The Basic Parabolic Cylinder Distribution and Its Multivariate Extension

Abstract

The basic parabolic cylinder (bpc) distributions of the firstBasic parabolic cylinder (bpc) distributionof the first kind and second kindsBasic parabolic cylinder (bpc) distributionof the second kind were derived by OgasawaraOgasawara (Commun Stat Theor Methods [17]) to obtain the absolute momentsMomentsabsolute of the normal variable underStripe truncation stripe (zebraicTruncationzebraic or tigerishTruncationtigerish) truncationTruncationstripe. The bpc distribution of the third kindBasic parabolic cylinder (bpc) distributionof the third kind or simply bpc is introduced as a general bpc, whose special cases are the bpc’s of the first and second kinds. TheWeighted or incomplete parabolic cylinder function weighted or incomplete parabolic cylinder functionParabolic cylinder functionincomplete is obtained using theWeighted or incomplete Kummer confluent hypergeometric function weighted or incomplete Kummer confluent hypergeometric functionKummer confluent hypergeometric function given by OgasawaraOgasawara [J Multivar Anal [18]) to have the absolute momentsMomentsabsolute. The multivariate bpc distributionMultivariate basic parabolic cylinder (bpc) distribution is also derived to obtain the absolute momentsMomentsabsolute of the normal vectorNormalvector underTruncationsectional sectional truncationSectional truncation, which is the multivariate version ofTruncationstripe stripe truncationStripe truncation.