Seeking help with question 2

1. Consider a random sample X1, .... X, from a Gaussian population with mean 0 and unknown variance o' > 0. The parameter of interest is T = 27. To simplify the notation, you can use X = EL, X? . (a) (15 pts) Find the method of moments estimator for + using the second non-central moment. (b) (15 pts) Write down the likelihood and log-likelihood for T. (c) (15 pts) Calculate the score function and the observed Fisher information function for T . (d) (15 pts) Prove that the maximum likelihood estimator for r is f = 1/X. (e) (15 pts) Consider the realisation of the random sample: 0.47, 0.07, 0.59, -0.16, -0.06, -0.37, 0.28, 0.18 Find an approximate 95% confidence interval for + around its maximum likelihood esti- mator. 2. (25 pts) Consider the statistical model of exercise 1. Investigate empirically the properties of f1= n and f2 = n ELIX? ELI( X?+ 1 ) assuming that T =2. In particular, estimate the bias, the expectation, the standard error, and the mean squared error for both estimators. Based on these results, discuss which estimator may be preferred and why. Use R to perform the simulations, and include the code used and any plot that you create