ISM Online Open House, 2020.10.27統計数理研究所オープンハウス（オンライン開催）、R2.10.27ポスター発

Koji Kanefuji

金藤 浩司

RISM (Repository of the Institute of Statistical Mathematics) / 統計数理研究所 学術研究リポジトリ

2020年10月27日　統計数理研究所　オープンハウス逆ガウス型分布の母中央値に関して金藤　浩司 データ科学研究系　教授ABSTRACTSome distributions for positive random variables have population median.But the median of the inverse Gaussian distribution cannot be expressed byexplicitly. We evaluate it by upper and lower limits.1 INTRODUCTIONLetX be an inverse Gaussian random variable IG(µ, c2) with the probabilityelement f (x)dxf (x)dx =1√2πc(xµ)−32exp[− 12c2(√xµ−√µx)2]dxµ,where 0 < x < ∞, 0 < µ < ∞ and 0 < c < ∞.The parameters µ and c are the population arithmetic mean µA and thecoefficient of variation of X , respectively.We have the cumulative distribution function of IG(µ, c2) as follows;F (x) = Φ(1c(√xµ−√µx))+ e2/c2Φ(−1c(√µx+√xµ)), where Φ(u) is the cumulative distribution function of the standard normaldistribution.Then, the population meadian µmedian for IG(µ, c2) is satisfied withF (µmedian) =12Therefore, the median for IG(µ, c2) can not be determined analytically.Let X be a lognormal random variable LN(µ, c2) with the probabilityelement, f (x)dx, expressed as follows:1√2πc(xµ)−1exp[− 12c2(ln(xµ))2] dxµ,where 0 < x < ∞, 0 < µ < ∞, and 0 < c < ∞.Here, µ is also a median, and c is a population standard deviation ofln(X/µ).2 POPULATION MEADIAN FOR IGWe have the cumulative distribution function of IG(µ, c2) as follows;F (x) = Φ(1c(√xµ−√µx))+ e2/c2Φ(−1c(√µx+√xµ)),where Φ(u) is the cumulative distribution function of the standard normaldistribution.Then, the population meadian µmedian for IG(µ, c2) is satisfied withF (µmedian) =12.Φ(0) =12.Φ(0)− Φ(1c(√µmedianµA−√µAµmedian))= e2/c2Φ(−1c(√µAµmedian+√µmedianµA)).The right hand side eqution is always positive. Then we have the followingformula.Φ(0) > Φ(1c(√µmedianµA−√µAµmedian))Φ(u) is the monotonically increasing function.√µmedianµA<√µAµmedian.Theorem X ∼ IG(µ, c2), thenµmedian < µA.12− e2c2Φ(−2c)≤ Φ(1c(√µmedianµA−√µAµmedian))<12.Φ(u) is the monotonically increasing function, then Φ−1(u) means the inversefunction of Φ(u).Φ−1(12− e2c2Φ(−2c))≤ 1c(√µmedianµA−√µAµmedian)< 0.k = Φ−1(12− e2c2Φ(−2c))< 0.0 >√µmedianµA−√µAµmedian≥ kc.µ2median − ((−k)2c2 + 2)µA · µmedian + µ2A ≤ 0.Theorem X ∼ IG(µ, c2), then its population medain µmedian is satisfiedwith the following formula.12(k2c2 + 2 + kc√k2c2 + 4)µA ≤ µmedian < µA.k = Φ−1(12− e2c2Φ(−2c))< 0.REFERENCESBarndorff-Nielsen, o. E. and Halgreen, C. (1977) “Infinite divisibility of thehyperbolic and generalized inverse Gaussian distributions”, Zeitschrift fürWahrscheinlichkeitstheorie und verwandte Gebiete, 38, 309-311.Gradshteyn, I. S, and Ryxhik, I. M. (2000) Table of Integrals, Series, andProducts(sixth edition), San Diego: Academic Press.Jørgensen, B. (1982) Statistical properties of the generalized inverse Gaus-sian distribution, New York: Springer-Verlag.Kanefuji, K. and , Iwase, K. (2020) On the Lower Limit of the PopulationMedian for the Inverse Gaussian Distribution, JSM 2020.Prudnikov, A. P., Brychokov, Y. A., and Marichev, O. I. (1986), Integralsand Series (volumes 1, 2, and 3), Amsterdam: Gordon and Breach SciencePublishers.Schröndinger, E. (1915) “Zur theorie der fall-und steigversuche an teilchenmit Brrownscher bewegung”, Physikalische Zeitschrift, 16, 289-295.※本研究は，岩瀬晃盛（広島大学名誉教授）との共同研究である．

逆ガウス型分布の母中央値に関して

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逆ガウス型分布の母中央値に関して

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逆ガウス型分布の母中央値に関して

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