Purely Sequential and k-Stage Procedures for Estimating the Mean of an Inverse Gaussian Distribution

Abstract

In the first part of this paper, we propose purely sequential and k-stage (k ≥ 3) procedures for estimation of the mean μ of an inverse Gaussian distribution having prescribed ‘proportional closeness’. The problem is constructed in such a manner that the boundedness of the expected loss is equivalent to the estimation of parameter with given ‘proportional closeness’. We obtain the associated second-order approximations for both the procedures. Second part of this paper deals with developing the minimum risk and bounded risk point estimation problems for estimating the mean μ of an inverse Gaussian distribution having unknown scale parameter λ. We propose an useful family of loss functions for both the problems and our aim is to control the associated risk functions. Moreover, we establish the failure of fixed sample size procedures to deal with these problems and hence propose purely sequential and k-stage (k ≥ 3) procedures to estimate the mean μ. We also obtain the second-order approximations associated with our sequential procedures. Further, we provide extensive sets of simulation studies and real data analysis to show the performances of our proposed procedures.

Appendix: Proofs of the Selected Results

Proof Proof of Theorem 1

N₁ from (6) can be written as, $N_{1} = inf\left \{n\geq m\geq 2 ; {\sum }_{j=1}^{n-1} Z_{j} \leq \frac {n^{2}}{n^{*}}\right \}$ where n^∗ comes from (5) and ${\sum }_{j=1}^{n-1} Z_{j} = \frac {n\lambda }{\widehat {\lambda }_{n}} \sim \chi ^{2}_{n-1}$ and $Z_{j} \sim {\chi ^{2}_{1}}$. N₁ can also be written as J + 1 w.p. 1 where,

$$ J = inf\left\{n \geq m-1; {\sum}_{j=1}^{n} Z_{j} \leq \frac{n^{2}}{n^{*}}\left( 1+\frac{1}{n}\right)^{2}\right\} $$

(38)

Now Comparing (38) with equation (1.1) of Woodroofe (1977), we get, α = 2, β = 1, c = n^∗− 1, μ = 1, λ = n^∗, $L(n)=1+\frac {2}{n}+\frac {1}{n^{2}}$, L₀ = 2, τ² = 2, a = 1/2. □

Result (7) now follows from his Theorem 2.4 for m > 2.

Now, using the fact that $\sqrt {n\lambda } (\overline {X}_{n} - \mu )/\mu \sqrt {\overline {X}_{n}}$ has a standard normal distribution, we can write,

$$ \begin{array}{@{}rcl@{}} E\left\{L(\mu, \overline{X}_{{N}_{1}})\right\} &= 1 - E\left[P\left\{{\chi^{2}_{1}} \leq N_{1}\lambda d^{2}\right\}\right] \\ &= 1 - E\left[{\Psi}\left\{N_{1}\lambda d^{2}\right\}\right] \end{array} $$

(39)

where Ψ(.) is the cdf of a ${\chi ^{2}_{1}}$ variate. We can expand equation (39) by Taylor Series expansion [see Hall (1981), page 1231] as follows:

$$ E\left\{L(\mu, \overline{X}_{{N}_{1}})\right\} = {\Psi}\left( \lambda d^{2}E(N_{1})\right) + \frac{\lambda^{2}d^{4}}{2} E\left\{N_{1} - E(N_{1})\right\}^{2}{\Psi}^{\prime\prime}\left( \lambda d^{2}E(N_{1})\right) + o(d^{2}) $$

(40)

where,

$$ \begin{array}{@{}rcl@{}} {\Psi}(x) &= \frac{1}{\sqrt 2\pi}{{\int}_{0}^{x}} e^{-y/2} y^{-1/2} dy \\ {\Psi}^{\prime}(x) &= \frac{1}{2\sqrt 2\pi} e^{-x/2} x^{-2} = \xi(x) \\ {\Psi}^{\prime\prime}(x) &= \frac{1}{2} \xi(x)\left\{-2x^{-2} - 1\right\} \end{array} $$

Now since we have,

${\Psi }\left (\lambda d^{2}E(N_{1})\right ) = \alpha + \left (\lambda d^{2}E(N_{1}) - z^{2}\right ){\Psi }^{\prime }(z^{2}) + o(d^{2})$ and

$\lambda d^{2}E(N_{1}) - z^{2} = \frac {n^{*}d^{4}}{z^{2}}\left \{n^{*}+\nu -3\right \} - z^{2}$

we can now easily obtain the result (8) by equation (40).

Proof Proof of Theorem 2

We want to employ Helmert transformation. Let

$Y_{i} = \frac {\lambda }{\mu ^{2}} \frac {\left ({\sum }_{i=1}^{n} X_{i} - n\mu \right )^{2}}{{\sum }_{i=1}^{n} X_{i}}, i = 1,2,3,...$ are independent ${\chi _{1}^{2}}$ r.v.’s. Denote, $\overline {Y}_{n} = n^{-1} {\sum }_{i=1}^{n} Y_{i}$ and $S_{n} = {\sum }_{i=1}^{n} Y_{i}$.

Let c₀ = c_k− 1 = 1, T₀ = m₀ − 1 and

$$ \begin{array}{@{}rcl@{}} T_{1} &=& max \left[\left\{c_{1}n^{*}\overline{Y}_{{T}_{0}}\right\}, T_{0}\right], L_{1} = \left\{c_{1}n^{*}\overline{Y}_{{T}_{0}}\right\} \\ &{\vdots} &{\vdots} \\ T_{k-2} &=& max \left[\left\{c_{k-2}n^{*}\overline{Y}_{{T}_{k-3}}\right\}, T_{k-3}\right], L_{k-2} = \left\{c_{k-2}n^{*}\overline{Y}_{{T}_{k-3}}\right\}\\ T_{k-1} &=& max \left[\left\{n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right\}, T_{k-2}\right], L_{k-1} = \left\{n^{*}\overline{\boldsymbol{Y}}_{{T}_{k-2}}+\eta\right\} \end{array} $$

It is clear that M_i = T_i + 1,i = 1, 2,...,k − 1. We want to prove following three results. These results will be helpful in proving the main theorem.

$$ E\left( T_{k-1}\right) = n^{*} - \frac{Var\left( Y_{1}\right)}{c_{k-2}} - \frac{1}{2} + \eta + o(1); $$

(41)

$$ Var\left( T_{k-1}\right) = \frac{n^{*} Var\left( Y_{1}\right)}{c_{k-2}} + o\left( n^{*}\right) $$

(42)

$$ E\mid T_{k-1} - E\left( T_{k-1}\right)\mid^{3} = o\left( n^{{*}^{2}}\right) $$

(43)

Now we state some important Lemmas and also give their proofs, wherever necessary. We will prove the results given in equations (41)-(43) with the help of these lemmas. □

Lemma 1

Let

$$ \begin{array}{@{}rcl@{}} A_{i}\left( \epsilon\right) &=& \left\{ \mid\overline{Y}_{{T}_{i}}-1\mid \geq \epsilon\right\}, i = 0,1,2,...,k-2, \\ B_{i} &=& \left\{T_{i} = T_{i-1}\right\}, i = 1,2,...,k-1,\\ C_{i}\left( \epsilon\right) &=& \left\{ \mid T_{i}-c_{i}n^{*}\mid \geq \epsilon c_{i}n^{*}\right\}, i = 1,2,...,k-2. \end{array} $$

