On purely sequential estimation of an inverse Gaussian mean

Abstract

The first part of this paper deals with developing a purely sequential methodology for the point estimation of the mean $\mu$ of an inverse Gaussian distribution having an unknown scale parameter $\lambda$. We assume a weighted squared error loss function and aim at controlling the associated risk function per unit cost by bounding it from above by a known constant $\omega$. We also establish first-order and second-order asymptotic properties of our stopping rule. The second part of this paper deals with obtaining a purely sequential fixed accuracy confidence interval for the unknown mean $\mu$, assuming that the scale parameter $\lambda$ is known. First-order asymptotic efficiency and asymptotic consistency properties are also built of our proposed procedures. We then provide extensive sets of simulation studies and real data analysis using data from fatigue life analysis to show encouraging performances of our proposed stopping strategies.

Appendix

Proof of Theorem 2.1

Part (i)

From (9) we have,

$$\begin{aligned} \frac{1}{\sqrt{c\omega }}\left( \frac{1}{\widehat{\lambda }_{N}}\right) ^{(k+1)/2}\le N\le \frac{1}{\sqrt{c\omega }}\left( \frac{1}{\widehat{\lambda }_{N-1}}\right) ^{(k+1)/2}+m. \end{aligned}$$

(20)

On dividing throughout by $n^{*}$ we get,

$$\begin{aligned} \left( \frac{\lambda }{\widehat{\lambda }_{N}}\right) ^{(k+1)/2}\le \frac{N}{n^{*}}\le \left( \frac{\lambda }{\widehat{\lambda }_{N-1}}\right) ^{(k+1)/2}+\frac{m}{n^{*}}. \end{aligned}$$

(21)

The proof follows by taking limits throughout and noting that $N\rightarrow \infty$ w.p.1, $\widehat{\lambda }_{N}\rightarrow \lambda$ w.p.1 and $m/n^{*}\rightarrow 0$ as $\omega \rightarrow 0$.

Part (ii)

From the right hand side of (21) we have (for sufficiently large $n^{*}$),

$$\begin{aligned} \frac{N}{n^{*}}\le \left( \frac{\lambda }{\widehat{\lambda }_{N-1}}\right) ^{(k+1)/2}+m. \end{aligned}$$

(22)

Now denoting $\sup _{n\ge 2}$ $\frac{1}{\widehat{\lambda }_{n}}$ by W, we can claim w.p.1:

$$\begin{aligned} N/n^{*}\le \left( \frac{\lambda }{W}\right) ^{(k+1)/2}+m. \end{aligned}$$

(23)

The right hand side of (23) is free from $\omega$ and using Wiener’s (1939) ergodic theorem we can claim the uniform integrability of all positive powers of $N/n^{*}$. The proof is hence complete by using part (i).

Part (iii)

As seen before, N from (9) can be expressed as $J+1$, where J is as defined in (10). Using Lemma 2.3 from Woodroofe (1977) with $b=1/2$ we claim:

$$\begin{aligned} P\left( J<\frac{1}{2}n^{*}\right) =O\left( n^{* ^{-\frac{2}{k+1}(m-1)}}\right) . \end{aligned}$$

(24)

Now since $N=J+1$ w.p.1, we have $0<\left( \frac{n^{*}}{N}\right) ^{s}<\left( \frac{n^{*}}{J}\right) ^{s}$ and it suffices to show that $\left( \frac{n^{*}}{J}\right) ^{s}$ is uniformly integrable.

