Question

4. Let Yi, . .. ,y, denote a random sample from the pdf 0-1 0Ky1, elsewhere. y"(1- y)0-1 0, (a) Find the method of moments estimator of θ. (b) Find a sufficient statistics for θ

Answer 1

Given that the dah S in the folbwin ohic is the pdf of beh clisthibution of -fust 13 The aran wl then the Seton teitial moment akout the O1l mean ๔-2. IS then χ~β, (m, n) if know Now ,Le mn 2 There fore 2 s9-+ 4 Now, we eauata the Seondl Sample moment about the mear to the Second theoartial ก0ment alot mean 80+4 80-+4

S2 49 niato,t, ) estimator of e is "The morrent ,' o is rhe Jeint density inchon of 0-1 0-1 (Ne) sef ciet. Ter(g.ふ...-3n) is the Statistic furd Such that or if fuctien 8 and tyd with the .fm İn-factor i2 ation heoem we get 'T = "N,' yi (1-d.) i- I

Pt the -factori2athion heorem, TI-d is a Sufficient Stahshc for 8