Then, for any 𝜖 > 0, there exists 0 < δ₀ < 1 and $0<\delta _{1}=\delta _{1}\left (\epsilon \right )<1$ such that for all sufficiently large n^∗,

$B_{1} \subseteq \left \{ \mid \overline {Y}_{{T}_{0}}-1\mid \geq \delta _{0}\right \} = A_{0}\left (\delta _{0}\right ),$

$C_{i}\left (\epsilon \right ) \subset B_{i} \cup A_{i-1}\left (\delta _{1}\right ),$

$B_{i+1} \subseteq c_{i}\left (\delta _{0}\right ) \cup \left \{ \mid \overline {Y}_{\left (1-\delta _{0}\right )c_{i}n^{*}}-1\mid \geq \delta _{0}\right \},$

$A_{i}\left (\epsilon \right ) \subset c_{i}\left (\delta _{1}\right ) \cup \left \{ \mid \overline {Y}_{\left (1-\delta _{1}\right )c_{i}n^{*}}-1\mid \geq \delta _{1}\right \} \cup \left \{ \mid \overline {Y}_{\left (1+\delta _{1}\right )c_{i}n^{*}}-1\mid \geq \delta _{1}\right \},$

for i = 1, 2, 3,...,k − 2. Using Markov inequality and the assumption $n^{*} = O\left ({m_{0}^{r}}\right )$ of Theorem 2, we have as $n^{*} \rightarrow \infty$, $P\left \{ \mid \overline {Y}_{{T}_{0}}-1\mid \geq \delta _{0}\right \} = O\left ({n^{*}}^{-1}\right )$ and $P\left \{ \mid \overline {Y}_{\left (1 \pm \delta _{1}\right )c_{i}n^{*}}-1\mid \geq \delta _{1}\right \} = O\left ({n^{*}}^{-1}\right )$ since $E\left (Y_{1}^{2r}\right )<\infty$. Now it follows from Lemma 1 that, for all sufficiently small 𝜖 > 0, $P\left (A_{i}\left (\epsilon \right )\right ) = O\left ({n^{*}}^{-1}\right ), P\left (B_{i}\right ) = O\left ({n^{*}}^{-1}\right )$ and $P\left (C_{i}\left (\epsilon \right )\right ) = O\left ({n^{*}}^{-1}\right )$ as $n^{*} \rightarrow \infty$ for all those $A_{i}\left (\epsilon \right )$, B_i and $C_{i}\left (\epsilon \right )$ defined in Lemma 1.

Lemma 2

For some constant C ₀

$E\left \{T_{k-3}\left (\overline {Y}_{{T}_{k-3}}-1\right )\right \} = 0,$

$E\left \{T_{k-3}\left (\overline {Y}_{{T}_{k-3}}-1\right )^{2}\right \} = Var\left (Y_{1}\right )+E\left [\left \{\left (S_{{T}_{k-4}}-T_{k-4}\right )^{2}-T_{k-4}Var\left (Y_{1}\right )\right \}/T_{k-3}\right ],$

$E\left [\left \{\surd {T_{i}}\left (\overline {Y}_{{T}_{i}}-1\right )\right \}^{4}\right ] \leq C_{0}, i=0,1,...,k-1,$

$E\left \{\left (S_{T}-T_{i}\right )^{2}-T_{i}Var\left (Y_{1}\right )\right \} = 0, i=0,1,...,k-1.$

Proof Proof of Lemma 2

The expectation on the right-hand side of the second assertion is taken to be 0 if k = 3. The first two assertions are clearly true if k = 3 and can be proved by using the conditional expectation formula conditioning on $Y_{1}, Y_{2},...,Y_{T_{k-4}}$ when k ≥ 4. We use mathematical induction to prove the third assertion. It is true when i = 0 and assume that it is true when $i=j \left (0 \leq j \leq k-2\right )$. So when i = j + 1, we have

$$ \begin{array}{@{}rcl@{}} &&E\left[\left\{\surd{T_{j+1}}\left( \overline{Y}_{{T}_{j+1}}-1\right)\right\}^{4}\right]\\ &=& E\left[E\left( \frac{1}{T_{j+1}^{2}}\left( S_{{T}_{j}}-T_{j}+S_{{T}_{j+1}}-S_{{T}_{j}}-\left( T_{j+1}-T_{j}\right)\right)^{4}\mid Y_{1},...,Y_{T}\right)\right] \\ &=& E\left[\frac{1}{T_{j+1}^{2}}\left( \left( S_{{T}_{j}}-T_{j}\right)^{4}+6\left( S_{{T}_{j}}-T_{j}\right)^{2}\left( T_{j+1}-T_{j}\right)Var\left( Y_{1}\right)\right.\right.\\ &&+\left.\left.4\left( S_{{T}_{j}}-T_{j}\right)\left( T_{j+1}-T_{j}\right)E\left( Y_{1}-1\right)^{3}+\left( T_{j-1}-T_{j}\right)E\left( Y_{1}-1\right)^{4}\right)\right] \\ &&\leq E\left[\left\{\surd{T_{j}}\left( \overline{Y}_{{T}_{j}}-1\right)\right\}^{4}\right] + 6E\left[\left\{\surd{T_{j}}\left( \overline{Y}_{{T}_{j}}-1\right)\right\}^{2}\right]Var\left( Y_{1}\right) \\ &&+ 4E\left( \surd{T_{j}} \mid \overline{Y}_{{T}_{j}}-1 \mid\right) E\left\{\left( Y_{1}-1\right)^{3}\right\} + E\left\{\left( Y_{1}-1\right)^{4}\right\}, \end{array} $$

which is bounded by the Cauchy-Schwartz inequality and the assumption that $E\left [\left \{\surd {T_{j}}\left (\overline {Y}_{{T}_{j}}-1\right )\right \}^{4}\right ]$ is bounded. The fourth assertion follows directly also from a conditional argument. □

Lemma 3

As $n^{*} \rightarrow \infty$, we have that for any 0 ≤ j ≤ k − 1 and 𝜖 > 0

$E\left \{{T_{j}^{3}}I\left (A_{i}\left (\epsilon \right )\right )\right \} = o(1), E\left \{{T_{j}^{3}}I\left (B_{i}\right )\right \} = o(1)$ and $E\left \{{T_{j}^{3}}I\left (C_{i}\left (\epsilon \right )\right )\right \} = o(1)$

for all those $A_{i}\left (\epsilon \right )$, B_i and $C_{j}\left (\epsilon \right )$defined in Lemma 1.

