Download cumulative distribution function distribution function cdf Exam C

cumulative distribution function
distribution function
cdf
survival function
Exam C
probability density function
density function
Exam C
probability function
probability mass function
Exam C
hazard rate
force of mortality
failure rate
Exam C
truncated
Exam C
censored
Exam C
k-th raw moment
Exam C
empirical model
Exam C
k-th central moment
Exam C
Exam C
SX (x) = P r(X > x) = 1 − FX (x)
Z
x
F (x) =
f (s)ds = P (X < x)
−∞
f (x) = F 0 (x) = −S 0 (x)
pX (x) = Pr(X = x)
An observation is truncated at d if when it is below
d it is below d it is not recorded but when it is above
d it is recorded at itsobserved value.
The kth raw moment of a random variable is the
expected value of the kth power of the variable, provided it exists. It is denoted by E(X k ) or by µ0k . The
first raw moment is called the mean of the random
variable and is usually denoted by µ.
Z ∞
X
0
k
xk f (x)dx =
xkj p(xj )
µk = E(X ) =
Z
−∞
hX (x) =
fX (x)
SX (x)
An observation is censored at u if, when it is above u
it is recorded as being equal to u but when it is below
u it is recorded at its observed value.
j
∞
E(x) =
S(x)dx
0
The kth central moment of a random variable is
the expected value of the kth power deviation of the
variable from its mean. It is denoted by E[(X − µ)k ]
or by µk . The second central moment is usually called
the variance and denoted σ 2 and the square root, σ,
is called the standard deviation.
Z ∞
µk = E[(X − µ)k ] =
(x − µ)k f (x)dx
−∞
X
=
(xj − µ)k p(xj )
j
The empirical model is a discrete distribution based
on a sample size n which assigns probability 1/n to
each data point.
coefficient of variation
skewness
Exam C
kurtosis
Exam C
symmetric distribution, then it has a skewness
Exam C
limited loss variable
right censored variable
Exam C
limited expected value
Exam C
left censored and shifted variable
And E(left censored and shifted variable)
Exam C
left censored and shifted variable
mean residual life function
complete expectation of life
left truncated and shifted variable
Exam C
mean excess loss function
Exam C
percentile
Exam C
Exam C
µ3
σ3
σ
µ
µ3
=0
σ3
µ4
σ4
½
E[X ∧ u]
Z
k
u
E[(X ∧ u) ] =
0
Z
=
X,
u,
Y =X ∧u=
X<u
X≥u
xk f (x) + uk [1 − F (u)]
u
S(x)dx
0
Y = X − d given that X > d
½
0,
X<d
X − d, X ≥ d
Y = (X − d)+ =
E[(X − d)+ ] = E(X) − E(X ∧ d)
Z ∞
=
S(x)dx
d
The 100pth percentile of a random variable is any
value πp such that F (πp −) ≤ p ≤ F (πp ).
eX (d) = e(d) = E(Y ) = E(X − d|X > d)
E(X − d|X > d) =
E(X) − E(X ∧ d)
1 − F (d)
Sk − E(Sk )
Distribution of lim p
k→∞
V ar(Sk )
median
Exam C
Exam C
moment generating function
E(X n ) =
moment generating function
Exam C
probability generating function
Exam C
pk from probability generating function
Exam C
Mean and Variance
using the probability generating function
Exam C
pgf/mgf of
Pn
j=1
Xj
Exam C
parametric distribution
Exam C
scale distribution
Exam C
Exam C
Normal distribution with mean 0 and variance 1.
(n)
E(X n ) = MX (0)
The 50th percentile, π0.5 is called the median.
MX (t) = E(etX )
(m)
P
(0)
pk = X
m!
The mth derivative of the pgf evalated at 0 devided by
m factorial.
MSk (t) =
k
Y
MXj (t)
j=1
PSk (z) =
k
Y
PXj (z)
X
PX (z) = E(z ) =
∞
X
pk z k
k=0
0
E(X) = PX
(1)
00
0
E[X 2 ] = PX
(1) + PX
(1)
00
0
0
Var(X) = PX
(1) + PX
(1) − [PX
(1)]2
j=1
A parametric distribution is a scale distribution if,
when a random variable from that set of distributions
is multiplied by a positive constant, the resulting random variable is also in that set of distributions.
A parametric distribution is a set of distribution
functions, each member of which is determined by
specifying one or more values called parameters. The
number of parameters is fixed and finite.
scale parameter
parametric distribution family
Exam C
k-point mixture
Exam C
variable-component mixture distribution
Exam C
data-dependant distribution
