Question

We observe a random sample of n values from the beta distribution with parameters c and 2.

(a) Find a general expression for the method of moments (MOM) estimate of c, and calculate this MOM estimate for the case where we observe two values, 0.96 and 0.84.

(b) Find the bias of the MOM estimator of c for the case where c = 1 and n = 1.

(c) Find a general expression for the maximum likelihood estimate (MLE) of c, and calculate this MLE for the case where we observe two values, 0.96 and 0.84.

Answer 1

Probability density function of the given beta distribution is

   if   

0 otherwise.

a) The two observed values are 0.96 and.84

therefore, mean = (.96+.84)/2 = .9

and, mean of beta dist. is

  

therefore ,The general expression for MOM estimate of c is:

Now,

The requried MOM estimate for is

b) Here bias of MOM estimate for is required and value of and n are given.n=1 here but the observation is not given.

So, we calculate the bias of the in (a) :

Bias=

= 18-1=17.

c)General expression for MLE of c:

ML function :

in this case taking log of likelihood function and derivatring wrt c and equating to 0 will give no answer.So here MLE does not exist.