Flashcards in Formulas Deck (17)
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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