Exam C
Equilibrium Distribution
Exam C
Survival function and hazard rate of Equilibrium
Distribution
Exam C
Survival function based on equilibrium distribution
Exam C
coherent risk measure
Exam C
Value-at-Risk
Exam C
Exam C
A parametric distribution family is a set of parametric distributions that are related in some meaningful way.
For random variables with nonnegative support, a
scale parameter is a parameter for a scale distribution that meets two conditions. First, when a member of the scale distribution is multiplied by a positive
constant, the scale parameter is multiplied by a positive constant, the scale parameter is multiplied by the
same constant. Second, when a member of the scale
distribution is multiplied by a positive constant, all
other parameters are unchanged.
A variable-component mixture distribution has
a distribution function that can be written as
A random variable Y is a k-point mixture of the
random variables X1 , X2 , · · · , Xk if its cdf is given by
F (x) =
K
X
aj Fj (x),
j=1
K
X
aj = 1, aj > 0
j=1
where all aj > 0 and a1 + a2 + · · · + ak = 1.
Assume X is a continuous distribution f (x) with a survival function S(x) and mean E(X). Then the equilibrium distribution is
fe (x) =
S(x) =
S(x)
,
E(X)
FY (y) = a1 FX1 (y) + a2 FX2 (y) + · · · + ak FXk (y)
A data-dependant distribution is at least as complex as the data or knowledge that produced it, and
the number of “parameters” increases as the number
of data points or amount of knowledge increase.
x≥0
1
E(x) − R0x [ e(t)
]dt
e
e(x)
Z
Se (x) =
fe (t)dt =
x
he (x) =
Let X denote a loss random variable. The Value-atRisk of X at the 100p% level, denoted VaRp (X) or πp ,
is the l00p percentile (or quantile) of the distribution
of X. And for continuous distributions it is the value
of πp satisfying
R∞
∞
x
S(t)dt
,
E(x)
x≥0
fe (x)
S(x)
1
= R∞
=
Se (x)
e(x)
S(t)dt
x
1. Subadditivity: ρ(X + Y ) ≤ ρ(X) + ρ(Y ).
2. Monotonicity: If X ≤ Y for all possible outcomes, then ρ(X) ≤ ρ(Y ).
Pr(X > πp ) = 1 − p
3. Positive homogeneity: For any positive constant
c. ρ(cX) = cρ(X).
VaR is not “coherent” as it does not meet the subaddivity requirement in some cases.
4. Translation invariance: For any positive constant c. ρ(X + c) = ρ(X) + c.
Given: F (x) and f (x) find FY (y) when
Y = θX
Tail-Value-at-Risk
Exam C
Given: F (x) and f (x) find FY (y) when
Y = X 1/τ
Exam C
transformed
Exam C
inverse
Exam C
inverse transformed
Exam C
Exam C
Given: F (x) and f (x) find FY (y) when
Y = eX
Given: F (x) and f (x) find FY (y) when
Y = g(X)
Exam C
Exam C
Raw Moments of a Mixture
Mixture Distribution (FX (x))
Exam C
Exam C
FY (y) = FX
³y ´
θ
³y ´
1
fY (y) = fX
θ
θ
Y = X 1/τ
τ >0
Let X denote a loss random variable. The TailValue-at-Risk of X at the 100p% security level, denoted TVaRp (X), is the expected loss given that the
loss exceeds the l00p percentile (or quantile) of the
distribution of X. And for continuous distributions it
can expressed as
R∞
xf (x)dx
π
TVaRp (X) = E(X|X > πp ) = p
1 − F (πp )
FY (y) = FX (y τ )
fY (y) = τ y τ −1 fX (y τ )
Y = X 1/τ
Y = X 1/τ
τ =< 0 and τ 6= −1
τ = −1
h(y) = g −1 (Y )
FY (y) = FX [h(y)]
fY (y) = |h0 (y)|fX [h(y)]
FY (y) = FX [ln(y)]
1
fY (y) = fX [ln(y)]
y
E(X k ) = E[E(X k | Λ)]
Z
FX (x) =
FX|Λ (x | λ)fΛ (λ)dλ
Important Mixtures:
Y = Poison(Λ)
Λ = Gamma(α, θ)
Variance of a Mixture
Exam C
Important Mixtures:
Y = Exponential(Λ)
Λ = Inv.Gamma(α, θ)
Exam C
Important Mixtures:
Y = Inv.Exponential(Λ)
Λ = Gamma(α, θ)
Exam C
Important Mixtures:
Y = Normal(Λ, σc2 )
Λ = Normal(µ, σd2 )
Exam C
k-component spliced distribution
Exam C
linear exponential family
Exam C
normalizing constant
Exam C
canonical parameter
Exam C
linear exponential family: Mean
Exam C
Exam C
Neg.Bin(r = α, β = θ)
Var(X) = E[V ar(X | Λ)] + V ar[E(X | Λ)]
Inv.Pareto(τ = α, θ)
Pareto(α, θ)
A k-component spliced distribution has a density
function that can be expressed as follows:

a1 f1 (x),
c0 < x < c 1



 a2 f2 (x),
c1 < x < c 2
fX (x) =
..
..

.
.



ak fk (x), ck−1 < x < ck
Normal(µ, σc2 + σd2 )
q(θ)
f (x; θ) =
f (x; θ) =
p(x)er(θ)x
q(θ)
E(X) = µ(θ) =
q 0 (θ)
r0 (θ)q(θ)
p(x)er(θ)x
q(θ)
r(θ)
f (x; θ) =
p(x)er(θ)x
q(θ)
linear exponential family: Variance
(a,b,0) class of distribution
Exam C
(a,b,1) class of distribution
Exam C
Poison (a,b,0) specification
Exam C
Binomial (a,b,0) specification
Exam C
Negative Binomial (a,b,0) specification
Exam C
Exam C
Geometric Relation to Negative Binomial
Geometric (a,b,0) specification
Exam C
memoryless
Exam C
E[(1 + r)X ∧ c]
Exam C
Exam C
Let pk be the pf of a discrete random variable. It is
member of the (a,b,0) class of distributions, provided that there exists constants a and b such that
pk
pk−1
=a+
b
k
Var(X) =
µ0 (θ)
r0 (θ)
k = 1, 2, 3, · · ·
a=0
b=λ
p0 = e−λ
Let pk be the pf of a discrete random variable. It is
member of the (a,b,0) class of distributions, provided that there exists constants a and b such that
pk
pk−1
=a+
b
k
k = 2, 3, 4, · · · p0
=1−
∞
X
pk
k=1
It is called truncated if p0 = 0.
It is called zero-modified if p0 > 0 and is a mixture
of an (a,b,0) class and a distribution where p0 = 1.
(AKA truncated with zeros)
a=
β
1+β
β
1+β
p0 = (1 + β)−r
b = (r − 1)
a=−
q
1−q
b = (m + 1)
q
1−q
p0 = (1 − q)m
Geomtric is Negative Binomial with parameter r = 1.
β
1+β
b=0
a=
p0 = (1 + β)−1
E[(1 + r)X ∧ c] = (1 + r)E [X ∧ c∗ ]
c
c∗ =
1+r
P (X > x + y|X > x) = P (Y > y)
The Geometric and Exponential distributions are
both examples of a memoryless distribution.
per-loss variable
per-payment variable
Exam C
Relationship between per-loss and per-payment
Exam C
Relationship between per-loss and per-payment
Variance
Exam C
franchise deductible
Expectation
Exam C
ordinary deductible
Expectation
Exam C
Exam C
Co-insurance
The loss elimination ratio
Exam C
Co-insurance, deductible and limits variable
Exam C
Exam C
Co-insurance, deductible and limits variable
Expectation
Exam C
½
YP =
undef ined,
X − d,
½
X≤d
X>d
¡
¢ µ
¶2
E [Y L ]2
E(Y L )
−
Var(Y ) =
SX (d)
SX (d)
P
YL =
0,
X≤d
X − d, X > d
If f meets certian conditions [not sure how to define
them, (linear?)] then
f (Y P ) =
f (Y L )
SX (d)
E(Y P ) =
E(Y L )
SX (d)
However,
Var(Y P ) 6=
E(X) − E(X ∧ d)
Var(Y L )
SX (d)
A franchise deductible modifies the ordinary deductible by adding the deductible whenever there is a
positive amount paid. For a franchise deductible the
expected cost per loss is
E(X) − E(X ∧ d) + d[1 − F (d)]
If co-insruance is the only modification, this changes
the loss variable X to the payment variable Y = αX.
E(X) − [E(X) − E(X ∧ d)]
E(X ∧ d)
=
E(X)
E(X)
E(Y L ) = α(1 + r) [E (X ∧ u∗ ) − E (X ∧ d∗ )]
Y
L

