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Download Package ghyp - Elementary Latin - Review Sheet | LATIN 1 and more Exams Latin language in PDF only on Docsity! Package ‘ghyp’ May 24, 2009 Type Package Version 1.5.1 Date 2009-05-20 Title A package on the generalized hyperbolic distribution and its special cases Author Wolfgang Breymann, David Luethi Maintainer David Luethi <luethid@gmail.com> LazyLoad yes LazyData no Depends R(>= 2.7), methods, numDeriv, graphics, stats, gplots Description This package provides detailed functionality for working with the univariate and multivariate Generalized Hyperbolic distribution and its special cases (Hyperbolic (hyp), Normal Inverse Gaussian (NIG), Variance Gamma (VG), skewed Student-t and Gaussian distribution). Especially, it contains fitting procedures, an AIC-based model selection routine, and functions for the computation of density, quantile, probability, random variates, expected shortfall and some portfolio optimization and plotting routines as well as the likelihood ratio test. In addition, it contains the Generalized Inverse Gaussian distribution. License GPL (>= 2) Encoding latin1 Repository CRAN Date/Publication 2009-05-24 08:36:11 R topics documented: ghyp-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 coef-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 fit.ghypmv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 fit.ghypuv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1 2 ghyp-package ghyp-constructors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ghyp-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ghyp-get . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ghyp-mle.ghyp-classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 ghyp-risk-performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 ghyp.moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 gig-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 hist-methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 lik.ratio.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 logLik-AIC-methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mean-vcov-skew-kurt-methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 pairs-methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 plot-lines-methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 portfolio.optimize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 qq-ghyp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 scale-methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 smi.stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 stepAIC.ghyp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 summary-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 transform-extract-methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Index 49 ghyp-package A package on the generalized hyperbolic distribution and its special cases Description This package provides detailed functionality for working with the univariate and multivariate Gen- eralized Hyperbolic distribution and its special cases (Hyperbolic (hyp), Normal Inverse Gaussian (NIG), Variance Gamma (VG), skewed Student-t and Gaussian distribution). Especially, it contains fitting procedures, an AIC-based model selection routine, and functions for the computation of den- sity, quantile, probability, random variates, expected shortfall and some portfolio optimization and plotting routines as well as the likelihood ratio test. In addition, it contains the Generalized Inverse Gaussian distribution. Details Package: ghyp Type: Package Version: 1.5.1 Date: 2009-05-20 License: GPL (GNU Public Licence), Version 2 or later Initialize: coef-method 5 • Once initialized the common functions belonging to a distribution can be called conveniently by passing the distribution object. A repeated input of the parameters is avoided. • Distributions returned from fitting procedures can be directly passed to, e.g., the density func- tion since fitted distribution objects add information to the distribution object and consequently inherit from the class of the distribution object. • Generic method dispatching can be used to provide a uniform interface to, e.g., plot the prob- ability density of a specific distribution like plot(distribution.object). Addition- ally, one can take advantage of generic programming since R provides virtual classes and some forms of polymorphism. Acknowledgement This package has been partially developed in the framework of the COST-P10 “Physics of Risk” project. Financial support by the Swiss State Secretariat for Education and Research (SBF) is gratefully acknowledged. Author(s) Wolfgang Breymann, David Luethi Institute of Data Analyses and Process Design (http://www.idp.zhaw.ch) Maintainer: David Luethi <luethid@gmail.com> References Quantitative Risk Management: Concepts, Techniques and Tools by Alexander J. McNeil, Ruediger Frey and Paul Embrechts Princeton Press, 2005 Intermediate probability: A computational approach by Marc Paolella Wiley, 2007 S-Plus and R Library for Quantitative Risk Management QRMlib by Alexander J. McNeil (2005) and Scott Ulman (R-port) (2007) http://www.math.ethz.ch/~mcneil/book/QRMlib.html and QRMlib coef-method Extract parameters of generalized hyperbolic distribution objects Description The function coef returns the parameters of a generalized hyperbolic distribution object as a list. The user can choose between the “chi/psi”, the “alpha.bar” and the “alpha/delta” parametrization. The function coefficients is a synonym for coef. 6 coef-method Usage ## S4 method for signature 'ghyp': coef(object, type = c("chi.psi", "alpha.bar", "alpha.delta")) ## S4 method for signature 'ghyp': coefficients(object, type = c("chi.psi", "alpha.bar", "alpha.delta")) Arguments object An object inheriting from class ghyp. type According to type the parameters of either the “chi/psi”, the “alpha.bar” or the “alpha/delta” parametrization will be returned. If type is missing, the