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3definition.The hierarchical structure among classes of H-functions isgiven through seven new theorems in Section 2.4.6.Every classof H-functions is wholly contained in mnny higher-order classesthrough the application of the duplication,of Figure 1informulasSection 3.5forthegammillustratesthishierarchical structure with a venn diagramn showing mony commiostatistical distributions as first and second order H-functiondistributions.Four new theoremgeneralizingconstantrepresentation.are given,in Section 2.4.7 show when and how anaybepresentanH-functionMany generalized H-function representationsincluding those of every cumulative distributionfunction of an H-function variate.isinThe generalizing constantalso possible in the H-function representations of powerfunctions,theerrorfunctionanditsincomplete gamma function and its ccmplemunt,beta function and its complement,conplement,thethe incmpletemny inverse trigonometricand hyperbolic functions, and the logarithmic functions.A number of new H-functionrepresentations ofcertainmathematical functions and statistical distributions are givenin Sections 2.5, 2.6, and 3.6.Several of these expand upon a

4previously unstatedfunctionlimitationrepresentationsofon the variablepowerfunctionsinthe H-and beta-typefunctions.The exact distribution of the general sum of independentErlang variates with different scale parameters, X, is derivedin Section 5.1.3.An Erlang variate is simply a gamma variatewith an integer shape parameter r.The derivation uses partialfractions to decaqpose the product of Laplace transforms of theindividual densities.This produces a sum of terms, each ofwhich can easily be inverted fran transform space, yielding thedesired density of the sum of independent variates.Since the H-function is not defined for zero or negativereal arguments,to continuousvalues.the scope of this research effort was limitedrandom variablesdefined only over positiveContinuous and doubly infinite distributions such asthe normal and Student's t are only represented as H-functionsin their folded forms.1.2.LITERATURE SURVEYRegrettably,littleresearch inhas been done in the United States.the field of H-functionsMuch of what is knownabout the H-function is due to Indian mthemticians.Mathaiand Saxena [1978] and Srivastava et al [1982] compiled manyresults of the early study of H-functions.In recent years,

5Soviet mathematicians[Prudnikov et al,considerable interestin1990] have shown athe H-function and have developedsignificant new results.The foundation of H-function theory is grounded in theintegral transform theory, complex analysis,gasmfunction,andstatisticallandmarkErd lyidistributionsuch as Abramowitzreferences[1953],theory.Erd lyi [1954],Therefore,severaland Stegun[1970),and Springer [1979],thoughsomewhat dated, have timeless value.Carter[1972]defined the H-function distribution and,using Mellin transform theory,results showing that products,gave startling and powerfulquotients, and rational powersof independent H-function variates are themselves H-functionvariates.Further, the p.d.f. of the new random variable caninuediately be written as an H-function distribution.Theusual techniques of conditioning on one of the random variablesand/or using the Jacobian of the transformation are no fulwhenonerealizes that nearly every common positive continuous randomvariableTherefore,products,canbewrittenthe p.d.f.quotients,asanH-functiondistribution.of any algebraic combination involvingor powers of any number of independent

6positive continuous random variables can imiediately be writtenas an H-function distribution.Carter [1972] also wrote a FCRTRAN computer program pendent H-function variates and approximate the p.d.f. andcumulative distribution function (c.d.f.) from these nunents.The approximation procedure was developed by Hill [1969] and,if possible, uses either a Grar-Charlier type A series (Hermitepolynomial)or a Laguerre polynomialseries.Ifa seriesapproxirration is not possible, the first four nmrents are usedto fit a probability distribution from the Pearson family.Carter [1972] himelf notes ".Asthere were nany situations inwhich the nethods did not work or in which the approxirationswere totally unsatisfactory."Springer [1979] literally wrote the book on the algebra ofrandom variables.He gives an excellent explanation of thevalue of integral transformin finding the distribution ofalgebraic combinations of random variables.He also gave theknown applications of the H-function in these problem.Cook [1981] gave a very thorough survey and an extensivebibliography of the literature related to H-functions and Hfunction distributions.He also presented a technique forfinding, in tabular form, the p.d.f. and c.d.f. of an algebraic

findependent H-function variates.Cook's technique [1981; Cook and Barnes, 1981] first usesCarter's [1972] results to find the H-function distribution ofany products, quotients, or powers of random chH-functionfunctionscorresponding evaluatedandThesumThesemultipliedof the transform variable,ofatyielding atabular representation of the Laplace transform in transformspace.This Laplace transform isusing Crurp's method.then numerically invertedHis FORTRAN computer program implementsthis technique and will plot the resulting p.d.f. and c.d.f.Bodenschatz and Boedigheimer [1983; Boedigheimer et al,1984] developed a technique to fit the H-function to a set ofdata using the method of moments. The technique can be used tocurve-fit a mathematical function or to estimate the density ofa particular probability distribution.program will accept known moments,pair data from a relativeTheir FORTRAN coMputerunivariate data,frequency,orderedor ordered pair datadirectly from the function.Kellogg[1984;Kelloggand Barnes,1987;KelloggandBarnes, 1989] studied the distribution of products, quotients,

