Reference no: EM132868476

1. Suppose θ > 0. Suppose Y1,...,Yn ∼ iid Uniform[0,θ]. Let

Y^– = 1/n ⁿ∑_i=1Y_i Y(n) = max(Y1,...,Yn)

Consider estimating θ with:

θˆ1 = 2Y¯              θ₂ = n+1/n Y(n)

(a) Find the probability density function of Y(n).
(b) Show that both θˆ1 and θˆ2 are unbiased.
(c) Derive the relative efficiency of θˆ1 compared to θˆ2. Which estimator do you prefer? (d) Are θˆ1 and θˆ2 consistent estimators? Explain.
(e) Write down the likelihood as a function of θ. (Hint: the likelihood is zero if θ is smaller than Y(n).)
(f) Find a sufficient statistic for θ.
(g) Derive the maximum likelihood estimator of θ.

2. Suppose a random sample Y1,...,Yn is collected from a distribution with the density function

f(y) = 3/λ y² e^-y3/λ, y > 0

where λ > 0 is an unknown parameter.
(a) Show that E(Y 3) = λ, where Y follows the above distribution.
(b) Find var(Y 3).
(c) Find the MLE λb of λ using a random sample Y1,...,Yn.
(d) Is this MLE an unbiased estimator? Give reasons.
(e) Find the asymptotic variance of λb.
(f) Hence write down the asymptotic normal distribution for λb by applying the MLE theory directly.
(g) You may also find the asymptotic normal distribution of λb by realizing that λb is a particular sample mean and then applying the CLT. Is this distribution the same as the one above?

(h) Write down a 95% CI for parameter λ.