In the paper, we consider the density functions of random variables that can be written as integrals of exponential functions of Gaussian random fields. In particular, we provide closed form asymptotic bounds for the density functions and under smoothness conditions we derive exact tail approximations of the density functions.

Convergence rates and central limit theorems for kernel estimators of the stationary density of a linear process have been obtained under the assumption that the innovation density is smooth (Lipschitz). We show that smoothness is not required. For example, it suffices that the innovation density has bounded variation.

Default priors for density estimation may be constructed through certain approximating sieves with certain natural default priors on these sieves. Finite sieves with metric approximation properties and convolution-sieves provide natural examples and lead to consistent posteriors. Best rate of convergence may be achieved with more reened constructions of sieves using bracketing or spline functio...

In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known or partially known noise density g. We construct goodness-of-fit testing procedures, which are adaptive with respect to the smoothness parameter of unknown density f , and also (in some cases) to some unknown parameter of the noise density g. For known polynomially ...

May [4] estimated some direct and inverse results for certain combinations of these operators. Very recently Finta and Gupta [1] studied some direct and inverse results for the second-order Ditzian-Totik modulus of smoothness. The rates of convergence in ordinary approximation for function of bounded variation for these operators and similar types of operators were estimated in [2, 3, 7]. Very ...

There are two kinds of estimates of the degree of approximation of continuous functions on [−1, 1] by algebraic polynomials, Nikolskii-type pointwise estimates and Jackson-type uniform estimates, involving either ordinary moduli of smoothness, or the DitzianTotik (DT) ones, or the recent estimates involving the weighted DTmoduli of smoothness. The purpose of this paper is to complete the table ...

Let f be a bounded function on the real line IF!. One may characterize the structural properties off by the modulus of smoothness w(t,f) = sup{lf (4 -f( y)l; x, y E 08, I x y I < t>, and, if w(t) is a continuous nondecreasing function of t > 0 such that w(O) = 0, by the Lipschitz class Lip(w) which is the set of all continuous functions such that su~~<~<i w(t, f)/o(t) < 00. It is possible to ex...

We consider Dirichlet series on convex polygons and their rate of approximation in AC(D). We show that the substitution of the respective Leont’ev coefficients by appropriate interpolating sums preserves the order of approximation up to a factor lnn. The estimates are given for moduli of smoothness of arbitrary order. This extends a result of Yu. I. Mel’nik in [4]. AMS Subject Classification (2...