Journal of Statistical Planning and Inference 137 (2007) 2336 – 2346 www.elsevier.com/locate/jspi Sequential risk-efficient estimation for the ratio of two binomial proportions Hokwon Cho ∗ Department of Mathematical Sciences, University of Nevada-Las Vegas, Las Vegas, NV 89154-4020, USA Received 16 May 2005; received in revised form 18 July 2006; accepted 1 August 2006 Available online 7 November 2006 Abstract A risk-efficient sequential point estimator is considered for the ratio of two independent binomial proportions based on maximum likelihood estimation under squared error loss and cost proportional to the observations. It is assumed that the cost per observation is constant. First-order asymptotic expansions are obtained for large-sample properties of the proposed procedure. Performance of the procedure is studied through the criteria of risk efficiency and regret analysis. Monte Carlo simulation is carried out to obtain the expected sample size that minimizes the risk and to examine its finite sample behavior. An example is provided to illustrate its use. © 2006 Elsevier B.V.All rights reserved. Keywords: Risk-efficient sequential point estimator; Ratio of two binomial proportions; First-order asymptotics; Risk efficiency; Regret analysis 1. Introduction Let X 1 ,X 2 ,... and Y 1 ,Y 2 ,... be two sequences of independent Bernoulli random variables with nonzero probabil- ities p 0 and p 1 , respectively. Moreover, let  = p 1 /p 0 . The problem addressed in this paper is to estimate the true ratio of the two binomial proportions when the loss incurred is of the form L n = ( ˆ  n - ) 2 + cn, (1) where ˆ  n is the maximum likelihood estimator (MLE) and c (> 0) is the known cost per unit of observations (X, Y ). Then, the risk is R n (c) = E( ˆ  n - ) 2 + cn = Var( ˆ  n ) + B 2 + cn, (2) where B represents the bias which is defined by B = E( ˆ  n ) - . The ratio of two binomial proportions arises in prospective studies, biological experiments or comparison of man- ufacturing processes for quality control in industry. It has been an important tool for measuring the risk ratio (Katz et al., 1978; Fleiss, 1981; Bailey, 1987) or the relative risk (Gart, 1985). In epidemiological problems, such as co- hort studies in two groups, the risk ratio or odds ratio is related to vaccine efficacy and attributable risk (Walter, 1976). However, all of these methods have dealt with approximate interval estimators based on the logarithmic method ∗ Tel.: +1 702 895 0393; fax: +1 702 895 4343. E-mail address: cho@unlv.nevada.edu. 0378-3758/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.08.005