Asymptotic properties of maximum likelihood estimators with sample size recalculation

Abstract

Consider an experiment in which the primary objective is to determine the significance of a treatment effect at a predetermined type I error and statistical power. Assume that the sample size required to maintain these type I error and power will be re-estimated at an interim analysis. A secondary objective is to estimate the treatment effect. Our main finding is that the asymptotic distributions of standardized statistics are random mixtures of distributions, which are non-normal except under certain model choices for sample size re-estimation (SSR). Monte-Carlo simulation studies and an illustrative example highlight the fact that asymptotic distributions of estimators with SSR may differ from the asymptotic distribution of the same estimators without SSR.

Appendix

Proof (Theorem 1)

The joint asymptotic density (8) is fully determined by $\left( \widehat{\theta }_{1},\widehat{\eta }_{1}, \widehat{\theta }_{2},\widehat{\eta }_{2}\right)$, and marginally under $\theta =h/\sqrt{n}$,

$$\begin{aligned} \xi _{n1} = \sqrt{n} \sigma _{\theta }^{-1} (\widehat{\theta }_1 - h/\sqrt{n}) \rightarrow \xi _1 \overset{d}{=} N(0,1). \end{aligned}$$

(16)

By the continuous mapping theorem and the consistency of MLEs, $\sigma _{\theta }$ evaluated as a function of $\widehat{\theta }_1$ and $\widehat{\eta }_1$ is a consistent estimator of $\sigma _{\theta }$ and using it in place of $\sigma _{\theta }$, (16) still holds. Similarly, conditionally on N,

$$\begin{aligned} \xi _{n2} = \sqrt{N-n} \cdot \sigma _{\theta }^{-1} (\widehat{\theta }_2 - h/\sqrt{n}) \rightarrow \xi _2 \overset{d}{=} N(0,1). \end{aligned}$$

(17)

Recalling that, with large n and $N-n$, the full sample MLE is approximately distributed as $(n/N)\widehat{\theta }_1 + (1-n/N)\widehat{\theta }_2$, a large sample approximation follows:

$$\begin{aligned} V&= \sqrt{N}\sigma _{\theta }^{-1} \left( \widetilde{\theta } - h/\sqrt{n}\right) \, \overset{d}{\approx }&\sqrt{\frac{n}{N}} \xi _1 + \sqrt{1 - \frac{n}{N}} \xi _2 \, \, \overset{d}{\rightarrow } \, \, \sqrt{\frac{\tau -1}{\tau }} \xi _1 + \sqrt{\frac{1}{\tau }} \xi _2 \end{aligned}$$

and

$$\begin{aligned} \text {Pr}\left( V< v\right)& \rightarrow \text {Pr}\left( \sqrt{\frac{\tau -1}{\tau }} \xi _1 + \sqrt{\frac{1}{\tau }} \xi _2< v\right) \nonumber \\& \quad =\, \text {E}_{\xi _1} \text {Pr}\left( \left. \xi _2 < \sqrt{\tau }v - \sqrt{\tau -1} y \right| \xi _1 = y \right) . \end{aligned}$$

(18)

It is possible that, for some values of $\widehat{\theta }_1$ and $\widehat{\eta }_1$, the total sample size N is not large enough to claim normality; but the probability of this event converges to zero. This follows directly from the equality

$$\begin{aligned} 1 = \text {Pr}\left( N\le n\right) + \text {Pr}\left( n < N \le n + \sqrt{n}\right) + \text {Pr}\left( N > n + \sqrt{n}\right) . \end{aligned}$$

The event $n < N \le n + \sqrt{n}$ includes all possible values of N where the asymptotic normality of V may not hold. The probability of this event

$$\begin{aligned} \text {Pr}\left( n< N \le n + \sqrt{n}\right) = \text {Pr}\left( 1 < \frac{N}{n} \le 1 + \frac{1}{\sqrt{n}}\right) \rightarrow 0 \end{aligned}$$

when N / n converges to a stationary random variable with a right-continuous distribution at 1.

Then, for large n,

$$\begin{aligned} \text {Pr}\left( V < v\right) \approx \int _{-\infty , \tau \ge 0}^{\infty } \varPhi \left( \sqrt{\tau } v - \sqrt{\tau -1}y\right) \phi (y) dy. \end{aligned}$$

(19)

Equations (18) and (19) are not defined for $\tau \le 1$ which corresponds to values of $\widehat{\theta }_1$ and $\widehat{\eta }_1$ associated with $N \le n$. When the bound is incorporated the proof is complete.

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Proof (Corollary 1)

The $\widehat{{{\text{CV}}}}^2$ is asymptotically distributed as a noncentral chi-squared random variable with 1 d.f. whose distribution is fully determined by ${{\text{CV}}}^2$. Then, asymptotically, under $H_0$, $\widehat{{{\text{CV}}}}^2$ is a central chi-squared random variable with 1 d.f. which completes the proof.

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