parame- ters belonging to the parametrization of the construction are returned. Details Internally, the “chi/psi” parametrization is used. However, fitting is only possible in the “alpha.bar” parametrization as it provides the most convenient parameter constraints. Value If type is “chi.psi” a list with components: lambda Shape parameter. chi Shape parameter. psi Shape parameters. mu Location parameter. sigma Dispersion parameter. gamma Skewness parameter. If type is “alpha.bar” a list with components: lambda Shape parameter. alpha.bar Shape parameter. mu Location parameter. sigma Dispersion parameter. gamma Skewness parameter. If type is “alpha.delta” a list with components: lambda Shape parameter. alpha Shape parameter. delta Shape parameter. mu Location parameter. Delta Dispersion matrix with a determinant of 1 (only returned in the multivariate case). beta Shape and skewness parameter. fit.ghypmv 7 Note A switch from either the “chi/psi” to the “alpha.bar” or from the “alpha/delta” to the “alpha.bar” parametrization is not yet possible. Author(s) David Luethi See Also ghyp, fit.ghypuv, fit.ghypmv, ghyp.fit.info, transform, [.ghyp Examples ghyp.mv <- ghyp(lambda = 1, alpha.bar = 0.1, mu = rep(0,2), sigma = diag(rep(1,2)), gamma = rep(0,2), data = matrix(rt(1000, df = 4), ncol = 2)) ## Get parameters coef(ghyp.mv, type = "alpha.bar") coefficients(ghyp.mv, type = "chi.psi") ## Simple modification (do not modify slots directly e.g. object@mu <- 0:1) param <- coef(ghyp.mv, type = "alpha.bar") param$mu <- 0:1 do.call("ghyp", param) # returns a new 'ghyp' object fit.ghypmv Fitting generalized hyperbolic distributions to multivariate data Description Perform a maximum likelihood estimation of the parameters of a multivariate generalized hyper- bolic distribution by using an Expectation Maximization (EM) based algorithm. Usage fit.ghypmv(data, lambda = 1, alpha.bar = 1, mu = NULL, sigma = NULL, gamma = NULL, opt.pars = c(lambda = T, alpha.bar = T, mu = T, sigma = T, gamma = !symmetric), symmetric = F, standardize = F, nit = 2000, reltol = 1e-8, abstol = reltol * 10, na.rm = F, silent = FALSE, save.data = T, trace = TRUE, ...) fit.hypmv(data, opt.pars = c(alpha.bar = T, mu = T, sigma = T, gamma = !symmetric), symmetric = F, ...) fit.NIGmv(data, 10 fit.ghypuv fit.ghypuv Fitting generalized hyperbolic distributions to univariate data Description This function performs a maximum likelihood parameter estimation for univariate generalized hy- perbolic distributions. Usage fit.ghypuv(data, lambda = 1, alpha.bar = 0.5, mu = median(data), sigma = mad(data), gamma = 0, opt.pars = c(lambda = T, alpha.bar = T, mu = T, sigma = T, gamma = !symmetric), symmetric = F, standardize = F, save.data = T, na.rm = T, silent = FALSE, ...) fit.hypuv(data, opt.pars = c(alpha.bar = T, mu = T, sigma = T, gamma = !symmetric), symmetric = F, ...) fit.NIGuv(data, opt.pars = c(alpha.bar = T, mu = T, sigma = T, gamma = !symmetric), symmetric = F, ...) fit.VGuv(data, lambda = 1, opt.pars = c(lambda = T, mu = T, sigma = T, gamma = !symmetric), symmetric = F, ...) fit.tuv(data, nu = 3.5, opt.pars = c(nu = T, mu = T, sigma = T, gamma = !symmetric), symmetric = F, ...) fit.gaussuv(data, na.rm = T, save.data = T) Arguments data An object coercible to a vector. lambda Starting value for the shape parameter lambda. alpha.bar Starting value for the shape parameter alpha.bar. nu Starting value for the shape parameter nu (only used in case of a student-t dis- tribution. It determines the degree of freedom and is defined as -2*lambda.) mu Starting value for the location parameter mu. sigma Starting value for the dispersion parameter sigma. gamma Starting value for the skewness parameter gamma. fit.ghypuv 11 opt.pars A named logical vector which states which parameters should be fitted. symmetric If TRUE the skewness parameter gamma keeps zero. standardize If TRUE the sample will be standardized before fitting. Afterwards, the param- eters and log-likelihood et cetera will be back-transformed. save.data If TRUE data will be stored within the mle.ghyp object. na.rm If TRUE missing values will be removed from data. silent If TRUE no prompts will appear in the console. ... Arguments passed to optim and to fit.ghypuv when fitting special cases of the generalized hyperbolic distribution. Details The general-purpose optimization routine optim is used to maximize the loglikelihood function. The default method is that of Nelder and Mead which uses only function values. Parameters of optim can be passed via the . . . argument of the fitting routines. Value An object of class mle.ghyp. Note The variance gamma distribution becomes singular when x − µ = 0. This singularity is catched and the reduced density function is computed. Because the transition is not smooth in the numerical implementation this can rarely result in nonsensical fits. Providing both arguments, opt.pars and symmetric respectively, can result in a conflict when opt.pars[’gamma’] and symmetric are TRUE. In this case symmetric will dominate and opt.pars[’gamma’] is set to FALSE. Author(s) Wolfgang Breymann, David Luethi References ghyp-package vignette in the doc folder or on http://cran.r-project.org/web/packages/ ghyp/. See Also fit.ghypmv, fit.hypmv, fit.NIGmv, fit.VGmv, fit.tmv for multivariate fitting rou- tines. ghyp.fit.info for information regarding the fitting procedure. 