{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:27:10Z","timestamp":1760239630387,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T00:00:00Z","timestamp":1607558400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>For a statistical distribution, the characteristic function (CF) is crucial because of the one-to-one correspondence between a distribution function and its CF and other properties. In order to avoid the calculation of contour integrals, the CFs of two popular distributions, the F and the skew-normal distributions, are derived by solving two ordinary differential equations (ODEs). The results suggest that the approach of deriving CFs by the ODEs is effective for asymmetric distributions. A much simpler approach is proposed to derive the CF of the multivariate F distribution in terms of a stochastic representation without using contour integration. For a special case of the multivariate F distribution where the variable dimension is one, its CF is consistent with that of the former (univariate) F distribution. This further confirms that the derivations are reasonable. The derivation is quite simple, and is suitable for presentation in statistics theory courses.<\/jats:p>","DOI":"10.3390\/sym12122041","type":"journal-article","created":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T22:15:36Z","timestamp":1607638536000},"page":"2041","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Simple New Proofs of the Characteristic Functions of the F and Skew-Normal Distributions"],"prefix":"10.3390","volume":"12","author":[{"given":"Jun","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Ningbo University, Ningbo 315000, China"}]},{"given":"Sung-Bum","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Konkuk University, Seoul 05029, Korea"}]},{"given":"Seog-Jin","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education, Konkuk University, Seoul 05029, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6014-5207","authenticated-orcid":false,"given":"Hyoung-Moon","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Konkuk University, Seoul 05029, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,10]]},"reference":[{"key":"ref_1","unstructured":"Bisgaard, T.M., and Sasv\u00e1ri, Z. (2000). Characteristic Functions and Moment Sequences, Nova Science Publishers, Inc."},{"key":"ref_2","unstructured":"Lukacs, E. (1983). Developments in Characteristic Function Theory, Macmillan Publishing Co., Inc."},{"key":"ref_3","first-page":"805","article-title":"On a distribution yielding the error functions of several well-known statistics","volume":"2","author":"Fisher","year":"1924","journal-title":"Proc. Int. Congr. Math."},{"key":"ref_4","first-page":"350","article-title":"On the characteristic function of the F- and t-distributions","volume":"32","author":"Ifram","year":"1970","journal-title":"Sankhya Ser. A"},{"key":"ref_5","unstructured":"Johnson, N.L., and Kotz, S. (1970). Continuous Univariate Distributions, 2, Houghton-Mifflin."},{"key":"ref_6","first-page":"128","article-title":"Remark on the characteristic function of the F distribution","volume":"42","author":"Awad","year":"1980","journal-title":"Sankhya Ser. A"},{"key":"ref_7","unstructured":"Abramowitz, M., and Stegun, I.A. (1983). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Chapter 26."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1093\/biomet\/69.1.261","article-title":"The true characteristic function of the F distribution","volume":"69","author":"Phillips","year":"1982","journal-title":"Biometrika"},{"key":"ref_9","unstructured":"Tricomi, F.G. (1954). Funzioni Ipergeometriche Confluenti, Edizioni Cremonese."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0378-3758(00)00208-1","article-title":"On the exact computation of the density and of the quantiles of linear combinations of t and F random variables","volume":"94","author":"Witkovsky","year":"2010","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"435","DOI":"10.4236\/tel.2012.25080","article-title":"Some exact results for an asset pricing test based on the average F distribution","volume":"2","author":"Hwang","year":"2012","journal-title":"Theor. Econ. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5614","DOI":"10.1016\/j.ijleo.2015.09.093","article-title":"Effectiveness of the Euclidean distance in high dimensional spaces","volume":"126","author":"Xia","year":"2015","journal-title":"Optik"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1093\/biomet\/63.1.201","article-title":"Bayes estimation subject to uncertainty about parameter constraints","volume":"63","author":"Leonard","year":"1976","journal-title":"Biometrika"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Cook, R.D., and Weisberg, S. (1994). An Introduction to Regression Graphics, John Wiley & Sons.","DOI":"10.1002\/9780470316863"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Azzilini, A., and Capitanio, A. (2014). The Skew-Normal and Related Families, Cambridge University Press.","DOI":"10.1017\/CBO9781139248891"},{"key":"ref_16","first-page":"189","article-title":"Multivariatedata analysis as a discriminating method of the origin of wines","volume":"25","author":"Forina","year":"1986","journal-title":"Vitis"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2459","DOI":"10.1080\/03610920008832616","article-title":"The wrapped skew-normal distribution on the circle","volume":"29","author":"Pewsey","year":"2000","journal-title":"Commun. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/BF02530549","article-title":"Characterization of the skew-normal distribution","volume":"56","author":"Gupta","year":"2004","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1105","DOI":"10.1016\/j.jmva.2011.03.004","article-title":"Characteristic functions of scale mixtures of multivariate skew-normal distributions","volume":"102","author":"Kim","year":"2011","journal-title":"J. Multivar. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1111\/j.1467-9469.2012.00822.x","article-title":"Characteristic function-based semiparametric inference for skew-symmetric models","volume":"40","author":"Potgieter","year":"2012","journal-title":"Scand. J. Stat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.csda.2015.03.015","article-title":"Fast goodness-of-fit tests based on the characteristic function","volume":"89","author":"Kim","year":"2015","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gaunt, R.E. (2019). A simple proof of the characteristic function of Student\u2019s t-distribution. Commun. Stat.","DOI":"10.1080\/03610926.2019.1702695"},{"key":"ref_23","unstructured":"Johnson, N.L., and Kotz, S. (1972). Continuous Multivariate Distributions, John Wiley."},{"key":"ref_24","unstructured":"Phillips, P.C.B. (1988). The Characteristic Function of the Dirichlet and Multivariate F Distributions. Cowles Foundation for Research in Economics, Yale University. Cowles Foundation Discussion Papers 865."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1093\/biomet\/83.4.715","article-title":"The multivariate skew-normal distribution","volume":"83","author":"Azzalini","year":"1996","journal-title":"Biometrika"},{"key":"ref_26","unstructured":"Olver, F.W.J., Lozier, D.W., Boisvert, R.F., and Clark, C.W. (2010). NIST Handbook of Mathematical Functions, Cambridge University Press."},{"key":"ref_27","first-page":"39","article-title":"\u00dcber die hypergeometrische Reihe F(a; b; x)","volume":"15","author":"Kummer","year":"1836","journal-title":"J. Die Reine Angew. Math."},{"key":"ref_28","unstructured":"Fok, V.A. (1961). \n            Tables of Values of the Function w(z) =\n            \n              \n                \n                  \n                    \n                      e\n                      \n                        \u2212\n                        \n                          z\n                          2\n                        \n                      \n                    \n                    (\n                    1\n                    +\n                    2\n                    i\n                    \n                      \u03c0\n                      \n                        \u2212\n                        1\n                        \/\n                        2\n                      \n                    \n                    \n                      \u222b\n                      0\n                      z\n                    \n                    \n                      e\n                      \n                        \n                          t\n                          2\n                        \n                      \n                    \n                    d\n                    t\n                    )\n                  \n                \n              \n            \n            for Complex Argument\n          . Pergamon Press. translated from the Russian by Fry. D. G."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/2041\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:43:07Z","timestamp":1760179387000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/12\/2041"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,10]]},"references-count":28,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["sym12122041"],"URL":"https:\/\/doi.org\/10.3390\/sym12122041","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,12,10]]}}}