##### Document Text Contents

Page 84

84 CHAPTER 4. MODEL SELECTION, VALIDATION, AND INFERENCE

As the above example suggests, there is flexibility with the method of moments. For example, we could

have matched the second and third moments instead of the first and second, yielding different estimators.

Furthermore, there is no guarantee that a solution will exist for each problem. You will also find that

matching moments is possible for a few problems where the data are censored or truncated, but in general,

this is a more difficult scenario. Finally, for distributions where the moments do not exist or are infinite,

method of moments is not available. As an alternative for the infinite moment situation, one can use the

percentile matching technique.

Percentile Matching

Under percentile matching, we approximate the quantiles or percentiles of the parametric distribution using

the empirical (nonparametric) quantiles or percentiles described in Section 4.1.1.

Show Example

Example – Property Fund. For the 2010 property fund, we illustrate matching on quantiles. In particular,

the Pareto distribution is intuitively pleasing because of the closed-form solution for the quantiles. Recall

that the distribution function for the Pareto distribution is

F (x) = 1 −

(

θ

x + θ

)α

Easy algebra shows that we can express the quantile as

F −1(q) = θ

(

(1 − q)−1/α − 1

)

for a fraction q, 0 < q < 1.

The 25th percentile (the first quartile) turns out to be 0.78853 and the 95th percentile is 50.98293 (both in

thousands of dollars). With two equations

0.78853 = θ

(

1 − (1 − .25)−1/α

)

and 50.98293 = θ

(

1 − (1 − .75)−1/α

)

and two unknowns, the solution is

α̂ = 0.9412076 and θ̂ = 2.205617.

We remark here that a numerical routine is required for these solutions as no analytic solution is available.

Furthermore, recall that the maximum likelihood estimates are α̂MLE = 0.9990936 and θ̂MLE = 2.2821147,

so the percentile matching provides a better approximation for the Pareto distribution than the method of

moments.

Exercise – Exam C Question 1. You are given:

(i) Losses follow a loglogistic distribution with cumulative distribution function:

F (x) =

(x/θ)γ

1 + (x/θ)γ

(ii) The sample of losses is:

10 35 80 86 90 120 158 180 200 210 1500

Calculate the estimate of θ by percentile matching, using the 40th and 80th empirically smoothed percentile

estimates.

Page 167

Bibliography

Bailey, R. A. and LeRoy, J. S. (1960). Two studies in automobile ratemaking. Proceedings of the Casualty

Actuarial Society Casualty Actuarial Society, XLVII(I).

Bishop, Y. M., Fienberg, S. E., and Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and

Practice. Cambridge [etc.]: MIT.

Blomqvist, N. (1950). On a measure of dependence between two random variables. The Annals of Mathe-

matical Statistic, pages 593–600.

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., and Nesbitt, C. J. (1986). Actuarial Mathematics.

Society of Actuaries Itasca, Ill.

Dabrowska, D. M. (1988). Kaplan-meier estimate on the plane. The Annals of Statistics, pages 1475–1489.

Dickson, D. C. M., Hardy, M., and Waters, H. R. (2013). Actuarial Mathematics for Life Contingent Risks.

Cambridge University Press.

Earnix (2013). 2013 insurance predictive modeling survey. Earnix and Insurance Services Office, Inc. [Re-

trieved on July 7, 2014].

Fechner, G. T. (1897). Kollektivmasslehre. Wilhelm Englemann, Leipzig.

Frees, E. and Valdez, E. (1998). Understanding relationships using copulas. North American Actuarial

Journal, 2(01):1–25.

Genest, C. and Mackay, J. (1986). The joy of copulas: Bivariate distributions with uniform marginals. The

American Statistician, 40:280–283.

Genest, C. and Neslohva, J. (2007). A primer on copulas for count data. Journal of the Royal Statistical

Society, pages 475–515.

Gorman, M. and Swenson, S. (2013). Building believers: How to expand the use of predictive analytics in

claims. SAS. [Retrieved on August 17, 2014].

Hettmansperger, T. P. (1984). Statistical inference based on ranks.

Hofert, M., Kojadinovic, I., Machler, M., and Yan, J. (2017). Elements of Copula Modeling with R. Springer.

Hougaard, P. (2000). Analysis of Multivariate Survival Data. Springer New York.

Insurance Information Institute (2015). International insurance fact book. Insurance Information Institute.

[Retrieved on May 10, 2016].

Joe, H. (2014). Dependence Modeling with Copulas. CRC Press.

Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, pages 81–93.

Lee Rodgers, J. and Nicewander, W. A. (1998). Thirteen ways to look at the correlation coeffeicient. The

American Statistician, 42(01):59–66.