12 ghyp-constructors Examples data(smi.stocks) nig.fit <- fit.NIGuv(smi.stocks[,"SMI"], opt.pars = c(alpha.bar = FALSE), alpha.bar = 1, control = list(abs.tol = 1e-8)) nig.fit summary(nig.fit) hist(nig.fit) ghyp-constructors Create generalized hyperbolic distribution objects Description Constructor functions for univariate and multivariate generalized hyperbolic distribution objects and their special cases in one of the parametrizations “chi/psi”, “alpha.bar” and “alpha/delta”. Usage ghyp(lambda = 0.5, chi = 0.5, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))), gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL) ghyp.ad(lambda = 0.5, alpha = 1.5, delta = 1, beta = rep(0, length(mu)), mu = 0, Delta = diag(rep(1, length(mu))), data = NULL) hyp(chi = 0.5, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))), gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL) hyp.ad(alpha = 1.5, delta = 1, beta = rep(0, length(mu)), mu = 0, Delta = diag(rep(1, length(mu))), data = NULL) NIG(chi = 2, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))), gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL) NIG.ad(alpha = 1.5, delta = 1, beta = rep(0, length(mu)), mu = 0, Delta = diag(rep(1, length(mu))), data = NULL) student.t(nu = 3.5, chi = nu - 2, mu = 0, sigma = diag(rep(1, length(mu))), gamma = rep(0, length(mu)), data = NULL) student.t.ad(lambda = -2, delta = 1, beta = rep(0, length(mu)), mu = 0, Delta = diag(rep(1, length(mu))), data = NULL) VG(lambda = 1, psi = 2*lambda, mu = 0, sigma = diag(rep(1, length(mu))), gamma = rep(0, length(mu)), data = NULL) ghyp-constructors 15 Parametrization Distribution “chi/psi” “alpha.bar” “alpha/delta” GH ghyp(...) ghyp(..., alpha.bar=x) ghyp.ad(...) hyp hyp(...) hyp(..., alpha.bar=x) hyp.ad(...) NIG NIG(...) NIG(..., alpha.bar=x) NIG.ad(...) Student-t student.t(..., chi=x) student.t(...) student.t.ad(...) VG VG(..., psi=x) VG(...) VG.ad(...) Have a look on the vignette of this package in the doc folder for further information regarding the parametrization and for the domains of variation of the parameters. Value An object of class ghyp. Note The Student-t parametrization obtained via the “alpha.bar” parametrization slightly differs from the common Student-t parametrization: The parameter sigma denotes the standard deviation in the univariate case and the variance in the multivariate case. Thus, set σ = √ ν/(ν − 2) in the univari- ate case to get the same results as with the standard R implementation of the Student-t distribution. In case of non-finite variance, the “alpha.bar” parametrization does not work because sigma is defined to be the standard deviation. In this case the “chi/psi” parametrization can be used by submitting the parameter chi. To obtain equal results as the standard R implmentation use student.t(nu = nu, chi = nu) (see Examples). Have a look on the vignette of this package in the doc folder for further information. Once an object of class ghyp is created the methods Xghyp have to be used even when the dis- tribution is a special case of the GH distribution. E.g. do not use dVG. Use dghyp and submit a variance gamma distribution created with VG(). Author(s) David Luethi References ghyp-package vignette in the doc folder or on http://cran.r-project.org/web/packages/ ghyp/ See Also ghyp-class for a summary of generic methods assigned to ghyp objects, coef for switch- ing between different parametrizations, d/p/q/r/ES/gyhp for density, distribution function et cetera, fit.ghypuv and fit.ghypmv for fitting routines. 16 ghyp-constructors Examples ## alpha.bar parametrization of a univariate GH distribution ghyp(lambda=2, alpha.bar=0.1, mu=0, sigma=1, gamma=0) ## lambda/chi parametrization of a univariate GH distribution ghyp(lambda=2, chi=1, psi=0.5, mu=0, sigma=1, gamma=0) ## alpha/delta parametrization of a univariate GH distribution ghyp.ad(lambda=2, alpha=0.5, delta=1, mu=0, beta=0) ## alpha.bar parametrization of a multivariate GH distribution ghyp(lambda=1, alpha.bar=0.1, mu=2:3, sigma=diag(1:2), gamma=0:1) ## lambda/chi parametrization of a multivariate GH distribution ghyp(lambda=1, chi=1, psi=0.5, mu=2:3, sigma=diag(1:2), gamma=0:1) ## alpha/delta parametrization of a multivariate GH distribution ghyp.ad(lambda=1, alpha=2.5, delta=1, mu=2:3, Delta=diag(c(1,1)), beta=0:1) ## alpha.bar parametrization of a univariate hyperbolic distribution hyp(alpha.bar=0.3, mu=1, sigma=0.1, gamma=0) ## lambda/chi parametrization of a univariate hyperbolic distribution hyp(chi=1, psi=2, mu=1, sigma=0.1, gamma=0) ## alpha/delta parametrization of a univariate hyperbolic distribution hyp.ad(alpha=0.5, delta=1, mu=0, beta=0) ## alpha.bar parametrization of a univariate NIG distribution NIG(alpha.bar=0.3, mu=1, sigma=0.1, gamma=0) ## lambda/chi parametrization of a univariate NIG distribution NIG(chi=1, psi=2, mu=1, sigma=0.1, gamma=0) ## alpha/delta parametrization of a univariate NIG distribution NIG.ad(alpha=0.5, delta=1, mu=0, beta=0) ## alpha.bar parametrization of a univariate VG distribution VG(lambda=2, mu=1, sigma=0.1, gamma=0) ## alpha/delta parametrization of a univariate VG distribution VG.ad(lambda=2, alpha=0.5, mu=0, beta=0) ## alpha.bar parametrization of a univariate t distribution student.t(nu = 3, mu=1, sigma=0.1, gamma=0) ## alpha/delta parametrization of a univariate t distribution student.t.ad(lambda=-2, delta=1, mu=0, beta=1) ## Obtain equal results as with the R-core parametrization ## of the t distribution: nu <- 4 standard.R.chi.psi <- student.t(nu = nu, chi = nu) standard.R.alpha.bar <- student.t(nu = nu, sigma = sqrt(nu /(nu - 2))) random.sample <- rnorm(3) dt(random.sample, nu) dghyp(random.sample, standard.R.chi.psi) # all implementations yield... dghyp(random.sample, standard.R.alpha.bar) # ...the same values random.quantiles <- runif(4) qt(random.quantiles, nu) ghyp-distribution 17 qghyp(random.quantiles, standard.R.chi.psi) # all implementations yield... qghyp(random.quantiles, standard.R.alpha.bar) # ...the same values ## If nu <= 2 the "alpha.bar" parametrization does not exist, but the ## "chi/psi" parametrization. The case of a Cauchy distribution: nu <- 1 standard.R.chi.psi <- student.t(nu = nu, chi = nu) dt(random.sample, nu) dghyp(random.sample, standard.R.chi.psi) # both give the same result pt(random.sample, nu) pghyp(random.sample, standard.R.chi.psi) # both give the same result ghyp-distribution The Generalized Hyperbolic Distribution Description Density, distribution function, quantile function, expected-shortfall and random generation for the univariate and multivariate generalized hyperbolic distribution and its special cases. Usage dghyp(x, object = ghyp(), logvalue = FALSE) pghyp(q, object = ghyp(), n.sim = 10000, subdivisions = 200, rel.tol = .Machine$double.eps^0.5, abs.tol = rel.tol, lower.tail = TRUE) qghyp(p, object = ghyp(), method = c("integration", "splines"), spline.points = 200, subdivisions = 200, root.tol = .Machine$double.eps^0.5, rel.tol = root.tol^1.5, abs.tol = rel.tol) rghyp(n, object = ghyp()) Arguments p A vector of probabilities. x A vector, matrix or data.frame of quantiles. q A vector, matrix or data.frame of quantiles. n Number of observations. object An object inheriting from class ghyp. logvalue If TRUE the logarithm of the density will be returned. 