 0,
α[(1 + r)X − d], d∗
=

α(u − d),
u∗
X
≤ X
≤ X
< d∗
< u∗
Co-insurance, deductible and limits variable
2nd Raw Moment
individual risk model
Exam C
collective risk model
OR
Compound Model
Exam C
Compound Model
Mean
Exam C
Compound Model
Variance
Exam C
Compound Model
N = Poisson(λ)
Exam C
Exam C
P r(a < S < b) = 0. Then, for a ≤ d ≤ b
E[(S − d)+ ] =
stop-loss insurance
Exam C
Theorem to calculate E[(S − d)+ ] using a discrete
probability function w(P (S = kh) = fk ≥ 0) with
equally spaced (h) nodes.
Exam C
Exam C
Convolution method (X1 + X2 )
Exam C
The individual risk model represents the aggregate
loss as a sum, S = X1 + . . . + Xn , of a fixed number, n, of insurance contracts. The loss amounts for
the n contracts are (X1 , . . . , Xn ), where the Xj s are
assumed to be independent but are not assumed to
be identically distributed. The distribution of the Xj s
usually has a probability mass at zero, corresponding
to the probability of no loss or payment
E[(Y L )2 ] =
= α2 (1 + r)2 {E[(X ∧ u∗ )2 ] − E[(X ∧ d∗ )2 ]
− 2d∗ E(X ∧ u∗ ) + 2d∗ E(X ∧ d∗ )}
S = X1 + . . . + XN with:
E[S] = E[E(S|N )]
= E(N )E(X) = µn µx
1. Conditional on N = n, the random variables
X1 , X2 , . . . , Xn are i.i.d random variables.
2. Conditional on N = n, the common distribution
of the random variables X1 , X2 , . . . , Xn , does not
depend on n.
3. The distribution of N does not depend in any
way on the values of X1 , X2 , . . .
E[S] = µn µx = λµx
V ar[S] =
λ(σx2
+
µ2x )
(N=Poisson)
2
= λE(X )
(N=Poisson)
V ar[S] = V ar[E(S|N )] + E[V ar(S|N )]
= Var(N )E(X)2 + E(N )Var(X)
= σn2 µ2x + µn σx2
b−d
E[(S − d)+ ] =
E[(S − a)+ ]
b−a
d−a
+
E[(S − b)+ ]
b−a
FS (k) =
X
fX1 (j)FX2 (k − j)
Insurance on the aggregate losses, subject to a deductible, is called stop-loss insurance. The expected
cost of this insurance is called the net stop-loss premium and can be computed as E[(S − d)+ ], where d
is the deductible and the notation (.)+ means to use
the value in parenthesis if it is positive but to use zero
otherwise
k = 0, 1, . . .. Then, provided d = jh, with j a nonnegative integer
all j
Z
fS (s) =
fX1 (t)fX2 (s − t)dt
Z
FS (s) =
fX1 (t)FX2 (s − t)dt
E[(S − d)+ ] = h
∞
X
{1 − FS [(m + j)h]}
m=0
E{[S − (j + 1)h]+ } = E[(S − jh)+ ] − h[1 − FS (jh)]
Convolution method (X1 + . . . + Xn )
Bias
Exam C
Exam C
consistent
asymptotically unbiased
Exam C
Exam C
uniformly minimum variance unbiased estimator
UMVUE
mean-squared error (MSE)
Exam C
confidence interval
Exam C
significance level
Exam C
uniformly most powerful
Exam C
p-value
Exam C
Exam C
biasθ̂ = E(θ̂ | θ) − θ
Z
∗n
FX
(x) =
Z
∗n
fX
(x) =
lim P r(|θ̂n − θ| > σ) = 0
n→∞
An estimator, θ̂, is called a uniformly minimum
variance unbiased estimator (UMVUE) if it is
unbiased and for any true value of θ there is no other
unbiased estimator that has a smaller variance.
x
0
∗(n−1)
FX
x
0
∗(n−1)
fX
(x − t)fX (t)dt
(x − t)fX (t)dt
lim E(θ̂n | θ) = θ
n→∞
M SEθ̂ (θ) = E[(θ̂ − θ)2 |θ]
= V ar(θ̂|θ) + [biasθ̂ (θ)]2
The significance levelof a hypothesis test is the probability of making a Type I error given that the null
hypothesis is true. If it can be true in more than one
way, the level of significance is the maximum of such
probabilities. The significance level is usually denoted
by the letter α.
A 100(1 − α)% confidence interval for a parameter
θ is a pair of random values, L and U , computed from
a random sample such that P r(L ≤ θ ≤ U ) ≥ 1 − α
for all θ.
The p-value is the smallest level of significance at
which H0 would be rejected when a specified test procedure is used on a given data set. Once the p-value
has been determined the conclusion at any particular
level α results from computing the p-value to α:
A hypothesis test is uniformly most powerful if no
other test exists that has the same or lower significance
level and for a particular value within the alternative
hypothesis has a smaller probability of making a Type
II error.
1. p-value ≤ α ⇒ reject H0 at level α.
2. p-value > α ⇒ do not reject H0 at level α.
[Probability and Statistics, Devore, 2000]
Log-Transformed Confidence Interval
empirical distribution
Fn (x) =
Sn (x)
Exam C
kernel smoothed distribution
Exam C
data set: variables
Exam C
data summary
Exam C
E[Sn (x)]
Exam C
Exam C
Sample Variance
Var[Sn (x)]
Exam C
Exam C
Empirical estimate of E[(X ∧ u)k ]
Empirical estimate of the variance
Exam C
Exam C
The empirical distribution is obtained by assigning
probability 1/n to each data point.
number of observations ≤ x
n
Sn (x) = 1 − Fn (x)
Fn (x) =
The 100(1 − α)% log-transformed confidence interval for Sn (t) is
³
´
Sn (t)U , Sn (t)(1/U )
where
q


