Question

2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter and rate parameter λ, Γ(a) Here α > 0 and λ > 0. Let θ sufficient statistic for the data (α, β). Determine the log-likelihood, I(0), and a 2-dimensional b) Suppose that xi,...,In are a random sample from a U(-0,) distribution, 1/(20) if- otherwise x-θ f(x;0)-' 0, Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood should take into account which values of θ give L(θ) 0 and which do not. You can get part marks for obtaining a two-dimensional sufficient statistic.

Answer 1

solution a) Gamma distribution Let Xl, X2 -- Xn be a r. s. from with pdf. Ex:(*) = I 29.19 e 2x xxo a BYO ; othenoise Therefore, the likelyhood function of epost BECA,A) for given param random sample [co-edra)? K)= FGC KR) scal, Heat? ki ? IN - az 시다 271ě LAM Therefore, the log-likelyhood of @ = (,x) is 1091 (0/2) = nalogt- log Ta +6-1) 109 (9 km) 21 109 CLC01x)) = n Calogs - 109 ra) + ck-1) logxi

Gratis ter @ Now, we want to find joint sufficient stats for (x,x) The joint clistribution of random Gample is FCE/ xx) = face = 344 )" (23*78** 243 = 9 (t/= , xi, tz = Biga, 1, ] (hex) The function g() is function of parameter 83 (116) and statistic Tac f xi, Exi) and hex) be function independent of a. parameter :. By Neyman's factorization theorem Txi, Exi) is sufficient statistic for G=(x,x). 2) Let X,,X2 ... Xn be a r-5. from Deo, 0] :Fx(X) = L ; exse othenoise.

The likelyhood function of o is LLO! Z) = f (24 a ) Ix;(-8,6) = G) Ex, Coa) The joint distnbution 078-5 is Fle) = + (21) = ()" if Ix; 60,0) = Go To Exp (-02 X00 4-.- Excause) 535a)? In t x; 6-& Xon) x, CX60); } {1} = g( t;= x«), t2=(Xc2)), o) h(x) g() is function of O, xal), Yess and h(x) is free from o :: By Neyman's factorization theorem T= min c xen, Xcho) is sufficient Statistic for a :