n}-\sigma \right| \le \left| a_{n}\right| \left| S_{n}-\sigma \right| +\sigma \left| a_{n}-1\right| \le 2\left| S_{n}-\sigma \right| +\sigma , \hbox { for } n\ge n_{0}. \end{aligned}$$
Now, \(W_{n}\) substituted by \(Q_{n}\) from (5.2) will satisfy all the stated conditions (C1)–(C7) which can be justified as we just pointed out underneath (5.1).

5.3 Illustration 2: \(\mathcal {P}\equiv \mathcal {P}_{2}\)

Next, we provide an illustration with \(W_{n}\) in the stopping rule (2.5) substituted by a suitable multiple of Gini’s Mean Difference ( GMD) defined as follows:
$$\begin{aligned} \hbox {GMD: } U_{n}=\left( {\begin{array}{c}n\\ 2\end{array}}\right) ^{-1}\Sigma \Sigma _{1\le i<j\le n}\left| X_{i}-X_{j}\right| . \end{aligned}$$
(5.5)
Under the normality assumption, we can accordingly construct the unbiased and consistent estimator for \(\sigma\) based on \(U_{n}\) where we may use the following estimator \(G_{n}\) for \(\sigma\):
$$\begin{aligned} G_{n}=\frac{\sqrt{\pi }}{2}U_{n}. \end{aligned}$$
(5.6)
As a robust estimator of the population standard deviation \(\sigma\), GMD was originally developed by Gini (1914, 1921). One should notice that \(G_{n}\) is indeed a U-statistic.
Again, one may refer to Hoeffding (1948, 1961), Sen and Ghosh (1981), Lee (1990), Mukhopadhyay and Hu (2017, 2018) for more details. Thus, we consider the following stopping rule:
$$\begin{aligned} N_{\mathcal {P}_{2}}\equiv N_{\mathcal {P}_{2}}\left( c\right) =\inf \left\{ n\ge m:n\ge \sqrt{A/c}\left( G_{n}+n^{-\lambda }\right) \right\} . \end{aligned}$$
(5.7)
so that \(\delta ^{2}=\frac{\pi +6\sqrt{3}-12}{3}\). Hence, we can immediately write:
$$\begin{aligned} \omega _{\mathcal {P}_{2}}\left( c\right) =\frac{\pi +6\sqrt{3}-12}{3} c+o(c)\approx 0.511c+o(c), \end{aligned}$$
(5.8)
under (C1)–(C7).

Now, (C1) follows from Lemma 3.4 since \(\{G_{k};\) \(2\le k\le n\}\) is location invariant for every fixed \(n(\ge 2)\). (C2)–(C3) follow from Hoeffding (1948). (C5)–(C6) will follow from the proof of Theorem 4.1 in Mukhopadhyay and Hu (2017, \(i=1\)). (C4) follows from (C5) to (C6) as shown in Lemma 3.2. (C7) follows from Mukhopadhyay and Hu (2017, Lemma 3.3). A direct proof of (5.8) will follow immediately from our main result, Theorem 4.1, since \(W_{n}\) substituted by \(G_{n}\) satisfies all the stated conditions (C1)–(C7).

