Volume 25, Issue 2 - 2

The purpose of this paper is to establish a functional large deviations principle (LDP) for L -statistics under some new tail conditions. The method is based on Sanov's theorem and on basic tools of large deviations theory. Our study includes a full treatment of the case of the uniform law and an example in which the rate function can be calculated very precisely. We extend our result by an LDP for normalized L -statistics. The case of the exponential distribution, which is not in the scope of the previous conditions, is completely treated with another method. We provide a functional LDP obtained via Gärtner–Ellis theorem.

We consider estimating a probability density p based on a random sample from this density by a Bayesian approach. The prior is constructed in two steps, by first constructing priors on a collection of models each expressing a qualitative prior guess on the true density, and next combining these priors in an overall prior by attaching prior weights to the models. The purpose is to show that the posterior distribution contracts to the true distribution at a rate that is (nearly) equal to the rate that would have been obtained had only the model that is most suitable for the true density been used. We study special model weights that yield this adaptation property in some generality. Examples include minimal discrete priors and finite-dimensional models, with special attention to scales of Banach spaces, such as Hölder spaces, spline models, and classes of densities that are not uniformly bounded away from zero or infinity.

Kernel functions K ( x ) are widely used for smoothing purposes in statistics, for instance see Silverman [22] or Wand and Jones [24] and also Schimek [21]. Usually, they have compact support and are of order ( v , k ), which means that the j -th moment M j is zero for j < k except j = v < k and is standardized appropriately for j = v . Here, M j means the integral over the product of K ( x ) and x j . In the case of v = 0, K is called a standard kernel. Kernels of order ( v , k ) are called optimal if they change the sign exactly k – 2 times and minimize the asymptotical integrated mean square error. In the case of k – v being even, Gasser et al. [3] have constructed polynomials K ( x ) of degree k with K (-1) = 0 = K (+1) which restricted to [-1,1] have exactly k – 2 sign changes and are of order ( v , k ). In some special cases those K could be proved to be optimal. Later on Granovsky and Müller [5] showed that optimal kernels are continuous functions with K (-1) = 0 = K (+1) and are polynomials on their support. Unfortunately, the converse is true only in the case k - v < 4. Pfeifer [19] in his diploma thesis constructed polynomials p i ( x ) of degree k and reals α i in [0,1] with α 1 < … < α m such that the restriction K i of p i to [-1, α i ] is of order ( v , k ) and fulfills the boundary condition p i (-1) = 0 = p i ( α i ), i = 1,…, m , where m is the integer part of ( k - v )/2; see also Granovsky et al. [6]. In the present paper we prove a long-standing conjecture in its most general form, which in the standard case has been verified by Mammitzsch [14]. By means of the theory of Gegenbauer (ultraspherical) polynomials we show that in the general case of arbitrary k , v with 0 ≤ v ≤ k – 2 the kernel K m is optimal and give its explicit form.