167

84 CHAPTER 4. MODEL SELECTION, VALIDATION, AND INFERENCE

As the above example suggests, there is flexibility with the method of moments. For example, we could

have matched the second and third moments instead of the first and second, yielding different estimators.

Furthermore, there is no guarantee that a solution will exist for each problem. You will also find that

matching moments is possible for a few problems where the data are censored or truncated, but in general,

this is a more difficult scenario. Finally, for distributions where the moments do not exist or are infinite,

method of moments is not available. As an alternative for the infinite moment situation, one can use the

percentile matching technique.

Percentile Matching

Under percentile matching, we approximate the quantiles or percentiles of the parametric distribution using

the empirical (nonparametric) quantiles or percentiles described in Section 4.1.1.

Show Example

Example – Property Fund. For the 2010 property fund, we illustrate matching on quantiles. In particular,

the Pareto distribution is intuitively pleasing because of the closed-form solution for the quantiles. Recall

that the distribution function for the Pareto distribution is

F (x) = 1 −

(

θ

x + θ

)α

Easy algebra shows that we can express the quantile as

F −1(q) = θ

(

(1 − q)−1/α − 1

)

for a fraction q, 0 < q < 1.

The 25th percentile (the first quartile) turns out to be 0.78853 and the 95th percentile is 50.98293 (both in

thousands of dollars). With two equations

0.78853 = θ

(

1 − (1 − .25)−1/α

)

and 50.98293 = θ

(

1 − (1 − .75)−1/α

)

and two unknowns, the solution is

α̂ = 0.9412076 and θ̂ = 2.205617.

We remark here that a numerical routine is required for these solutions as no analytic solution is available.

Furthermore, recall that the maximum likelihood estimates are α̂MLE = 0.9990936 and θ̂MLE = 2.2821147,

so the percentile matching provides a better approximation for the Pareto distribution than the method of

moments.

Exercise – Exam C Question 1. You are given:

(i) Losses follow a loglogistic distribution with cumulative distribution function:

F (x) =

(x/θ)γ

1 + (x/θ)γ

(ii) The sample of losses is:

10 35 80 86 90 120 158 180 200 210 1500

Calculate the estimate of θ by percentile matching, using the 40th and 80th empirically smoothed percentile

estimates.

Page 167

Bibliography

Bailey, R. A. and LeRoy, J. S. (1960). Two studies in automobile ratemaking. Proceedings of the Casualty

Actuarial Society Casualty Actuarial Society, XLVII(I).

Bishop, Y. M., Fienberg, S. E., and Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and

Practice. Cambridge [etc.]: MIT.

Blomqvist, N. (1950). On a measure of dependence between two random variables. The Annals of Mathe-

matical Statistic, pages 593–600.

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., and Nesbitt, C. J. (1986). Actuarial Mathematics.

Society of Actuaries Itasca, Ill.

Dabrowska, D. M. (1988). Kaplan-meier estimate on the plane. The Annals of Statistics, pages 1475–1489.

Dickson, D. C. M., Hardy, M., and Waters, H. R. (2013). Actuarial Mathematics for Life Contingent Risks.

Cambridge University Press.

Earnix (2013). 2013 insurance predictive modeling survey. Earnix and Insurance Services Office, Inc. [Re-

trieved on July 7, 2014].

Fechner, G. T. (1897). Kollektivmasslehre. Wilhelm Englemann, Leipzig.

Frees, E. and Valdez, E. (1998). Understanding relationships using copulas. North American Actuarial

Journal, 2(01):1–25.

Genest, C. and Mackay, J. (1986). The joy of copulas: Bivariate distributions with uniform marginals. The

American Statistician, 40:280–283.

Genest, C. and Neslohva, J. (2007). A primer on copulas for count data. Journal of the Royal Statistical

Society, pages 475–515.

Gorman, M. and Swenson, S. (2013). Building believers: How to expand the use of predictive analytics in

claims. SAS. [Retrieved on August 17, 2014].

Hettmansperger, T. P. (1984). Statistical inference based on ranks.

Hofert, M., Kojadinovic, I., Machler, M., and Yan, J. (2017). Elements of Copula Modeling with R. Springer.

Hougaard, P. (2000). Analysis of Multivariate Survival Data. Springer New York.

Insurance Information Institute (2015). International insurance fact book. Insurance Information Institute.

[Retrieved on May 10, 2016].

Joe, H. (2014). Dependence Modeling with Copulas. CRC Press.

Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, pages 81–93.

Lee Rodgers, J. and Nicewander, W. A. (1998). Thirteen ways to look at the correlation coeffeicient. The

American Statistician, 42(01):59–66.

167