MSO4311 - Probability Density Functions - Financial Accounting Assignment Help

Assignment Task :

1. Consider the one-parameter family of probability density functions

where b > 0.
(a) Write R code to plot this pdf for various values of b > 0. 
(b) Determine the method of moments estimator bˆMoM for the parameter b.

(c) Find the Maximum Likelihood Estimator bˆMLE for the parameter b.

On the MyLearning page, there is a data file for use in this question.
Load the data into R using the command
D <- read.csv(‘‘path to file’’)$x
This data contains 1000 independent draws from the probability distribution with pdf FB(x).
(d) Write R code to calculate an appropriate Method of Moments Estimate and a Maximum Likelihood Estimate for the parameter b, given this data. 

2. Let X1, X2, . . . , Xn be non-constant iid random variables with finite mean and variance.
Suppose that μˆ is an unbiased estimator for E (X).
Show that (ˆμ)is a biased estimator of (E (X))2

3. Let X1, . . . , Xn be iid random variables with Xi ∼ N0, σ2

, where σ > 0 is unknown.

Using the theory of MLEs, determine an approximate distribution for the MLE σMLE in terms of σ and n, justifying
your answers.

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