20 ghyp-get Description These functions simply return data stored within generalized hyperbolic distribution objects, i.e. slots of the classes ghyp and mle.ghyp. ghyp.fit.info extracts information about the fit- ting procedure from objects of class mle.ghyp. ghyp.data returns the data slot of a gyhp object. ghyp.dim returns the dimension of a gyhp object. ghyp.name returns the name of the distribution of a gyhp object. Usage ghyp.fit.info(object) ghyp.data(object) ghyp.name(object, abbr = FALSE, skew.attr = TRUE) ghyp.dim(object) Arguments object An object inheriting from class ghyp. abbr If TRUE the abbreviation of the ghyp distribution will be returned. skew.attr If TRUE an attribute will be added to the name of the ghyp distribution stating whether the distribution is symmetric or not. Value ghyp.fit.info returns list with components: logLikelihood The maximized log-likelihood value. aic The Akaike information criterion. fitted.params A boolean vector stating which parameters were fitted. converged A boolean whether optim converged or not. n.iter The number of iterations. error.code Error code from optim. error.message Error message from optim. parameter.variance Parameter variance (only for univariate fits). trace.pars Trace values of the parameters during the fitting procedure. ghyp.data returns NULL if no data is stored within the object, a vector if it is an univariate generalized hyperbolic distribution and matrix if it is an multivariate generalized hyperbolic dis- tribution. ghyp.name returns the name of the ghyp distribution which can be the name of a special case. Depending on the arguments abbr and skew.attr one of the following is returned. abbr == FALSE & skew.attr == TRUE abbr == TRUE & skew.attr == TRUE (A)symmetric Generalized Hyperbolic (A)symm ghyp (A)symmetric Hyperbolic (A)symm hyp ghyp-get 21 (A)symmetric Normal Inverse Gaussian (A)symm NIG (A)symmetric Variance Gamma (A)symm VG (A)symmetric Student-t (A)symm t Gaussian Gauss abbr == FALSE & skew.attr == FALSE abbr == TRUE & skew.attr == FALSE Generalized Hyperbolic ghyp Hyperbolic hyp Normal Inverse Gaussian NIG Variance Gamma VG Student-t t Gaussian Gauss ghyp.dim returns the dimension of a ghyp object. Note ghyp.fit.info requires an object of class mle.ghyp. In the univariate case the parameter variance is returned as well. The parameter variance is defined as the inverse of the negative hesse- matrix computed by optim. Note that this makes sense only in the case that the estimates are asymptotically normal distributed. The class ghyp contains a data slot. Data can be stored either when an object is initialized or via the fitting routines and the argument save.data. Author(s) David Luethi See Also coef, mean, vcov, logLik, AIC for other accessor functions, fit.ghypmv, fit.ghypuv, ghyp for constructor functions, optim for possible error messages. Examples ## multivariate generalized hyperbolic distribution ghyp.mv <- ghyp(lambda = 1, alpha.bar = 0.1, mu = rep(0, 2), sigma = diag(rep(1, 2)), gamma = rep(0, 2), data = matrix(rt(1000, df = 4), ncol = 2)) ## Get data ghyp.data(ghyp.mv) ## Get the dimension ghyp.dim(ghyp.mv) ## Get the name of the ghyp object ghyp.name(ghyp(alpha.bar = 0)) ghyp.name(ghyp(alpha.bar = 0, lambda = -4), abbr = TRUE) 22 ghyp-mle.ghyp-classes ## 'ghyp.fit.info' does only work when the object is of class 'mle.ghyp', ## i.e. is created by 'fit.ghypuv' etc. mv.fit <- fit.tmv(data = ghyp.data(ghyp.mv), control = list(abs.tol = 1e-3)) ghyp.fit.info(mv.fit) ghyp-mle.ghyp-classes Classes ghyp and mle.ghyp Description The class “ghyp” basically contains the parameters of a generalized hyperbolic distribution. The class “mle.ghyp” inherits from the class “ghyp”. The class “mle.ghyp” adds some additional slots which contain information about the fitting procedure. Namely, these are the number of iterations (n.iter), the log likelihood value (llh), the Akaike Information Criterion (aic), a boolean vector (fitted.params) stating which parameters were fitted, a boolean converged whether the fitting procedure converged or not, an error.code which stores the status of a possible error and the corresponding error.message. In the univariate case the parameter variance is also stored in parameter.variance. Objects from the Class Objects should only be created by calls to the constructors ghyp, hyp, NIG, VG, student.t and gauss or by calls to the fitting routines like fit.ghypuv, fit.ghypmv, fit.hypuv, fit.hypmv et cetera. Slots Slots of class ghyp: call: The function-call of class call. lambda: Shape parameter of class numeric. alpha.bar: Shape parameter of class numeric. chi: Shape parameter of an alternative parametrization. Object of class numeric. psi: Shape parameter of an alternative parametrization. Object of class numeric. mu: Location parameter of lass numeric. sigma: Dispersion parameter of class matrix. gamma: Skewness parameter of class numeric. model: Model, i.e., (a)symmetric generalized hyperbolic distribution or (a)symmetric special case. Object of class character. dimension: Dimension of the generalized hyperbolic distribution. Object of class numeric. expected.value: The expected value of a generalized hyperbolic distribution. Object of class numeric. variance: The variance of a generalized hyperbolic distribution of