Estimating the unknown sample size

Abstract Let us consider a random experiment repeated n times with outcomes y =(y 1 ,…,y n ) , and that we are given the value of a statistic, T( y ) , but not the sample size n . Our aim is to estimate the unknown sample size n with the information provided by T( y ) . If T is a discrete random variable such that the corresponding family of p.m.f's, {P n } n⩾1 , is uniparametric, n being the unknown parameter, and if a certain recursive relation holds, we obtain unbiased estimators for functions of the sample size, h(n) , and also we characterize the maximum likelihood estimators (m.l.e.) of n in terms of the modes of the family of p.m.f's {P n } n⩾1 . As an application, we estimate the unknown sample size when we only know the number of r -records in a random sample from an absolutely continuous distribution. Also we present other applications related with the Poissonian binomial sampling.