Estimating the Total Probability of the Unobserved Outcomes of an Experiment
An experiment has the possible outcomes E 1, E 2, … with unknown probabilities p 1, p 2, … ; p i ≧ 0, Σ i p i = 1. In n independent trials suppose that E i occurs x i times, i = 1, 2, …, with \(\sum {_i{x_i}} = n.\) Let φ i = 1 or 0 according as x i = 0 or x i ≠ 0. Then the random variable u = \(\sum {_i{p_i}{\varphi _i}} \) the sum of the probabilities of the unobserved outcomes. How can we “estimate” u? (The quotation marks appear because u is not a parameter in the usual statistical sense