5.4 Illustration 3: \(\mathcal {P}\equiv \mathcal {P}_{3}\)

In this Section, we provide an application with \(W_{n}\) in the stopping rule (2.5) substituted by a multiple of Mean Absolute Deviation (MAD). As an alternative estimator of the population standard deviation \(\sigma\), MAD came under practical scrutiny with regard to robustness issues. The MAD is defined as follows:
$$\begin{aligned} \hbox {MAD: } V_{n}=n^{-1}\Sigma _{i=1}^{n}\left| X_{i}-\overline{X} _{n}\right| . \end{aligned}$$
(5.9)
Under the normality assumption, we can accordingly construct the unbiased and consistent estimator for \(\sigma\) based on MAD denoted by \(M_{n}\), where
$$\begin{aligned} M_{n}=\sqrt{\frac{\pi n}{2\left( n-1\right) }}V_{n}. \end{aligned}$$
(5.10)
Thus, we consider the following stopping rule:
$$\begin{aligned} N_{\mathcal {P}_{3}}\equiv N_{\mathcal {P}_{3}}\left( c\right) =\inf \left\{ n\ge m:n\ge \sqrt{A/c}\left( M_{n}+n^{-\lambda }\right) \right\} . \end{aligned}$$
(5.11)
That is, \(\delta ^{2}=\frac{\pi -2}{2}\). We denote this estimation strategy \(\mathcal {P}_{3}\). Hence, we can immediately write:
$$\begin{aligned} \omega _{\mathcal {P}_{3}}\left( c\right) =\frac{\pi -2}{2}c+o(c)\approx 0.571c+o(c), \end{aligned}$$
(5.12)
under (C1)–(C7).
Now, (C1) follows from Lemma 3.4 since \(\{M_{k};\) \(2\le k\le n\}\) is location invariant for every fixed \(n(\ge 2)\). (C2) follows from Mukhopadhyay and Hu (2017, Lemma 3.2). With regard to (C3), more specifically, we have:
$$\begin{aligned} \sqrt{n}\left( \sigma ^{-1}M_{n}-1\right) \overset{\pounds }{\rightarrow } N\left( 0,\frac{\pi -2}{2}\right) \text { as }n\rightarrow \infty . \end{aligned}$$
See Babu and Rao (1992). (C5)–(C6) can be proved along the lines of the proof of Theorem 4.1 from Mukhopadhyay and Hu (2017, \(i=2\)) with minor modifications. (C4) follows from (C5) to (C6) as shown in Lemma 3.2. (C7) can be verified in the spirit of the proof of Lemma 3.3 in Mukhopadhyay and Hu (2017). A direct proof of (5.12) will follow from our main result, Theorem 4.1, since \(W_{n}\) substituted by \(M_{n}\) will satisfy all the stated conditions (C1)–(C7).

Remark 5.1

Obviously, there can be a large number of choices of suitable \(W_{n}\). We have exhibited four different choices of \(W_{n}\) leading to the methodologies \(\mathcal {P\equiv P}_{0},\mathcal {P}_{1}, \mathcal {P}_{2},\) and \(\mathcal {P}_{3}\) respectively. Which one stands out? There is no simple answer. By comparing the second-order expansions of the regret functions alone, we feel that (i) \(\mathcal {P}_{0},\mathcal {P}_{1}\) would perform nearly identically, but (ii) \(\mathcal {P}_{2}\) may be marginally preferable to \(\mathcal {P}_{3}\). On the other hand, if it is important to require a methodology that would withstand some possible outliers, then (i) \(\mathcal {P}_{0},\mathcal {P}_{1}\) are not very desirable, (ii) \(\mathcal {P}_{2},\mathcal {P}_{3}\) would perform nearly identically, whereas \(\mathcal {P}_{2}\) has a slight edge, and (iii) both \(\mathcal {P}_{2}, \mathcal {P}_{3}\) are more robust than \(\mathcal {P}_{0},\mathcal {P}_{1}\). From a practical point of view, one should lean in favor of using \(W_{n}\)’s from either \(\mathcal {P}_{2}\) or \(\mathcal {P}_{3}\). More pertinent details with regard to robustness issues are found in Mukhopadhyay and Hu (2017).