Proof Proof of Lemma 3

Let H be either $A_{i}\left (\epsilon \right )$ or B_i or $C_{j}\left (\epsilon \right )$. We want to show that $E\left \{{T_{j}^{3}}I\left (H\right )\right \} = o(1)$ by using mathematical induction on j. This is true when j = 0 by Lemma 1 and we assume this is true when $j=1 \left (0 \leq l \leq k-2\right )$. Thus, when j = l + 1, it follows from the inductive assumption and Lemma 1 that

$$ \begin{array}{@{}rcl@{}} E\left\{T_{l+1}^{3}I\left( H\right)\right\} &\leq& E\left\{\left( c_{l+1}n^{*}\overline{Y}_{{T}_{l}}+\eta+T_{l}\right)^{3}I(H)\right\} \\ &\leq& 9E\left\{\left( c_{l+1}^{3}{n^{*}}^{3}\left( \overline{Y}_{{T}_{l}}\right)^{3}+\eta^{3}+{T_{l}^{3}}\right)I(H)\right\} \\ &\leq& 9c_{l+1}^{3}{n^{*}}^{3}E\left\{\left( \overline{Y}_{{T}_{l}}\right)^{3}I(H)\right\}+o(1). \end{array} $$

Now,

$$ \begin{array}{@{}rcl@{}} {n^{*}}^{3}E\left\{\left( \overline{Y}_{{T}_{l}}\right)^{3}I(H)\right\} &\leq& 4{n^{*}}^{3}E\left\{\mid \overline{Y}_{{T}_{l}}-1 \mid^{3}I(H)\right\} + 4{n^{*}}^{3}P(H) \\ &\leq& 4{n^{*}}^{3}\left[E\left\{\left( \overline{Y}_{{T}_{l}}-1\right)^{4}\right\}\right]^{3/4}\left\{P(H)\right\}^{1/4} + o(1) \\ &=& o(1), \end{array} $$

from Lemmas 1 and 2 and the Lemma 3 follows. □

Lemma 4

As $n^{*} \rightarrow \infty , E\left (\overline {Y}_{{T}_{k-2}}\right ) = n^{*} - Var\left (Y_{1}\right )/c_{k-2} + o(1).$

Proof Proof of Lemma 4

Let

$$ D_{k-3} = \left\{\begin{array}{ll} B_{k-2} \cup A_{k-3}\left( \epsilon\right) \cup C_{k-3}\left( \epsilon\right) &if k\geq4 \\ B_{k-2} \cup A_{k-3}\left( \epsilon\right) &if k=3 \end{array}\right. $$

with 𝜖 > 0. We shall now show that as $n^{*} \rightarrow \infty$

$$ \begin{array}{@{}rcl@{}} n^{*}E\left\{\overline{Y}_{{T}_{k-2}}I\left( D_{k-3}\right)\right\} &=& o(1) \\ n^{*}E\left\{\overline{Y}_{{T}_{k-2}}I\left( {D^{c}}_{k-3}\right)\right\} &=& n^{*} - Var\left( Y_{1}\right)/c_{k-2} + o(1), \end{array} $$

(44)

from which the lemma follows. First we have

$$ \begin{array}{@{}rcl@{}} n^{*}E\left\{\overline{Y}_{{T}_{k-2}}I\left( D_{k-3}\right)\right\} &=& n^{*}E\left\{I\left( D_{k-3}\right)E\left( \overline{Y}_{{T}_{k-2}} \mid Y_{1},...,Y_{{T}_{k-3}}\right)\right\} \\ &= &n^{*}E\left[I\left( D_{k-3}\right)\left\{\frac{T_{k-3}}{T_{k-2}}\left( \overline{Y}_{{T}_{k-3}}-1\right)+1\right\}\right] \\ &\leq &n^{*}E\left\{I\left( D_{k-3}\right)\mid \overline{Y}_{{T}_{k-3}}-1 \mid\right\} + n^{*}P\left( D_{k-3}\right) \\ &\leq& n^{*}\left\{P\left( D_{k-3}\right)E\left( \surd{T_{k-3}}\mid \overline{Y}_{{T}_{k-3}}-1 \mid\right)^{2}\right\}^{1/2} + n^{*}P\left( D_{k-3}\right), \end{array} $$

which is o(1) since ${n^{*}}^{2}P\left (D_{k-3}\right )=o(1)$ by Lemma 1 and $E\left (\surd {T_{k-3}}\mid \overline {Y}_{{T}_{k-3}}-1 \mid \right )^{2}$ is bounded by Lemma 2. To prove the result given in equation (44) we note that

$$ \begin{array}{@{}rcl@{}} &&c_{k-2}n^{*}E\left\{\overline{Y}_{{T}_{k-2}}I\left( {D^{c}}_{k-3}\right)\right\}\\ &=& c_{k-2}n^{*}E\left\{I\left( D_{k-3}^{c}\right)E\left( \overline{Y}_{{T}_{k-2}} \mid Y_{1},...,Y_{{T}_{k-3}}\right)\right\} \\ &=& E\left\{I\left( D_{k-3}^{c}\right)c_{k-2}n^{*}\frac{T_{k-3}}{T_{k-2}}\left( \overline{Y}_{{T}_{k-3}}-1\right)\right\} + c_{k-2}n^{*}P\left( D_{k-3}^{c}\right) \\ &=& E\left\{I\left( D_{k-3}^{c}\right)c_{k-2}n^{*}\frac{T_{k-3}}{T_{k-2}}\left( \overline{Y}_{{T}_{k-3}}-1\right)\right\} + c_{k-2}n^{*} + o(1) \\ &=& E\left\{I\left( D_{k-3}^{c}\right)\left( 1-\left( \overline{Y}_{{T}_{k-3}}-1\right)+R_{k-3}\right)T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)\right\} \\ &+& c_{k-2}n^{*} + o(1), \end{array} $$

(45)

where $R_{k-3} = \frac {c_{k-2}n^{*}}{T_{k-2}} - 1 + \left (\overline {Y}_{{T}_{k-3}}-1\right )$. Now on $D_{k-3}^{c}$, we have $R_{k-3} = \frac {c_{k-2}n^{*}}{c_{k-2}n^{*}\overline {Y}_{{T}_{k-3}}} - 1 + \left (\overline {Y}_{{T}_{k-3}}-1\right )$ and it is straight forward that $\mid R_{k-3} \mid \leq C_{0}\left \{\left (\overline {Y}_{{T}_{k-3}}-1\right )^{2}+\left (c_{k-2}n^{*}\right )^{-1}\right \}$ for some constant C₀. Consequently,

$\mid E\left \{I\left (D_{k-3}^{c}\right )R_{k-3}T_{k-3}\left (\overline {Y}_{{T}_{k-3}}-1\right )\right \} \mid$

$$ \begin{array}{@{}rcl@{}} &\leq& C_{0}E\left[I\left( D_{k-3}^{c}\right)\left\{\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}+\left( c_{k-2}n^{*}\right)^{-1}\right\} T_{k-3} \mid \overline{Y}_{{T}_{k-3}} \mid \right] \\ &\leq& C_{0}\epsilon E\left[I\left( D_{k-3}^{c}\right)\left\{\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}+\left( c_{k-2}n^{*}\right)^{-1}\right\} T_{k-3}\right] \\ &\leq& C_{0}\epsilon \left[E\left\{T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}\right\}+\left( 1+\epsilon\right)\frac{c_{k-3}}{c_{k-2}}\right] \\ &\rightarrow& 0 \text{as} \epsilon \rightarrow 0, \end{array} $$

(46)

since $E\left \{T_{k-3}\left (\overline {Y}_{{T}_{k-3}}-1\right )^{2}\right \}$ is bounded by Lemma 2. We also have

$\mid E\left \{I\left (D_{k-3}^{c}\right )T_{k-3}\left (\overline {Y}_{{T}_{k-3}}-1\right )\right \} \mid$

$$ \begin{array}{@{}rcl@{}} &=& \mid E\left\{T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)\right\} - E\left\{I\left( D_{k-3}\right)T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)\right\} \mid \\ &=& \mid E\left\{I\left( D_{k-3}\right)\surd{T_{k-3}}\surd{T_{k-3}}\left( \overline{Y}_{{T}_{k-3}}-1\right)\right\} \mid \\ &\leq& \left[E\left\{I\left( D_{k-3}\right)T_{k-3}\right\}E\left\{T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}\right\}\right]^{1/2} \\ &=& o(1), \end{array} $$

