Creating New Distributions

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1 Creating New Distributions Section 5.2 Stat Loss Models Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 1 / 18

2 Generating new distributions Some methods to generate new distributions There are many methods to generate new distributions; some of these methods allow us to give in-depth interpretation to the distributions. Among the methods used can be sub-divided into: 1 Addition of several random variables For example, sums of (independent) Exponentials give a Gamma. This method will not be further explored. 2 Transformation of random variables scalar multiplication power operations exponentiation (or logarithmic transformation) 3 Mixing of distributions frailty models 4 Spliced distributions Section 5.2 of Klugman, et al. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 2 / 18

3 Generating new distributions general theory The general theory of transformation Suppose we are interested in deriving the distribution of Y = g(x), where X has a known distribution function. Assume that the function g is a one-to-one transformation (i.e. invertible). It can be shown that the distribution function of Y can be expressed as F Y (y) = F X (g 1 (y)), in the case of increasing transformation. If decreasing, we have F Y (y) = 1 F X (g 1 (y)). Its density can be explicitly written as f Y (y) = f X (g 1 (y)) dg 1 (y) dy. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 3 / 18

4 Some examples scalar transformations Scalar transformations In the case where Y = ax for some a > 0, then this is called a scalar transformation and its density function can be expressed as f Y (y) = 1 a f X(y/a). Insurance interpretation: if X denotes claims, then scalar transformation can be interpreted as applying inflation factor across all levels of claims. A family of distributions that is closed under scalar multiplication (i.e. after scalar transformation, the new random variable remains in the same family) is called a scale family of distributions. Some scale families are: Normal Exponential (Example 5.1) Pareto Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 4 / 18

5 Some examples power transformations Power transformations A power transformation involves raising the random variable by a power such as Y = X 1/τ or Y = X 1/τ, where τ > 0. In the first case, we have a transformed X distribution; the other case, we have an inverse transformed X distribution. In the special case where Y = X 1, we have an inverse X distribution. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 5 / 18

6 Some examples power transformations Distribution and density functions of power transformations It is easy to show the following results: In the transformed case where Y = X 1/τ, we have F Y (y) = F X (y τ ) and f Y (y) = τy τ 1 f X (y τ ). In the inverse transformed case where Y = X 1/τ, we have F Y (y) = 1 F X (y τ ) and f Y (y) = τy τ 1 f X (y τ ). In the inverse case where Y = X 1, we have F Y (y) = 1 F X (y 1 ) and f Y (y) = 1 y 2 f X(1/y). Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 6 / 18

7 Some examples power transformations Example 5.2 Let X be exponentially distributed with mean parameter 1. Derive the cumulative distribution and density functions of the transformed, inverse transformed and inverse random variables. Note that we derive: Inverse Exponential distribution: for the case of the inverse distribution after a scale transformation Weibull distribution: for the case of the transformed distribution after a scale transformation Inverse Weibull distribution: for the case of the inverse transformed distribution after a scale transformation Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 7 / 18

8 Some examples power transformations Illustrative example 1 Suppose X is a random variable with cumulative distribution function F X (x) = 1 (1 + x c ) γ, x > 0, c > 0, γ > 0. Derive the probability density function of the inverse of X, i.e. Y = 1/X. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 8 / 18

9 Some examples power transformations Illustrative example 2 Suppose X is an exponential random variable with mean parameter equal to 1. Derive the distribution of Y = αx 1/β, for α > 0, β > 0. Specify its density function. The distribution of Y is called a Weibull and we can write Y Weibull(α, β), where α is obviously a scale parameter. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 9 / 18

10 Some examples exponentiation Exponentiation Suppose Y = exp(x). For y > 0, its distribution function can be expressed as F Y (y) = F X (log y) and its corresponding density as f Y (y) = 1 y f X(log y) Derive the distribution/density functions corresponding to the exponential transformation Y = exp(x) when: X N(µ, σ 2 ) X Exp(1) Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 10 / 18

