(a) Suppose that 𝜃̂is an estimator for a parameter 𝜃 and 𝐸[𝜃̂] = 𝑎𝜃 + …
(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)𝑋n] /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.
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