(47)

by Lemmas 2 and 3. Finally, we show that

$$ E\left\{I\left( D_{k-3}^{c}\right)T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}\right\} = Var\left( Y_{1}\right) + o(1), $$

(48)

and result given in equation (44) follows from equations (45)-(48). Now it follows from Lemma 2 that

$E\left \{I\left (D_{k-3}^{c}\right )T_{k-3}\left (\overline {Y}_{{T}_{k-3}}-1\right )^{2}\right \}$

$$ \begin{array}{@{}rcl@{}} &=& E\left\{T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}\right\} - E\left\{I\left( D_{{T}_{k-3}}\right)T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}\right\} \\ &=& E\left\{T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}\right\} + o(1) \\ &=& Var\left( Y_{1}\right) + E\left[\left\{\left( S_{{T}_{k-4}}-T_{k-4}\right)^{2}-T_{k-4}Var\left( Y_{1}\right)\right\}/T_{k-3}\right] \\ &+& o(1), \end{array} $$

and so equation (48) is obviously true when k = 3. Now we show that when k ≥ 4

$E\left [\left \{\left (S_{{T}_{k-4}}-T_{k-4}\right )^{2}-T_{k-4}Var\left (Y_{1}\right )\right \}/T_{k-3}\right ] = o(1)$

Let

$$ H = \left\{\begin{array}{ll} C_{k-3}\left( \epsilon\right) &if k=4 \\ C_{k-3}\left( \epsilon\right) \cup C_{k-4}\left( \epsilon\right) &if k\geq5 \end{array}\right. $$

with 𝜖 > 0, then it follows from Lemmas 1 and 2 that

$E\left [I(H)\left \{\left (S_{{T}_{k-4}}-T_{k-4}\right )^{2}-T_{k-4}Var\left (Y_{1}\right )\right \}/T_{k-3}\right ]$

$$ \begin{array}{@{}rcl@{}} &\leq& T_{0}E\left[I(H)\left\{T_{k-4}\left( \overline{Y}_{{T}_{k-4}}-1\right)^{2}+Var\left( Y_{1}\right)\right\}\right] \\ &\leq& \left[E\left\{\left( T_{k-4}\left( \overline{Y}_{{T}_{k-4}}-1\right)^{2}\right)^{2}\right\}{T_{0}^{2}}P(H)\right]^{1/2} + Var\left( Y_{1}\right)T_{0}P(H) \\ &=& o(1). \end{array} $$

(49)

Also, for all sufficiently large n^∗, $E\left [I(H^{c})\left \{\left (S_{{T}_{k-4}}-T_{k-4}\right )^{2}-T_{k-4}Var\left (Y_{1}\right )\right \}/T_{k-3}\right ]$ is

$$ \begin{array}{@{}rcl@{}} &\leq& E\left[I(H^{c})\left( S_{{T}_{k-4}}-T_{k-4}\right)^{2}/\left\{\left( 1-\epsilon\right)c_{k-3}n^{*}\right\}\right]\\ &-& E\left[I(H^{c})T_{k-4}Var\left( Y_{1}\right)/\left\{\left( 1+\epsilon\right)c_{k-3}n^{*}\right\}\right]\\ &+&E\left[I(H^{c})T_{k-4}Var\left( Y_{1}\right)2\epsilon/\left\{\left( 1-\epsilon\right)\left( 1+\epsilon\right)c_{k-3}n^{*}\right\}\right]\\ \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=& E\left[I(H)\left( \left( S_{{T}_{k-4}}-T_{k-4}\right)^{2}-T_{k-4}Var\left( Y_{1}\right)\right)/\left\{\left( 1-\epsilon\right)c_{k-3}n^{*}\right\}\right] \\ &+&E\left[I(H^{c})T_{k-4}Var\left( Y_{1}\right)2\epsilon/\left\{\left( 1-\epsilon\right)\left( 1+\epsilon\right)c_{k-3}n^{*}\right\}\right] \\ &\leq& E\left[I(H)\left( \left( S_{{T}_{k-4}}-T_{k-4}\right)^{2}-T_{k-4}Var\left( Y_{1}\right)\right) /\left\{\left( 1-\epsilon\right)c_{k-3}n^{*}\right\}\right] \\ &+&c_{k-4}Var\left( Y_{1}\right)2\epsilon/\left\{\left( 1-\epsilon\right)\left( 1+\epsilon\right)c_{k-3}\right\} \\ &=& c_{k-4}Var\left( Y_{1}\right)2\epsilon/\left\{\left( 1-\epsilon\right)\left( 1+\epsilon\right)c_{k-3}\right\}+o(1) \\ &\rightarrow& 0 \text{as} \epsilon \rightarrow 0, \end{array} $$

(50)

where the second equality follows from Lemma 2 and the last equality follows from the Cauchy-Schwartz inequality and Lemmas 2 and 3. From a similar argument we establish that

$E\left [I(H^{c})\left \{\left (S_{{T}_{k-4}}-T_{k-4}\right )^{2}-T_{k-4}Var\left (Y_{1}\right )\right \}/T_{k-3}\right ]$

$$ \begin{array}{@{}rcl@{}} &\geq& o(1) - \frac{c_{k-4} Var\left( Y_{1}\right)2\epsilon}{\left\{\left( 1-\epsilon\right)\left( 1+\epsilon\right)c_{k-3}\right\}} \\ &\rightarrow& 0 \text{as} \epsilon \rightarrow 0. \end{array} $$

(14)

Now equation (48) follows clearly from equations (49)-(51) and thus the proof is completed. □

Lemma 5

As $n^{*} \rightarrow \infty , E\left (T_{k-1}\right ) = E\left (L_{k-1}\right ) + o(1)$.The Lemma 5 can be proved easily by using Lemma 3.

Lemma 6

As $n^{*} \rightarrow \infty , U_{n^{*}} \equiv n^{*}\overline {Y}_{{L}_{k-2}} + \eta - \left [n^{*}\overline {Y}_{{L}_{k-2}} + \eta \right ]$is asymptotically uniform on (0, 1).

Proof Proof of Lemma 6

Let $J \equiv \left [c_{k-2}n^{*}\overline {Y}_{{T}_{k-3}}\right ], V \equiv c_{k-2}n^{*}\overline {Y}_{{T}_{k-3}}-J$. An argument similar to Hall (1981, p. 1237) shows that for 0 < x < 1,J > N_k− 3 and V ∈ (0, 1),

$P\left \{U_{n^{*}} \leq x \mid J,V,T_{k-3}\right \} = x + r_{4n^{*}}$,

where $\mid r_{4n^{*}} \mid \leq C_{0}\left (J^{1/2}/n^{*} + J/{n^{*}}^{2}\right )$ as $n^{*} \rightarrow \infty$ uniformly in V ∈ (0, 1) and $J>\left (1+\epsilon \right )T_{k-3}$ for 𝜖 > 0 and some constant C₀. Consequently

$$ \begin{array}{@{}rcl@{}} P\left\{U_{n^{*}} \leq x\right\} &=& E\left\{P\left( U_{n^{*}} \leq J,V,T_{k-3}\right)\right\} \\ &=& E\left\{P\left( U_{n^{*}} \leq x \mid J,V,T_{k-3}\right)I\left( J>\left( 1+\epsilon\right)T_{k-3}\right)\right\} \\ &+& E\left\{P\left( U_{n^{*}} \leq x \mid J,V,T_{k-3}\right)I\left( J \leq \left( 1+\epsilon\right)T_{k-3}\right)\right\} \\ &=& E\left\{\left( x+r_{4n^{*}}\right)I\left( J>\left( 1+\epsilon\right)T_{k-3}\right)\right\} \\ &+& E\left\{P\left( U_{n^{*}} \leq x \mid J,V,T_{k-3}\right)I\left( J \leq \left( 1+\epsilon\right)T_{k-3}\right)\right\} \\ &=& x + r_{5n^{*}}, \end{array} $$

where $\mid r_{5n^{*}} \mid \leq P\left \{J \leq \left (1+\epsilon \right )T_{k-3}\right \}+C_{0}E\left (J^{1/2}/n^{*} + J/{n^{*}}^{2}\right )+P\left \{J > \left (1+\epsilon \right )T_{k-3}\right \}$.It remains to show that as $n^{*} \rightarrow \infty$ and for small 𝜖 > 0

