Cumulative Distribution Functions

Discrete

A discrete random variable is defined by the values it can have, \{x_j\}, and the probability it takes on those values, P(X = x_j) = p_j, where p_j > 0 and \sum_j p_j = 1. It has cdf F(x) = \sum_j 1(x\le x_j) p_j and density f(x) = \sum_j δ_{x_j}(x) p_j where δ_a is the delta function, or point mass, at a. The Esscher transform of a discrete random variable is discrete and takes the same values with P(X_s = x_j) ε_s(x_j) p_j.

Normal

The standard normal random variable X has density φ(x) = \exp(-x^2/2)/\sqrt{2\pi}, -\infty < x < \infty. The cdf can be expressed in terms of error functions as Φ(x) = (1 + \operatorname{erf}(x/\sqrt{2}))/2 = 1 - \operatorname{erfc}(x/\sqrt{2})/2. Since \exp(sx - s^2/2) \exp(-x^2/2) = \exp(-(x - s)^2/2) the Esscher transformed density is φ_s(x) = φ(x - s).

The derivatives of the density are φ^{(n)}(x) = (-1)^nφ(x)H_n(x) where H_n are the Hermite polynomials. They satisfy the recurrence H_0(x) = 1, H_1(x) = x and H_{n+1}(x) = x H_n(x) - n H_{n-1}(x), n\ge 1.

Poisson

The Poisson distribution with parameter λ has density P(X = n) = e^{-λ}λ^n/n!, n\ge 0 and moment generating function M(s) = E[e^{s X}] = \exp(λ(e^s - 1)), s < λ. Since \exp(sn - λ(e^s - 1)) e^{-λ}λ^n = \exp(-λe^s) (λ e^s)^n the Esscher transformed distribution is also Poisson with parameter λe^s.

Taking a derivative with respect to s we have ∂_s E_s[g(X_λ)]] = λ e^s E[g(X_{λ e^s} + 1) - g(X_{λ e^s})] so ∂_s E_s[1(X_λ \le x)] = λ e^s e^{-λ e^s} (λe^s)^n/n! where λ e^s < n \le λ e^s + 1.

Exponential

The density of an exponential with parameter λ is f(x) = λ\exp(-λ x), x\ge 0. The moment generating function is M(s) = E[\exp sX] = \int_0^\infty \exp(sx) λ\exp(-λ x)\,dx = λ/(λ - s), s < λ. The Esscher transformed density is also a an exponential distribution with parameter λ with parameter λ - s.

Generalized Logistic

A generalized logistic random variate has probability density function f(α,b;x) = c e^{-βx}/(1 + e^{-x})^{α + β}, -\infty < x < \infty, where c = 1/B(α,β) is the Beta function. If α = 1 and β = 1 this is the standard logistic function. The Esscher transformed density is also a generalized logistic with parameters α + s, β - s.

Using u = 1/(1 + e^{-x}), so e^x = u/(1 - u) and dx = du/u(1-u) the moment generating function is \begin{aligned} E[e^{sX}] &= c \int_{-\infty}^\infty e^{sx} e^{-βx}/(1 + e^{-x})^{α + β}\,dx \\ &= c \int_0^1 (u/(1 - u))^{s-β} u^{α + β}\,du/u(1 - u) \\ &= c \int_0^1 u^{α + s - 1} (1 - u)^{β - s - 1}\,du \\ &= c B(α + s, β - s)\\ \end{aligned} where B(α,β) is the Beta function. Since 1 = cB(α, β) M(s) = B(α + s, β - s)/B(α, β). A similar calculation shows the incomplete moment generating function is M(s,x) = B(α + s, β - s; u)/B(α, β) = I_u(α + s, β - s) where u = 1/(1 + e^{-x}), B(α, β; u) is the incomplete Beta function, and I_u(α, β) is the regularized incomplete Beta function.

The Esscher transformed cumulative distribution function is F_s(x) = B(α + s, β - s; u)/B(α + s, β - s) = I_u(α + s, β - s).