8and powers of dependent random variables with bivariate Hfumction distributions.Jacobs [1986;Jacobs et al,1987]presented a method of obtaining parameter estimates for the Hfunction distribution using the method of maximum likelihood orthe method of moments.Prudnikov et al [1990] gave extensive tables of H-functionresults and Mellin transform.Their books,although moreterse than the series by Erd6lyi [1953 and 1954], are at leastas complete and, likely, will become the new standard referencefor special functions.1.3.INTERAL TRANSFCRS AND TRANSFORM PAIRSIntegral transforms are frequently encountered in severalareas of mathematics,probability,and statistics.Althoughvarious integral transforms exist, certain characteristics arecamxmiamong them.The function to be transformed is usuallymultiplied by another function (called the kernel)integrated over an appropriate range.various transformintegration,are the kerneland thenWhat distinguishes thefunction,and the type of integrationthe limits of(e.g.Riemaunn orLebesgue).Oftendifficultthe use n simplify ausuallyfirstencountered in the solution of systems of linear differential

9equations.In probability and statistics, certain integraltransforms are often known by other names such as the mom-entgenerating function,characteristicfunction,or probabilitygenerating function.Thedefinitionsstandard.of mostintegraltransformsare notIt is important, therefore, to explicitly state theform of the definition to be used.Listed below are ute a transform pair.1.3.1.LAPLACE TRANSFORMConsider a function f(t) which is sectionally continuousand defined for all positive values of the variable t withf(t) O for tO.A sectionally continuous function may not havean infinite number of discontinuities nor any positive verticalasymptotes.Iff(t) grows no faster than an exponentialfunction, then the Laplace transform of f(t) will exist.Theremust exist two positive nunbers M and T such that for all t Tand for same real number a,f (t )t(1.1).

10The definition of the Laplace transform of the functionf(t), zsf f(t)}, isZ (tJoe-}(1.2)f (t) dt0In general,s is a complex variable.f(t) willexist(Re(s)fortherealpartThe Laplace transform ofofsgreaterthan a a).The inversion integralor inverse Laplacetransform isgiven byJf(t)where-fest ZsftM)} isf(t)ds(1.3)an analytic function for Re(s) w.function is analytic at s s0 if its derivative exists at sat every point in some neighborhood of s o .AandThe Taylor seriesexpansion of an analytic function of a ccrplex variable willexist,converge,argument.andequalthe functionevaluatedat theFor all practical purposes, a function f(t) and itsLaplace transform (if it exists) uniquely determine each other.In probability and statistics, if f(t) is the p.d.f. of arandom variable defined only for positive values, its momentgenerating function issinply the Laplace transform with r

11replacing -s in Eq (1.2).1.3.2.FOURIER TRANSFMThe form of the exponential Fourier transform used in thisthesis is,s{f(t)}e stf(t)dt(1.4)The Fourier transform is a function of the coniplex variable s.If f(t) is a p.d.f.,this definition corresponds to thedefinition of the characteristic function in probability andstatistics.The characteristic function of a p.d.f. willalways exist but the nmnnt generating function of a p.d.f. nayor ay not exist.The inversion integralor inverse Fourier transform isgiven bywheref ()*1imf(t)tt 0 f(t) 1imtlt 0t t(1.6)t t oIf f(t) exists and is continuous at tthe inverse Fourier

12transform ofY,(f(t) )willgive f(t).Iff(t) iscontinuous at to, the inverse Fourier transform of Ynotf(t)}will produce the average of the limits of f(t) from the left ofto and the right of to.1.3.3.MELLIN TRANPOINBecause the Mellin transform is perhaps less well knownthan the Laplace or Fourier transform and because the Mellintransform isso crucialinthe study of H-functions,com n sets of transform pairs will be presented.transform uses a powerbothThe Mellinfunction instead of an exponentialfunction as its ycontinuous and defined for all positive values of the variablet with f(t) O for t,o.Using what will be regarded in thisthesis as the primary definition of the Mellin transform, theMellin transform of f(t), AA(f(t) ) f(t)},isJotS 1 f(t) dt(1.7)0The Mellin transform is related to the Fourier and Laplacetransforms as follows [Erd6lyi, 1954, p.305]:

13f f(t)} f(et)f-i{{-Again,sisf(et)(1.8)} }f(e-)Z{a carplex variable.(1.9)The Mellin transforminversion integral, or inverse Mellin transform, is given as1,(O iCOc1-iAs mentioned earlier, there is another important transformpairalsoreferredalternateto asdefinitiondefinitionofthewilla MellinariseH-functiontransform efinition isi f(t)} Jt-rl f(t) dt(1.11)0with inverse transformf(t)JVVC xr{J f(t)}dr(1.12)u-iCD1.4.TRANSFUA canmnfindtheTICIS OF INDEHWIT RANDC4 VARIABLESproblem in statistical distribution theory is todistributionofanalgebraiccombinationof

14The algebraic cumbination couldindependent random tsand/orindependent random variables or their powers.Itisofimportantto recognize that the algebraic combination is itself a randomThevariable and, therefore, has a probability distribution.task is to find this distribution.itUsing the properties of mathematical expectation,isrelatively easy to find the mean, variance, and other nxmentsof the algebraic cumbination of independent random variables.the mean of the sum of two independent randomFor example,variablesissimplythe meansthe sum ofofthe randomFinding the completely specified distribution ofvariables.the algebraic combination is usually much more difficult.For simple ccubinaticns of independent randum variables,the method of Jacobiansone-to-onetransformationis often employed.betweentheAn appropriateindependentrandumvariables in the algebraic cumbination and a set of new setransformation functions, the Jacobian (the determinant of thematrix of first partial derivatives of the inverse functions)may be computed.The joint p.d.f. of the newly defined random variables isthe absolute value of the Jacobian multiplied by the product of

15theoriginaldensitieswiththeinversefunctions substituted for the variables.transformationA great deal of caremust be used in determining the values of the new variables forwhich the joint density is nonzero.Cnce this is done, thedesired marginal density can be obtained by integrating overthe complete ranges of the other new variables.An example of the method of Jacobians will be presentedbelow.In most of the following sections, however,only theresult using integral transforms will be given.1.4.1.LetX1DISTRIBUTION OF a SMand Xbeindependentrespective densities f i(x)2.variableswithand f 2 (x), each nonzero only forpositive values of the variable.of Y X XrandnSuppose we want the densityUsing the method of Jacobians, we define W X 2 sothe inverse transformations are XI Y-W and X--W.The Jacobianis: n:10(1.13)1The joint density of Y and W isfywCY,w) f1 (y-w) f 2 (w)The marginal p.d.f.0 W y C(1.14)of Y can be obtained by integrating thejoint p.d.f. with respect to W over the range of W.

16 J1 0Eq (1.15)integralfl(Y-w) f2 (w)dw(1.15)nay be recognized as the Fourier convolution(Springer,1979,p. 47].This isno accident orcoincidence as the Fourier or Laplace transform could also havebeen used to find the distribution of Y.It is well known thatthe Laplace (or Fourier) transform of the p.d.f. of the sum oftwo independent random variables defined for positive values isthe product of the Laplace (or Fourier)individual densities.Further,transforms of thethe product of two transformfunctions, upon inversion, yields a convolution integral.If the product of transform functions can be recognized asthe transform of some function, then the convolution inversionisnot necessary.transform isThe p.d.f.the product.of Y isthe function whoseWhen statisticians use the momentgenerating function of each density to find the p.d.f.of Y,this recognition approach isusually taken.using transform functions isthat the procedure easily extendsone advantage ofwhen the distribution of the sum of three or more independentrandom variables is desired.Finding the distribution of the sum of independent randomvariableswithcertainconsiderably sinplified.specialdistributionalformsisSeveral of these cases are covered in

17the following subsections.1.4.1.1.INFINITE DIVISIBILITYAlthough a ccmplete discussion of infinite divisibility isbeyond the scope of this thesis, its definition is given below[Petrov, 1975, p. 25).A distribution function F(x) and the correspondingcharacteristic function f(t) are said to beinfinitely divisible if for every positive integer nthere exists a characteristic function fn(t) suchthatf(t) (fn(t))nIn other words, the distribution F is infinitelydivisible if for every positive integer n thereexists a distribution function Fn such that F F*n .Here F*n is the n-fold convolution of the functionnF.nCcmnon exanples of infinitely divisible distributions includethe normal and Poisson distributions.1.4.1.2.TheSPECIAL les with certain distributional form is well known andimrediately available.For exanple, the sum of n independentand identically distributed randmn variables with a Bernoullidistribution with parameter p has a Binanial distribution withparametersn andgeometricallyp.Similarly,distributedrandomthe sum ofvariableswithindependentacaot

18parameter has a negative BinaTial (or Pascal) distribution.the sum of n independentIn continuous random mdistribution with parameters n and X.1.4.1.3.AREPRODUCTIVE svariablespositiveisaddition"reproductive"ofwith the same distributionalindependentifitrandomThe normalform.distribution is reproduct

Mellin transform of an H-function is also easily obtained. The H-function exactly represents the probability density function and cmarulative distribution function of nearly all continuous . Table Page 1 Convergence Types for H-Functions of Eq (2.1) . 30 2 Convergence Types f or H-Funct