$P\left \{J \leq \left (1+\epsilon \right )T_{k-3}\right \} = o(1)$ and $E\left (J/{n^{*}}^{2}\right ) = o(1).$

Now, $E\left (J/{n^{*}}^{2}\right ) \leq E\left (c_{k-2}n^{*}\overline {Y}_{{T}_{k-3}}/{n^{*}}^{2}\right ) \leq c_{k-2}{n^{*}}^{-1}\left (E \mid \overline {Y}_{{T}_{k-3}}-1 \mid +1\right ) = o(1)$ by Lemma 2 and for all sufficiently large n^∗, small 𝜖 > 0 and k ≥ 4,

$$ \begin{array}{@{}rcl@{}} P\left\{J \leq \left( 1+\epsilon\right)T_{k-3}\right\} &\leq& P\left\{c_{k-2}n^{*}\overline{Y}_{{T}_{k-3}} \leq \left( 1+\epsilon\right)T_{k-3}\right\} \\ &\leq& P\left\{c_{k-2}n^{*}\overline{Y}_{{T}_{k-3}} \leq c_{k-2}n^{*}\left( 1-\epsilon\right)\right\} \\ &+& P\left\{c_{k-2}n^{*}\left( 1-\epsilon\right) \leq \left( 1+\epsilon\right)T_{k-3}+1\right\}\\ &\leq& P\left\{\overline{Y}_{{T}_{k-3}}-1 \leq -\epsilon\right\} \\ &+& P\left\{T_{k-3}-c_{k-3}n^{*} \geq \epsilon c_{k-3}n^{*}\right\}\left( c_{k-2}>c_{k-3}\right) \\ &\leq& P\left\{A_{k-3}\left( \epsilon\right)\right\} + P\left\{c_{k-3}\left( \epsilon\right)\right\}, \end{array} $$

(52)

which is o(1) by Lemma 1. When k = 3, it follows directly from equation (52) that$P\left \{J \leq \left (1+\epsilon \right )T_{k-3}\right \} = o(1)$ by noting that $\lim \sup T_{0}/{c_{1}n^{*}} < 1$. The proof is thus completed. We can now prove the result given in equation (41). It follows from Lemmas 4-6 that

$$ \begin{array}{@{}rcl@{}} E\left( T_{k-1}\right) &=& E\left( L_{k-1}\right) + o(1) \\ &=& E\left( \left[n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right]\right) + o(1)\\ &=& E\left( \left[n^{*}\overline{Y}_{{L}_{k-2}}+\eta\right]-\left( n^{*}\overline{Y}_{{L}_{k-2}}+\eta\right)\right)\\ &+& E\left( n^{*}\overline{Y}_{{L}_{k-2}}+\eta-n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right)\\ &+& E\left( n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right) \\ &+& E\left( \left[n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right]-\left[n^{*}\overline{Y}_{{L}_{k-2}}+\eta\right]\right) + o(1) \\ & =& -\frac{1}{2} + E\left( n^{*}\overline{Y}_{{L}_{k-2}}-n^{*}\overline{Y}_{{T}_{k-2}}\right) + n^{*} - \frac{Var\left( Y_{1}\right)}{c_{k-2}} + \eta \\ &+& E\left( \left[n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right]-\left[n^{*}\overline{Y}_{{L}_{k-2}}+\eta\right]\right) + o(1), \end{array} $$

and so equation (41) will follow if we show that

$$ \begin{array}{@{}rcl@{}} E\left( n^{*}\overline{Y}_{{L}_{k-2}}-n^{*}\overline{Y}_{{T}_{k-2}}\right) &=& o(1), \\ E\left( \left[n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right]-\left[n^{*}\overline{Y}_{{L}_{k-2}}+\eta\right]\right) &=& o(1). \end{array} $$

Let B_k− 2 be the set defined in Lemma 1. Then

$$ \begin{array}{@{}rcl@{}} \Bigl| E\left( n^{*}\overline{Y}_{{L}_{k-2}}-n^{*}\overline{Y}_{{T}_{k-2}}\right) \Bigr| &=& \Bigl| {\int}_{B_{k-2}} \left( n^{*}\overline{Y}_{{L}_{k-2}}-n^{*}\overline{Y}_{{T}_{k-2}}\right) dP \Bigr| \\ &\leq& {\int}_{B_{k-2}} n^{*}\overline{Y}_{{L}_{k-2}} dP + {\int}_{B_{k-2}} n^{*}\overline{Y}_{{T}_{k-2}}dP\\ &\leq& {\int}_{B_{k-2}} n^{*}S_{{T}_{k-2}}dP + {\int}_{B_{k-2}} n^{*}\overline{Y}_{{T}_{k-2}}dP \end{array} $$

which is o(1) by the Cauchy-Schwartz inequality and Lemmas 2 and 3 as before. Hence equation (41) is proved.

The result given in equation (11) follows directly from equation (41) since M_k− 1 = T_k− 1 + 1. Now we prove the results given in equations (42) and (43) and we need some more Lemmas for this. □

Lemma 7

As $n^{*} \rightarrow \infty$, $Var\left (T_{k-1}\right ) = Var\left (L_{k-1}\right ) + o(1)$.Note that we can prove this lemma by using Lemma 3.

Lemma 8

A simple conditional argument by conditioning on $Y_{1},...,Y_{{T}_{k-3}}$ gives

$$ \begin{array}{@{}rcl@{}} Var\left( \overline{Y}_{{T}_{k-2}}\right) &=& Var\left\{\frac{T_{k-3}}{T_{k-2}}\left( \overline{Y}_{{T}_{k-3}}-1\right)\right\} - Var\left( Y_{1}\right)E\left( \frac{T_{k-3}}{T_{k-2}^{2}}\right) \\ &+& Var\left( Y_{1}\right)E\left( \frac{1}{T_{k-2}}\right). \end{array} $$

Lemma 9

The following are true as $n^{*} \rightarrow \infty$

$$ E\left( T_{k-2}^{-1}\right) = \left( c_{k-2}n^{*}\right)^{-1}\left( 1+o(1)\right), $$

(16)

$$ E\left( T_{k-3}\right)/n^{*} \leq C_{0}, $$

(17)

$$ E\left( T_{k-2}^{-1}{\sum}_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right) = o\left( {n^{*}}^{-1/2}\right), $$

(18)

$$ E\left( \mid \overline{Y}_{{T}_{k-2}}-1 \mid^{3}\right) = o\left( {n^{*}}^{-1}\right). $$

(19)