d n (t)]
zα/2 Var[S

U = exp 
Sn (t) ln Sn (t)
A kernel smoothed distribution is obtained by replacing each data point with a continuous random
variable and then assigning probability 1/n to each
such random variable. The random variable used must
be identical except for a location or scale change that
is related to its associated data point.
1. n - insureds
2. di - entry time
3. xi - death time
4. ui - censored time
Y = number greater than x in sample
E[Sn (x)] = S(x) =
Y
n
1. m - death points
2. yj - death point time
3. sj - deaths @ time yj
4. rj - number @ risk @ time yj
n
1 X
(xi − x̄)2
n − 1 i=1


1 X k
xi − uk · [number of xi ’s > u]
n
xi ≤u
Y = number greater than x in sample
Y
n
S(x)[1 − S(x)]
E[Sn (x)] =
n
S(x) =
n
1X
(xi − x̄)2
n i=1
cumulative hazard rate function
Nelson-Åalen estimate
Exam C
variance of the Nelson-Åalen estimate
Exam C
Kaplan-Meier product-limit Estimator
Exam C
Greenwood Approximation
Variance of the Kaplan-Meier product-limit
Estimator
Exam C
log-transformed interval for the Nelson-Åalen
estimate
Exam C
method of moments
Exam C
percentile matching
Exam C
Exam C
likelihood function
log-likelihood function
smoothed empirical estimate
Exam C
Exam C