class matrix. ghyp-risk-performance 25 Arguments alpha A vector of confidence levels. L A vector of threshold levels. object A univarite generalized hyperbolic distribution object inheriting from class ghyp. distr Whether the ghyp-object specifies a return or a loss-distribution (see Details). ... Arguments passed from ESghyp to qghyp and from ghyp.omega integrate. Details The parameter distr specifies whether the ghyp-object describes a return or a loss-distribution. In case of a return distribution the expected-shortfall on a confidence level α is defined as ESα := E(X|X ≤ F−1X (α)) while in case of a loss distribution it is defined on a confidence level α as ESα := E(X|X > F−1X (α)). Omega is defined as the ratio of a European call-option price divided by a put-option price with strike price L (see References): Ω(L) := C(L)P (L) . Value ESghyp gives the expected shortfall and ghyp.omega gives the performance measure Omega. Author(s) David Luethi References Omega as a Performance Measure by Hossein Kazemi, Thomas Schneeweis and Raj Gupta University of Massachusetts, 2003 See Also ghyp-class definition, ghyp constructors, univariate fitting routines, fit.ghypuv, portfolio.optimize for portfolio optimization with respect to alternative risk measures, integrate. Examples data(smi.stocks) ## Fit a NIG model to Credit Suisse and Swiss Re log-returns cs.fit <- fit.NIGuv(smi.stocks[, "CS"], silent = TRUE) swiss.re.fit <- fit.NIGuv(smi.stocks[, "Swiss.Re"], silent = TRUE) ## Confidence levels for expected shortfalls es.levels <- c(0.001, 0.01, 0.05, 0.1) 26 ghyp.moment cs.es <- ESghyp(es.levels, cs.fit) swiss.re.es <- ESghyp(es.levels, swiss.re.fit) ## Threshold levels for Omega threshold.levels <- c(0, 0.01, 0.02, 0.05) cs.omega <- ghyp.omega(threshold.levels, cs.fit) swiss.re.omega <- ghyp.omega(threshold.levels, swiss.re.fit) par(mfrow = c(2, 1)) barplot(rbind(CS = cs.es, Swiss.Re = swiss.re.es), beside = TRUE, names.arg = paste(100 * es.levels, "percent"), col = c("gray40", "gray80"), ylab = "Expected Shortfalls (return distribution)", xlab = "Level") legend("bottomright", legend = c("CS", "Swiss.Re"), fill = c("gray40", "gray80")) barplot(rbind(CS = cs.omega, Swiss.Re = swiss.re.omega), beside = TRUE, names.arg = threshold.levels, col = c("gray40", "gray80"), ylab = "Omega", xlab = "Threshold level") legend("topright", legend = c("CS", "Swiss.Re"), fill = c("gray40", "gray80")) ## => the higher the performance, the higher the risk (as it should be) ghyp.moment Compute moments of generalized hyperbolic distributions Description This function computes moments of arbitrary orders of the univariate generalized hyperbolic dis- tribution. The expectation of f(X − c)k is calculated. f can be either the absolute value or the identity. c can be either zero or E(X). Usage ghyp.moment(object, order = 3:4, absolute = FALSE, central = TRUE, ...) Arguments object A univarite generalized hyperbolic object inheriting from class ghyp. order A vector containing the order of the moments. absolute Indicate whether the absolute value is taken or not. If absolute = TRUE then E(|X − c|k) is computed. Otherwise E((X − c)k). c depends on the argument central. absolute must be TRUE if order is not integer. central If TRUE the moment around the expected value E((X −E(X))k) is computed. Otherwise E(Xk). ... Arguments passed to integrate. gig-distribution 27 Details In general ghyp.moment is based on numerical integration. For the special cases of either a “ghyp”, “hyp” or “NIG” distribution analytic expressions (see References) will be taken if non- absolute and non-centered moments of integer order are requested. Value A vector containing the moments. Author(s) David Luethi References Moments of the Generalized Hyperbolic Distribution by David J. Scott, Diethelm Wuertz and Thanh Tam Tran Working paper, 2008 See Also mean, vcov, Egig Examples nig.uv <- NIG(alpha.bar = 0.1, mu = 1.1, sigma = 3, gamma = -2) # Moments of integer order ghyp.moment(nig.uv, order = 1:6) # Moments of fractional order ghyp.moment(nig.uv, order = 0.2 * 1:20, absolute = TRUE) gig-distribution The Generalized Inverse Gaussian Distribution Description Density, distribution function, quantile function, random generation, expected shortfall and ex- pected value and variance for the generalized inverse gaussian distribution. 30 hist-methods hist-methods Histogram for univariate generalized hyperbolic distributions Description The function hist computes a histogram of the given data values and the univariate generalized hyperbolic distribution. Usage ## S4 method for signature 'ghyp': hist(x, data = ghyp.data(x), gaussian = TRUE, log.hist = F, ylim = NULL, ghyp.col = 1, ghyp.lwd = 1, ghyp.lty = "solid", col = 1, nclass = 30, plot.legend = TRUE, location = if (log.hist) "bottom" else "topright", legend.cex = 1, ...) Arguments x Usually a fitted univariate generalized hyperbolic distribution of class mle.ghyp. Alternatively an object of class ghyp and a data vector. data An object coercible to a vector. gaussian If TRUE the probability density of the normal distribution is plotted as a refer- ence. log.hist If TRUE the logarithm of the histogramm is plotted. ylim The “y” limits of the plot. ghyp.col The color of the density of the generalized hyperbolic distribution. ghyp.lwd The line width of the density of the generalized hyperbolic distribution. ghyp.lty The line type of the density of the generalized hyperbolic distribution. col The color of the histogramm. nclass A single number giving the number of cells for the histogramm. plot.legend If TRUE a legend is drawn. location The location of the legend. See legend for possible values. legend.cex The character expansion of the legend. ... Arguments passed to plot and qqghyp. Value No value is returned. Author(s) David Luethi indices 31 See Also qqghyp, fit.ghypuv, hist, legend, plot, lines. Examples data(smi.stocks) univariate.fit <- fit.ghypuv(data = smi.stocks[,"SMI"], opt.pars = c(mu = FALSE, sigma = FALSE), symmetric = TRUE) hist(univariate.fit) indices Monthly returns of five indices Description Monthly returns of indices representing five asset/investment classes Bonds, Stocks, Commodities, Emerging Markets and High Yield Bonds. Usage data(indices) Format hy.bond JPMorgan High Yield Bond A (Yahoo symbol “OHYAX”). emerging.mkt Morgan Stanley Emerging Markets Fund Inc. (Yahoo symbol “MSF”). commodity Dow Jones-AIG Commodity Index (Yahoo symbol “DJI”). bond Barclays Global Investors Bond Index (Yahoo symbol “WFBIX”). stock Vanguard Total Stock Mkt Idx (Yahoo symbol “VTSMX”). See Also smi.stocks Examples data(indices) pairs(indices) 32 lik.ratio.test lik.ratio.test Likelihood-ratio test Description This function performs a likelihood-ratio test on fitted generalized hyperbolic distribution objects of class mle.ghyp. Usage lik.ratio.test(x, x.subclass, conf.level = 0.95) Arguments x An object of class mle.ghyp. x.subclass An object of class mle.ghyp whose parameters form a subset of those of x. conf.level Confidence level of the test. Details The likelihood-ratio test can be used to check whether a special case of the generalized hyperbolic distribution is the “true” underlying distribution. The likelihood-ratio is defined as Λ = sup{L(θ|X) : θ ∈ Θ0} sup{L(θ|X) : θ ∈ Θ} . Where L denotes the likelihood function with respect to the parameter θ and data X, and Θ0 is a subset of the parameter space Θ. The null hypothesis H0 states that θ ∈ Θ0. Under the null hypothesis and under certain regularity conditions it can be shown that −2 log(Λ) is asymtotically chi-squared distributed with ν degrees of freedom. ν is the number of free parameters specified by Θ minus the number of free parameters specified by Θ0. The null hypothesis is rejected if −2 log(Λ) exceeds the conf.level-quantile of the chi-squared distribution with ν degrees of freedom. Value A list with components: statistic The value of the L-statistic. p.value The p-value for the test. df The degrees of freedom for the L-statistic. H0 A boolean stating whether the null hypothesis is TRUE or FALSE. Author(s) David Luethi mean-vcov-skew-kurt-methods 35 mean-vcov-skew-kurt-methods Expected value, variance-covariance, skewness and kurtosis of gener- alized hyperbolic distributions Description The function mean returns the expected value. The function vcov returns the variance in the uni- variate case and the variance-covariance matrix in the multivariate case. The functions ghyp.skewness and ghyp.kurtosis only work for univariate generalized hyperbolic distributions. Usage ## S4 method for signature 'ghyp': mean(x) ## S4 method for signature 'ghyp': vcov(object) ghyp.skewness(object) ghyp.kurtosis(object) Arguments x, object An object inheriting from class ghyp. Details The functions ghyp.skewness and ghyp.kurtosis are based on the function ghyp.moment. Numerical integration will be used in case a Student.t or variance gamma distribution is submitted. Value Either the expected value, variance, skewness or kurtosis. Author(s) David Luethi See Also ghyp, ghyp-class, Egig to compute the expected value and the variance of the generalized inverse gaussian mixing distribution distributed and its special cases. 36 pairs-methods Examples ## Univariate: Parametric vg.dist <- VG(lambda = 1.1, mu = 10, sigma = 10, gamma = 2) mean(vg.dist) vcov(vg.dist) ghyp.skewness(vg.dist) ghyp.kurtosis(vg.dist) ## Univariate: Empirical vg.sim <- rghyp(10000, vg.dist) mean(vg.sim) var(vg.sim) ## Multivariate: Parametric vg.dist <- VG(lambda = 0.1, mu = c(55, 33), sigma = diag(c(22, 888)), gamma = 1:2) mean(vg.dist) vcov(vg.dist) ## Multivariate: Empirical vg.sim <- rghyp(50000, vg.dist) colMeans(vg.sim) var(vg.sim) pairs-methods Pairs plot for multivariate generalized hyperbolic distributions Description This function is intended to be used as a graphical diagnostic tool for fitted multivariate generalized hyperbolic distributions. An array of graphics is created and qq-plots are drawn into the diagonal part of the graphics array. The upper part of the graphics matrix shows scatter plots whereas the lower part shows 2-dimensional histogramms. Usage ## S4 method for signature 'ghyp': pairs(x, data = ghyp.data(x), main = "'ghyp' pairwise plot", nbins = 30, qq = TRUE, gaussian = TRUE, hist.col = c("white", topo.colors(40)), spline.points = 150, root.tol = .Machine$double.eps^0.5, rel.tol = root.tol, abs.tol = root.tol^1.5, ...) Arguments x Usually a fitted multivariate generalized hyperbolic distribution of class mle.ghyp. Alternatively an object of class ghyp and a data matrix. data An object coercible to a matrix. main The title of the plot. plot-lines-methods 37 nbins The number of bins passed to hist2d. qq If TRUE qq-plots are drawn. gaussian If TRUE qq-plots with the normal distribution are plotted. hist.col A vector of colors passed to hist2d. spline.points The number of support points when computing the quantiles used by the qq-plot. Passed to qqghyp. root.tol The tolerance of the quantiles. Passed to uniroot via qqghyp. rel.tol The tolerance of the quantiles. Passed to integrate via qqghyp. abs.tol The tolerance of the quantiles. Passed to integrate via qqghyp. ... Arguments passed to plot and axis. Author(s) David Luethi See Also pairs, fit.ghypmv, qqghyp, hist2d Examples data(smi.stocks) fitted.smi.stocks <- fit.NIGmv(data = smi.stocks[1:200, ]) pairs(fitted.smi.stocks) plot-lines-methods Plot univariate generalized hyperbolic densities Description These functions plot probability densities of generalized hyperbolic distribution objects. Usage ## S4 method for signature 'ghyp, missing': plot(x, range = qghyp(c(0.001, 0.999), x), length = 1000, ...) ## S4 method for signature 'ghyp': lines(x, range = qghyp(c(0.001, 0.999), x), length = 1000, ...) 