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# A characterization of the arcsine distribution

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4YH, UK

Statistics [?] Probability Letters (Impact Factor: 0.6). 12/2009; 79(24):2451-2455. DOI: 10.1016/j.spl.2009.08.018 Source: RePEc

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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.Journal of the American Statistical Association 02/2013; 105(491):1093-1103. DOI:10.1214/12-AOS1079 · 1.98 Impact Factor - [Show abstract] [Hide abstract]

**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 04/2013; 76(3). DOI:10.1007/s00184-012-0391-y · 0.52 Impact Factor