x < y1
 0,
Pj−1 si
Ĥ(x) =
i=1 ri , yj−1 ≤ x < yj , j = 2, . . . , k

 Pk si , x ≥ y
i
k
i=1 ri
Sn (t) =

1,


³
 Q



H(x) = − ln S(x)
d Ĥ(yj )] =
Var[
0 ≤ t < y1
´
j−1 ri −si
i=1
ri
Qk ³ ri −si ´
i=1
ri
j
X
si
r2
i=1 i
yj−1 ≤ t < yj , j = 2, . . . , k
t ≥ yk
j
= Ĥ(t)U,

where
q

d Ĥ(yj )]
zα/2 Var[

U = exp ±
Ĥ(t)
πgk (θ) = π̂gk , k = 1, 2, . . . , p
X
.
2
d n (yj )] =
Var[S
[Sn (yj )]
i=1
si
ri (ri − si )
µ0k (θ) = µ̂0k , k = 1, 2, . . . , p
F (π̂gk | θ) = gk , k = 1, 2, . . . , p
Here b.c indicates the greatest integer function and
x(1) ≤ x(2) ≤ . . . ≤ x(n) are the order statistics from
the sample.
L(θ) =
n
Y
Pr(Xj ∈ Aj | θ)
j=1
l(θ) = ln L(θ)
π̂g = (1 − h)x(j) + hx(j+1) , where
j = b(n + 1)gc and h = (n + 1)g − j
Information function
Variance of the likelihood estimate (θ)
Exam C
Delta Method (single variable)
Exam C
Delta Method (general)
Exam C
Non-normal confidence interval
Exam C
p-p plot
Exam C
Exam C
Kolmogorov-Smirnov Test
D(x) plot
Exam C
Anderson-Darling test
Exam C
Chi-Square Goodness-of-fit
Exam C
Exam C
−1
·
¸
∂2
I(θ) = −E
ln L(θ)
∂θ2
"µ
¶2 #
∂2
ln L(θ)
=E
∂θ2
Z
∂2
= −n f (x; θ) 2 ln[f (x; θ)]dx
∂θ
Var(θ̂) = [I(θ)]
Let θ̂ n = (θ̂1n , . . . , θ̂kn )T be a multivariate parameter
vector of dimension k based on a sample size of n.
Assume that θ̂ n is asymptotically normal with mean
of θ̂ and covariance matrix Ω/n. Then g(θ1 , . . . , θk ) is
asymptotically normal with
Let θ̂n be a parameter estimated using a sample size
of n. Assume that θ̂n is asymptotically normal with
mean of µ and variance of σ 2 /n. Then g(θ̂n ) is asymptotically normal with
E[g(θ)] = g(θ̂n ))
E[g(θ)] = g(θ̂ n )
Var[g(θ)] = (∂g)T
Var[g(θ))] = [g 0 (θ̂n )]2
Ω
∂g
n
(
(Fn (xj ), F ∗ (xj )), where
j
Fn (xj ) =
n+1
θ : l(θ) ≥ l(θ̂) −
χ2α/2
)
2
where the first term is the loglikelihood value at the
maximum likelihood estimate and the second term is
the 1 − α percentile from the chi-square distribution
with degrees of freedom equal to the number of estimated parameters.
D = max |Fn (x) − F ∗ (x)|
D(x) = Fn (x) − F ∗ (x), where
j
Fn (xj ) =
n
t≤x≤u
Z
k
X
n(p̂j − pnj )2
χ =
p̂j
j=1
2
u
[Fn (x) − F ∗ (x)]2 ∗
f (x)dx
∗
∗
t F (x)[1 − F (x)]
= −nF ∗ (u)
A =n
2
k
X
(Ej − Oj )2
=
Ej
j=1
σ2
n
+n
k
X
[1 − Fn (yj )]2 (ln[1 − F ∗ (yj )] − ln[1 − F ∗ (yj+1 )])
j=0
The critical values for this test comes from the chisquare distribution with degrees of freedom of (k − 1 −
r). Where k is the number of terms in the sum and r
is the number of estimated parameters values.
+n
k
X
j=0
Fn (yj )2 [ln F ∗ (yj+1 ) − ln F ∗ (yj )]
likelihood ratio test
Schwarz Bayesian Criterion
Exam C
Full credibility
(Single Variable Case)
