"Frontmatter". In: Analysis of Financial Time Series

214 HIGH-FREQUENCY DATA∫= βm ∞Ɣ(κ) 0y κ+m−1 e −y dy = βm Ɣ(κ + m).Ɣ(κ)In particular, the mean and variance of X are E(X) = κβ and Var(X) =κβ 2 .Whenβ = 1, the distribution is called a standard Gamma distribution with parameter κ.We use the notation G ∼ Gamma(κ) to denote that G follows a standard Gammadistribution with parameter κ. The moments of G areE(G m ) =Ɣ(κ + m), m > 0. (5.54)Ɣ(κ)Weibull distributionA random variable X has a Weibull distribution with parameters α and β (α >0,β>0) if its pdf is given byf (x | α, β) ={ αβ α x α−1 e −(x/β)α if x ≥ 00 if x < 0,where β and α are the scale and shape parameters of the distribution. The mean andvariance of X are(E(X) = βƔ 1 + 1 )(, Var(X) = β{Ɣ2 1 + 2 ) [ (− Ɣ 1 + 1 )] } 2αααand the CDF of X is{0 if x < 0F(x | α, β) =1 − e −(x/β)α if x ≥ 0.When α = 1, the Weibull distribution reduces to an exponential distribution.Define Y = X/[βƔ(1 +α 1 )].WehaveE(Y ) = 1 and the pdf of Y is⎧ [ (⎨α Ɣ 1 + 1 )] α { [ (yf (y | α) =α−1 exp − Ɣ 1 + 1 ) ] α }y if y ≥ 0α α⎩0 otherwise,(5.55)where the scale parameter β disappears due to standardization. The CDF of the standardizedWeibull distribution is⎧⎨F(y | α) =⎩1 − exp{ [− Ɣ0(if y < 01 + 1 ) ] α }y if y > 0,αand we have E(Y ) = 1andVar(Y ) = Ɣ(1 + 2 α )/[Ɣ(1 + 1 α )]2 − 1. For a durationmodel with Weibull innovations, the prior pdf is used in the maximum likelihoodestimation.