We can now split $\left( \frac{n^{*}}{J}\right) ^{s}$ into two parts as $\left( \frac{n^{*}}{J}\right) ^{s}I(J>\frac{1}{2}n^{*})$ and $\left( \frac{n^{*}}{J}\right) ^{s}I(J<\frac{1}{2}n^{*})$ and hope to show that both these parts are $1+o(1)$ accounting for restrictions on m. Now we have,

$$\begin{aligned} \left( \frac{n^*}{J}\right) ^s I\left( J>\frac{1}{2}n^*\right) <2^s, \end{aligned}$$

(25)

and so $\left( \frac{n^*}{J}\right) ^s I(J>\frac{1}{2}n^*)$ becomes uniformly integrable. However, $\left( \frac{n^*}{J}\right) ^s I(J>\frac{1}{2}n^*)\overset{P}{\rightarrow }$ 1 and hence on applying the dominated convergence theorem we must have,

$$\begin{aligned} E\left[ \left( \frac{n^*}{J}\right) ^s I\left( J>\frac{1}{2}n^*\right) \right] =1+o(1). \end{aligned}$$

(26)

Also from (31) we have,

$$\begin{aligned} E\left[ \left( \frac{n^*}{J}\right) ^s I\left( J<\frac{1}{2}n^*\right) \right] \le \left( \frac{n^*}{m-1}\right) ^sP\left( J<\frac{1}{2}n^*\right) =O\left( n^{* ^{-\frac{2}{k+1}(m-1)+s}}\right) , \end{aligned}$$

(27)

which is o(1) if $m>1+\frac{k+1}{2}s$. Hence combining (26) and (27) we have that $\left( \frac{n^*}{J}\right) ^s$ is uniformly integrable, which gives,

$$\begin{aligned} E\left[ (n^*/J)^s\right] =1+o(1), \end{aligned}$$

(28)

if $m>1+\frac{k+1}{2}s$ and $s>0$.

Part (iv)

From (2), the associated loss function is given by:

$$\begin{aligned} L_{N}=\frac{(\overline{X}_{N}-\mu )^{2}}{\mu ^{3}}. \end{aligned}$$

Now, recalling $\hbox {RPUC}_{N}$ from (5) and utilizing the facts that N is an observable finite random variable and the event $N=n$ is measurable only with respect to $\{\widehat{\lambda }_{j};$ $m\le j\le n\}$ for all fixed $n\ge m$, we can evaluate $E\left[ \text {RPUC}_{N}\right]$ as:

$$\begin{aligned}&E\left[ \text {RPUC}_{N}\right] \\&\quad =\underset{m\le n<\infty }{\sum }E\left[ \frac{L_{N}}{C_{N} }\mid N=n\right] P(N=n)\\&\quad =\underset{m\le n<\infty }{\sum }E\left[ \frac{L_{n}}{C_{n} }\mid N=n\right] P(N=n).\\&\quad =\underset{m\le n<\infty }{\sum }E\left[ \frac{L_{n}}{C_{n} }\right] P(N=n)\\&\quad =\underset{m\le n<\infty }{\sum }\frac{E[L_{n}]}{C_{n}} P(N=n)\\&\quad =\underset{m\le n<\infty }{\sum }\frac{R_{n}}{C_{n}}P(N=n)\\&\quad =\underset{m\le n<\infty }{\sum }\frac{1}{cn^{2}\lambda ^{k+1}}\\&\quad =\underset{m\le n<\infty }{\sum }\left( \frac{n^{*}}{n}\right) ^{2}\omega . \end{aligned}$$

Thus we have:

$$\begin{aligned} \omega ^{-1}E\left[ \text {RPUC}_{N}\right] =E\left[ \left( \frac{n^{*} }{N}\right) ^{2}\right] , \end{aligned}$$

which is clearly o(1) by utilizing part (iii), when $m>k+2$. $\square$

Proof of Theorem 2.2

Let us first prove the following:

$$\begin{aligned} E\underset{}{\left[ \left( J/n^{*}\right) ^{t}\right] =1+\left\{ t\eta _{k}+\frac{1}{2}t(t-1)p\right\} n^{*-1}+o\left( n^{*-1}\right) }, \end{aligned}$$

(29)

for different values of t, where J comes from (10). By utilizing Theorem 3 in Ghosh and Mukhopadhyay (1979) and Theorem 2.3 in Woodroofe (1977), we define V where,

$$\begin{aligned} V=(J-n^*)^2/n^*\overset{\mathcal {L}}{\rightarrow }p\chi ^2_1, \end{aligned}$$

(30)

and V is also uniformly integrable if $m>k+1$. Also, from Theorem 2.4 of Woodroofe (1977) we have,

$$\begin{aligned} E(J)=n^*+\eta +o(1), \end{aligned}$$

(31)

if $m>k+1$.

