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Extra resources for Applied Statistics and the SAS Programming Language

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Numerical Recipes, (1992). 1) plot (x, dhyp(x, alpha=1, beta=0)) xmpDistDFnig: Although the fbasics library contains a fast compiled version of dnig, let us write a functions, dnig2, for the evaluation of the inverse Gaussian distribution function. "dnig2" <- function(x, alpha, beta, delta=1, mu=0) { # Density: result <- (delta*exp(delta*sqrt(alpha^2-beta^2)+beta*(x-mu)) * xK1(alpha* sqrt(delta*delta+(x-mu)^2)) / (delta^2+(x-mu)^2)/pi) # Return Value: result} Let us write functions ehyp(), enig() for the MLE of the distribution parameters.

Rte() for the Reuters financial market data service doing the same job. So it becomes very easy to import data from databases and/or from the internet and/or from professional data providers. 32 . 2 Distribution Functions in Finance Much of the real world is controlled as much by the “tails” of distributions as by means or averages: by the exceptional, not the mean; by the catastrophe, not the steady drip; by the very rich, not the middle class. We need to free ourselves from “average” thinking.

50 USD-DEM 6 hour returns / 1373 obs. 1 1 10 100 Empirical Normal Hyperbolic NIG GH 50 density 100 150 200 250 300 USD-DEM 6 hour returns / 1373 obs. 01 100 Empirical Normal Hyperbolic NIG GH 50 density 100 150 200 250 S&P500 1 hour returns / 1703 obs. 9: Densities and log-densities of high frequency USDDEM exchange rate and SP500 stock market index. Source: Prause (1999). α2 − β 2 , 2nd parameterization: ζ = δ = β/α 3rd parameterization: ξ = (1 + ζ)−1/2 , χ = ξ 4th parameterization: α = αδ, β = βδ Note, that for the symmetric distributions β = β = = χ = 0 holds.

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