Table 1

Explanation of the set of notation used in Table 2:

\(n_{l}\) : sample size in \(l^{\text {th}}\) run;

\(\overline{n}=L^{-1}\Sigma _{l=1}^{L}n_{l}\) : should estimate \(n^{*}\)

\(s(\overline{n})=\left\{ (L^{2}-L)^{-1}\Sigma _{l=1}^{L}(n_{l}-\overline{n})^{2}\right\} ^{1/2}\) : estimated standard error (s.e.) of \(\overline{n}\)

\(s_{n_{l}}^{2}\) : sample variance from observed data \(x_{1},\ldots ,x_{n_{l}}\) in \(l^{\text {th}}\) run;

\(\widehat{R}_{n_{l}}=As_{n_{l}}^{2}/n_{l}+cn_{l}\) : estimated risk in \(l^{\text {th}}\) run;

\(\overline{R}=L^{-1}\Sigma _{l=1}^{L}\widehat{R}_{n_{l}}\) : should estimate \(R_{n^{*}}(c)\);

\(\widehat{\xi }=\overline{R}/R_{n^{*}}(c)\) : should estimate \(\xi (c)(\equiv \xi _{\mathcal {P}}(c))\);

\(s(\widehat{\xi })=\left\{ (L^{2}-L)^{-1}\Sigma _{l=1}^{L}\left( \widehat{R}_{n_{l}}-\overline{R}\right) ^{2}\right\} ^{1/2}/R_{n^{*}}(c)\) : estimated s.e. of \(\widehat{\xi }\);

\(\widehat{r}_{n_{l}}=c(n_{l}-n^{*})^{2}/n_{l}\) : estimated regret in \(l^{\text {th}}\) run;

\(\widehat{\omega }=cL^{-1}\Sigma _{l=1}^{L}\widehat{r}_{n_{l}}\) : should estimate \(\omega (c)(\equiv \omega _{\mathcal {P}}(c))\);

\(s(\widehat{\omega })=\left\{ (L^{2}-L)^{-1}\Sigma _{l=1}^{L}\left( \widehat{r}_{n_{l}}-\widehat{\omega }\right) ^{2}\right\} ^{1/2}\) : estimated s.e. of \(\widehat{\omega }\);

\(\delta ^{2}c\) : theoretical approximation of \(\omega (c)\).

6 Simulations

In the spirit of Mukhopadhyay and Hu (2017), we implemented the purely sequential MRPE methodologies based on various stopping rules given by (5.3), (5.7), and (5.11) respectively under the normal case. To be more specific, we generated pseudorandom samples from a \(N\left( 5,4\right)\) population. We also fixed the weight function \(A=100\), the pilot sample size \(m=10,\) and \(\lambda =2,\) while selecting a wide range of values of c including 0.16, 0.04, 0.01,  and 0.0025 so that the optimal sample sizes \(n^{*}\) can be determined to be 50,100,200, and 400 accordingly by (2.3).

Throughout this section, we continue to use the following sets of notation to designate a specific methodology under implementation.
$$\begin{aligned} \begin{array}{ll} \mathcal {P}_{0}: &{} \hbox {Sample standard deviation-based stopping rule } (2.4) \\ &{} \hbox {with } \delta ^{2}=\frac{1}{2}; \\ \mathcal {P}_{1}: &{} \hbox {Sample standard deviation-based stopping rule } (5.3) \\ &{} \hbox {with } \delta ^{2}=\frac{1}{2}; \\ \mathcal {P}_{2}: &{} \hbox {GMD-based stopping rule } (5.7) \hbox { with } \delta ^{2}=\frac{ \pi +6\sqrt{3}-12}{3}; \\ \mathcal {P}_{3}: &{} \hbox {MAD-based stopping rule } (5.11) \hbox { with } \delta ^{2}=\frac{ \pi -2}{2}. \end{array} \end{aligned}$$
(6.1)
An explanation is to make it clear how we implemented the methodology \(\mathcal {P}_{1}\) from (5.3). In its boundary condition, we use the statistic \(Q_{n}\equiv a_{n}S_{n}\) with \(a_{n}\) defined in (5.2). Since \(a_{n}\) involves a ratio of gamma functions, we evaluated it as follows: We used the exact expression of \(a_{n}\) for \(n=m,m+1,\ldots ,342,\) but we replaced \(a_{n}\) with \((1-\frac{1}{n})^{1/2}(1+\frac{3}{4n})\) for \(n=343,344,\ldots\) as we ran through the stopping algorithm with \(n=m,m+1,\ldots\).
Table 2