Next we show ∂_x^n e^{-β x}(1 + e^{-x})^{- α - β} = \sum_{k=0}^n A_{n,k} (e^{-x})^{β + k}(1 + e^{-x})^{- α - β - k} for coefficients A_{n,k}, 0\le k\le n, not depending on x. Clearly A_{0,0} = 1. \begin{aligned} ∂_x^n e^{-β x}(1 + e^{-x})^{- α - β} &= ∂_x\sum_{k=0}^{n-1} A_{n-1,k} (e^{-x})^{β + k}(1 + e^{-x})^{- α - β - k} \\ &= \sum_{k=0}^{n-1} A_{n-1,k} \left((β + k)(e^{-x})^{β + k - 1}(-e^{-x})(1 + e^{-x})^{- α - β - k} + (e^{-x})^{β + k}(- α - β - k)(1 + e^{-x})^{- α - β - k - 1} (-e^{-x})\right)\\ &= \sum_{k=0}^{n-1} A_{n-1,k} \left(-(β + k)(e^{-x})^{β + k}(1 + e^{-x})^{- α - β - k} + (α + β + k) (e^{-x})^{β + k + 1}(1 + e^{-x})^{- α - β - k - 1}\right)\\ &= \sum_{k=0}^{n-1} A_{n-1,k} (-(β + k))(e^{-x})^{β + k}(1 + e^{-x})^{- α - β - k} + \sum_{k=1}^n A_{n-1,k-1} (α + β + k - 1) (e^{-x})^{β + k}(1 + e^{-x})^{- α - β - k}\\ &= A_{n-1,0} (-β)(e^{-x})^β(1 + e^{-x})^{- α - β} + \sum_{k=1}^{n-1} A_{n-1,k} (-(β + k)) + A_{n-1,k-1}(α + β + k - 1)) (e^{-x})^{β + k}(1 + e^{-x})^{- α - β - k} + A_{n-1, n-1} (α + β + n - 1) (e^{-x})^{β + n}(1 + e^{-x})^{- α - β - n}\\ \end{aligned} so A_{n,0} = (-b)^n A_{n-1,0}, A_{n,k} = -(β + k)A_{n-1,k} + (α + β + k - 1) A_{n-1,k-1}, 0 < k < n, and A_{n,n} = (α + β + n - 1) A_{n-1, n-1}. If we define A_{n,-1} = 0 = A_{n,n+1} A_{n,k} = -(β + k)A_{n-1,k} + (α + β + k - 1) A_{n-1,k-1} for n > 0 and A_{0,0} = 1.

The cumulant of the generalized logistic is κ(s) = \log B(α + s, β - s)/B(α, β) = \log Γ(α + s) - \log Γ(α) + \log Γ(β - s) - \log Γ(β) using the fact B(α,β) = Γ(α)Γ(β)/Γ(α + β)

Recall the digamma function ψ(s) = Γ'(s)/Γ(s) is the derivative of the log of the Gamma function so κ^{(n+1)}(s) = ψ^{(n)}(α + s) - (-1)^n ψ^{(n)}(β - s) for n\ge 0. In particular the mean is κ'(0) = ψ(α) - ψ(β) and variance is κ''(0) = ψ'(α) + ψ'(β).

Let the subscripts 1 and 2 indicate the partial derivatives with respect to the first and second parameter respectively. Recall B_1(α,β;u) = B(α,β;u)\log u - u^a\;_3F_2(α, α, 1 - β; α + 1, α + 1; u) so B_1(u)/B(u) - B_1/B = \log u omitting the parameters α and β.

Using B(α,β;u) = 1 - B(β,α;1 - u) we have B_2(α,β;u) = -(\log (1 - u) - ψ(β) + ψ(β + α))B(β, α;1 - u) so B_2(u)/B(u) - B_2/B = -\log (1 - u). Since F_s(x) = B(α + s, β - s; u)/B(α + s, β - s) = B(u)/B \begin{aligned} ∂_s F_s(x) &= \frac{B (B_1(u) - B_2(u)) - B(u)(B_1 - B_2)}{B^2} \\ &= \frac{B(u)}{B}\left[\left(\frac{B_1(u)}{B(u)} - \frac{B_1}{B}\right) - \left(\frac{B_2(u)}{B(u)} - \frac{B_2}{B}\right)\right] \\ &= F_s(x) (\log u + \log (1 - u)). \end{aligned}

Recall B(α,β;u) = \frac{u^α}{α}\,_2F_1(α, 1 - β, α + 1;u). where \,_2F_1(a,b;c;x) = \sum_{n=0}^\infty\frac{(a)_n (b)_n}{(c)_n} x^n/n! is the hypergeometric function.

The derivatives of the hypergeometic function are ∂_x^n\,_2F_1(a, b; c; x) = \frac{(a)_n (b)_n}{(c)_n}\,_2F_1(a + n, b + n; c + n;x).