Proof Proof of Lemma 9

We first prove the result given in equation (16). Let $c_{k-2}\left (\epsilon \right )$ be the set defined in Lemma 1 with 𝜖 > 0. Then it follows from Lemma 1 that

$$ \begin{array}{@{}rcl@{}} \Biggl| E\left( \frac{c_{k-2}n^{*}}{T_{k-2}}\right)-1 \Biggr| &\leq& {\int}_{{c_{k-2}}\left( \epsilon\right)} \Biggl| \frac{c_{k-2}n^{*}}{T_{k-2}}-1 \Biggr| dp + {\int}_{{c_{k-2}}\left( \epsilon\right)} \frac{c_{k-2}n^{*}}{T_{k-2}} dp + {\int}_{{c_{k-2}}\left( \epsilon\right)} 1 dp \\ &\leq& C_{0}\epsilon + c_{k-2}n^{*}P\left( c_{k-2}\left( \epsilon\right)\right) + P\left( c_{k-2}\left( \epsilon\right)\right) \rightarrow 0, \end{array} $$

as $n^{*} \rightarrow \infty$ and then $\epsilon \rightarrow 0$. This proves equation (56). The assertion (54) is true when k = 3 and can be proved in a similar way as (53) by using Lemma 3 when k ≥ 4. Assertion (55) follows from Lemmas 1-3 by noting that

$$ \begin{array}{@{}rcl@{}} \left| E\left\{\frac{\surd{n^{*}}}{T_{k-2}}{\sum}_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right\} \right| &=& \left| E\left\{\left( \frac{\surd{n^{*}}}{T_{k-2}}-\frac{1}{c_{k-2}\surd{n^{*}}}\right){\sum}_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right\} \right| \\ &=& \left| E\left[\left\{I\left( c_{k-2}^{c}\left( \epsilon\right)\right)+I\left( c_{k-2}\left( \epsilon\right)\right)\right\}\left( \frac{\surd{n^{*}}}{T_{k-2}}-\frac{1}{c_{k-2}\surd{n^{*}}}\right) \right.\right. \\ &\times&\left.\left. {\sum}_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right] \right| \\ &\leq& C_{0}\epsilon E\left( \surd{T_{k-3}} \mid \overline{Y}_{{T}_{k-3}}-1 \mid\right) \\ &+& C_{0}\surd{n^{*}}E\left\{I\left( c_{k-2}\left( \epsilon\right)\right)\left( \surd{T_{k-3}} \mid \overline{Y}_{{T}_{k-3}}-1 \mid\right)\right\} \\ &+& C_{0}/\surd{n^{*}}E\left\{\surd{T_{k-3}}\surd{T_{k-3}} \mid \overline{Y}_{{T}_{k-3}}-1 \mid I\left( c_{k-2}\left( \epsilon\right)\right)\right\} \\ &\leq& C_{0}\epsilon E\left( \surd{T_{k-3}} \mid \overline{Y}_{{T}_{k-3}}-1 \mid\right) \\ &+& C_{0}\surd{n^{*}}\left\{P\left( c_{k-2}\left( \epsilon\right)\right)\left( T_{k-3} \mid \overline{Y}_{{T}_{k-3}}-1 \mid^{2}\right)\right\}^{1/2} \\ &+& C_{0}/\surd{n^{*}}\left\{E\left( T_{k-3}I\left( c_{k-2}\left( \epsilon\right)\right)\right)E\left( T_{k-3} \mid \overline{Y}_{{T}_{k-3}}-1 \mid^{2}\right)\right\}^{1/2} \\ &=& C_{0}\epsilon E\left( \surd{T_{k-3}} \mid \overline{Y}_{{T}_{k-3}}-1 \mid\right) + o(1) \rightarrow 0 \text{as} \epsilon \rightarrow 0. \end{array} $$

To prove equation (56), note that

$$ \begin{array}{@{}rcl@{}} E\left( \mid \overline{Y}_{{T}_{k-2}}-1 \mid^{3}\right) &=& E\left\{\mid \overline{Y}_{{T}_{k-2}}-1 \mid^{3}I\left( c_{k-2}^{c}\left( \epsilon\right)\right)\right\} \\ &+& E\left\{\mid \overline{Y}_{{T}_{k-2}}-1 \mid^{3}I\left( c_{k-2}\left( \epsilon\right)\right)\right\} \\ &\leq& \frac{1}{\left\{\left( 1-\epsilon\right)c_{k-2}n^{*}\right\}^{3/2}}E\left\{\left( \surd{T_{k-2}} \mid \overline{Y}_{{T}_{k-2}}-1 \mid\right)^{3}\right\} \\ &+& \left\{E\left( \overline{Y}_{{T}_{k-2}}-1\right)^{4}\right\}^{3/4}\left\{P\left( c_{k-2}\left( \epsilon\right)\right)\right\}^{1/4} \\ &=& o(1). \end{array} $$

by Lemmas 1 and 2. This completes the proof of Lemma 9. □

Lemma 10

As $n^{*} \rightarrow \infty$

$$ \Bigl| Var\left\{\frac{1}{T_{k-2}}\sum\nolimits_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right\} - Var\left( Y_{1}\right)E\left( \frac{T_{k-3}}{T_{k-2}^{2}}\right) \Bigr| = o\left( {n^{*}}^{-1}\right) $$

Proof Proof of Lemma 10

From Lemma 9, LHS of the above equation can be written as

$$ \Biggl| E\left\{\frac{1}{T_{k-2}^{2}}\left( \sum\nolimits_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right)^{2}\right\} - \left\{E\left( \frac{1}{T_{k-2}}\sum\nolimits_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right)\right\}^{2} - Var\left( Y_{1}\right)E\left( \frac{T_{k-3}}{T_{k-2}^{2}}\right) \Biggr| $$

$$ = \Biggl| E\left\{\frac{1}{T_{k-2}^{2}}\left( \sum\nolimits_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right)^{2}\right\} - Var\left( Y_{1}\right)E\left( \frac{T_{k-3}}{T_{k-2}^{2}}\right) + o\left( {n^{*}}^{-1}\right) \Biggr|, $$

and so it remains to show that

$$E\left\{\frac{1}{T_{k-2}^{2}}\left( \sum\nolimits_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right)^{2}\right\} - Var\left( Y_{1}\right)E\left( \frac{T_{k-3}}{T_{k-2}^{2}}\right) = o\left( {n^{*}}^{-1}\right).$$

It follows from Lemmas 1-3 and Wald’s equation that

$E\left \{\frac {1}{T_{k-2}^{2}}\left ({\sum }_{i=1}^{T_{k-3}}\left (Y_{i}-1\right )\right )^{2}\right \} - Var\left (Y_{1}\right )E\left (\frac {T_{k-3}}{T_{k-2}^{2}}\right )$

$$ \begin{array}{@{}rcl@{}} &\leq& E\left\{T_{k-3}\left( \overline{Y}_{{T}_{k-3}}-1\right)^{2}I\left( c_{k-2}\left( \epsilon\right)\right)\right\} + \frac{E\left( {\sum}_{i=1}^{T_{k-3}}\left( Y_{i}-1\right)\right)^{2}}{\left\{\left( 1-\epsilon\right)c_{k-2}n^{*}\right\}^{2}} \\ &+& Var\left( Y_{1}\right)P\left( c_{k-2}\left( \epsilon\right)\right) - \frac{Var\left( Y_{1}\right)}{\left\{\left( 1+\epsilon\right)c_{k-2}n^{*}\right\}^{2}}E\left\{T_{k-3}I\left( c_{k-2}^{c}\left( \epsilon\right)\right)\right\} \\ &=& \frac{Var\left( Y_{1}\right)}{\left\{\left( 1-\epsilon\right)c_{k-2}n^{*}\right\}^{2}}E\left( T_{k-3}\right) - \frac{Var\left( Y_{1}\right)}{\left\{\left( 1+\epsilon\right)c_{k-2}n^{*}\right\}^{2}}E\left( T_{k-3}\right) + o\left( n{^{*}}^{-1}\right) \\ &=& \frac{4\epsilon Var\left( Y_{1}\right)}{\left\{\left( 1-\epsilon\right)\left( 1+\epsilon\right)c_{k-2}\right\}^{2}}\frac{E\left( T_{k-3}\right)}{{n^{*}}^{2}} + o\left( {n^{*}}^{-1}\right), \end{array} $$

which is $o\left ({n^{*}}^{-1}\right )$ since 𝜖 can be arbitrary small and $E\left (T_{k-3}\right )/n^{*} \leq C_{0}$ by Lemma 9. A similar argument shows that