11 Mixtures of distributions Mixtures of distributions A random variable X is said to be a mixture of distributions if its distribution function has one of the following forms: 1 Discrete mixture: F X (x) = i a if Xi (x), for some sequence of random variables X 1, X 2,... and some sequence of positive numbers a 1, a 2,... satisfying i a i = 1. 2 Continuous mixture: F X (x) = F X Λ=λ(x)f Λ (λ)dλ, for some random variable Λ satisfying f Λ(λ)dλ = 1. In terms of actuarial/insurance applications: Discrete mixtures arise in situations where the risk class of a policyholder is uncertain, and the number of possible risk classes is discrete. Continuous mixtures arise when a risk parameter from the loss distribution is uncertain and the uncertain parameter is continuous. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 11 / 18

12 Mixtures of distributions examples Illustrative example 1 An insurer has two groups of policyholders: the good and the bad risks. The insurer has a portfolio where 75% are considered good risks. The claim distributions for both groups of risks are Exponential. The average claim amount of a good risk policyholder is $100 while for bad risk, it is twice that. A new customer whose risk class is not known with certainty, has just recently purchased a policy from the insurer. Calculate the probability that this new customer will claim an amount exceeding $150. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 12 / 18

13 Mixtures of distributions examples Illustrative example 2 Consider a claims random variable X that, given a risk classification (random) parameter Λ, can be modeled as an Exponential random variable with P (X x Λ = λ) = 1 e x/λ, for x > 0. Assume that Λ has a Gamma(α, θ) distribution. 1 Show that the unconditional distribution of X is a Pareto. 2 Derive its mean and variance. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 13 / 18

14 Mixtures of distributions mean and variance of mixtures Deriving the mean and variance of mixtures Suppose that X is a mixture with mixing variable Λ. Then the unconditional mean and variance can be determined using the following formulas: Law of iterated expectations or in general, we have Conditional variance formula E(X) = E Λ [E(X Λ)], E(X k ) = E Λ [E(X k Λ)]. Var(X) = E Λ [Var(X Λ)] + Var Λ [E(X Λ)]. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 14 / 18

15 Mixtures of distributions mean and variance of mixtures Illustration The policyholders of an insurance company fall into one of two classes. The claims distributions for each class are given in the following table: Class 1 Class 2 claim size probability claim size probability 1, , , , , , There are 30% of policyholders in class 1, while the remaining policyholders are in class 2. Denote by L 1 the claim incurred by a randomly selected policyholder from class 1, while L 2 from class 2. Let L denote the claim incurred by a randomly selected policyholder whose risk class is unknown. Calculate the mean and variance of L. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 15 / 18

16 Frailty models Frailty models Define Λ to be a positive, frailty random variable such that, conditional on Λ = λ, the hazard rate of X is of the form h X Λ (x λ) = λa(x), for some specified and known function a(x) of x. It can be shown that the conditional survival function of X Λ is ( x ) S X Λ (x λ) = exp h X Λ (z λ)dz = e λa(x) where A(x) = x 0 a(z)dz. The (unconditional) survival function can then be derived as ( S X (x) = E e λa(x)) = M Λ ( A(x)), where M Λ is the mgf of the frailty. 0 Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 16 / 18

17 Spliced distributions Spliced distributions A spliced distribution is one whose form of distribution is different in different portions of the domain of the random variable. An interpretation in insurance claims is that the distributions vary by size of claims. To illustrate, consider a two-spliced distribution: { p1 f f X (x) = 1 (x), for 0 < x < c p 2 f 2 (x), for c x <. where p 1 + p 2 = 1 and f 1 and f 2 are both legitimate density functions on the corresponding intervals. This concept can be extended to a k-component spliced distributions. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 17 / 18

18 Spliced distributions SOA question SOA question An actuary for a medical device manufacturer initially models the failure time for a particular device with an Exponential distribution with mean 4 years. This distribution is replaced with a spliced model whose density function: is Uniform over [0,3] is proportional to the initial modeled density function after 3 years is continuous Calculate the probability of failure in the first 3 years under the revised distribution. Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 18 / 18

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