Saddlepoint approximation method - Wikipedia

The saddlepoint approximation method, initially proposed by Daniels (1954) is a specific example of the mathematical saddlepoint technique applied to ... Saddlepointapproximationmethod FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Thesaddlepointapproximationmethod,initiallyproposedbyDaniels(1954)isaspecificexampleofthemathematicalsaddlepointtechniqueappliedtostatistics.ItprovidesahighlyaccurateapproximationformulaforanyPDForprobabilitymassfunctionofadistribution,basedonthemomentgeneratingfunction.ThereisalsoaformulafortheCDFofthedistribution,proposedbyLugannaniandRice(1980). Definition[edit] Ifthemomentgeneratingfunctionofadistributioniswrittenas M ( t ) {\displaystyleM(t)} andthecumulantgeneratingfunctionas K ( t ) = log ⁡ ( M ( t ) ) {\displaystyleK(t)=\log(M(t))} thenthesaddlepointapproximationtothePDFofadistributionisdefinedas: f ^ ( x ) = 1 2 π K ″ ( s ^ ) exp ⁡ ( K ( s ^ ) − s ^ x ) {\displaystyle{\hat{f}}(x)={\frac{1}{\sqrt{2\piK''({\hat{s}})}}}\exp(K({\hat{s}})-{\hat{s}}x)} andthesaddlepointapproximationtotheCDFisdefinedas: F ^ ( x ) = { Φ ( w ^ ) + ϕ ( w ^ ) ( 1 w ^ − 1 u ^ ) for  x ≠ μ 1 2 + K ‴ ( 0 ) 6 2 π K ″ ( 0 ) 3 / 2 for  x = μ {\displaystyle{\hat{F}}(x)={\begin{cases}\Phi({\hat{w}})+\phi({\hat{w}})({\frac{1}{\hat{w}}}-{\frac{1}{\hat{u}}})&{\text{for}}x\neq\mu\\{\frac{1}{2}}+{\frac{K'''(0)}{6{\sqrt{2\pi}}K''(0)^{3/2}}}&{\text{for}}x=\mu\end{cases}}} where s ^ {\displaystyle{\hat{s}}} isthesolutionto K ′ ( s ^ ) = x {\displaystyleK'({\hat{s}})=x} , w ^ = sgn ⁡ s ^ 2 ( s ^ x − K ( s ^ ) ) {\displaystyle{\hat{w}}=\operatorname{sgn}{\hat{s}}{\sqrt{2({\hat{s}}x-K({\hat{s}}))}}} and u ^ = s ^ K ″ ( s ^ ) {\displaystyle{\hat{u}}={\hat{s}}{\sqrt{K''({\hat{s}})}}} References[edit] Butler,RonaldW.(2007),Saddlepointapproximationswithapplications,Cambridge:CambridgeUniversityPress,ISBN 9780521872508 Daniels,H.E.(1954),"SaddlepointApproximationsinStatistics",TheAnnalsofMathematicalStatistics,25(4):631–650,doi:10.1214/aoms/1177728652 Daniels,H.E.(1980),"ExactSaddlepointApproximations",Biometrika,67(1):59–63,doi:10.1093/biomet/67.1.59,JSTOR 2335316 Lugannani,R.;Rice,S.(1980),"SaddlePointApproximationfortheDistributionoftheSumofIndependentRandomVariables",AdvancesinAppliedProbability,12(2):475–490,doi:10.2307/1426607,JSTOR 1426607 Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Saddlepoint_approximation_method&oldid=1053265278" Categories:AsymptoticanalysisPerturbationtheory Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages Addlinks