Simulations from \(N\left( 5,4\right)\) with \(A=100,m=10,\lambda =2\) under 1000 runs implementing \(\mathcal {P}_{0}\) from (2.4), \(\mathcal {P}_{1}\) from (5.3), \(\mathcal {P}_{2}\) from (5.7), and \(\mathcal {P}_{3}\) from (5.11): Associated \({\delta ^{2}}\) are summarized in (6.1)

\(n^{*}\)

100c

\(\mathcal {P}\)

\(\overline{n}\)

\(s(\overline{n})\)

\(\widehat{\xi }\)

\(s(\widehat{\xi })\)

\(\delta ^{2}\)

\(\widehat{\omega }/c\)

\(s(\widehat{\omega })\)

50

16

\({\mathcal {P}}_{0}\)

50.012

0.1671

0.9880

0.003340

0.5

0.593131

0.005126

\({\mathcal {P}}_{1}\)

50.199

0.1729

0.9861

0.003449

0.5

0.647494

0.007424

\({\mathcal {P}}_{2}\)

50.313

0.1703

0.9879

0.003377

0.5113

0.612431

0.005171

\({\mathcal {P}}_{3}\)

50.259

0.1778

0.9872

0.003339

0.5708

0.666431

0.005623

100

4

\({\mathcal {P}}_{0}\)

99.955

0.2408

0.9932

0.002404

0.5

0.599650

0.001153

\({\mathcal {P}}_{1}\)

100.107

0.2378

0.9922

0.002373

0.5

0.581725

0.001146

\({\mathcal {P}}_{2}\)

100.335

0.2347

0.9943

0.002306

0.5113

0.561200

0.001107

\({\mathcal {P}}_{3}\)

100.332

0.2495

0.9939

0.002327

0.5708

0.636125

0.001183

200

1

\({\mathcal {P}}_{0}\)

200.012

0.3325

0.9969

0.001660

0.5

0.561100

0.000254

\({\mathcal {P}}_{1}\)

200.132

0.3330

0.9962

0.001667

0.5

0.563800

0.000264

\({\mathcal {P}}_{2}\)

200.255

0.3380

0.9971

0.001661

0.5113

0.580400

0.000264

\({\mathcal {P}}_{3}\)

200.026

0.3562

0.9962

0.001659

0.5708

0.643800

0.000300

400

0.25

\({\mathcal {P}}_{0}\)

399.931

0.4588

0.9983

0.001146

0.5

0.531200

0.000068

\({\mathcal {P}}_{1}\)

400.225

0.4531

0.9984

0.001132

0.5

0.520800

0.000069

\({\mathcal {P}}_{2}\)

400.282

0.4508

0.9984

0.001114

0.5112

0.514000

0.000067

\({\mathcal {P}}_{3}\)

400.232

0.4873

0.9985

0.001145

0.5708

0.598800

0.000070

The large-sample approximation (for \(n\ge 343\)) of \(a_{n}\) is based on the well-known (Abramowitz and Stegun 1972, Result 6.1.47, p. 257) result:
$$\begin{aligned} \frac{\Gamma (z+\alpha )}{\Gamma (z+\beta )}=z^{\alpha -\beta }\left\{ 1+ \frac{(\alpha -\beta )(\alpha +\beta -1)}{2z}+O(z^{-2})\right\} , \end{aligned}$$
with \(z=\frac{1}{2}n\), \(\alpha =-\frac{1}{2}\), and \(\beta =0\).

Table 2 presents the average simulated performances obtained from \(L(=1000)\) independent replications in the construction of each row. For specific entities shown in Table 2, we have used the set of precisely defined notation explained in Table 1.