$E\left \{\frac {1}{T_{k-2}^{2}}\left ({\sum }_{i=1}^{T_{k-3}}\left (Y_{i}-1\right )\right )^{2}\right \} - Var\left (Y_{1}\right )E\left (\frac {T_{k-3}}{T_{k-2}^{2}}\right )$

$$ \begin{array}{@{}rcl@{}} &\geq& -\frac{4\epsilon Var\left( Y_{1}\right)}{\left\{\left( 1-\epsilon\right)\left( 1+\epsilon\right)c_{k-2}\right\}^{2}}\frac{E\left( T_{k-3}\right)}{{n^{*}}^{2}} + o\left( {n^{*}}^{-1}\right) \\ &=& o\left( {n^{*}}^{-1}\right) \text{as} \epsilon \rightarrow 0. \end{array} $$

The proof of Lemma 10 is thus completed. Now we prove the result given in equation (42). From equation (47), it suffices to show that

$Var\left (L_{k-1}\right ) = n^{*}Var\left (Y_{1}\right )/c_{k-2} + o\left (n^{*}\right ).$

From Lemmas 8-10, we have

$$ \begin{array}{@{}rcl@{}} {n^{*}}^{-2}Var\left( n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right) &=& Var\left( \overline{Y}_{{T}_{k-2}}\right) \\ &=& Var\left\{\frac{T_{k-3}}{T_{k-2}}\left( \overline{Y}_{{T}_{k-3}}-1\right)\right\} - Var\left( Y_{1}\right)E\left( \frac{T_{k-3}}{T_{k-2}^{2}}\right) \\ &+& Var\left( Y_{1}\right)E\left( \frac{1}{T_{k-2}}\right) \\ &=& o\left( {n^{*}}^{-1}\right) + Var\left( Y_{1}\right)\frac{1}{c_{k-2}n^{*}}\left( 1+o(1)\right) \\ &=& Var\left( Y_{1}\right)\frac{1}{c_{k-2}n^{*}} + o\left( {n^{*}}^{-1}\right), \end{array} $$

and so it remains to show that

$Var\left (L_{k-1}\right ) - Var\left (n^{*}\overline {Y}_{{T}_{k-2}}+\eta \right ) = o\left (n^{*}\right ),$

This follows directly from the following three facts:

$$ \begin{array}{@{}rcl@{}} Var\left( L_{k-1}\right) &=& Var\left\{\left( n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right) + \left( L_{k-1}-n^{*}\overline{Y}_{{T}_{k-2}}-\eta\right)\right\} \\ &=& Var\left( n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right) + Var\left( L_{k-1}-n^{*}\overline{Y}_{{T}_{k-2}}-\eta\right) \\ &+& Cov\left( n^{*}\overline{Y}_{{T}_{k-2}}, L_{k-1}-n^{*}\overline{Y}_{{T}_{k-2}}\right), \end{array} $$

$Var\left (L_{k-1}-n^{*}\overline {Y}_{{T}_{k-2}}-\eta \right ) \leq 1$ since $\mid L_{k-1}-n^{*}\overline {Y}_{{T}_{k-2}}-\eta \mid \leq 1,$and

$$ \begin{array}{@{}rcl@{}} Cov\left( n^{*}\overline{Y}_{{T}_{k-2}}, L_{k-1}-n^{*}\overline{Y}_{{T}_{k-2}}\right) &\leq& \left\{Var\left( n^{*}\overline{Y}_{{T}_{k-2}}\right)Var\left( L_{k-1}-n^{*}\overline{Y}_{{T}_{k-2}}-\eta\right)\right\}^{1/2} \\ &\leq& \left\{Var\left( n^{*}\overline{Y}_{{T}_{k-2}}\right)\right\}^{1/2} \\ &=& \left\{n^{*}Var\left( Y_{1}\right)/c_{k-2} + o\left( n^{*}\right)\right\}^{1/2} \\ &=& o\left( n^{*}\right). \end{array} $$

The proof of Eq. 42 is thus completed. To prove the result given in Eq. 43, note that

$E\mid T_{k-1}-E\left (T_{k-1}\right ) \mid ^{3} = E\mid T_{k-1}-n^{*}+Var\left (Y_{1}\right )/c_{k-2}+1/2-\eta +o(1)\mid ^{3},$

and so it suffices to show that $E\mid T_{k-1}-n^{*} \mid ^{3} = o\left ({n^{*}}^{2}\right )$. Let B_k− 1 be the set defined in Lemma 1, then

${\int \limits }_{B_{k-1}}\mid T_{k-1}-n^{*} \mid ^{3}dP \leq {\int \limits }_{B_{k-1}}\left (T_{k-1}+n^{*}\right )^{3}dP = o\left ({n^{*}}^{2}\right )$

by Lemmas 1 and 3 and

$$ \begin{array}{@{}rcl@{}} {\int}_{B_{k-1}}\mid T_{k-1}-n^{*} \mid^{3}dP &=& {\int}_{B_{k-1}}\mid \left[n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right]-n^{*} \mid^{3}dP \\ &\leq& E\mid \left[n^{*}\overline{Y}_{{T}_{k-2}}+\eta\right]-n^{*} \mid^{3} \\ &\leq& E\left\{\left( \mid n^{*}\overline{Y}_{{T}_{k-2}}-n^{*} \mid+\eta+1\right)^{3}\right\} \\ &=& o\left( {n^{*}}^{2}\right). \end{array} $$

Since $E\left \{\left ({n^{*}}\mid \overline {Y}_{{T}_{k-2}}-1 \mid \right )^{3}\right \} = o\left ({n^{*}}^{2}\right )$ by Lemma 9, So $E\mid T_{k-1}-n^{*} \mid ^{3} = o\left ({n^{*}}^{2}\right )$ and the proof is completed.Now to prove result (12) of Theorem 2, we proceed as follows:

$$ \begin{array}{@{}rcl@{}} E\left\{L(\mu, \overline{X}_{M_{k-1}})\right\} &=& 1 - E\left[P\left\{{\chi^{2}_{1}} \leq M_{k-1}\lambda d^{2}\right\}\right] \\ &=& 1 - E\left[{\Psi}\left\{M_{k-1}\lambda d^{2}\right\}\right] \end{array} $$

