Exam C. Exam C. Exam C. Exam C. Exam C

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1 cumulative distribution function distribution function cdf survival function probability density function density function probability function probability mass function hazard rate force of mortality failure rate truncated censored k-th raw moment empirical model k-th central moment

2 x S X (x) = P r(x > x) = 1 F X (x) F (x) = f(s)ds = P (X < x) p X (x) = Pr(X = x) f(x) = F (x) = S (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. h X (x) = f X(x) S X (x) 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 µ k. The first raw moment is called the mean of the random variable and is usually denoted by µ. 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. µ k = E(X k ) = E(x) = x k f(x)dx = j 0 S(x)dx x k j p(x j ) 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 µ k. The second central moment is usually called the variance and denoted σ 2 and the square root, σ, is called the standard deviation. µ k = E[(X µ) k ] = = j (x µ) k f(x)dx (x j µ) k p(x j ) The empirical model is a discrete distribution based on a sample size n which assigns probability 1/n to each data point.

3 coefficient of variation skewness kurtosis symmetric distribution, then it has a skewness limited loss variable right censored variable limited expected value left censored and shifted variable And E(left censored and shifted variable) left censored and shifted variable mean residual life function complete expectation of life left truncated and shifted variable mean excess loss function percentile

4 µ 3 σ 3 σ µ µ 3 σ 3 = 0 µ 4 σ 4 E[(X u) k ] = = E[X u] u 0 u 0 x k f(x) + u k [1 F (u)] S(x)dx Y = X u = { X, X < u u, X u Y = X d given that X > d { 0, X < d Y = (X d) + = X d, X d E[(X d) + ] = E(X) E(X d) = d S(x)dx The 100pth percentile of a random variable is any value π p such that F (π p ) p F (π p ). e X (d) = e(d) = E(Y ) = E(X d X > d) E(X d X > d) = E(X) E(X d) 1 F (d)

5 median S k E(S k ) Distribution of lim k V ar(sk ) moment generating function moment generating function E(X n ) = probability generating function p k from probability generating function Mean and Variance using the probability generating function pgf/mgf of n j=1 X j parametric distribution scale distribution

6 Normal distribution with mean 0 and variance 1. The 50th percentile, π 0.5 is called the median. E(X n ) = M (n) X (0) M X(t) = E(e tx ) p k = P (m) X (0) m! The mth derivative of the pgf evalated at 0 devided by m factorial. P X (z) = E(z X ) = p k z k k=0 M Sk (t) = P Sk (z) = k M Xj (t) j=1 k P Xj (z) j=1 E(X) = P X(1) E[X 2 ] = P X(1) + P X(1) Var(X) = P X(1) + P X(1) [P X(1)] 2 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.

7 scale parameter parametric distribution family k-point mixture variable-component mixture distribution data-dependant distribution Equilibrium Distribution Survival function and hazard rate of Equilibrium Distribution Survival function based on equilibrium distribution coherent risk measure Value-at-Risk

8 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 X 1, X 2,, X k if its cdf is given by K F (x) = a j F j (x), j=1 K a j = 1, a j > 0 j=1 F Y (y) = a 1 F X1 (y) + a 2 F X2 (y) + + a k F Xk (y) where all a j > 0 and a 1 + a a k = 1. Assume X is a continuous distribution f(x) with a survival function S(x) and mean E(X). Then the equilibrium distribution is f e (x) = S(x) E(X), x 0 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. S(x) = E(x) x e(x) e 0 [ e(t)]dt 1 S e (x) = x f e (t)dt = x S(t)dt, x 0 E(x) h e (x) = f e(x) S e (x) = S(x) x S(t)dt = 1 e(x) Let X denote a loss random variable. The Value-at- Risk of X at the 100p% level, denoted VaR p (X) or π p, is the l00p percentile (or quantile) of the distribution of X. And for continuous distributions it is the value of π p satisfying Pr(X > π p ) = 1 p VaR is not coherent as it does not meet the subaddivity requirement in some cases. 1. Subadditivity: ρ(x + Y ) ρ(x) + ρ(y ). 2. Monotonicity: If X Y for all possible outcomes, then ρ(x) ρ(y ). 3. Positive homogeneity: For any positive constant c. ρ(cx) = cρ(x). 4. Translation invariance: For any positive constant c. ρ(x + c) = ρ(x) + c.

