Reference no: EM132432400

Problem: Assume that the data s = (x1, . . . , xn) are a realization of a random sample X1, . . . , Xn from a Galenshore distribution with parameters α > 0 and θ > 0, where α is known to be equal to α0 while θ is unknown. The probability density function of a Galenshore distribution is given by f(x) = ( 2 Γ(α) θ 2αx 2α-1 exp{-θ 2x 2} if x > 0 0 otherwise, where the gamma function Γ(α) is defined for any α > 0 as Γ(α) = R ∞ 0 y α-1 e -ydy.

Required:

Question 1: Write a likelihood function of θ given s.

Question 2: Find a sufficient statistic for s.

Question 3: Find the maximum likelihood estimator for θ.

Question 4: Assume that α0 = 1, n = 5 and s = (2.1, 3.5, 3.1, 2.9, 2.8). Find the MLE estimate of θ given s.