An Inventory of Continuous Distributions

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1 Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α > 0. t α 1 e t dt, > 0. At times we will need this integral for nonpositive values of α. Integration by parts produces the relationship G(α; ) = α e + 1 G(α +1;) α α This can be repeated until the first argument of G is α + k, a positive number. Then it can be evaluated from G(α + k; ) =Γ(α + k)[1 Γ(α + k; )]. The incomplete beta function is given by β(a, b; ) = Γ(a + b) Γ(a)Γ(b) Z 0 t a 1 (1 t) b 1 dt, a > 0, b>0, 0 <<1. 1

2 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 2 A.2 Transformed beta family A.2.2 A Three-parameter distributions Generalized Pareto (beta of the second kind) α,, τ Γ(α + τ) α τ 1 ( + ) α+τ F () =β(τ,α; u), u = + E[X k ] = k Γ(τ + k)γ(α k), τ <k<α E[X k ] = k τ(τ +1) (τ + k 1), if k is an integer (α 1) (α k) E[(X ) k ] = k Γ(τ + k)γ(α k) β(τ + k, α k; u)+ k [1 F ()], k > τ mode = τ 1 α +1, τ > 1, else0 A Burr (Burr Type XII, Singh-Maddala) α,, γ αγ(/) γ [1 + (/) γ ] α+1 F () =1 u α 1, u = 1+(/) γ E[X k ] = k Γ(1 + k/γ)γ(α k/γ), γ <k<αγ VaR p (X) = [(1 p) 1/α 1] 1/γ E[(X ) k ] = k Γ(1 + k/γ)γ(α k/γ) β(1 + k/γ, α k/γ;1 u)+ k u α, k > γ µ 1/γ γ 1 mode =, γ > 1, else 0 αγ +1 A Inverse Burr (Dagum) τ,,γ τγ(/) γτ [1 + (/) γ ] τ+1 F () =u τ, u = (/)γ 1+(/) γ E[X k ] = k Γ(τ + k/γ)γ(1 k/γ), τγ<k<γ VaR p (X) = (p 1/τ 1) 1/γ E[(X ) k ] = k Γ(τ + k/γ)γ(1 k/γ) β(τ + k/γ, 1 k/γ; u)+ k [1 u τ ], µ 1/γ τγ 1 mode =, τγ > 1, else 0 γ +1 k > τγ

3 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 3 A.2.3 A Two-parameter distributions Pareto (Pareto Type II, Loma) α, α α µ α ( + ) α+1 F () =1 + E[X k ] = k Γ(k +1)Γ(α k), 1 <k<α E[X k ] = k k!, (α 1) (α k) if k is an integer VaR p (X) = [(1 p) 1/α 1] (1 p) 1/α TVaR p (X) = VaR p (X)+, α > 1 α 1 " µ # α 1 E[X ] = 1, α 6= 1 α 1 + µ E[X ] = ln, α =1 + µ α E[(X ) k ] = k Γ(k +1)Γ(α k) β[k +1,α k; /( + )] + k, all k + mode = 0 A Inverse Pareto τ, τ τ 1 µ τ ( + ) τ+1 F () = + E[X k ] = k Γ(τ + k)γ(1 k), τ <k<1 E[X k k ( k)! ] =, if k is a negative integer (τ 1) (τ + k) VaR p (X) = [p 1/τ 1] 1 Z /(+) µ τ E[(X ) k ] = k τ y τ+k 1 (1 y) k dy + 1 k, k > τ 0 + mode = τ 1, τ > 1, else 0 2 A Loglogistic (Fisk) γ, γ(/) γ [1 + (/) γ ] 2 F () =u, u = (/)γ 1+(/) γ E[X k ] = k Γ(1 + k/γ)γ(1 k/γ), γ <k<γ VaR p (X) = (p 1 1) 1/γ E[(X ) k ] = k Γ(1 + k/γ)γ(1 k/γ)β(1 + k/γ, 1 k/γ; u)+ k (1 u), k > γ µ 1/γ γ 1 mode =, γ > 1, else 0 γ +1

4 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 4 A Paralogistic α, This is a Burr distribution with γ = α. α 2 (/) α [1 + (/) α ] α+1 F () =1 u α 1, u = 1+(/) α E[X k ] = k Γ(1 + k/α)γ(α k/α), α <k<α 2 VaR p (X) = [(1 p) 1/α 1] 1/α E[(X ) k ] = k Γ(1 + k/α)γ(α k/α) β(1 + k/α, α k/α;1 u)+ k u α, k > α µ 1/α α 1 mode = α 2, α > 1, else 0 +1 A Inverse paralogistic τ, This is an inverse Burr distribution with γ = τ. τ 2 (/) τ 2 [1 + (/) τ ] τ+1 F () =u τ, u = (/)τ 1+(/) τ E[X k ] = k Γ(τ + k/τ)γ(1 k/τ), τ 2 <k<τ VaR p (X) = (p 1/τ 1) 1/τ E[(X ) k ] = k Γ(τ + k/τ)γ(1 k/τ) β(τ + k/τ, 1 k/τ; u)+ k [1 u τ ], k > τ 2 mode = (τ 1) 1/τ, τ > 1, else 0 A.3 Transformed gamma family A.3.2 A Two-parameter distributions Gamma α, (/)α e / F () =Γ(α; /) M(t) = (1 t) α, t < 1/ E[X k ]= k Γ(α + k), k > α E[X k ] = k (α + k 1) α, if k is an integer E[(X ) k ] = k Γ(α + k) Γ(α + k; /)+ k [1 Γ(α; /)], k > α = α(α +1) (α + k 1) k Γ(α + k; /)+ k [1 Γ(α; /)], k an integer mode = (α 1), α > 1, else 0