9 Tail-Value-at-Risk Given: F (x) and f(x) find F Y (y) when Y = θx Given: F (x) and f(x) find F Y (y) when Y = X 1/τ transformed inverse inverse transformed Given: F (x) and f(x) find F Y (y) when Y = e X Given: F (x) and f(x) find F Y (y) when Y = g(x) Mixture Distribution (F X (x)) Raw Moments of a Mixture

10 ( y ) F Y (y) = F X θ f Y (y) = 1 ( y ) θ f X θ Let X denote a loss random variable. The Tail- Value-at-Risk of X at the 100p% security level, denoted TVaR p (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 TVaR p (X) = E(X X > π p ) = π p xf(x)dx 1 F (π p ) Y = X 1/τ τ > 0 F Y (y) = F X (y τ ) f Y (y) = τy τ 1 f X (y τ ) Y = X 1/τ τ =< 0 and τ 1 Y = X 1/τ τ = 1 h(y) = g 1 (Y ) F Y (y) = F X [h(y)] f Y (y) = h (y) f X [h(y)] F Y (y) = F X [ln(y)] f Y (y) = 1 y f X[ln(y)] E(X k ) = E[E(X k Λ)] F X (x) = F X Λ (x λ)f Λ (λ)dλ

11 Variance of a Mixture Important Mixtures: Y = Poison(Λ) Λ = Gamma(α, θ) Important Mixtures: Y = Exponential(Λ) Λ = Inv.Gamma(α, θ) Important Mixtures: Y = Inv.Exponential(Λ) Λ = Gamma(α, θ) Important Mixtures: Y = Normal(Λ, σ 2 c ) Λ = Normal(µ, σ 2 d ) k-component spliced distribution linear exponential family normalizing constant canonical parameter linear exponential family: Mean

12 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: a 1 f 1 (x), c 0 < x < c 1 a 2 f 2 (x), c 1 < x < c 2 f X (x) =.... a k f k (x), c k 1 < x < c k Normal(µ, σ 2 c + σ 2 d ) q(θ) f(x; θ) = p(x)er(θ)x q(θ) f(x; θ) = p(x)er(θ)x q(θ) E(X) = µ(θ) = q (θ) r (θ)q(θ) r(θ) f(x; θ) = p(x)er(θ)x q(θ)

13 linear exponential family: Variance (a,b,0) class of distribution (a,b,1) class of distribution Poison (a,b,0) specification Binomial (a,b,0) specification Negative Binomial (a,b,0) specification Geometric (a,b,0) specification Geometric Relation to Negative Binomial memoryless E[(1 + r)x c]

14 Let p k 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 Var(X) = µ (θ) r (θ) p k p k 1 = a + b k k = 1, 2, 3, a = 0 b = λ p 0 = e λ Let p k 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 p k p k 1 = a + b k k = 2, 3, 4, p 0 = 1 k=1 p k It is called truncated if p 0 = 0. It is called zero-modified if p 0 > 0 and is a mixture of an (a,b,0) class and a distribution where p 0 = 1. (AKA truncated with zeros) a = β 1 + β β b = (r 1) 1 + β p 0 = (1 + β) r a = q 1 q q b = (m + 1) 1 q p 0 = (1 q) m Geomtric is Negative Binomial with parameter r = 1. a = β 1 + β b = 0 p 0 = (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.

15 per-loss variable per-payment variable Relationship between per-loss and per-payment Relationship between per-loss and per-payment Variance franchise deductible Expectation ordinary deductible Expectation The loss elimination ratio Co-insurance Co-insurance, deductible and limits variable Co-insurance, deductible and limits variable Expectation

16 { Y P undefined, X d = X d, X > d { Y L 0, X d = X d, X > d Var(Y P ) = E ( [Y L ] 2) S X (d) ( E(Y L ) 2 ) If f meets certian conditions [not sure how to define S X (d) them, (linear?)] then f(y P ) = f(y L ) S X (d) However, E(Y P ) = E(Y L ) S X (d) Var(Y P ) Var(Y L ) S X (d) E(X) E(X 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) = E(X d) E(X) E(Y L ) = α(1 + r) [E (X u ) E (X d )] 0, X < d Y L = α[(1 + r)x d], d X < u α(u d), u X

17 Co-insurance, deductible and limits variable 2nd Raw Moment individual risk model collective risk model OR Compound Model Compound Model Mean Compound Model Variance Compound Model N = Poisson(λ) stop-loss insurance P r(a < S < b) = 0. Then, for a d b E[(S d) + ] = Theorem to calculate E[(S d) + ] using a discrete probability function w(p (S = kh) = f k 0) with equally spaced (h) nodes. Convolution method (X 1 + X 2 )

