Question

Question 1. Let Xi, X2, X3, X4 be a random sample from a population X with mean E(X)-? and standard deviation Sd(X)-. Consider the following two estimators for (a) Compute the bias for and ?2 respectively. (c) Which estimator is better? Why?

Answer 1

Let x_1,x_2, x_3, x_4 be a rs from a population x with E(x) = mu and sd(x) = sigma i.e v(x) = (sd(c))^2 = sigma^2 mu_1 = 2x_1 - x_2 and mu_2 = (5/8) x_1 + (- 1/8) x_2 + (1/8) x_3 + 1/8) x_4 (a) we have to complete basic mu_1 and mu_2 E[mu_1] = 2E[x_1) - E(x_2) = 2mu - mu = mu bias(mu_1) = E(mu_1) - mu = mu - mu \r\n bias (mu_1) = 0 E(mu_2) 5/8 E(x_1) + 1/8 E(x_2) + 1/8 E(x_3) + 1/8 E(x_4) = 5/8 mu + 1/8 mu + 1/8 mu + 1/8 mu = mu(5 + 1 + 1 + 1/8) = mu 8/8 E(mu_2) = mu bias(mu_2) = E(mu_2) - mu = mu = mu bias(mu_2) = 0 Therefore mu_1 and mu_2 is unbiased variance for mu_1 and mu_2 v(mu_1) = v(2x_1 - x_4) = 4v(x_1) + v(x_4) = 4 sigma^2 + sigma^2 v(mu_1) = 5 sigma^2 v(mu_1) = v[5/8x_1 + 1/8 x_2 + 1/8 x_3 + 1/8 x_4] = 25/64 v(x_1) + 1/64 v(x_2) + 1/64 v(x_3) + 1/64 v(x_4) = 1/64 sigma^2 [25 + 1 + 1+1] \r\n v(mu_2) = 28/64 sigms^2 v(mu_2) = 7/16 sigma^2