Infill statistics, that is, statistical inference based on very dense observations over a fixed domain has become of late a subject of growing importance. On the other hand, it is a known phenomenon that in many cases infill statistics do not provide optimal rates. The degree of sub-optimality...

<Para ID="Par1">We study the asymptotic behavior of the weighted least squares estimators of the unknown parameters of bifurcating integer-valued autoregressive processes. Under suitable assumptions on the immigration, we establish the almost sure convergence of our estimators, together with a quadratic strong...</para>

<Para ID="Par1">In this paper we investigate the large-sample behaviour of the maximum likelihood estimate (MLE) of the unknown parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\theta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">θ</mi> </math> </EquationSource> </InlineEquation> for processes following the model <Equation ID="Equ38"> <EquationSource Format="TEX">$$\begin{aligned} d\xi _{t}=\theta f(t)\xi _{t}\,dt+d\mathrm {B}_t, \end{aligned}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink" display="block"> <mrow> <mtable columnspacing="0.5ex"> <mtr> <mtd columnalign="right"> <mrow> <mi>d</mi> <msub> <mi mathvariant="italic">ξ</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="italic">θ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="italic">ξ</mi>...</msub></mrow></mtd></mtr></mtable></mrow></math></equationsource></equationsource></equation></equationsource></equationsource></inlineequation></para>

The objective of the paper is to identify and investigate all possible types of asymptotic behavior for the maximum likelihood estimators of the unknown parameters in the second-order linear stochastic ordinary differential equation driven by Gaussian white noise. The emphasis is on the...

We consider two problems of constructing of goodness of fit tests for ergodic diffusion processes. The first one is concerned with a composite basic hypothesis for a parametric class of diffusion processes, which includes the Ornstein–Uhlenbeck and simple switching processes. In this case we...

In this paper we consider a wide class of truncated stochastic approximation procedures. These procedures have three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function. We establish...

A problem of goodness-of-fit test for ergodic diffusion processes is presented. In the null hypothesis the drift of the diffusion is supposed to be in a parametric form with unknown shift parameter. Two Cramer–von Mises type test statistics are studied. The first test uses the local time...

Identifying a model by the penalized contrast procedure, we give an analytical estimation of misfitting subsets in the specific case of a least squares contrast. Then, specifying the statistical model, this allows to determine penalization rates ensuring a consistent identification. Applications...

We consider the model selection problem for ergodic diffusion processes based on sampled data. The adaptive estimators for parameters of drift and diffusion coefficients are used in order to construct Akaike’s information criterion (AIC) type model selection statistics. Asymptotic properties...

One-dimensional indexed bilinear (BL) models are widely used for modeling non Gaussian time series. Extending BL models to multidimensional indexed (spatial) SBL one, yields a novel class of models which are capable of taking into account the important characteristic of non Gaussianity and...