Question

Let Xi ∼ N(iθ,1), i = 1,...,n and they are independent. (a) Obtain the MLE of θ. (b) Find its bias.

Answer 1

To find the MLE of θ for Xi ~ N(iθ,1), compute the likelihood function and maximize it. The bias of the MLE can be found by taking the expected value of the MLE and subtracting θ.

Step-by-step explanation:

(a) To find the maximum likelihood estimator (MLE) of θ, we need to compute the likelihood function and maximize it. The likelihood function is given by:

L(θ) = ∏(i=1 to n) [1/(√(2π))] * exp[-(xi - iθ)^2/2]

Take the natural logarithm of the likelihood function and differentiate it with respect to θ. Set the derivative equal to zero and solve for θ to get the MLE.

(b) The bias of the MLE can be found by taking the expected value of the MLE and subtracting the true parameter value θ. If the bias is zero, the estimator is unbiased.