Statistica Sinica 19 (2009), 315-323

ASYMPTOTIC OVERSHOOTS FOR ARITHMETIC

I.I.D. RANDOM VARIABLES

I-Ping Tu

Academia Sinica, Taipei

Abstract: The Sequential Probability Ratio Test (SPRT) has been widely applied in quality control and clinical studies. There are two important quantities in SPRT: $[1-E(e^{-\alpha S_{\tau_+}})]/E(S_{\tau_ +})$ for calculating the p-value and ${E(S_{\tau_+}^2)/2E(S_{\tau_+})}$ for estimating the sample size, where $S_n$ is the i.i.d. summation of random variables and $\tau_+$ refers to the first time that $S_n$ becomes positive. For non-arithmetic i.i.d. random variables, Woodroofe (1979) provided computation formulas for these two quantities. To find the threshold for the IBD score statistics in testing genetic linkage, Tu and Siegmund (1999) provided a computation formula to calculate $[1-E(e^{-\alpha S_{\tau_+}})]/(1-e^{-\alpha h}) E(S_{\tau_ +})$ for arithmetic i.i.d. random variables when $\alpha$ is not too small. This paper gives another computation formula to calculate $[1-E(e^{-\alpha S_{\tau_+}})]/(1-e^{-\alpha h}) E(S_{\tau_ +})$ for arithmetic i.i.d. random variables, which can be applied for any positive $\alpha$ including $\alpha \downarrow 0$. We also provide a computation formula for $E(S_{\tau+}^2) /2E(S_{\tau_+})$ to estimate the overshoot for arithmetic i.i.d. random variables. Furthermore, we show that these two formula reproduce Woodroofe's non-arithmetic formula by letting the span $h$ go to zero, and we derive a computation formula to calculate $E(S_{\tau_+})$, that can be applied to estimate the number of 'new-high' points in reaching a threshold.

Key words and phrases: Arithmetic, ladder height, overshoot, sample size, second moment, sequential analysis.