Case 1 $t=-1$

$$\begin{aligned} E\{(n^*/J)\}=n^{*^{-1}}\{E((J-n^*)^2/J)-E(J-n^*)+n^*\}. \end{aligned}$$

(32)

One can now follow along similar lines as part (iii) of Theorem 2.1, and split $E[(J-n^*)^2/J]$ into two parts and show:

$$\begin{aligned} E\left\{ \frac{(J-n^*)^2}{J}I\left( J>\frac{1}{2}n^*\right) \right\} =p+o(1), \end{aligned}$$

(33)

if $m>(k+1)$ and,

$$\begin{aligned} E\left\{ \frac{(J-n^*)^2}{J}I\left( J<\frac{1}{2}n^*\right) \right\} =o(1), \end{aligned}$$

(34)

if $m>2(k+1)$, by Lemma 2.3 of Woodroofe (1977). Now combining (31)–(34) the expansion follows, if $m>2(k+1)$.

Case 2 $t\in (-\infty , 0)-\{-1\}$

Let $t=-r$, where $r>0$. Now,

$$\begin{aligned} E\{(J/n^*)^t\}=1-rE(J-n^*)n^{*-1}+\frac{1}{2}r(r+1)n^{*{-1}}E\{V/Q^{r+2}\}, \end{aligned}$$

(35)

where V comes from (30) and Q is a suitable random variable between $J/n^*$ and 1. As before, one can now easily show:

$$\begin{aligned} E\{VQ^{-r-2}I(J>\frac{1}{2}n^*)\}=p+o(1), \end{aligned}$$

(36)

if $m>(k+1)$ and,

$$\begin{aligned} E\left\{ VQ^{-r-2}I\left( J<\frac{1}{2}n^*\right) \right\}&\le \left\{ n^{*r+1}m^{-r}+n^{*r+3}m^{-r+2}P\left( J<\frac{1}{2}n^*\right) \right\} \nonumber \\&=o(1), \end{aligned}$$

(37)

if $m>(r+3)(k+1)$, by Lemma 2.3 of Woodroofe (1977). Now combining (35)–(37) the expansion follows, if $m>(r+3)(k+1)$ which is nothing but $m>(3-t)(k+1)$.

Case 3 $0<t\le 2$

If $t=1$ or $t=2$, the result readily follows by noting (30)–(31) and from Woodroofe (1977), if $m>k+1$. So we can restrict to $t\in (0,2)-\{1\}$. Now following along the lines of Case 2 one can show, $E\{UQ^{t-2}I(J>\frac{1}{2}n^*)\}=p+o(1)$, if $m>(k+1)$ and $E\{UQ^{t-2}I(J<\frac{1}{2}n^*)\}=o(1)$, if $m>(3-t)(k+1)$. Hence the result follows under this case as well.

Case 4 $t>2$

Following along the lines of Case 2 one can show, $E\{UQ^{t-2}I(J<2n^*)\}=p+o(1)$, if $m>(k+1)$ and $E\{UQ^{t-2}I(J>2n^*)\}=o(1)$, which follows from Lemma 2.2 of Woodroofe (1977). Hence the result is true under this case also.

This proves (29) under all possible cases, with different conditions on m. Finally by noting $N=J+1$ completes the proof of Theorem 2.2. $\square$

For proving Theorem 2.3 we first note that $\omega ^{-1}E\left[ \text {RPUC}_{N}\right] =E\left[ \left( n^{*}/N\right) ^{2}\right]$, from part (iv) of Theorem 2.1. One can now exploit Theorem 2.2 by replacing $t=-2$.

