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1

MGF

E(e^tx) = Mx(t)

2

Franchise deductible in terms of regular deductible

E((X-d)+) +dS(d)

3

TVaR

E(X | X > VaRp(X)) =
VaRp(X) + (E(X) - E(X min VaRp(X))/(1-p)

4

Tail Weight Measures

1. more positive moments -> lower tail weight
2. if lim S1(x)/S2(x) > 1 or lim f1(x)/f2(x) > 1 then numerator has higher tail weight
3. increasing h(x) -> lighter tail
4. increasing ex(d) -> heavier tail

5

Consistency

theta hat is consistent if:

1. lim pr( | theta hat - theta| < delta ) = 1 for all delta > 0, or
2. bias -> 0 and Var (theta hat) -> 0

6

Cov (Fx, Fy - Fx)

= -Fx(Fy-Fx)/n, x< y

7

variance of exact exposure

var(qj) = (1-qj)^2 * dj/ej^2

8

var of actuarial exposure

qj(1-qj)/(ej/n)

9

percentile matching with incomplete data: censored/truncated

censored -> select percentiles within the range of uncensored observations

truncated -> match percentiles of the conditional distribution

10

MLE of grouped data btw d and cj and left-truncated from below at d:

(F(cj)-F(d))/S(d)

11

MLE = MOM

Poisson
Binomial
NB (r known)
Gamma (a known)
Normal mean/SD

12

Hypothesis tests - fitted distribution with deductible

F*(x) = 1- S(x)/S(d)

13

5 points about K-S

1. only for individual data
2. lower critical value if u < infinity
3. If params are fitted, critical value should be lowered
4. Larger sample size has lower critical value
5. Uniform weight on all parts of distribution

14

5 points about Chi-Sq

1. May be used for individual or grouped data
2. No adjustments on critical value if u < infinity
3. If parameters are fitted, critical value is automatically adjusted
4. Critical value is independent of sample size
5. Higher weight on intervals with low fitted probability

15

Loss functions

Type of loss/bayesian estimate
squared error/mean
absolute/median
zero-one/mode

16

lambda k

sk = -ln(1-uk)/lambda k

poisson = lambda
binomial = -mln(1-q) + k ln(1-q)
NB = r ln(1+B) + k ln (1+B)

sum from 0 - n until sum > 1, result is n.

'time between' type questions.

17

Polar method

v = 2u - 1
s = v1^2 + v2^2 <= 1 then T = sqrt(-2ln(S)/S)