Sequential Estimation of Conditional Odds Ratio

Abstract

This paper deals with fixed-width confidence interval of conditional odds ratio in the context of clinical trial experiment with matched pair-type of data structure. Some related asymptotic result is also obtained. The procedure is evaluated by simulation followed by a data study.

Appendix

(i):

$P(N<\infty )=1$ for every $d>0$.

Proof

To proof the above claim it is sufficient to show $P(N>k)\rightarrow 0$, as $k\rightarrow \infty$.

For every fixed $d>0$,

$$\begin{aligned} P(N>k)=P(k<\frac{a^{2}\hat{\sigma _{k}^{2}}}{d^{2}})\rightarrow P(\phi )=0 \end{aligned}$$

(ii):

$\hat{N}/N\rightarrow 1$ almost surely as $d\rightarrow 0$.

$\square$

Proof

Let us write $\hat{N }$ as N(d).

By definition, it can be written that, $N(d)\rightarrow \infty$ almost surely as $d\rightarrow 0$.

So,

$$\begin{aligned} \frac{a^{2}\hat{\sigma _{N(d)}^{2}}}{d^{2}}\le {N(d)}< {n_0}+ \frac{a^{2}\hat{\sigma ^{2}}_{N(d)-1}}{d^{2}} \end{aligned}$$

This implies,

$$\begin{aligned} \frac{{{N(d)}d^{2}}}{a^{2}}\rightarrow \,\sigma ^{2}, \end{aligned}$$

almost surely as $d\downarrow 0$.

Also,

$$\begin{aligned} \frac{{{N}d^{2}}}{a^{2}}\rightarrow \,\sigma ^{2}, \end{aligned}$$

almost surely as $d\downarrow 0$. Hence,

$\hat{N}/N\rightarrow 1$ almost surely as $d\rightarrow 0$. $\square$