Question

In: Statistics and Probability

Let 𝑋1, 𝑋2,…, 𝑋𝑛 be a random sample of a gamma distribution with parameters 𝛼 (known) and 𝛽 (unknown), prove that 𝑇 (𝑿) = 𝑋̅ is a sufficient estimator for 𝛽.

Expert Solution

Let be a random sample from the pdf or pmf . Then the statistics is sufficient for , iff we can factorize the joint PDF and PMF of as
Where depends on but not on and depends on and on only through

In our problem with pdf :

Join density function is given by:

Now let,

Cleraly depends on but not on
But depends depends on and on only through

Hence factorization theorem implies is sufficient Statistics for estimator

orchestra answered 2 years ago

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