## Estimators in General

We may sometimes want to estimate a parameter of a distribution other than a moment. An example is the probability (pi) that the ace will turn up in a roll of a die. A “natural” estimator in this case is the ratio of the number of times the ace appears in n rolls to n—denote it by p. In general, we estimate a parameter 0 by some function of the sample. Mathematically we express it as

(7.1.1) 0 = ф(Х], X2, . . . , Xn).

We call any function of a sample by the name statistic. Thus an estimator is a statistic used to estimate a parameter. Note that an estimator is a random variable. Its observed value is called an estimate.

The pi just defined can be expressed as a function of the sample. Let Xi be the outcome of the zth roll of a die and define У, = 1 if X* = 1 and Yi = 0 otherwise...

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