STAT30280 Adv Data Analytics (online) - Assignment 3 Submission deadline: Monday 19th of April 2021 at 9pm. To complete the assignment, you should submit on the Brightspace assignment page up to two...

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STAT30280 Adv Data Analytics (online) - Assignment 3 Submission deadline: Monday 19th of April 2021 at 9pm. To complete the assignment, you should submit on the Brightspace assignment page up to two pdf files. One file should contain all your solutions, while the other should be the completed School of Mathematics and Statistics’ honour code (available from module information folder). Scanned documents are accepted but it is the student’s responsibility to ensure that they are readable and that they can be graded. 1. Consider a Gaussian random sample X1, . . . , Xn IID∼ N ( 0, eθ ) where n = 60 and the parameter of interest is θ ∈ R. A realisation of the random sample gives: ¯̄x = 1 n n∑ i=1 x2i = 1.20 (a) (15pts) Construct the method of moments estimator for θ, and calculate its numerical value. (b) (15pts) Write down the likelihood and the log-likelihood functions for θ. (c) (15pts) Write down the score and observed Fisher information functions for θ. (d) (15pts) Find the maximum likelihood estimator for θ. (e) (15pts) Explain how you can use a Gaussian distribution to approximate the distribution of the maximum likelihood estimator for θ. (f) (25pts) Assume that the true parameter is θ = 0. Investigate empirically the properties of the maximum likelihood estimator for θ using R. Compare a histogram of the empirical distribution with the approximate density obtained in question (e). Note: to get full marks, you need to show and clearly explain every step. When writing pmfs or pdfs, you should always define the sample spaces for the random variables, and the space where the parameters of the distributions are defined. 1