(57)

where Ψ(.) is the cdf of a ${\chi ^{2}_{1}}$ variate. We can expand this equation by Taylor Series expansion [see Hall (1981), page 1231] as follows:

$$ \begin{array}{@{}rcl@{}} E\left\{L(\mu, \overline{X}_{{M}_{k-1}})\right\} &=& {\Psi}\left( \lambda d^{2}E(M_{k-1})\right)\\ &&+ \frac{\lambda^{2}d^{4}}{2} E\left\{M_{k-1} - E(M_{k-1})\right\}^{2}{\Psi}^{\prime\prime}\left( \lambda d^{2}E(M_{k-1})\right)\\ &&+ o(d^{2}) \end{array} $$

(58)

where,

$$ \begin{array}{@{}rcl@{}} {\Psi}(x) &=& \frac{1}{\sqrt 2\pi}{{\int}_{0}^{x}} e^{-y/2} y^{-1/2} dy \\ {\Psi}^{\prime}(x) &=& \frac{1}{2\sqrt 2\pi} e^{-x/2} x^{-2} = \xi(x) \\ {\Psi}^{\prime\prime}(x) &=& \frac{1}{2} \xi(x)\left\{-2x^{-2} - 1\right\} \end{array} $$

Now since we have,

${\Psi }\left (\lambda d^{2}E(M_{k-1})\right ) = \alpha + \left (\lambda d^{2}E(M_{k-1}) - z^{2}\right ){\Psi }^{\prime }(z^{2}) + o(d^{2})$,

$\lambda d^{2}E(M_{k-1}) - z^{2} = \frac {n^{*}d^{4}}{z^{2}}\left \{n^{*}-\frac {2}{c_{k-2}}+\frac {1}{2}+\eta \right \} - z^{2}$ and

$\frac {\lambda ^{2}d^{4}}{2}E\left \{M_{k-1}-E\left (M_{k-1}\right )\right \}^{2} = \frac {\lambda ^{3}d^{2}z^{2}}{2c_{k-2}} + o(d^{2})$Using these equations and Eq. 58, we can obtain result (12) of Theorem 2. □

Proof Proof of Theorem 3

N₂ from (19) can be written as,

$$ N_{2} = inf\left\{n\geq m\geq 2 ; {\sum}_{j=1}^{n-1} Z_{j} \leq \frac{n^{\frac{2t+s}{2p+s}+1}}{n_{0}^{\frac{2t+s}{2p+s}}}\right\} $$

where n₀ comes from (18) and ${\sum }_{j=1}^{n-1} Z_{j} = \frac {n\lambda }{\widehat {\lambda }_{n}} \sim \chi ^{2}_{n-1}$ and $Z_{j} \sim {\chi ^{2}_{1}}$. N₂ can also be written as J + 1 w.p. 1 where,

$$ J = \inf\left\{n \geq m-1; {\sum}_{j=1}^{n} Z_{j} \leq \frac{n^{\frac{2t+s}{2p+s}+1}}{{n_{0}}^{\frac{2t+s}{2p+s}}}\left( 1+\frac{1}{n}\right)^{\frac{2t+s}{2p+s}}\right\} $$

(59)

Now Comparing (59) with equation (1.1) of Woodroofe (1977), we get,

α = (2t + s)/(2p + s) + 1, β = 2p + s/2t + s, $c={n_{0}}^{-(2t+s/2p+s)}$, μ = 1, λ = n₀, $L(n)=1+\frac {2t+s}{(2p+s)n}+o\left (\frac {1}{n}\right )$, L₀ = 2t + s/2p + s, τ² = 2, a = 1/2.Result (20) now follows from his Theorem 2.4 for m > (4p + 2t)/(2t + s).Now we have,

$$ R_{N_{2}}(c) = \frac{2ct\lambda^{p}{n_{0}^{t}}}{s} E\left\{\left( \frac{n_{0}}{N_{2}}\right)^{s/2}\right\} + c{n_{0}^{t}}\lambda^{p} E\left\{\left( \frac{N_{2}}{n_{0}}\right)^{t}\right\} $$

(60)

Let us solve $E\left \{\left (\frac {n_{0}}{N_{2}}\right )^{s/2}\right \}$. Consider $\frac {N_{2}}{n_{0}} = x$, so that $\frac {n_{0}}{N_{2}} = x^{-1}$. We can write,

$$ E\left\{\left( \frac{n_{0}}{N_{2}}\right)^{s/2}\right\} = E(x^{-s/2}) $$

Expanding f(x) around x = 1 by Taylor series, we obtain,

$$ E\left\{\left( \frac{n_{0}}{N_{2}}\right)^{s/2}\right\} = 1 - \frac{s}{2n_{0}} E(N_{2}-n_{0}) + \frac{s}{4}\left( \frac{s}{2}+1\right) E\left\{\frac{(N_{2}-n_{0})^{2}}{{n_{0}^{2}}}\right\} $$

From Theorem 3 of Ghosh and Mukhopadhyay (1979) and Theorem 2.3 of Woodroofe (1977), we define V, where

$$ V = \frac{(J-n_{0})^{2}}{n_{0}} \rightarrow \frac{(2p+s)^{2}}{4t+2s} {\chi_{1}^{2}} $$

(61)

for J defined in (59) and V is also uniformly integrable if m > 4p + 2s/2t + s and it suffices our job. Moreover, following Chaturvedi etal (2019), we can easily obtain

$E\left [\frac {(J-n_{0})^{2}}{J}I\left (J > \frac {n_{0}}{2}\right )\right ] = \frac {(2p+s)^{2}}{2(2t+s)} +o(1),$$E\left [\frac {(J-n_{0})^{2}}{J}I\left (J < \frac {n_{0}}{2}\right )\right ] = o(1)$Now for m > 2p + s, by Lemma 2.3 of Woodroofe (1977), we can obtain

$$ E\left\{\left( \frac{n_{0}}{N_{2}}\right)^{s/2}\right\} = 1 - \frac{s}{2n_{0}}\!\left[\frac{2p+s}{2t+s}\nu - \frac{2p+s}{2t+s}\left( \!1+\frac{2p+s}{2t+s}\!\right)\!\right] + \frac{(2p+s)^{2}}{2n_{0}(2t+s)}\frac{s}{4}\left( \frac{s+2}{2}\right) $$

(62)

Similarly, we can compute

$$ E\left\{\left( \frac{N_{2}}{n_{0}}\right)^{t}\right\} = 1 + \frac{t}{n_{0}}\left[\frac{2p+s}{2t+s}\nu - \frac{2p+s}{2t+s}\left( 1+\frac{2p+s}{2t+s}\right)\right] + \frac{(2p+s)^{2}}{2n_{0}(2t+s)}\frac{t(t-1)}{2} $$

(63)

Result (21) of Theorem 3 can now be obtained by putting (62) and (63) in (60). □

Proof Proof of Theorem 4

Result (24) of Theorem 4 can be obtained by using Lemmas 1-6 of Theorem 2 discussed above with some obvious changes in notations. Also proof of result (25) of Theorem 4 is similar to the proof of result (21) of Theorem 3 except the fact that in Theorem 4 we are working on a k-stage procedure (Subsection 3.2) instead of a purely sequential procedure (Subsection 3.1). Hence we omit the proof. □

Proof Proof of Theorem 5

Proof of this theorem can be obtained along the lines of Theorem 3 with obvious changes in notations. We can do this by replacing the notations and preliminaries of Section 3 (Subsection 3.1) by Section 4 (Subsection 4.1). Hence we omit the proof. □

Proof Proof of Theorem 6

Proof of this theorem will be exactly similar to the proof of Theorem 4 with some obvious changes in notations and preliminaries. Hence we omit the proof. □