As reflected in Table 2, the average estimated sample sizes \(\overline{n}\) lie within a narrow band of the optimal sample size \(n^{*}\). Additionally, all values of \(\widehat{\xi }\), the estimated risk efficiency, found in Column 6 are close to 1, whereas larger (or smaller) the sample size (or c) is, the closer \(\widehat{\xi }\) is to 1. This empirically verifies the asymptotic first-order risk efficiency properties of the purely sequential MRPE methodologies (2.4), (5.3), (5.7), and (5.11).

Furthermore, the values of \(\widehat{\omega }\), the estimated regrets, shown in Column 9 are extremely close to the corresponding theoretical approximations provided in Column 8 corresponding to the sequential methodologies (2.4), (5.3), (5.7), and (5.11). This empirically validates the asymptotic second-order expansion of the regret across the purely sequential MRPE methodologies (2.4), (5.3), (5.7), and (5.11).

7 Some concluding thoughts

We should point out that (i) the statistic \(W_{n}\) used in (2.4) is not unbiased for \(\sigma\) but it is consistent for \(\sigma\), whereas (ii) the statistics \(W_{n}\) used in (5.3), (5.7), and (5.11) are all unbiased for \(\sigma\) and are consistent for \(\sigma\). That is a major difference between the MRPE methodologies \(\mathcal {P}_{0}\) and \(\mathcal {P}_{1}-\mathcal {P}_{3}\) for estimating the normal mean.

Next, \(W_{n}\) used in (2.4) also satisfies all conditions (C1)–(C7) leading up to the asymptotic second-order regret expansion shown in (5.1). Mukhopadhyay and Hu (2017) also proposed analogous MRPE methodologies with both GMD-based and MAD-based boundaries involving suitable \(W_{n}\) where \(W_{n}^{2}\) estimated \(\sigma ^{2}\) unbiasedly and consistently. They emphasized asymptotic first-order risk efficiency properties for their individual MRPE methodologies in the context of separate problems.

In this present work, however, the statistics \(W_{n}\) used in (5.3), (5.7), and (5.11) are all unbiased for \(\sigma\) and are consistent for \(\sigma\). We began with a general unified theory with substantial illustrations newer than those in Mukhopadhyay and Hu (2017), each giving rise to appropriate asymptotic second-order expansions of the associated regret functions.

Indeed, the earlier proposed GMD-based and MAD-based MRPE methodologies of Mukhopadhyay and Hu (2017) do enjoy the exact same second-order expansions of their associated regret functions as shown in (5.8) and (5.12). Mukhopadhyay and Hu (2017) were not in a position to claim asymptotic second-order expansions of their regret functions. Now, we have them all, by the courtesy of our general theoretical treatment and Theorem 4.1.

From Table 2, it is clear that all MRPE methodologies \(\mathcal {P}_{0}- \mathcal {P}_{3}\) have nearly comparable \(\widehat{\xi }\) and \(\widehat{ \omega }\) values, however, the newer MRPE methodologies \(\mathcal {P}_{2}- \mathcal {P}_{3}\) may be associated with very marginally larger regret values compared to those of \(\mathcal {P}_{0}-\mathcal {P}_{1}\). On a positive note, our newer MRPE methodologies \(\mathcal {P}_{2}-\mathcal {P}_{3}\) tend to be more robust under possibilities of observing outliers than the MRPE methodologies \(\mathcal {P}_{0}-\mathcal {P}_{1}\).

Notes

Acknowledgements

The comments received from two anonymous reviewers, the Associate Editor, and the Executive Editor on our earlier version have genuinely helped us in preparing this revised manuscript. We express our gratitude to all of them and thank them.

Compliance with ethical standards

Conflict of interest statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© Japanese Federation of Statistical Science Associations 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA
  2. 2.Department of StatisticsUniversity of ConnecticutStorrsUSA

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