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ACTL30004 Actuarial Statistics

创建日期：2024-06-01 17:19:35

所属分类：会计Accounting

标签： ACTL30004

Actuarial Statistics

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ACTL30004 Actuarial Statistics

Assignment 1, COVER SHEET
Due by 11:59 PM on Sunday 3 September 2023. Submission via LMS
This assignment contributes 15% of the total university assessment of this subject.
Please attach this cover sheet on top of your answers to your submission. Include your
name and student ID number in the table provided below
Student Number Name in full Signature
Declaration
I declare that this assignment is our own work and does not involve plagiarism or collu-
sion. I understand that penalties will be imposed if the instructions accompanying the
assignment are not followed.
Plagiarism and Collusion
Plagiarism is the presentation by a student of an assignment which has been copied in whole or
in part from another student’s work, or from any other source without due acknowledgement.
Collusion is the presentation by a student of an assignment as his or her own which is the
result, in whole or in part, of unauthorised collaboration, with another person(s). Allowing your
work to be seen or used by other students is also collusion, as is any form of discussion, before
submission, with any other student. A student who assists another student in any way is also
colluding.
Page 1 of 5
ACTL30004 Actuarial Statistics
Assignment 1, 2023
Instructions:
1. Complete the ACT30004 assignment cover sheet and include it in your
submission.
2. Write your answers to the questions below. You must show full working
in each question.
3. The total number of marks is 100. Up to 20% marks can be deducted if
your solutions are poorly presented.
4. You may include part of your code in your submission. You should
submit sufficient working so that the process you have implemented in
each question can be followed.
5. Your submission should be no longer than 12 pages excluding cover sheet.
You are reminded that heavy penalties apply to students who plagiarise or collude. These
terms are defined on the assignment cover sheet.
Part I
An actuarial student wishes to examine the relationship between the number of vehicle
accidents and the unemployment rate in a particular region. For this purpose, you use the
dataset traffic2 in the wooldridge R package and take as response and explanatory
variables totacc and unem respectively.
(i) Determine the simple linear regression model and estimate all three parameters of the
model. Plot the scatterplot and the best fitting line.
[10 marks ]
(ii) Predict a value of the response variable for an unemployment rate of 9%. Provide
the 90% and 95% confidence intervals of this prediction.
[5 marks ]
Page 2 of 5
(iii) Quantify the degree of association between the two variables by using the Pearson’s
correlation coefficient. Do not use in-built function cor(). Give comments on the
result. Test the hypothesis that the Pearson’s population coefficient ρ = 0 against
ρ ̸= 0 at the 5% significance level.
[10 marks ]
(iv) Quantify the degree of association between the two variables by using the Spearman’s
rank correlation coefficient. Do not use in-built function cor(). Give comments on
the result.
[10 marks ]
(v) Calculate the Kendall’s tau rank correlation coefficient, the number of concordant
and discordant pairs, and the number of ties. Do not use in-built function cor().
Use a modification of the two expressions given at the bottom of slide 8 of Lecture 1-
2. Give comments on the result. Test the hypothesis that Kendall’s rank correlation
coefficient τ = 0 against τ ̸= 0 at the 5% significance level.
[15 marks ]
Part II
The the danish dataset in the SMPracticals R package contains losses on major
insurance claims due to fires in Denmark, 1980-1990 adjusted to reflect 1985 values. It
consists of 2492 measurements in millions DKK. The values of the claims have been
rescaled for commercial reasons. For this question we will use the lognormal (LN) and
inverse Gaussian (IG) distributions whose probability density functions are given in the
Appendix.
(a) Use R to fit the two distributions by the method of moments to this set of data.
Comment your results.
[5 marks ]
(b) Use the estimates obtained in part (a) as initial values to compute the maximum
likelihood estimates for each model. Comment your results.
[10 marks ]
Page 3 of 5
(c) Plot a histogram of the data and superimpose the graphs of the probability den-
sity functions of the two distributions by using the maximum likelihood estimates
computed in Part (b).
[5 marks ]
(d) Use parametric bootstrap with B = 5, 000 to show that:
(i) For each distribution, the maximum likelihood estimates computed in part (b)
are consistent. You must consider
the lognormal distribution, LN(µˆ, σˆ), where µˆ, σˆ are the estimates ob-
tained in part (b);
the inverse Gaussian distribution, IG(µˆ, λˆ), where µˆ, λˆ are the estimates
obtained in part (b).
(ii) For each distribution, the maximum likelihood estimates computed in part (b)
are more efficient than the moment estimates obtained in part (a).
Use sample sizes of n := 20, 50, 100 to complete this question.

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MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 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