Modern Stochastics: Theory and Applications

Finite mixtures with different regression models for different mixture components naturally arise in statistical analysis of biological and sociological data. In this paper a model of mixtures with varying concentrations is considered in which the mixing probabilities are different for different observations. The modified local linear regression estimator (mLLRE) is considered for nonparametric estimation of the unknown regression function for the given component of mixture. The asymptotic normality of the mLLRE is proved in the case when the regressor’s probability density function has jumps. Theoretically optimal bandwidth is derived. Simulations were made to estimate the accuracy of the normal approximation.

Let ${({\xi _{k}},{\eta _{k}})_{k\ge 1}}$ be independent identically distributed random vectors with arbitrarily dependent positive components and ${T_{k}}:={\xi _{1}}+\cdots +{\xi _{k-1}}+{\eta _{k}}$ for $k\in \mathbb{N}$. The random sequence ${({T_{k}})_{k\ge 1}}$ is called a (globally) perturbed random walk. Consider a general branching process generated by ${({T_{k}})_{k\ge 1}}$ and let ${Y_{j}}(t)$ denote the number of the jth generation individuals with birth times $\le t$. Assuming that $\mathrm{Var}\hspace{0.1667em}{\xi _{1}}\in (0,\infty )$ and allowing the distribution of ${\eta _{1}}$ to be arbitrary, a law of the iterated logarithm (LIL) is proved for ${Y_{j}}(t)$. In particular, an LIL for the counting process of ${({T_{k}})_{k\ge 1}}$ is obtained. The latter result was previously established in the article by Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that $\mathbb{E}{\eta _{1}^{a}}\lt \infty $ for some $a\gt 0$. In this paper, it is shown that the aforementioned additional assumption is not needed.

Given a compound mixed renewal process S under a probability measure P, we provide a characterization of all progressively equivalent martingale probability measures Q on the domain of P, that convert S into a compound mixed Poisson process. This result extends earlier works of Delbaen and Haezendonck, Lyberopoulos and Macheras, and the authors, and enables us to find a wide class of price processes satisfying the condition of no free lunch with vanishing risk. Implications to the ruin problem and to the computation of premium calculation principles in an arbitrage-free insurance market are also discussed.

The projected normal distribution, with isotropic variance, on the 2-sphere is considered using intrinsic statistics. It is shown that in this case, the expectation commutes with the projection, and that the covariance of the normal variable has a 1-1 correspondence with the intrinsic covariance of the projected normal distribution. This allows us to estimate, after the model identification, the parameters of the underlying normal distribution that generates the data.

The paper is devoted to the study of the short rate equation of the form with deterministic functions $F,{G_{1}},\dots ,{G_{d}}$ and a multivariate Lévy process $Z=({Z_{1}},\dots ,{Z_{d}})$ with possibly dependent coordinates. This equation is assumed to have a nonnegative solution which generates an affine term structure model. Under some mild assumptions on the Lévy measure of Z it is shown that the same term structure is generated by an equation with affine drift term and noise being a one-dimensional α-stable process with index of stability $\alpha \in (1,2)$. For this case the shape of possible simple forward curves is characterized. A precise description of normal, inverse and humped profiles in terms of the equation coefficients and the stability index α is provided.

In this paper, new closed form formulae for moments of the (generalized) Student’s t-distribution are derived in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the moments of the Student’s t-distribution can be written in terms of the familiar Gamma function, Kummer’s confluent hypergeometric function, and the hypergeometric function. This work aims to provide a concise and unified treatment of the moments for this important distribution.

The expected exit time from the interval $[-1,1]$ is investigated for an autoregressive process defined recursively by Here, ε is a small positive parameter, $f:\mathbb{R}\mapsto \mathbb{R}$ is usually a contractive function and ${\{{\xi _{n}}\}_{n\ge 1}}$ is a sequence of i.i.d. random variables. In this paper, previous results for a linear function $f(x)=ax$ are extended to more general cases, with the main focus on piecewise linear functions.