Chapter 16: Statistical distributions

Question 1

Features of a normal distribution MASTAM

Answer 1

• Mathematically tractable
• Approximation for Bin and Poi distribution
• Sysmetrictal
• Mesokurtic
• Any mean has a normal distribution – CLT
• Test statistics exist to check for normality

Question 2

Features of a t-distribution LSD TTT

Answer 2

• Symmetric
• Leptokurtic
• Derived as an MVN
• Tails of distribution follow power law
• Tends to normality for large values of gamma
• Test statistic

Question 3

Features of Log Normal distribution PIS FR

Answer 3

• Positive values only
• Insurance claims
• Skewed as a result
• Financial variable modelling
• Assume ln of variable will follow a random walk

Question 4

Features of Wald distribution PRAP

Answer 4

• Positively skewed
• Random walk with drift – time taken to reach a certain level
• Aggregates easily
• Positive values only

Question 5

Features of Exponential PEM IP

Answer 5

• Positively skewed
• Exponential tail decreases
• Monotonically decreasing
• Inflexible – only one parameter
• Poor fit to data

Question 6

Features of the Gamma FAF

Answer 6

• Flexible
• Aggregates easily
• Fitting is easy

Question 7

Features of a Fréchet MIES

Answer 7

• Monotonically decreasing
• Inflexible
• Extreme value theory usage
• Skewness to right

Question 8

Features of the Pareto MP

Answer 8

• Monotonically decreasing

* Power law in tails

Question 9

Testing for multivariate normality DEM CASM

Answer 9

1. Describe observations
2. Estimators created
3. Mahalanobis distance created
4. Compare to QQ plot of Chi-square
5. Angle of Mahalanobis created
6. Skewness and kurtosis created
7. Mardia’s test stat and kurtosis test stat created

Question 10

Jacque-Bera Test

Answer 10

T/6(ω^2-κ^2/4)