Asymptotic behavior of statistical estimates of the shift parameter for samples with unbounded density

The present paper is a continuation of the authors' work [1]. As in [1], we consider a sample from a general collection with distribution density f(x– ) depending on the shift parameter . In contrast to [1] it is assumed that the function f(x) is unbounded in a neighborhood of points x₁,h., x_N where it can be represented in the form (1.1). The main results assert that for the Bayes estimates t_n the normalized differences n^1/(1+)(t_n–) have a proper limit distribution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 175–184, 1976.     

Asymptotic theory of nonparametric regression estimates with censored data

For regression analysis, some useful information may have been lost when the responses are right censored. To estimate nonparametric functions, several estimates based on censored data have been proposed and their consistency and convergence rates have been studied in literature, but the optimal rates of global convergence have not been obtained yet. Because of the possible information loss, one may think that it is impossible for an estimate based on censored data to achieve the optimal rates of global convergence for nonparametric regression, which were established by Stone based on complete data. This paper constructs a regression spline estimate of a general nonparametric regression function based on right_censored response data, and proves, under some regularity conditions, that this estimate achieves the optimal rates of global convergence for nonparametric regression. Since the parameters for the nonparametric regression estimate have to be chosen based on a data driven criterion, we also obtain the asymptotic optimality of AIC, AICC, GCV, Cp and FPE criteria in the process of selecting the parameters.     

Computational aspects of statistical intervals based on two Type-II censored samples

In this paper, we propose an efficient branch and bound procedure to compute exact nonparametric statistical intervals based on two Type-II right censored data sets. The procedure is based on some recurrence relations for the distribution and density functions of progressively Type-II censored order statistics which can be applied to compute the coverage probabilities. We illustrate the method for both confidence and prediction intervals of a given level.     

Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models

We first show asymptotic L ² bounds for a class of equations, which includes the Burger-Sivashinsly model for odd solutions with periodic boundary conditions. We consider the conditional stability of stationary solutions of Kuramoto-Sivashinsky equation in the periodic setting. We establish rigorously the general structure of the spectrum of the linearized operator, in particular the linear instability of steady states. In addition, we show conditional asymptotic stability with asymptotic phase, under a natural spectral hypothesis for the corresponding linearized operator. For the zero solution, we have more precise results. Namely, in the non-resonant regime L ≠ n π, we prove conditional asymptotic stability, provided one considers only mean value zero data. If, however, L = n ₀ π (but still ò\nolimits_-L^L u₀(x) dx=0{\int\nolimits_{-L}^L u_0(x) dx=0}), then we have conditional orbital stability. More specifically, the solutions relax to a small (but generally non-zero) function as long as the initial data are small and lie on a center-stable manifold of codimension 2(n ₀ − 1).     

Asymptotic statistical properties of spectral estimates with different tapers for discrete time processes

This paper concerns the problem of estimating the spectral density, spectral distribution and autocovariance functions on non-overlapped and overlapped intervals for a discrete parameter stationary time series with different tapers. Also, we obtain another estimate of spectral density function via different weight functions. Asymptotic statistical properties of these estimates and their limits will be considered.     

Sequences of maximal terms and central exponents of derivatives of Dirichlet series

For the Dirichlet series corresponding to a functionF with positive exponents increasing to ∞ and with abscissa of absolute convergenceA ∈ (−∞, +∞], it is proved that the sequences (μ(σ, F ^(m) )) of maximal terms and (Λ(σ, F ^(m) )) of central exponents are nondecreasing to ∞ asm → ∞ for any givenσ <A, and

Asymptotic normality of the nearest neighbor hazard estimates

1.IntroductionSupposethatXI)')Xu,'beani.i.d.sequenceofrandomvariableswithdistributionfunctionF(x)anddensityfunctionf(x).TOestimatethedensityfunctionatxbasedonthefirstnobservations,thekerneltypeestimategiveswhereK')isakernelfunction,R.(x)isabandwidthsequenceandFi')isanestimateofF,usuallytakentobetheempiricaldistributionfunction.Formoredetails,see[21.Inmedicalapplications,itisoftenmoreimportanttoestimatethehazardfunctiondefinedbyA(x)~f(x)/(1--F(x)).IfXrepresentsalife-time,thenA(x)repre…     

Asymptotic estimates of the pion charge form factor

The asymptotic behavior of the pion charge from factor is obtained in the relativistic Hamiltonian dynamic framework in the limit of infinite momentum transfer and zero mass of the constituent quark. This asymptotic behavior agrees with the perturbative QCD results, is determined only by relativistic kinematics, and is independent of the constituent-quark interaction in the pion. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 2, pp. 215–224, August, 1998.     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.