Exam C
Full credibility
(Poisson)
Exam C
Exam C
Full Credibility
Compound Distributions
n
X
Si
Full Credibility
Compound Distributions
# of Si ’s
i=1
Exam C
Full Credibility
Compound Distributions
n
X
Si
µ
i=1 y
Exam C
Full Credibility
Compound Distributions
N = P oisson(λ)
# of Si ’s
Exam C
Exam C
Full Credibility
Compound Distributions
N = P oisson(λ)
n
X
Si
Full Credibility
Compound Distributions
N = P oisson(λ)
n
X
Si
µ
i=1 y
i=1
Exam C
Exam C
Recommends that when ranking models a deduction of
(r/2) ln n should be made from the loglikelihood value,
where r is the number of estimated parameters and n
is the sample size.
L0 = L(θ 0 )
L1 = L(θ 1 )
µ ¶
L1
T = 2 ln
= 2 (ln L1 − ln L0 )
L0
The critical values come from a chi-square distribution
with degrees of freedom equal to the number of free
parameters in L1 less the number of free parameters
in L0 .
n ≥ n0
1
λ
W n ≥ n0
µ ¶2
σ
V ar[W ]
2
= n0
= n0 CVW
µ
(E[W ])2
V ar[W ]
σ2
2
= n0
= n0 E[W ]CVW
W n ≥ n0
µ
E[W ]
n ≥ n0
σn2 µ2y + µn σy2
µn µy
n0
σn2 µ2y + µn σy2
(µn µy )2
"
#
σy2
1
1+ 2
n0
λ
µy
n0
σn2 µ2y + µn σy2
µn µ2y
n0
"
n0
σy2
1+ 2
µy
#
"
n0
σy2
µy +
µy
#
Partial Credibility
Model distribution
Exam C
Joint distribution
Exam C
Prior distribution
Exam C
marginal distribution
Exam C
posterior distribution
Exam C
Exam C
Bayes Premium
E(xn+1 )
predictive distribution
Exam C
Exam C
Bayes Premium
E(xn+1 )
with: E[X|θ] = θ
(eg. X Poisson)
Z
∞
xa e−cx dx
0
Exam C
Exam C
The model distribution (mx|θ ) is the probability
distribution for the data as collected given a particular value for the parameter. Its pdf is denoted
mx|θ = fX|Θ (x|θ).
mx|θ = fX|Θ (x|θ) =
n
Y
fX|Θ (xi |θ)
Q = ZW + (1 − Z)M Where,
Z=
Ãs
!
information available
min
,1
information required for full credibility
i=0
The prior distribution (πθ ) is a probability over
the space of possible parameter values. It is denoted
π(θ) and represents our opinion concerning the relative chances that various values of θ are the true value
of the parameter.
The joint distribution (jx,θ ) has pdf
The posterior distribution (pθ|x ) is the conditional
probability distribution of the parameters given the
observed data. Its pdf is
The marginal distribution (gx ) of x has pdf
Z
Z
gx = fX (x) = jx,θ dθ = fX|Θ (x|θ)π(θ)dθ
jx,θ = fX,Θ (x, θ) = mx|θ π(θ) = fX|Θ (x|θ)π(θ)
jx,θ
gx
fX|Θ (x|θ)π(θ)
=R
fX|Θ (x|θ)π(θ)dθ
pθ|x = πΘ|X (θ|x) =
Z
E(xn+1 |x) =
E(x|θ)pθ|x dθ
The predictive distribution is the conditional probability distribution of a new observation y given the
data x = x1 , . . . , xn . Its pdf
Z
fY |X (y|x) = fY |Θ (y|θ)pθ|x dθ
Z
= fY |Θ (y|θ)πΘ|x (θ|x)dθ
Z
Z
∞
0
Γ(a + 1)
ca+1