40 portfolio.optimize Value A list with components: portfolio.dist An univariate generalized hyperbolic object of class ghyp which represents the distribution of the optimal portfolio. risk.measure The risk measure which was used. risk The risk. opt.weights The optimal weights. converged Convergence returned from optim. message A possible error message returned from optim. n.iter The number of iterations returned from optim. Note In case object denotes a non-elliptical distribution and the risk measure is either “value.at.risk” or “expected.shortfall”, then the type “tangency” optimization problem is not supported. Constraints like avoiding short-selling are not supported yet. Author(s) David Luethi See Also transform, fit.ghypmv Examples data(indices) t.object <- fit.tmv(-indices, silent = TRUE) gauss.object <- fit.gaussmv(-indices) t.ptf <- portfolio.optimize(t.object, risk.measure = "expected.shortfall", type = "minimum.risk", level = 0.99, distr = "loss", silent = TRUE) gauss.ptf <- portfolio.optimize(gauss.object, risk.measure = "expected.shortfall", type = "minimum.risk", level = 0.99, distr = "loss") qq-ghyp 41 par(mfrow = c(1, 3)) plot(c(t.ptf$risk, gauss.ptf$risk), c(-mean(t.ptf$portfolio.dist), -mean(gauss.ptf$portfolio.dist)), xlim = c(0, 0.035), ylim = c(0, 0.004), col = c("black", "red"), lwd = 4, xlab = "99 percent expected shortfall", ylab = "Expected portfolio return", main = "Global minimum risk portfolios") legend("bottomleft", legend = c("Asymmetric t", "Gaussian"), col = c("black", "red"), lty = 1) plot(t.ptf$portfolio.dist, type = "l", xlab = "log-loss ((-1) * log-return)", ylab = "Density") lines(gauss.ptf$portfolio.dist, col = "red") weights <- cbind(Asymmetric.t = t.ptf$opt.weights, Gaussian = gauss.ptf$opt.weights) barplot(weights, beside = TRUE, ylab = "Weights") qq-ghyp Quantile-Quantile Plot Description This function is intended to be used as a graphical diagnostic tool for fitted univariate generalized hyperbolic distributions. Optionally a qq-plot of the normal distribution can be added. Usage qqghyp(object, data = ghyp.data(object), gaussian = TRUE, line = TRUE, main = "Generalized Hyperbolic Q-Q Plot", xlab = "Theoretical Quantiles", ylab = "Sample Quantiles", ghyp.pch = 1, gauss.pch = 6, ghyp.lty = "solid", gauss.lty = "dashed", ghyp.col = "black", gauss.col = "black", plot.legend = TRUE, location = "topleft", legend.cex = 0.8, spline.points = 150, root.tol = .Machine$double.eps^0.5, rel.tol = root.tol, abs.tol = root.tol^1.5, add = FALSE, ...) Arguments object Usually a fitted univariate generalized hyperbolic distribution of class mle.ghyp. Alternatively an object of class ghyp and a data vector. data An object coercible to a vector. 42 qq-ghyp gaussian If TRUE a qq-plot of the normal distribution is plotted as a reference. line If TRUE a line is fitted and drawn. main An overall title for the plot. xlab A title for the x axis. ylab A title for the y axis. ghyp.pch A plotting character, i.e., symbol to use for quantiles of the generalized hyper- bolic distribution. gauss.pch A plotting character, i.e., symbol to use for quantiles of the normal distribution. ghyp.lty The line type of the fitted line to the quantiles of the generalized hyperbolic distribution. gauss.lty The line type of the fitted line to the quantiles of the normal distribution. ghyp.col A color of the quantiles of the generalized hyperbolic distribution. gauss.col A color of the quantiles of the normal distribution. plot.legend If TRUE a legend is drawn. location The location of the legend. See legend for possible values. legend.cex The character expansion of the legend. spline.points The number of support points when computing the quantiles. Passed to qghyp. root.tol The tolerance of the quantiles. Passed to uniroot. rel.tol The tolerance of the quantiles. Passed to integrate. abs.tol The tolerance of the quantiles. Passed to integrate. add If TRUE the points are added to an existing plot window. The legend argument then becomes deactivated. ... Arguments passed to plot. Author(s) David Luethi See Also hist, fit.ghypuv, qghyp, plot, lines Examples data(smi.stocks) smi <- fit.ghypuv(data = smi.stocks[, "Swiss.Re"]) qqghyp(smi, spline.points = 100) qqghyp(fit.tuv(smi.stocks[, "Swiss.Re"], symmetric = TRUE), add = TRUE, ghyp.col = "red", line = FALSE) stepAIC.ghyp 45 Arguments data A vector, matrix or data.frame. dist A character vector of distributions from where the best fit will be identified. symmetric Either NULL, TRUE or FALSE. NULL means that both symmetric and asymmet- ric models will be fitted. For symmetric models select TRUE and for asymmetric models select FALSE. ... Arguments passed to fit.ghypuv or fit.ghypmv. Value A list with components: best.model The model minimizing the AIC. all.models All fitted models. fit.table A data.framewith columns model, symmetric, lambda, alpha.bar, aic, llh (log-Likelihood), converged, n.iter (number of iterations) sorted according to the aic. In the univariate case three additional columns containing the parameters mu, sigma and gamma are added. Author(s) David Luethi See Also lik.ratio.test, fit.ghypuv and fit.ghypmv. Examples data(indices) # Multivariate case: aic.mv <- stepAIC.ghyp(indices, dist = c("ghyp", "hyp", "t", "gauss"), symmetric = NULL, control = list(maxit = 500), silent = TRUE, nit = 500) summary(aic.mv$best.model) # Univariate case: aic.uv <- stepAIC.ghyp(indices[, "stock"], dist = c("ghyp", "NIG", "VG", "gauss"), symmetric = TRUE, control = list(maxit = 500), silent = TRUE) # Test whether the ghyp-model provides a significant improvement with # respect to the VG-model: lik.ratio.test(aic.uv$all.models[[1]], aic.uv$all.models[[3]]) 46 transform-extract-methods summary-method mle.ghyp summary Description Produces a formatted output of a fitted generalized hyperbolic distribution. Usage ## S4 method for signature 'mle.ghyp': summary(object) Arguments object An object of class mle.ghyp. Value Nothing is returned. Author(s) David Luethi See Also Fitting functions fit.ghypuv and fit.ghypmv, coef, mean, vcov and ghyp.fit.info for accessor functions for mle.ghyp objects. Examples data(smi.stocks) mle.ghyp.object <- fit.NIGmv(smi.stocks[, c("Nestle", "Swiss.Re", "Novartis")]) summary(mle.ghyp.object) transform-extract-methods Linear transformation and extraction of generalized hyperbolic distri- butions Description The transform function can be used to linearly transform generalized hyperbolic distribution objects (see Details). The extraction operator [ extracts some margins of a multivariate generalized hyperbolic distribution object. transform-extract-methods 47 Usage ## S4 method for signature 'ghyp': transform(`_data`, summand, multiplier) ## S3 method for class 'ghyp': x[i = c(1, 2)] Arguments _data An object inheriting from class ghyp. summand A vector. multiplier A vector or a matrix. x A multivariate generalized hyperbolic distribution inheriting from class ghyp. i Index specifying which dimensions to extract. ... Arguments passed to transform. Details If X ∼ GH , transform gives the distribution object of “multiplier * X + summand”, where X is the argument named _data. If the object is of class mle.ghyp, iformation concerning the fitting procedure (cf. ghyp.fit.info) will be lost as the return value is an object of class ghyp. Value An object of class ghyp. Author(s) David Luethi See Also scale, ghyp, fit.ghypuv and fit.ghypmv for constructors of ghyp objects. Examples ## Mutivariate generalized hyperbolic distribution multivariate.ghyp <- ghyp(sigma=var(matrix(rnorm(9),ncol=3)), mu=1:3, gamma=-2:0) ## Dimension reduces to 2 transform(multivariate.ghyp, multiplier=matrix(1:6,nrow=2), summand=10:11) ## Dimension reduces to 1 transform(multivariate.ghyp, multiplier=1:3) ## Simple transformation 50 INDEX [.ghyp (transform-extract-methods), 45 AIC, 3, 20, 22, 31 AIC,mle.ghyp-method (logLik-AIC-methods), 32 AIC.mle.ghyp (logLik-AIC-methods), 32 axis, 35 coef, 3, 14, 20, 22, 44 coef,ghyp-method (coef-method), 5 coef-method, 5 coef.ghyp (coef-method), 5 coefficients,ghyp-method (coef-method), 5 d/p/q/r/ES/gyhp, 14 dghyp, 2, 14 dghyp (ghyp-distribution), 15 dgig, 3 dgig (gig-distribution), 26 Egig, 25, 34 Egig (gig-distribution), 26 ESghyp, 3, 17 ESghyp (ghyp-risk-performance), 23 ESgig, 4 ESgig (gig-distribution), 26 fit.gaussmv, 3 fit.gaussmv (fit.ghypmv), 7 fit.gaussuv, 2 fit.gaussuv (fit.ghypuv), 9 fit.ghypmv, 3, 6, 7, 11, 14, 17, 20, 22, 28, 32, 35, 38, 43, 44, 46 fit.ghypuv, 2, 6, 9, 9, 14, 17, 20, 22, 24, 28, 29, 31, 32, 41, 43, 44, 46 fit.hypmv, 3, 11, 20 fit.hypmv (fit.ghypmv), 7 fit.hypuv, 2, 9, 20 fit.hypuv (fit.ghypuv), 9 fit.NIGmv, 3, 11 fit.NIGmv (fit.ghypmv), 7 fit.NIGuv, 2, 9 fit.NIGuv (fit.ghypuv), 9 fit.tmv, 3, 11 fit.tmv (fit.ghypmv), 7 fit.tuv, 2, 9 fit.tuv (fit.ghypuv), 9 fit.VGmv, 3, 11 fit.VGmv (fit.ghypmv), 7 fit.VGuv, 2, 9 fit.VGuv (fit.ghypuv), 9 gauss, 2, 20, 22 gauss (ghyp-constructors), 11 ghyp, 2, 5, 6, 13, 14, 16–25, 28, 33–38, 40, 41, 45, 46 ghyp (ghyp-constructors), 11 ghyp-class, 14, 17, 24, 34 ghyp-class (ghyp-mle.ghyp-classes), 20 ghyp-constructors, 11 ghyp-distribution, 15 ghyp-get, 18 ghyp-mle.ghyp-classes, 20 ghyp-package, 2 ghyp-risk-performance, 23 ghyp.data, 3, 8 ghyp.data (ghyp-get), 18 ghyp.dim, 3 ghyp.dim (ghyp-get), 18 ghyp.fit.info, 3, 6, 8, 9, 11, 32, 44, 45 ghyp.fit.info (ghyp-get), 18 ghyp.kurtosis, 3 ghyp.kurtosis (mean-vcov-skew-kurt-methods), 33 ghyp.moment, 3, 24, 33 ghyp.name, 3 ghyp.name (ghyp-get), 18 ghyp.omega, 3, 17 ghyp.omega (ghyp-risk-performance), 23 ghyp.skewness, 3 ghyp.skewness (mean-vcov-skew-kurt-methods), 33 gig-distribution, 26 hist, 3, 22, 29, 36, 41 hist,ghyp-method (hist-methods), 28 hist-methods, 28 hist.ghyp (hist-methods), 28 hist2d, 35 INDEX 51 hyp, 2, 17, 20, 22 hyp (ghyp-constructors), 11 indices, 29, 42 integrate, 16, 17, 23–25, 27, 28, 35, 40 legend, 29, 40 lik.ratio.test, 3, 30, 32, 43 lines, 3, 22, 29, 36, 41 lines,ghyp-method (plot-lines-methods), 36 lines-methods (plot-lines-methods), 36 lines.ghyp (plot-lines-methods), 36 logLik, 3, 20, 22, 31 logLik,mle.ghyp-method (logLik-AIC-methods), 32 logLik-AIC-methods, 32 logLik.mle.ghyp (logLik-AIC-methods), 32 mean, 3, 20, 22, 25, 41, 44 mean,ghyp-method (mean-vcov-skew-kurt-methods), 33 mean-methods (mean-vcov-skew-kurt-methods), 33 mean-vcov-skew-kurt-methods, 33 mean.ghyp (mean-vcov-skew-kurt-methods), 33 mle.ghyp, 8, 10, 18, 19, 28, 32, 35, 40, 44, 45 mle.ghyp-class, 32 mle.ghyp-class (ghyp-mle.ghyp-classes), 20 NIG, 2, 17, 20, 22 NIG (ghyp-constructors), 11 optim, 8, 10, 19, 20, 22, 37, 38 pairs, 3, 17, 22, 35, 36 pairs,ghyp-method (pairs-methods), 34 pairs-methods, 34 pairs.ghyp (pairs-methods), 34 pghyp, 2 pghyp (ghyp-distribution), 15 pgig, 3 pgig (gig-distribution), 26 plot, 3, 22, 29, 35, 36, 40, 41 plot,ghyp,missing-method (plot-lines-methods), 36 plot-lines-methods, 36 plot-methods (plot-lines-methods), 36 plot.ghyp (plot-lines-methods), 36 portfolio.optimize, 3, 17, 24, 37 qghyp, 2, 23, 36, 40, 41 qghyp (ghyp-distribution), 15 qgig, 3 qgig (gig-distribution), 26 qq-ghyp, 39 qqghyp, 3, 17, 29, 35, 36 qqghyp (qq-ghyp), 39 rghyp, 2 rghyp (ghyp-distribution), 15 rgig, 4, 17 rgig (gig-distribution), 26 scale, 3, 22, 46 scale,ghyp-method (scale-methods), 41 scale-methods, 41 scale.ghyp (scale-methods), 41 show,ghyp-method (ghyp-mle.ghyp-classes), 20 show,mle.ghyp-method (ghyp-mle.ghyp-classes), 20 show.ghyp (ghyp-mle.ghyp-classes), 20 show.mle.ghyp (ghyp-mle.ghyp-classes), 20 smi.stocks, 30, 42 spline, 17, 28 stepAIC.ghyp, 3, 31, 43 student.t, 2, 17, 20, 22 student.t (ghyp-constructors), 11 subsettting, 17 summary, 3, 22 summary,mle.ghyp-method (summary-method), 44 summary-method, 44 52 INDEX summary-methods (summary-method), 44 summary.mle.ghyp (summary-method), 44 transform, 3, 6, 22, 38, 41 transform,ghyp-method (transform-extract-methods), 45 transform-extract-methods, 45 transform.ghyp (transform-extract-methods), 45 transformation, 17 uniroot, 16, 17, 27, 28, 35, 40 vcov, 3, 20, 22, 25, 41, 44 vcov,ghyp-method (mean-vcov-skew-kurt-methods), 33 vcov-methods (mean-vcov-skew-kurt-methods), 33 vcov.ghyp (mean-vcov-skew-kurt-methods), 33 VG, 2, 17, 20, 22 VG (ghyp-constructors), 11 xxx.ad, 22