Download 6 Solved Problems on Actuarial Statistics - Exam | MATH 3621 and more Exams Mathematics in PDF only on Docsity!
Problem 1:
The following exercise proves the decomposition formula:
n
i= 1
(Yi − Y)^2 ︸ ︷︷ ︸ Total SS
n
i= 1
(Yi − Ŷ i ︸ ︷︷ ︸
)^2
Error SS
n
i= 1
( Ŷ i − Y)^2 ︸ ︷︷ ︸ Regression SS
where ’SS’ stands for sum of squares.
(a) Expand the square to get n
i= 1
(Yi − Y)^2 =
n
i= 1
(Yi − Ŷ i)^2 +
n
i= 1
( Ŷ i − Y)^2 + 2
n
i= 1
̂ εi( Ŷ i − Y).
(b) Show that Ŷ i = Y + b 1 (Xi − X).
(c) Use part (b) to prove that n
i= 1
̂ εi = 0 and that
n
i= 1
̂ εi Ŷ i = 0.
(d) Finally, use part (c) to confirm that n
i= 1
̂ εi( Ŷ i − Y) = 0.
Problem 2:
In a study by Schmit and Roth (1990) about the cost effectiveness of risk management practices, a survey of 73 risk managers of large U.S. based organizations was conducted. The purpose was to re- late cost effectiveness to management’s philosophy of controlling the company’s exposure to various property and casualty losses, after adjusting for exogenous company factors such as size and industry type. For this exercise, we will simply be interested with understanding cost effectiveness, called firm- cost, defined to be total property and casualty premiums and uninsured losses as a percentage of total assets, and using the variable logassets, the logarithm of total assets as a possible explanatory vari- able. In the study, other controlling variables were present considering the company size measured in terms of assets is not usually within the control of the risk managers.
The following provides summary statistics of the firmcost and logassets variables:
risk.mng <- read.csv("C:/.../Math238-Fall2007/Data/C7-SURVY.csv") attach(risk.mng)
names(risk.mng) [1] "firmcost" "assume" "cap" "logassets" "indcost" "central" [7] "soph"
summary(firmcost) Min. 1st Qu. Median Mean 3rd Qu. Max.
0.20 3.51 6.08 10.97 12.71 97.
sd(firmcost) [1] 16.
summary(logassets) Min. 1st Qu. Median Mean 3rd Qu. Max. 5.270 7.650 8.270 8.332 8.950 10. sd(logassets) [1] 0.
The correlation between firmcost and logassets is:
cor(firmcost,logassets) [1] -0.
and this relationship is depicted in the figure below.
plot(firmcost˜logassets) abline(lm(firmcost˜logassets))
l
l ll l
l
l
l l l ll ll
l
l
ll l l
l l l
l ll
l
l
l l
l
l
l l (^) l l l l ll l (^) l l l l
l l
l
l l
l (^) l
l l l l l l
l l l l ll
l l
l l
l l
l
l
l
6 7 8 9 10
0
20
40
60
80
100
logassets
firmcost
(a) You are to fit the regression model
firmcost = β 0 + β 1 logassets + ε
to the data. Using all the available information from above, give the least squares estimates for the intercept and slope.
(b) Provide an interpretation of these estimates.
(c) Compute the coefficient of determination R^2 and interpret this number.
(d) Construct the analysis of variance table.
(Intercept) 30.396 3.674 8.273 1e-10 (^) *** eps 4.381 1.084 4.042 0.000195 (^) ***
Signif. codes: 0 ’’ 0.001 ’’ 0.01 ’’ 0.05 ’.’ 0.1 ’ ’ 1
Residual standard error: 18.63 on 47 degrees of freedom Multiple R-Squared: 0.2579, Adjusted R-squared: 0. F-statistic: 16.34 on 1 and 47 DF, p-value: 0.
anova(lm1) Analysis of Variance Table
Response: stkprice Df Sum Sq Mean Sq F value Pr(>F) eps 1 5667.7 5667.7 16.338 0.0001952 (^) *** Residuals 47 16304.7 346.
Signif. codes: 0 ’’ 0.001 ’’ 0.01 ’’ 0.05 ’.’ 0.1 ’ ’ 1
The fitted line to the data is also depicted in the figure below.
l
l
l l
l
l
l
l
l l
l
l
l l
ll
l
l
l
l
l
l
l
l
l
l l
l
l
ll
l
l
l
l l
l l
l
l
l
ll
l l
l
l l
l
0 5 10
20
40
60
80
100
eps
stkprice
(a) Write the fitted regression equation.
(b) Interpret the coefficients of this regression equation.
(c) Give the value of R^2 , explain how this can be obtained from the anova table, and interpret this value.
(d) Does the data support eps is an important predictor of stkprice? Justify your response.
Problem 4:
Large software packages comprise of a large number of modules, or subroutines. The modules often undergo careful testing to remove any “bugs”. Specification parameters are recorded for a random selection of modules to assess how the number of faults vary with these specification parameters.
The data in the table below record the number of faults, Y, together with the corresponding values of the specification parameters, SPEC, for 12 software modules written by the same programmer.
Y SPEC 3 14. 10 31. 4 2. 24 22. 5 8. 4 2. 43 53. 3 6. 2 2. 26 34. 3 2. 2 6.
(a) Using R, fit the following model to the data:
Y = β 0 + β 1 SPEC + ε.
Comment on the apparent fit of the model to the data.
(b) Now, fit the following model to the same set of data:
Y = β 0 + β 1 log(SPEC) + ε,
where ‘log’ refers to the natural logarithm. Comment on whether you think this is a more sensi- ble idea than the previous model.
