Answer to Question #295171 in Statistics and Probability for john k

Question

(a) Suppose that 𝜃̂is an estimator for a parameter 𝜃 and 𝐸[𝜃̂] = 𝑎𝜃 + 𝑏 for some nonzero constant 𝑎 and 𝑏. (i) Determine the biasness of the estimator.

(ii) Find a function of 𝜃̂, say 𝜃̂∗ that is an unbiased estimator for 𝜃.

(b) Given that a 𝑋1,𝑋2, … , 𝑋𝑛 denoted a random sample from an exponential distribution with parameter 𝛽 = 1 /𝜃 . Consider two estimators 𝜃̂₁ = 𝑋̅ 𝑎𝑛𝑑 𝜃̂₂= [𝑋₁ + (𝑛 − 1)𝑋_n] /𝑛 Show that both 𝜃̂₁ and 𝜃̂₂ are unbiased estimator of 𝜃.

(c) If 𝑋₁, 𝑋₂, … 𝑋₁₀ is random sample of size 𝑛 from a gamma distribution with parameter 𝛼 and 𝛽.

(i) Use method of moment to estimate 𝛼 and 𝛽.

(ii) Determine the maximum likelihood estimate of 𝛽 if 𝛼 is known.