1. ## Estimators

Let x1,...xn be a sample random from pdf
f(x\t)=tx^(t-1) , 0<x<1 , t>0
1) to show that the pdf belongs to the exponential family
2)To find the lower limit of variance of two estimators unbiased to t
3)Find a enough statistics to t and its distribution
4)Suggest a estimator unbiased to t that be from unbiased function and check if it is efficient

Thank you

2. 1)

the function belongs to a exponential family if its pdf can be written as:

$f(x|t) = \bigg\{exp\bigg(c(t) \; T(x) \; + b(t)+S(x)\bigg)\bigg\}$

$t.x^{t-1} = \mbox{exp}(log(t)).\mbox{exp}((t-1)log(x)) = \mbox{exp}\bigg(logt+(t-1)log(x)\bigg) =...$