Definition 1.1 (Parametric family of distributions) A parametric distribution is a set of distribution functions, each of which is determined by speci

Definition 1.1 (Parametric family of distributions) A parametric distribution is a set of distribution functions, each of which is determined by specifying one or more values called parameters. The number of parameters is fixed and finite. A family of distributions can be quite simple such as for instance the exponential Exp(λ) and normal N(µ, σ 2 ). On the other hand we can have X F (θ 1, θ 2,..., θ n ) that is much more complicated. Definition 1.2 (Scale family of distributions) A risk X with cdf F(x; σ) where σ > 0 is said to belong to a scale family of distributions if F (x; σ) = F (x/σ; 1). Here, σ is called the scale parameter. Edward Furman Risk theory 4280 2 / 87

Definition 1.3 (Location-scale family of distributions) A risk X with cdf F(x; µ, σ) where < µ <, σ > 0 is said to belong to a location-scale family of distributions if F (x; µ, σ) = F((x µ)/σ; 0, 1). Here, µ and σ are the location and scale parameters, respectively. Example 1.1 Edward Furman Risk theory 4280 3 / 87

Definition 1.4 (A family of parametric distributions) A family of parametric distributions is a set of parametric distributions that are related in a meaningful way. Edward Furman Risk theory 4280 4 / 87

Definition 1.4 (A family of parametric distributions) A family of parametric distributions is a set of parametric distributions that are related in a meaningful way. Example 1.2 Think of X Ga(γ, α). This can be seen as the family of gamma distributions. Setting γ = 1, for instance, we obtain the Exp(α). Considering integer γ only, we have the Erlang distribution. Also, setting α = 1/2 and γ = ν/2, we have the X 2 (ν) distribution. When we look at gamma family, we do not know the number of parameters to work with. When we concentrate on gamma distributions, we restrict our attention to the two parameter case. Edward Furman Risk theory 4280 4 / 87

Definition 1.5 (Mixed distributions) A risk Y is said to be an n point mixture of the risks X 1, X 2,..., X n if its cdf is F Y (y) = for α 1 + α n = 1, α k > 0. n α k F Xk k=1 Edward Furman Risk theory 4280 5 / 87

Definition 1.5 (Mixed distributions) A risk Y is said to be an n point mixture of the risks X 1, X 2,..., X n if its cdf is F Y (y) = for α 1 + α n = 1, α k > 0. n α k F Xk k=1 Definition 1.6 (Variable-component mixture distributions) A variable-component mixture distribution has a cdf F Y (y) = N α k F Xk, k=1 for α 1 + + α N = 1, α k > 0. Here N is random. Edward Furman Risk theory 4280 5 / 87

Example 1.3 Let X k Exp(λ k ), where k = 1, 2,.... The n point mixture of these risks has the cdf Edward Furman Risk theory 4280 6 / 87

Example 1.3 Let X k Exp(λ k ), where k = 1, 2,.... The n point mixture of these risks has the cdf F (x) = 1 α 1 e λ 1x α 2 e λ 2x α n e λnx Edward Furman Risk theory 4280 6 / 87

Example 1.3 Let X k Exp(λ k ), where k = 1, 2,.... The n point mixture of these risks has the cdf F (x) = 1 α 1 e λ 1x α 2 e λ 2x α n e λnx = (α 1 + + α n ) α 1 e λ 1x α 2 e λ 2x α n e λnx. The pdf is then Edward Furman Risk theory 4280 6 / 87

Example 1.3 Let X k Exp(λ k ), where k = 1, 2,.... The n point mixture of these risks has the cdf F (x) = 1 α 1 e λ 1x α 2 e λ 2x α n e λnx = (α 1 + + α n ) α 1 e λ 1x α 2 e λ 2x α n e λnx. The pdf is then f (x) = α 1 λ 1 e λ 1x + α 2 λ 2 e λ 2x + + α n λ n e λnx. The hazard function is Edward Furman Risk theory 4280 6 / 87

A distribution must not be parametric. Definition 1.7 (Empirical model) An empirical model is a discrete distribution based on a sample of size n that assigns probability 1/n to each data point. Example 1.4 Let us say we have a sample of 8 claims with the values {3, 5, 6, 6, 6, 7, 7, 8}. Then the empirical model is p(x) = 0.125 x = 3, 0.125 x = 5, 0.375 x = 6, 0.250 x = 7, 0.125 x = 8. Edward Furman Risk theory 4280 7 / 87

Proposition 1.1 Let X be a continuous rv having pdf f and cdf F. Let Y = θx with θ > 0. Then F Y (y) = F X (y/θ) and f Y (y) = 1 θ f X (y/θ). Proof. F Y (y) = Edward Furman Risk theory 4280 8 / 87

