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In Course Assessment: STAT0020

Course Assessment代写 This assessment counts for 10% of your final mark for STAT0020.You should work in groups of no more than 4 students.

• Thisassessment counts for 10% of your final mark for STAT0020.Course Assessment代写

• You should work in groups of no more than 4 students. You may work on the project alone if you wish, but note that this is not efficient. It is up to you to form your own groups.
• I will request the student ID number of each group member on Tuesday 5^th of March at the end of lecture.
• You may use R, Matlab , or any other statistical software of your choice in order to solve some parts of the questions in this assessment.Course Assessment代写
• Non-submission of this assessment may mean that your overall examination mark is recorded as ”non-complete”, i.e., you may not obtain a pass mark for the course.
• The report should be uploaded to Moodle, and officially ‘submitted’ by 4pm on Tuesday , 19^th March.
• Each student must provide a statement of their contribution to the project signed with the submission.
• Please note: Each group should submit a single pdf file and a single R or Matlab script.Course Assessment代写
• The R or Matlab script will include your code and comments which may also refer- ence to which question the section of code relates.

• The pdf file will consist of your answers to the non-coding parts of the questions and the required output plots.

• One member of every group should upload the final version of the report, using the relevant submission link on Moodle (which will appear closer to the deadline), and press ‘submit’. This will require you to read and accept a plagiarism statement.
• All remaining members of the group should press their ‘submit’ button; again this will require each student to read and accept a plagiarism statement.
• All groups will be able to see which of of their members are yet to press‘submit’.
• The report won’t be considered submitted until every member has officially submitted the report, and this needs to be complete by the deadline above.
• You will work together within your group and the usual plagiarism and collusion regulations do not apply to this form of interaction. However, they do apply to collusion with other groups or plagiarism of work from other groups or from other sources.Course Assessment代写
• Any plagiarism will normally result in zero marks for all students involved, and may also mean that your overall examination mark is recorded as non-complete.
• Guidelines as to what constitutes plagiarism may be found in Departmental Student Handbooks. The relevant excerpt from the Statistical Science handbook is also posted in the ICA section of the Moodle page.

How will the report be marked?

Your report will be marked out of 100, with 30 percent of the marks allocated for presen- tation of report (including clarity of expression and presentation of proofs, figures, plots), and 70 percent of the marks available for the analyses and interpretation.

I will be looking for:

• Clarity of writing the expressions, proofs etc;Course Assessment代写

(Please remember you Do Not have to type all your report.)

• Abilityto select the most important points from an analysis;
• Sensibleuse and display of plots and tables (if necessary);
• Accuracy of interpretation of results
• Clarity and readability of computer

The mark you will receive is your group mark – everyone in the group will be awarded the same mark, except in exceptional circumstances (e.g. a member of a group did not contribute fairly to the project).Course Assessment代写

1 Compound process models under LDA frameworkCourse Assessment代写

Question A

(a)Consider a severity model for losses in a given business unit risk type givenby

X_i(t) ∼ Gamma(α, β) (1)

Write the expression for the distribution and density in closed form for the sum of n-losses, considering all X_i’s are independent and identically distributed random variables with Gamma distribution as mentioned in equation 1. [4]

(b)Writethe closed-form expression for the distribution and density in closed form for the compound process LDA model with Poisson frequency with intensity parameter λ and severity model characterised by Gamma distribution, as equation 1. [4]

(c)Create a Matlab or R script to plot the distribution and density function of the compoundprocess given by the Poisson-Gamma model, labelling clearly all compo- nents of the plots. Consider a range of parameter values for LDA model that you may choose, and specify the choices made in the reporting of the  [8]

(d)Write the steps you would take to simulate the annual loss distribution from this LDAmodel (Poisson-Gamma) using a Monte Carlo  [6]Course Assessment代写

(e)Implementyour Monte Carlo algorithm in R to Matlab for 10,000 simulated annual years for the same range of parameter values for the LDA model(Poisson-Gamma) that you choose (specify the choices made in the reporting of the results). Plot the histogram estimates of the annual loss distribution and density and compare them to the exact results  [8]

Question BCourse Assessment代写

(a)Consider a severity model for losses in a given business unit risk type givenby

X_i(t) ∼ S_α(β, γ, δ; 0) (2)

Write the expression for the distribution and density in closed form for the sum of n-losses when α = 0.5 and β = 1. [4]

(b)Writethe expression for the distribution and density in closed form for the compoundprocess LDA model with Poisson frequency with intensity parameter λ and severity model characterised by distribution in equation 2. [4]Course Assessment代写

(c)Create a Matlab or R script to plot the distribution and density function of the compoundprocess given by the Poisson-Stable model, labelling clearly all compo- nents of the plots. Consider a range of parameter values for LDA model that you may choose, and specify the choices made in the reporting of the  [9]

(d)Write the steps you would take to simulate the annual loss distribution from this LDAmodel (Poisson-Stable)by Monte Carlo  [5]Course Assessment代写

(e)Implementyour Monte Carlo algorithm in R to Matlab for 10,000 simulated annual years for the same range of parameter values for the LDA model(Poisson-Stable) that you choose (specify the choices made in the reporting of the results). Plot the histogram estimates of the annual loss distribution and density and compare them to the exact results  [8]

2 Heavy tail loss models and quantilesCourse Assessment代写

Question A

(a)Provide a detailed definition of a generic heavy tailed severity model, withdistri- bution F that admits as density f, from the perspective of the properties of the hazard function and hazard  Start by defining the hazard function and hazard rate, then explain its properties and link this to the heavy tailedness of the distri- bution. Provide a worked example to illustrate this point using one of the one or two parameter family of severity models. [10]Course Assessment代写

(b)Considerthe annual loss random variable Z given by compound process LDA model

NZ = X_n (3)n=0

for frequency random variable N ∼ G(n) and independent severity random variables that are independent identically distributed according to X_n F (x)for all losses.

Find an expression for the mean and variance of the annual loss in terms of moments of N and X_n. In the process show the following bound holds

V ar[Z] ≤ E[X²] max(E[N ], V ar[N ]) (4) [13]Course Assessment代写

3 Risk measure calculations and asymptotic approx- imations Course Assessment代写

Question A

(a)DefineVaR measure in a sufficiently general form to deal with differers types of supports for the loss random  [2]

(b)Providea definition for the Expected shortfall risk  [2]

Question B

• ConsiderPoisson-Lognormal LDA model, provide a first order asymptotic approxi- mation for the Poisson-Lognormal model tail  [3]
• Considerthe Poisson -Lognormal LDA model, using the first order asymptotic ap- proximation for the Poisson-Lognormal model tail  Create an R or Matlab script to plot this function for a range of parameters λ, µ and σ that you choose. Include the commented code in the answer and detail clearly the plots and comment on the results. [10]Course Assessment代写

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