Estimator definition

Estimator definition

by Gabin Paul Jacques Leroy -
Number of replies: 3
Hello,

Let \Omega be our sample space. Consider a model in which we have a random vector (X_1,…,X_n) with the X_i all distributed with p a probability distribution on some finite alphabet \xi=\{1,…,k\} , and independent. I have seen in several statistical courses that an estimator is any function of the random variables X_i that do not depend on unknown parameter. In the course, when we say that q is an estimator, do we mean that q(X_1,…,X_n): \Omega \rightarrow \Delta_k ?


Best regards,

GL
In reply to Gabin Paul Jacques Leroy

Re: Estimator definition

by Gabin Paul Jacques Leroy -
So in our course, q(X_1,...,X_n) is an estimator if, when evaluated in a point of our sample space, its component sum up to 1 and are non-negative correct ? Is that why we say that this estimator estimates a distribution of p ?
In reply to Gabin Paul Jacques Leroy

Re: Estimator definition

by Thomas Weinberger -
Hello Gabin,

If we are doing distribution estimation, you can indeed assume (or, if you construct the estimator yourself, you should always require) that the output will be in the simplex (non-neg. and sums up/integrates to 1). This should be considered the minimum requirement whenever we do distribution estimation.

Of course, this is only a special case of estimation. The empirical mean estimator for example can a-priori return anything.

Cheers,
Thomas