Suppose X1, X2, X3 constitute a random sample from an exponentially distributed population given by f(x) = (1/θ)e^(-x/θ) for x > 0 and 0 elsewhere.
Note that for the above exponential distribution the mean is μ = E(X) = θ and σ^2 = Var(X) = θ^2. Consider the following estimators of θ:
θ̂1 = X1, θ̂2 = (X1 + X2)/2, θ̂3 = (X1 + 2X2)/3, θ̂4 = X̄, θ̂5 = 5,
where X̄ = ∑(i=1 to 3) Xi/3.
(a) Which of these estimators are unbiased?
(b) Calculate the variances of these estimators. Which estimator has the smallest variance?
(c) Among the unbiased estimators, which one(s) has/have the smallest variance?
To discuss in class: Do you have a preference for one of the estimators?