18 The individual risk model represents the aggregate loss as a sum, S = X X n, of a fixed number, n, of insurance contracts. The loss amounts for the n contracts are (X 1,..., X n ), where the X j s are assumed to be independent but are not assumed to be identically distributed. The distribution of the X j s usually has a probability mass at zero, corresponding to the probability of no loss or payment E[(Y L ) 2 ] = = α 2 (1 + r) 2 {E[(X u ) 2 ] E[(X d ) 2 ] 2d E(X u ) + 2d E(X d )} E[S] = E[E(S N)] = E(N)E(X) = µ n µ x S = X X N with: 1. Conditional on N = n, the random variables X 1, X 2,..., X n are i.i.d random variables. 2. Conditional on N = n, the common distribution of the random variables X 1, X 2,..., X n, does not depend on n. 3. The distribution of N does not depend in any way on the values of X 1, X 2,... E[S] = µ n µ x = λµ x V ar[s] = λ(σ 2 x + µ 2 x) = λe(x 2 ) (N=Poisson) (N=Poisson) V ar[s] = V ar[e(s N)] + E[V ar(s N)] = Var(N)E(X) 2 + E(N)Var(X) = σ 2 nµ 2 x + µ n σ 2 x E[(S d) + ] = b d b a E[(S a) +] + d a b a E[(S b) +] 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 F S (k) = f X1 (j)f X2 (k j) all j f S (s) = f X1 (t)f X2 (s t)dt F S (s) = f X1 (t)f X2 (s t)dt k = 0, 1,.... Then, provided d = jh, with j a nonnegative integer E[(S d) + ] = h {1 F S [(m + j)h]} m=0 E{[S (j + 1)h] + } = E[(S jh) + ] h[1 F S (jh)]

19 Convolution method (X X n ) Bias asymptotically unbiased consistent mean-squared error (MSE) uniformly minimum variance unbiased estimator UMVUE confidence interval significance level uniformly most powerful p-value

20 biasˆθ = E(ˆθ θ) θ F n X (x) = f n X (x) = x 0 x 0 F (n 1) X (x t)f X (t)dt f (n 1) X (x t)f X (t)dt lim P r( ˆθ n θ > σ) = 0 n lim E(ˆθ n θ) = θ n An estimator, ˆθ, 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. MSEˆθ(θ) = E[(ˆθ θ) 2 θ] = V ar(ˆθ θ) + [biasˆθ(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 H 0 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 H 0 at level α. 2. p-value > α do not reject H 0 at level α. [Probability and Statistics, Devore, 2000]

21 Log-Transformed Confidence Interval empirical distribution F n (x) = S n (x) kernel smoothed distribution data set: variables data summary E[S n (x)] Var[S n (x)] Sample Variance Empirical estimate of the variance Empirical estimate of E[(X u) k ]

22 The empirical distribution is obtained by assigning probability 1/n to each data point. number of observations x F n (x) = n S n (x) = 1 F n (x) The 100(1 α)% log-transformed confidence interval for S n (t) is ( S n (t) U, S n (t) (1/U)) where U = exp z α/2 Var[S n (t)] S n (t) ln S n (t) 1. n - insureds 2. d i - entry time 3. x i - death time 4. u i - censored time 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. Y = number greater than x in sample E[S n (x)] = S(x) = Y n 1. m - death points 2. y j - death point time 3. s j - time y j 4. r j - time y j 1 n 1 n Y = number greater than x in sample (x i x) 2 i=1 S(x) = Y n S(x)[1 S(x)] E[S n (x)] = n 1 x k i u k [number of x i s > u] n xi u 1 n n (x i x) 2 i=1

23 cumulative hazard rate function Nelson-Åalen estimate variance of the Nelson-Åalen estimate Kaplan-Meier product-limit Estimator Greenwood Approximation Variance of the Kaplan-Meier product-limit Estimator log-transformed interval for the Nelson-Åalen estimate method of moments percentile matching smoothed empirical estimate likelihood function log-likelihood function

