{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:19:29Z","timestamp":1760059169877,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T00:00:00Z","timestamp":1748304000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Edgeworth\u2013Cornish\u2013Fisher expansions are hugely important, as they give the distribution, density and quantiles of any standard estimate. Here we show that the sample mean of a univariate or multivariate stationary process is a standard estimate, so that all the known results for standard estimates can be applied. We also show how to allow for missing data and weighted means.<\/jats:p>","DOI":"10.3390\/axioms14060406","type":"journal-article","created":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T09:29:36Z","timestamp":1748338176000},"page":"406","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Distribution and Quantiles of the Sample Mean from a Stationary Process"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0694-1883","authenticated-orcid":false,"given":"Christopher S.","family":"Withers","sequence":"first","affiliation":[{"name":"Callaghan Innovation (Formerly Industrial Research Ltd.), 101 Allington Road, Wellington 6012, New Zealand"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"307","DOI":"10.2307\/1400905","article-title":"Moments and cumulants in the specification of distributions","volume":"5","author":"Cornish","year":"1937","journal-title":"Rev. de l\u2019Inst. Int. de Statist."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1080\/00401706.1960.10489895","article-title":"The percentile points of distributions having known cumulants","volume":"2","author":"Fisher","year":"1960","journal-title":"Technometrics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1111\/j.2517-6161.1984.tb01310.x","article-title":"Asymptotic expansions for distributions and quantiles with power series cumulants","volume":"46","author":"Withers","year":"1984","journal-title":"J. R. Statist. Soc. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1214\/aos\/1176346163","article-title":"Expansions for the distribution and quantiles of a regular functional of the empirical distribution with applications to non-parametric confidence intervals","volume":"11","author":"Withers","year":"1983","journal-title":"Annals Statist."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/S0167-7152(99)00153-4","article-title":"A simple expression for the multivariate Hermite polynomials","volume":"47","author":"Withers","year":"2020","journal-title":"Stat. Prob. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s00362-008-0135-2","article-title":"Expansions for log densities of asymptotically normal estimates","volume":"51","author":"Withers","year":"2010","journal-title":"Stat. Pap."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1007\/s11009-016-9488-5","article-title":"Expansions for log densities of multivariate estimates","volume":"18","author":"Withers","year":"2016","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_8","first-page":"271","article-title":"Charlier and Edgeworth expansions via Bell polynomials","volume":"29","author":"Withers","year":"2009","journal-title":"Probab. Math. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF02481007","article-title":"The distribution and quantiles of a function of parameter estimates","volume":"34","author":"Withers","year":"1982","journal-title":"Ann. Inst. Statist. Math. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1214\/10-BJPS126","article-title":"Cornish-Fisher expansions for sample autocovariances and other functions of sample moments of linear processes","volume":"26","author":"Withers","year":"2012","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_11","unstructured":"Ibragimov, I.A., and Linnik, Y.V. (1971). Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1214\/aoms\/1177700517","article-title":"Estimating the current mean of a normal distribution which is subjected to changes in time","volume":"35","author":"Chernoff","year":"1964","journal-title":"Ann. Math. Statist."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Comtet, L. (1974). Advanced Combinatorics, Reidel.","DOI":"10.1007\/978-94-010-2196-8"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1007\/s10463-008-0206-0","article-title":"Tilted Edgeworth expansions for asymptotically normal vectors","volume":"62","author":"Withers","year":"2010","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Withers, C.S. (2024). 5th-Order multivariate Edgeworth expansions for parametric estimates. Mathematics, 12.","DOI":"10.20944\/preprints202403.0195.v1"},{"key":"ref_16","unstructured":"Arfken, G.B., Weber, H.J., and Harris, F.E. (2012). Mathematical Methods for Physicists, Academic Press. [7th ed.]."},{"key":"ref_17","unstructured":"Abramowitz, M., and Stegun, I.A. (1964). Handbook of Mathematical Functions, Dover Publications. U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.2307\/1912315","article-title":"A general theorem in the theory of asymptotic expansions as an approximation to the finite sample distributions of econometric estimators","volume":"45","author":"Phillips","year":"1977","journal-title":"Econometrika"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1017\/S0266466600004126","article-title":"Asymptotic expansions in nonstationary vector autoregressions","volume":"3","author":"Phillips","year":"1987","journal-title":"Econom. Theory"},{"key":"ref_20","unstructured":"Hannan, E.J. (1962). Time Series Analysis, Wiley."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Hannan, E.J. (1970). Multiple Time Series, Wiley.","DOI":"10.1002\/9780470316429"},{"key":"ref_22","unstructured":"Kendall, M.G., and Ord, K. (1990). Time Series, Griffin. [3rd ed.]."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Taniguchi, M., and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series, Springer.","DOI":"10.1007\/978-1-4612-1162-4"},{"key":"ref_24","first-page":"281","article-title":"Edgeworth and Cornish Fisher expansions and confidence intervals for the distribution, density and quantiles of kernel density estimates, and confidence intervals for densities","volume":"68","author":"Withers","year":"2008","journal-title":"Statistica"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/406\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:41:30Z","timestamp":1760031690000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/406"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,27]]},"references-count":24,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060406"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060406","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,27]]}}}