# Question

With reference to Exercise 10.72, check whether the following estimators are maximum likelihood estimators of θ:

(a) 1/2 (Y1 + Yn);

(b) 1/3 (Y1 + 2Y2).

In exercise

Let X1, X2, . . . , Xn be a random sample of size n from the uniform population given by

Show that if Y1 and Yn are the first and nth order statistic, any estimator Θ such that

Can serve as a maximum likelihood estimator of θ. This shows that maximum likelihood estimators need not be unique.

(a) 1/2 (Y1 + Yn);

(b) 1/3 (Y1 + 2Y2).

In exercise

Let X1, X2, . . . , Xn be a random sample of size n from the uniform population given by

Show that if Y1 and Yn are the first and nth order statistic, any estimator Θ such that

Can serve as a maximum likelihood estimator of θ. This shows that maximum likelihood estimators need not be unique.

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