24 0, x < y 1 j 1 s Ĥ(x) = i i=1 r i, y j 1 x < y j, j = 2,..., k x i y k k i=1 s i r i, H(x) = ln S(x) S n (t) = 1, 0 t < y 1 ( ) j 1 ri s i i=1 r i y j 1 t < y j, j = 2,..., k ( ) k ri s i i=1 r i t y k j Var[Ĥ(y j)] = s i r 2 i=1 i = Ĥ(t)U, where U = exp ± z Var[Ĥ(y α/2 j)] Ĥ(t) Var[S n (y j )]. = [S n (y j )] 2 j i=1 s i r i (r i s i ) π gk (θ) = ˆπ gk, k = 1, 2,..., p F (ˆπ gk θ) = g k, k = 1, 2,..., p Here. indicates the greatest integer function and x (1) x (2)... x (n) are the order statistics from the sample. µ k(θ) = ˆµ k, k = 1, 2,..., p n L(θ) = Pr(X j A j θ) j=1 l(θ) = ln L(θ) ˆπ g = (1 h)x (j) + hx (j+1), where j = (n + 1)g and h = (n + 1)g j

25 Information function Variance of the likelihood estimate (θ) Delta Method (single variable) Delta Method (general) Non-normal confidence interval p-p plot D(x) plot Kolmogorov-Smirnov Test Anderson-Darling test Chi-Square Goodness-of-fit

26 Var(ˆθ) = [I(θ)] 1 [ ] 2 I(θ) = E ln L(θ) θ2 [ ( ) 2 2 ] = E ln L(θ) θ2 = n f(x; θ) 2 ln[f(x; θ)]dx θ2 Let ˆθ n = (ˆθ 1n,..., ˆθ kn ) T be a multivariate parameter vector of dimension k based on a sample size of n. Assume that ˆθ n is asymptotically normal with mean of ˆθ and covariance matrix Ω/n. Then g(θ 1,..., θ k ) is asymptotically normal with E[g(θ)] = g(ˆθ n ) Var[g(θ)] = ( g) T Ω n g Let ˆθ n be a parameter estimated using a sample size of n. Assume that ˆθ n is asymptotically normal with mean of µ and variance of σ 2 /n. Then g(ˆθ n ) is asymptotically normal with E[g(θ)] = g(ˆθ n )) Var[g(θ))] = [g (ˆθ n )] 2 σ2 n (F n (x j ), F (x j )), where F n (x j ) = j n + 1 { θ : l(θ) l(ˆθ) χ2 α/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 t x u F n(x) F (x) D(x) = F n (x) F (x), where F n (x j ) = j n χ 2 = = k n(ˆp j p nj ) 2 j=1 ˆp j k (E j O j ) 2 j=1 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. E j +n +n u A 2 [F n (x) F (x)] 2 = n t F (x)[1 F (x)] f (x)dx = nf (u) k [1 F n (y j )] 2 (ln[1 F (y j )] ln[1 F (y j+1 )]) j=0 k F n (y j ) 2 [ln F (y j+1 ) ln F (y j )] j=0

27 likelihood ratio test Schwarz Bayesian Criterion Full credibility (Single Variable Case) Full credibility (Poisson) Full Credibility Compound Distributions # of S i s Full Credibility Compound Distributions n i=1 S i Full Credibility Compound Distributions n S i µ i=1 y Full Credibility Compound Distributions N = P oisson(λ) # of S i s Full Credibility Compound Distributions N = P oisson(λ) n i=1 S i Full Credibility Compound Distributions N = P oisson(λ) n S i µ i=1 y

28 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. T = 2 ln ( L1 L 0 = L(θ 0 ) L 1 = L(θ 1 ) L 0 ) = 2 (ln L 1 ln L 0 ) The critical values come from a chi-square distribution with degrees of freedom equal to the number of free parameters in L 1 less the number of free parameters in L 0. n n 0 1 λ W n n 0 n n 0 W n n 0 σ 2 ( ) 2 σ V ar[w ] = n 0 µ (E[W ]) 2 = n 0CVW 2 µ = n 0 V ar[w ] E[W ] = n 0 E[W ]CV 2 W σnµ 2 2 y + µ n σy 2 σnµ 2 2 y + µ n σy 2 n 0 n 0 µ n µ y (µ n µ y ) 2 [ ] 1 n σ2 y λ µ 2 y n 0 σ 2 nµ 2 y + µ n σ 2 y µ n µ 2 y [ ] n σ2 y µ 2 y [ ] n 0 µ y + σ2 y µ y

29 Partial Credibility Model distribution Joint distribution Prior distribution marginal distribution posterior distribution predictive distribution Bayes Premium E(x n+1 ) Bayes Premium E(x n+1 ) with: E[X θ] = θ (eg. X Poisson) 0 x a e cx dx

