Sufficiency

Abstract

In any statistical analysis we attempt to use all the data. The reader will already be accustomed to the idea of condensing the raw data into one or more summary statistics without perhaps dwelling overlong on the possibility that valuable information may thereby have been cast away. Suppose, for instance, that a coin is tossed independently n times and that the data are the sequence of heads and tails obtained. If it is desired to estimate the probability, 0, of a head on an individual toss and if r is the total number of heads, then it is known that R/n is an unbiased estimator of θ. There is, however, a more pervasive and general property enjoyed by the statistic R. In a sense it contains all the information about θ that the sample affords. More precisely, the conditional probability of any feature that may be displayed by the sample, given that it contains just r heads, does not depend on θ at all.