Proposition 1.1 Let X be a continuous rv having pdf f and cdf F. Let Y = θx with θ > 0. Then F Y (y) = F X (y/θ) and f Y (y) = 1 θ f X (y/θ). Proof. F Y (y) = P[Y y] = Edward Furman Risk theory 4280 8 / 87

Proposition 1.1 Let X be a continuous rv having pdf f and cdf F. Let Y = θx with θ > 0. Then F Y (y) = F X (y/θ) and f Y (y) = 1 θ f X (y/θ). Proof. F Y (y) = P[Y y] = P[X y/θ] = Edward Furman Risk theory 4280 8 / 87

Proposition 1.1 Let X be a continuous rv having pdf f and cdf F. Let Y = θx with θ > 0. Then F Y (y) = F X (y/θ) and f Y (y) = 1 θ f X (y/θ). Proof. F Y (y) = P[Y y] = P[X y/θ] = F X (y/θ). Also f Y (y) = Edward Furman Risk theory 4280 8 / 87

Proposition 1.1 Let X be a continuous rv having pdf f and cdf F. Let Y = θx with θ > 0. Then F Y (y) = F X (y/θ) and f Y (y) = 1 θ f X (y/θ). Proof. F Y (y) = P[Y y] = P[X y/θ] = F X (y/θ). Also f Y (y) = d dy F Y (y) = 1 θ f X (y/θ). Edward Furman Risk theory 4280 8 / 87

Proposition 1.2 Let X be a continuous rv having pdf f and cdf F, F(0) = 0. Let Y = X 1/τ. Then if τ > 0 F Y (y) = F X (y τ ) and f Y (y) = τy τ 1 f X (y τ ), y > 0 And if τ < 0, then F Y (y) = 1 F X (y τ ) and f Y (y) = τy τ 1 f X (y τ ). Edward Furman Risk theory 4280 9 / 87

Proposition 1.2 Let X be a continuous rv having pdf f and cdf F, F(0) = 0. Let Y = X 1/τ. Then if τ > 0 F Y (y) = F X (y τ ) and f Y (y) = τy τ 1 f X (y τ ), y > 0 And if τ < 0, then Proof. If τ > 0, then F Y (y) = 1 F X (y τ ) and f Y (y) = τy τ 1 f X (y τ ). F Y (y) = P[Y y] = P[X y τ ] = F X (y τ ), and the pdf follows by differentiation. Edward Furman Risk theory 4280 9 / 87

Proposition 1.3 Let X be a continuous rv having pdf f and cdf F. Let Y = e X. Then, for y > 0 F Y (y) = F X (log(y)) and f Y (y) = 1 y f X (log(x)). Edward Furman Risk theory 4280 10 / 87

Proposition 1.3 Let X be a continuous rv having pdf f and cdf F. Let Y = e X. Then, for y > 0 F Y (y) = F X (log(y)) and f Y (y) = 1 y f X (log(x)). Proof. We have that P[Y y] = P[e X y] = P[X log(y)] = F X (log(y)). The density follows by differentiation. Edward Furman Risk theory 4280 10 / 87

Solution For the cdf, we have that for positive y F Y (y) = Edward Furman Risk theory 4280 11 / 87

Solution For the cdf, we have that for positive y F Y (y) = F X (log(y)) = Edward Furman Risk theory 4280 11 / 87

Solution For the cdf, we have that for positive y F Y (y) = F X (log(y)) = Φ((log(y) µ)/σ). Also, for the pdf f Y (y) = Edward Furman Risk theory 4280 11 / 87

Solution For the cdf, we have that for positive y F Y (y) = F X (log(y)) = Φ((log(y) µ)/σ). Also, for the pdf f Y (y) = 1 y f X (log(y)) = Edward Furman Risk theory 4280 11 / 87

Solution For the cdf, we have that for positive y Also, for the pdf F Y (y) = F X (log(y)) = Φ((log(y) µ)/σ). f Y (y) = 1 y f X (log(y)) = 1 φ((log(y) µ)/σ), σy which of course reduces to Edward Furman Risk theory 4280 11 / 87

Solution For the cdf, we have that for positive y Also, for the pdf F Y (y) = F X (log(y)) = Φ((log(y) µ)/σ). f Y (y) = 1 y f X (log(y)) = 1 φ((log(y) µ)/σ), σy which of course reduces to { f Y (y) = 1 1 exp 1 σy 2π 2 that is a log-normal distribution. ( ) } log(y) µ 2 σ Edward Furman Risk theory 4280 11 / 87

Proposition 1.4 Let X be a continuous risk having cdf F X and pdf f X, and let h : R R be a strictly monotone function. Also, let Y = h(x), then the cdf of Y denoted by F Y is { FX (h F Y (y) = 1 (y)), if h is strictly increasing 1 F X (h 1 (y)), if h is strictly decreasing Moreover, if x = h 1 (y) is differentiable, then f Y (y) = f X (h 1 (y)) d dy h 1 (y). Proof. Edward Furman Risk theory 4280 12 / 87