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Severity Models - Special Families of Distributions

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Severity Models - Special Families of Distributions Severity Models - Special Families of Distributions Sections 5.3-5.4 Stat 477 - Loss Models Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 1 / 1 Introduction Introduction Given that a claim occurs, the (individual) claim size X is typically referred to as claim severity. While typically this may be of continuous random variables, sometimes claim sizes can be considered discrete. When modeling claims severity, insurers are usually concerned with the tails of the distribution. There are certain types of insurance contracts with what are called long tails. Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 2 / 1 Introduction parametric class Parametric distributions A parametric distribution consists of a set of distribution functions with each member determined by specifying one or more values called \parameters". The parameter is fixed and finite, and it could be a one value or several in which case we call it a vector of parameters. Parameter vector could be denoted by θ. The Normal distribution is a clear example: 1 −(x−µ)2=2σ2 fX (x) = p e ; 2πσ where the parameter vector θ = (µ, σ). It is well known that µ is the mean and σ2 is the variance. Some important properties: Standard Normal when µ = 0 and σ = 1. X−µ If X ∼ N(µ, σ), then Z = σ ∼ N(0; 1). Sums of Normal random variables is again Normal. If X ∼ N(µ, σ), then cX ∼ N(cµ, cσ). Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 3 / 1 Introduction some claim size distributions Some parametric claim size distributions Normal - easy to work with, but careful with getting negative claims. Insurance claims usually are never negative. Gamma/Exponential - use this if the tail of distribution is considered `light'; applicable for example with damage to automobiles. Lognormal - somewhat heavier tails, applicable for example with fire insurance. Burr/Pareto - used for heavy-tailed business, such as liability insurance. Inverse Gaussian - not very popular because complicated mathematically. Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 4 / 1 Some familiar distributions Lognormal The Lognormal distribution If Y ∼ Normal(µ, σ), then X = exp(Y ) is lognormal and we write X ∼ Lognormal(µ, σ). Its density can be expressed as 1 −(log x−µ)2=2σ2 fX (x) = p e ; for x > 0: xσ 2π Thus, if X is lognormal, then log(X) is Normal. Moments: E(Xk) = exp(kµ + k2σ2=2) Mean: E(X) = eµ+σ2=2 Variance: Var(X) = (eσ2 − 1)e2µ+σ2 Derive the mode. Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 5 / 1 Some familiar distributions Lognormal Lognormal densities for various σ's Lognormal density functions 2.0 µ = 0 1.5 σ = 5 σ = 3 2 σ = 1 1.0 σ = 1 2 σ = 1 4 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 6 / 1 Some familiar distributions Gamma The Gamma distribution We shall write X ∼ Gamma(α; θ) if density has the form 1 f (x) = xα−1e−x/θ; for x > 0; α; θ > 0; X θαΓ(α) with α, the shape parameter, and θ, the scale parameter. Γ(α + k) Higher moments: E(Xk) = θk Γ(α) Mean: E(X) = αθ Variance: Var(X) = αθ2 −α Moment generating function: MX (t) = (1 − θt) , provided t < 1/θ. Special cases: Exponential: When α = 1, we have X ∼ Exp (θ). Chi-square: When α = n=2 and θ = 2, we have a chi-squared distribution with n degrees of freedom. Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 7 / 1 Some familiar distributions scale parameters A scale parameter For random variables with non-negative support, a (single-valued) parameter is said to be a scale parameter if it meets two conditions: for a member of the distribution multiplied by a positive constant, say c, then the scale parameter is also multiplied by the same constant c; and for a member of the distribution multiplied by a positive constant, the rest of the parameters remain unchanged. Demonstrate that in the Gamma distribution, the parameter θ is a scale parameter. Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 8 / 1 Some familiar distributions Pareto The Pareto distribution We shall write X ∼ Pareto(α; θ) if density has the form αθα f (x) = ; for x > 0; X (x + θ)α+1 where α > 0 and θ > 0. α, the shape parameter; θ, the scale parameter. θ α CDF: F (x) = 1 − . X x + θ θ Mean: E(X) = α − 1 Γ(α − k) Higher moments: E(Xk) = θkΓ(k + 1); for − 1 < k < α. Γ(α) Variance: (derive it!) Mgf: does not exist Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 9 / 1 Some familiar distributions Pareto Pareto densities for various α's Pareto density functions 3.0 2.5 θ = 1 α = 1 α = 2 2.0 α = 3 1.5 1.0 0.5 0.0 0 1 2 3 4 5 x Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 10 / 1 Some familiar distributions Pareto Some important properties A positive scalar multiple of a Pareto is again a Pareto. If X ∼ Pareto(α; θ) and c is a positive constant, then cX ∼ Pareto(α; cθ). This also explains why θ is called the scale parameter. The Pareto distribution is a continuous mixture of exponentials with Gamma mixing weights. If (XjΛ = λ) ∼ Exp(1/λ) and Λ ∼ Gamma(α; 1/θ), then unconditionally, X ∼ Pareto(α; θ). To be derived in lecture - also part of generating new distributions. Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 11 / 1 Some familiar distributions Burr The Burr distribution We shall write X ∼ Burr(α; θ; γ) if density has the form αγ(x/θ)γ f (x) = ; for x > 0; X x[1 + (x/θ)γ]α+1 where α > 0, θ > 0 and γ > 0. α, the shape parameter; θ, the scale parameter. Sometimes more precisely called Burr Type XII distribution and the Pareto is a special case when γ = 1. Γ(1 + k/γ)Γ(α − k/γ) Higher moments: E(Xk) = θk, provided Γ(α) −γ < k < αγ. Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 12 / 1 Some familiar distributions Burr Burr densities for various parameter values Burr density functions 2.5 θ = 1 α = 1, γ = 1 2.0 α = 2, γ = 1 α = 3, γ = 1 α = 1, γ = 2 1.5 α = 1, γ = 3 α = 2, γ = 0.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 13 / 1 Some familiar distributions linear exponential family The linear exponential family A random variable X has a distribution from the linear exponential family if its density has the form 1 f (x; θ) = p(x)er(θ)x; X q(θ) where p(x) depends only on x and q(θ) is a normalizing constant. The support of X must not depend on θ. Mean: q0(θ) E(X) = µ(θ) = r0(θ)q(θ) Variance: µ0(θ) Var(X) = v(θ) = r0(θ) Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 14 / 1 Some familiar distributions linear exponential family Some members of the linear exponential family Continuous: Normal Gamma (and all special cases e.g. Exponential, Chi-squared) Discrete: Poisson Binomial Negative Binomial Sections 5.3-5.4 (Stat 477) Claim Severity Models Brian Hartman - BYU 15 / 1.
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