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Originally Posted by bobboufl11
Let Y_1, Y_2,...Y_n denote a random sample from the density function given by
Find a sufficient statistic for theta. pretty sure I got this(sum of the y's from i = 1 to i = n)
You need to write the density for fn(y|theta) by multiplying n densities for the y1,y2,....yn. Then to find a sufficient statistic, use the factorization critereon. That is, identify one factor of fn(y|theta) that can depend on only the y's but not theta, and the other factor that depends on theta but only depends on the y's through the sufficient statistic r(y) where y = y1,y2,..yn:
fn(y|theta) = u(y)*v(theta,r(y)).
Can you do that if the sufficient statistic r(y) is the sum of the y's, or does it need to be something else?
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Find the Maximum Likelihood estimator of theta. Tried taking the derivative of the likelihood function and setting it equal to 0.
You want to maximize the fn(y|theta) that you got above from multiplying the n densities. But maximizing fn(y|theta) is the same as maximizing the log[fn(y|theta)], and that makes it easier. Remember that a log of a product is a sum of logs, and log(1/theta^n) = -n*log(theta). The part that doesn't depend on theta just becomes an added log term that goes to zero when you take the derivative.
Last edited by BruceZ; 04-20-2013 at 07:11 PM.