Theorem 2.4 follows readily from an application found in Ghosh and Mukhopadhyay (1975) where we have:

$$\begin{aligned} U=n^{*^{-1/2}}(N-n^*)\overset{\mathcal {L}}{\rightarrow }N(0,p), \end{aligned}$$

where p comes from (12). One may also refer to Theorem 2.4.3 or Theorem 2.4.8, part (ii) in Mukhopadhyay and Solanky (1994).

Proof of Theorem 3.1

Part (i)

From (18) one can get,

$$\begin{aligned} \frac{\widehat{\mu }_{N}}{\lambda }\left( \frac{z_{\alpha /2}}{\log d}\right) ^{2}\le N\le \frac{\widehat{\mu }_{N-1}}{\lambda }\left( \frac{z_{\alpha /2} }{\log d}\right) ^{2}+m. \end{aligned}$$

(38)

Dividing throughout by $n^{*}$ we have,

$$\begin{aligned} \frac{\widehat{\mu }_{N}}{\mu }\le \frac{N}{n^{*}}\le \frac{\widehat{\mu }_{N}}{\mu }+\frac{m}{n^{*}}. \end{aligned}$$

(39)

The proof follows by taking limits throughout and noting that $N\rightarrow \infty$ w.p.1, $\widehat{\mu }_{N}\rightarrow \mu$ w.p.1 and $m/n^{*}\rightarrow 0$ as $d\rightarrow 1$.

Part (ii)

From the right hand side of (39) we have (for sufficiently large $n^{*}$),

$$\begin{aligned} N/n^{*}\le \frac{\widehat{\mu }_{N}}{\mu }+m. \end{aligned}$$

(40)

Now denoting $\sup _{n\ge 2}(\widehat{\mu }_{n})$ by U, we can claim w.p.1:

$$\begin{aligned} N/n^{*}\le \frac{U}{\mu }+m. \end{aligned}$$

(41)

Since the right hand side of (41) is free from d, one can apply Wiener’s (1939) ergodic theorem to conclude the uniform integrability of $N/n^{*}$. The proof is hence complete by using part (i).

Part (iii)

From (19), the confidence interval for $\mu$ is $C_{N}=\left[ d^{-1} \widehat{\mu }_{N},\text { }d\widehat{\mu }_{N}\right]$. We may now write,

$$\begin{aligned} P_{\mu }(\mu \in C_{N})= & {} P_{\mu }(d^{-1}\widehat{\mu }_{N}\le \mu \le d\widehat{\mu }_{N})\nonumber \\= & {} P_{\mu }(\log \widehat{\mu }_{N}-\log d\le \log \mu \le \log \widehat{\mu }_{N}+\log d\nonumber \\= & {} P_{\mu }\left( \left| \log \widehat{\mu }_{N} -\log \mu \right| \le \log d\right) \end{aligned}$$

(42)

We now invoke Anscombe’s (1952) random CLT and can thus write,

$$\begin{aligned} \frac{\sqrt{N}(\widehat{\mu }_{N}-\mu )}{\sigma }\overset{D}{\rightarrow }N(0,1) \hbox { and } \frac{\sqrt{n^{*}}(\widehat{\mu }_{N}-\mu )}{\sigma }\overset{D}{\rightarrow }N(0,1) \hbox { as } d\rightarrow 1. \end{aligned}$$

(43)

Using (43) and the Mann–Wald theorem, we introduce $V_{N}$, where,

$$\begin{aligned} V_{N}=\frac{z_{\alpha /2}(\log \widehat{\mu }_{N}-\log \mu )}{\log d}\overset{D}{\rightarrow }N(0,1) \hbox { as } d\rightarrow 1. \end{aligned}$$

(44)

The proof is hence complete by using (42) and noting,

$$\begin{aligned} P_{\mu }(d^{-1}\widehat{\mu }_{N}\le \mu \le d\widehat{\mu }_{N})=P_{\mu } (|V_{N}|\le z_{\alpha /2})\rightarrow 1-\alpha \hbox { as } d\rightarrow 1. \end{aligned}$$

(45)