Article

# A characterization of the arcsine distribution

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4YH, UK
(Impact Factor: 0.6). 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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Available from: Anatoly Zhigljavsky, Dec 23, 2013
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##### Article: Optimal design for linear models with correlated observations

Full-text · Article · Sep 2011
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##### Article: A New Approach to Optimal Design for Linear Models With Correlated Observations
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ABSTRACT: In the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated. A necessary condition for the optimality of a given design is provided, which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data. If the regression functions are eigenfunctions of an integral operator defined by the covariance kernel, it is shown that the corresponding measure defines a universally optimal design. For several models universally optimal designs can be identified explicitly. In particular, it is proved that the uniform distribution is universally optimal for a class of trigonometric regression models with a broad class of covariance kernels and that the arcsine distribution is universally optimal for the polynomial regression model with correlation structure defined by the logarithmic potential. To the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations, which are not restricted to the location scale model.
Full-text · Article · Feb 2013 · Journal of the American Statistical Association
• ##### Article: An extremal property of the generalized arcsine distribution
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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 $${\gamma=\frac12}$$).
No preview · Article · Apr 2013 · Metrika