30 The model distribution (m x θ ) is the probability distribution for the data as collected given a particular value for the parameter. Its pdf is denoted m x θ = f X Θ (x θ). n m x θ = f X Θ (x θ) = f X Θ (x i θ) i=0 Q = ZW + (1 Z)M Where, Z = ( ) information available min information required for full credibility, 1 The prior distribution (π θ ) 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 (j x,θ ) has pdf j x,θ = f X,Θ (x, θ) = m x θ π(θ) = f X Θ (x θ)π(θ) The posterior distribution (p θ x ) is the conditional probability distribution of the parameters given the observed data. Its pdf is The marginal distribution (g x ) of x has pdf g x = f X (x) = j x,θ dθ = f X Θ (x θ)π(θ)dθ p θ x = π Θ X (θ x) = j x,θ g x = f X Θ(x θ)π(θ) fx Θ (x θ)π(θ)dθ E(x n+1 x) = E(x θ)p θ x dθ The predictive distribution is the conditional probability distribution of a new observation y given the data x = x 1,..., x n. Its pdf f Y X (y x) = f Y Θ (y θ)p θ x dθ = f Y Θ (y θ)π Θ x (θ x)dθ 0 x a e cx dx = Γ(a + 1) c a+1 = a! c a+1 a is an integer E(x n+1 x) = θp θ x dθ or the mean of the posterior distribution.

31 0 e c x x k dx Conjugate Prior Poisson-Gamma λ = gamma(α, θ) x = Poisson(λ) Conjugate Prior Exponential-Inverse Gamma λ = gamma 1 (α, θ) x = exp(λ) Conjugate Prior Binomial-Beta q = beta(a, b, 1) x = bin(m, q) Conjugate Prior Inverse Exponential-Gamma λ = gamma(α, θ) x = exp 1 (λ) Conjugate Prior Normal-Normal λ = normal(µ, a 2 ) x = normal(λ, σ 2 ) Conjugate Prior Uniform-Pareto λ = single.pareto(α, θ) x = uniform(0, λ) hypothetical mean or collective premium process variance expected value of the hypothetical means EVHM

32 ( gamma α + ) θ x i, nθ e c x Γ(k 1) dx = xk c k 1 k > 1 (k 2)! = c k 1 k 2 beta(a + x i, b + km x i, 1) gamma 1 (α + n, θ + x i ) ([ xi normal σ 2 + µ ] [ n a 2 / σ ] a 2, 1 n σ a 2 ) gamma ( α + n, [ 1 θ + ] ) 1 1 x i µ(θ) = E(X ij Θ i = θ) single.pareto(α + n, max(x, θ)) µ = E[µ(θ)] v(θ) = m ij V ar(x ij Θ i = θ)

33 expected value of the process variance EVPV variance of the hypothetical means VHM Bühlmann s k or credibility coefficient Bühlmann credibility factor Bühlmann credibility premium Var(X) = f(evpv,vhm) Non-Paramtric estimation: µ Non-Paramtric estimation: v Non-Paramtric estimation: a Method using c = Non-Paramtric estimation: a Loss models technique

34 a = V ar[µ(θ)] v = E[v(θ)] Z i = m i m i + v/a k = v a Var(X) = a + v = EV P V + V HM Z i X + (1 Zi )µ v = 1 r i=1 (n i 1) r n i ( ) 2 m ij Xij X i i=1 j=1 µ = X ( a = m 1 m r i=1 m 2 i ) 1 [ r ] m i ( X i X) 2 v(r 1) i=1 [ r c = r 1 m ( i r m i=1 ( [ r r a = c r 1 ( a = c i=1 m i Var(X i ) vr m 1 m i m ) ] 1 m (X i X) 2 ) ] ) vr m

35 Non-Paramtric estimation: a µ is given Non-Paramtric estimation: a If µ is given only data available is for policy holder i inverse transformed method bootstrap estimate of the mean squared error Chi-Square Test for number of claims is the result of a sum of a number (n) of i.i.d random variables (x)

36 v i = ni j=1 m ij(x ij X i ) 2 n i 1 a i = (X i µ) 2 v i m i a = r i=1 m i m (X i µ) 2 r m v Data: y = {y 1,..., y n } A statistic: θ from the empirical distribution function. x ij = y randi (1,n) i = 1,..., m; j = 1,..., n ˆθ i = g(x i ) MSE(ˆθ) = 1 m ) 2 (ˆθi θ = Var(ˆθ) + bias m 2ˆθ i=1 x = F 1 X (rand(0, 1)) (Continuous) F (x j 1 ) rand(0, 1) < F (x j ) (Discrete) k χ 2 (E j O j ) 2 = j=1 E j = ne(x) V j = nvar(x) V j