Question

Let X1​,…,Xn​ be a random sample from a population with pdf f(x∣θ)=(1/2)​(1+θx), −1(a) Derive the method of moments estimator of θ based on the first moment.

(b) Show that this estimator is consistent.
(c) Derive the asymptotic distribution of this estimator.

Answer 1

To derive the method of moments estimator of θ based on the first moment, equate the first moment of the sample to the first moment of the population. Show that this estimator is consistent by showing its convergence in probability. Derive the asymptotic distribution of this estimator using the Central Limit Theorem.

Step-by-step explanation:

(a) To derive the method of moments estimator of θ based on the first moment, we equate the first moment of the sample to the first moment of the population. The first moment of a random variable X is E(X), and the first moment of the population is given by the integral of xf(x|θ) over its support.

(b) To show that this estimator is consistent, we need to show that it converges in probability to the true parameter value as the sample size n approaches infinity.

(c) To derive the asymptotic distribution of this estimator, we need to use the Central Limit Theorem, which states that as the sample size goes to infinity, the distribution of the estimator approaches a normal distribution.