Abstract
The main result of the paper is the following characterization of the generalized arcsine density p γ (t) = t γ−1(1 − t)γ−1/B(γ, γ) with \({t \in (0, 1)}\) and \({\gamma \in(0,\frac12) \cup (\frac12,1)}\) : a r.v. ξ supported on [0, 1] has the generalized arcsine density p γ (t) if and only if \({ {\mathbb E} |\xi- x|^{1-2 \gamma}}\) has the same value for almost all \({x \in (0,1)}\) . Moreover, the measure with density p γ (t) is a unique minimizer (in the space of all probability measures μ supported on (0, 1)) of the double expectation \({ (\gamma-\frac12 ) {\mathbb E} |\xi-\xi^{\prime}|^{1-2 \gamma}}\) , where ξ and ξ′ are independent random variables distributed according to the measure μ. These results extend recent results characterizing the standard arcsine density (the case \({\gamma=\frac12}\)).
Similar content being viewed by others
References
Abramowitz M, Stegun IA (1964) Handbook of mathematical functions. Dover, New York
Durrett R (2005) Probability: theory and examples. Brooks/Cole–Thomson Learning, Belmont
Fahmy MH, Abdou MA, Darwish MA (1999) Integral equations and potential-theoretic type integrals of orthogonal polynomials. J Comput Appl Math 106(2): 245–254
Hille E (1962) Analytic function theory, vol II. Ginn & Co, Boston
Nadarajah S, Gupta AK (2004) Characterizations of the beta distribution. Commun Stat Theory Methods 33(11–12): 2941–2957
Saff EB, Totik V (1997) Logarithmic potentials with external fields. Springer, New York
Schmidt KM, Zhigljavsky A (2009) A characterization of the arcsine distribution. Stat Probab Lett 79(24): 2451–2455
Zhigljavsky A, Dette H, Pepelyshev A (2010) A new approach to optimal design for linear models with correlated observations. J Am Stat Assoc 105(491): 1093–1103
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schmidt, K.M., Zhigljavsky, A. An extremal property of the generalized arcsine distribution. Metrika 76, 347–355 (2013). https://doi.org/10.1007/s00184-012-0391-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-012-0391-y