a!
= a+1 ‘a’ is an integer
c
xa e−cx dx =
E(xn+1 |x) =
θpθ|x dθ
or the mean of the posterior distribution.
Z
0
∞
Conjugate Prior
Poisson-Gamma
λ = gamma(α, θ)
x = Poisson(λ)
−c
x
e
dx
xk
Exam C
Conjugate Prior
Exponential-Inverse Gamma
λ = gamma−1 (α, θ)
x = exp(λ)
Exam C
Conjugate Prior
Binomial-Beta
q = beta(a, b, 1)
x = bin(m, q)
Exam C
Conjugate Prior
Inverse Exponential-Gamma
λ = gamma(α, θ)
x = exp−1 (λ)
Exam C
Conjugate Prior
Normal-Normal
λ = normal(µ, a2 )
x = normal(λ, σ 2 )
Exam C
Conjugate Prior
Uniform-Pareto
λ = single.pareto(α, θ)
x = uniform(0, λ)
Exam C
hypothetical mean
or collective premium
Exam C
Exam C
expected value of the hypothetical means
EVHM
process variance
Exam C
Exam C
µ
X
gamma α +
xi ,
θ
nθ + 1
¶
Z
∞
0
beta(a +
X
xi , b + km −
X
normal
gamma−1 (α + n, θ +
xi , 1)
µ· P
¸ ·
¸
xi
µ
n
1
+ 2 / 2+ 2 ,
σ2
a
σ
a
n
σ2
1
+
−c
ex
Γ(k − 1)
dx =
k
x
ck−1
(k − 2)!
=
ck−1
¶
1
a2
Ã
·
k>1
k≥2
X
1 X 1
gamma α + n,
+
θ
xi
xi )
¸−1 !
µ(θ) = E(Xij |Θi = θ)
single.pareto(α + n, max(x, θ))
µ = E[µ(θ)]
v(θ) = mij V ar(Xij |Θi = θ)
expected value of the process variance
EVPV
variance of the hypothetical means
VHM
Exam C
Bühlmann’s k
or credibility coefficient
Exam C
Bühlmann credibility factor
Exam C
Bühlmann credibility premium
Exam C
Var(X) = f(EVPV,VHM)
Exam C
Non-Paramtric estimation: “µ”
Exam C
Non-Paramtric estimation: “v”
Exam C
Non-Paramtric estimation: “a”
Method using c =
Exam C
Non-Paramtric estimation: “a”
Loss models technique
Exam C
Exam C
a = V ar[µ(θ)]
Zi =
v = E[v(θ)]
mi
mi + v/a
k=
Var(X) = a + v = EV P V + V HM
r
Zi X̄ + (1 − Zi )µ
n
i
XX
¢2
¡
1
mij Xij − X i
(n
−
1)
i=1 i
i=1 j=1
µ=X
v = Pr
Ã
a=
r
1 X 2
m−
m
m i=1 i
!−1 "
r
X
i=1
#
2
mi (X̄i − X̄) − v(r − 1)
v
a
" r
#−1
mi ´
r − 1 X mi ³
1−
c=
r
m
m
i=1
Ã
" r
!
#
X mi
r
vr
2
a=c
(X i − X) −
r − 1 i=1 m
m
³
´
vr
a = c Var(X i ) −
m
Non-Paramtric estimation: ”a”
If µ is given
only data available is for policy holder i
Non-Paramtric estimation: ”a”
µ is given
Exam C
inverse transformed method
Exam C
bootstrap estimate of the mean squared error
Exam C
Chi-Square Test for
number of claims is the result of a sum of a number
(n) of i.i.d random variables (x)
Exam C
Exam C
Pni
vi =
j=1
a=
2
mij (Xij − X i )
r
X
mi
i=1
ni − 1
vi
ai = (X i − µ)2 −
mi
Data: y = {y1 , . . . , yn }
A statistic: θ from the empirical distribution function.
m
(X i − µ)2 −
−1
x = FX
(rand(0, 1))
(Continuous)
F (xj−1 ) ≤ rand(0, 1) < F (xj )
xij = yrandi (1,n)
i = 1, . . . , m;
j = 1, . . . , n
θ̂i = g(xi )
m
´2
1 X³
M SE(θ̂) =
θ̂i − θ = Var(θ̂) + bias2θ̂
m i=1
k
X
(Ej − Oj )2
χ =
Vj
j=1
2
Ej = nE(x)
Vj = nVar(x)
r
v
m
(Discrete)