5 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 5 A Inverse gamma (Vinci) α, (/)α e / F () =1 Γ(α; /) E[X k ] = k Γ(α k), k < α E[X k k ]=, if k is an integer (α 1) (α k) E[(X ) k ] = k Γ(α k) [1 Γ(α k; /)] + k Γ(α; /) = k Γ(α k) G(α k; /)+ k Γ(α; /), all k mode = /(α +1) A Weibull, τ τ(/)τ e (/)τ F () =1 e (/)τ E[X k ] = k Γ(1 + k/τ), k > τ VaR p (X) = [ ln(1 p)] 1/τ E[(X ) k ] = k Γ(1 + k/τ)γ[1 + k/τ;(/) τ ]+ k e (/)τ, k > τ µ 1/τ τ 1 mode =, τ > 1, else 0 τ A Inverse Weibull (log Gompertz), τ τ(/)τ e (/)τ E[X k ] = k Γ(1 k/τ), k < τ VaR p (X) = ( ln p) 1/τ F () =e (/)τ E[(X ) k ] = h i k Γ(1 k/τ){1 Γ[1 k/τ;(/) τ ]} + k 1 e (/)τ, all k = h i k Γ(1 k/τ)g[1 k/τ;(/) τ ]+ k 1 e (/)τ mode = µ 1/τ τ τ +1

6 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 6 A.3.3 A One-parameter distributions Eponential e / F () =1 e / M(t) = (1 t) 1 E[X k ]= k Γ(k +1), k > 1 E[X k ] = k k!, if k is an integer VaR p (X) = ln(1 p) TVaR p (X) = ln(1 p)+ E[X ] = (1 e / ) E[(X ) k ] = k Γ(k +1)Γ(k +1;/)+ k e /, k > 1 = k k!γ(k +1;/)+ k e /, k an integer mode = 0 A Inverse eponential A.5 Other distributions e / 2 E[X k ] = k Γ(1 k), k < 1 VaR p (X) = ( ln p) 1 F () =e / E[(X ) k ] = k G(1 k; /)+ k (1 e / ), all k mode = /2 A Lognormal μ, (μ can be negative) 1 2π ep( z2 /2) = φ(z)/(), E[X k ] = ep(kμ + k 2 2 /2) E[(X ) k ] = µ ln μ k ep(kμ + k /2)Φ mode = ep(μ 2 ) z = ln μ + k [1 F ()] F () =Φ(z)

7 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 7 A Inverse Gaussian μ, A µ 1/2 2π 3 ep µ z2, z = μ 2 μ " µ # 1/2 µ " µ # 1/2 2 F () = Φ z +ep Φ y, y = + μ μ μ " Ã r!# M(t) = ep 1 1 2tμ2, t < μ 2μ 2, E[X] =μ, Var[X] =μ3 / " µ # 1/2 µ " µ # 1/2 2 E[X ] = μzφ z μy ep Φ y μ log-t r, μ, (μ can be negative) Let Y have a t distribution with r degrees of freedom. Then X = ep(y + μ) has the log-t distribution. Positive moments do not eist for this distribution. Just as the t distribution has a heavier tail than the normal distribution, this distribution has a heavier tail than the lognormal distribution. µ r +1 Γ ³ r " πrγ r µ ln μ F () = F r F () = 2 µ # 2 (r+1)/2, ln μ with F r (t) the cdf of a t distribution with r d.f., 1 2 β r 2, 1 2 ; r µ 2 ln μ, 0 < eμ, r β r 2, 1 2 ; r µ 2 ln μ, eμ. r + A Single-parameter Pareto α, αα α+1, > F() =1 (/)α, > VaR p (X) = (1 p) 1/α α(1 p) 1/α TVaR p (X) =, α > 1 α 1 E[X k ] = αk α k, k < α E[(X )k ]= αk α k k α, (α k)α k mode = Note: Although there appears to be two parameters, only α is a true parameter. The value of must be set in advance.

8 APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS 8 A.6 Distributions with finite support For these two distributions, the scale parameter is assumed known. A Generalized beta a, b,, τ Γ(a + b) Γ(a)Γ(b) ua (1 u) b 1 τ, 0 <<, u=(/)τ F () = β(a, b; u) E[X k ] = k Γ(a + b)γ(a + k/τ) Γ(a)Γ(a + b + k/τ), k > aτ E[(X ) k ] = k Γ(a + b)γ(a + k/τ) Γ(a)Γ(a + b + k/τ) β(a + k/τ, b; u)+k [1 β(a, b; u)] A beta a, b, Γ(a + b) Γ(a)Γ(b) ua (1 u) b 1 1, 0 <<, u= / F () = β(a, b; u) E[X k ] = k Γ(a + b)γ(a + k) Γ(a)Γ(a + b + k), k > a E[X k ] = E[(X ) k ] = k a(a +1) (a + k 1), if k is an integer (a + b)(a + b +1) (a + b + k 1) k a(a +1) (a + k 1) β(a + k, b; u) (a + b)(a + b +1) (a + b + k 1) + k [1 β(a, b; u)]