F79MA: Statistical Model Assignment, HWU, Malaysia Suppose that you are a trainee actuary working in the mathematical modeling team for a non-governmental organization that is rolling out a micro-credit scheme to support rural communities in developing countries

Project description

In a manner akin to Project 1, suppose that you are a trainee actuary working in the mathematical modeling team for a non-governmental organization that is rolling out a micro-credit scheme to support rural communities in developing countries. As a reminder, the scheme al- lows members of the community to borrow the equivalent of 50 Euros to support or develop a commercial activity (e.g., repair tools, buy consumables, expand production capacity, etc.). The micro-credit scheme only has one active loan within the community at any given time and operates the first-come-first-served queue. This introduces a form of community-based peer pressure that keeps the loan default rate low and that allows the micro-credit scheme to bypass formal credit checks and issue loans without charging a fee.

The scheme that you studied in Project 1 has been running for several months successfully. However, because of a change in economic conditions, the model needs to be updated. You decide to consider the following model for the time that an individual stays in the queue:

X ∼ Exp(θ),

where θ ∈ R  is unknown. The likelihood function associated with an observation x is

L(θ; x) = θ exp(−θx).

You decide to perform Bayesian inference on θ given a realization x = {xi} N
i=1 of an i.i.d. sample X1, . . . , XN of size N = 100. You do not wish to take into consideration any prior knowledge about θ, so you choose to use Jeffrey’s prior for θ, given by

πJ (θ) ∝ 1/θ.

You are required to perform the following analyses:

1. Derive the posterior density function π(θ|X = x). Check that this density function is proper. State the expression of the posterior mean and the posterior variance. Justify your answers (you are not required to derive the mean and variance from first principles).

2. Let P100
i=1 xi = 20. Plot the posterior density π(θ|X = x). Calculate the value of the posterior mean and the posterior variance. Using R, or otherwise, calculate an equi-tailed 95% credible interval for (θ|X = x).

3. Use simulation in R to compute the predictive distribution of a future observation X101 given the observations X1 = x1, . . . , X100 = x100, when P100 i=1 xi = 20. Plot the predictive density (e.g., use a histogram representation).

4. Use simulation in R to calculate a lower 95% predictive credible interval for X101 given
X1 = x1, . . . , X100 = x100 when P100 i=1 xi = 20.

Your findings should be presented in the form of a report for your line manager, who is also part of the statistical modeling team but has not worked directly on this problem. Your report should:

• have a clear and logical structure; include an introduction and clearly stated conclusions that can be understood by any numerate scientist;

• include detail of your mathematical calculations so that your results could be reproduced by another statistician;

• include clearly labeled and correctly referenced tables and diagrams, as appropriate;

• include the R code you used in an appendix (you do not need to explain individual R commands but some comments should be included to indicate the purpose of each section of code). Do not include figures, references, or mathematical developments in your appendix, only R code with comments. The R code should be included in text format that can be copied into R (not presented as an image file).

• include a citation and reference for any material (books, papers, websites, etc) used. When possible, use reliable sources, produced by respected and well-known authors, published by recognized publishers, and associated with the well-established government, academic, or educational institutions. Note that some web pages, YouTube videos, blog posts, and Wikipedia pages might include errors.

• Your report should be produced using a word-processing program, such as Microsoft Word, or LaTeX, and uploaded as a single file in PDF format. All equations should be produced digitally (typeset) by using an equation editor (we recommend avoiding scanned or handwritten equations). You may use the Microsoft Word template provided, but this is not compulsory.

• maximum page limit of four (4) pages (2 double-sided sheets of paper, 11-point font, A4 size). The R code in the appendix does not count toward this page limit. Only the R code should be presented in the appendix, there should be no other material in the appendix.

• Please complete the Standard Declaration of Student Authorship form and attach a copy of the completed form at the end of your report (you should submit a single file with this form attached at the end, after the appendix). This form does not count toward the page limit.

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