A characterization of the arcsine distribution

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4YH, UK
Statistics [?] Probability Letters (Impact Factor: 0.53). 12/2009; 79(24):2451-2455. DOI: 10.1016/j.spl.2009.08.018
Source: RePEc

ABSTRACT The following characterization of the arcsine density is established. Let [xi] be a r.v. supported on (-1,1); then [xi] has the arcsine density , -1<t<1, if and only if has the same value for almost all x[set membership, variant][-1,1].

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    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 ${\gamma=\frac12}$ ).
    Metrika 01/2013; 76(3). · 0.45 Impact Factor

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