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Abstract The transition density of the diffusion process plays an important role in calculating the dynamic characteristics of the underlying variables as well as the model estimation. In this article, we combine the idea of conditional probability density function with non parametric kernel regression, and introduce the kernel estimation of joint density function and marginal density function, then construct the non parametric kernel estimator of the transition density of second-order diffusion process based on discrete observational samples. In order to obtain the asymptotic properties of the new kernel estimator, we analyze the asymptotic expectation and asymptotic variance of the proposed estimator under some mild conditions. Finally, the consistency and asymptotic normality of the new proposed non parametric estimator of the transition density function are proved. [ABSTRACT FROM AUTHOR]

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively. [ABSTRACT FROM AUTHOR]

Weak convergence and moment convergence issues are investigated for the New Better than Average Failure Rate (NBAFR) family (introduced by Loh (1984)). We explore the validity of these results in the context of a more general ageing class that we introduce. We prove some new properties of this class and derive its interrelationships with other non-monotonic ageing families. Reliability and moment bounds are obtained and an interesting characterization of exponentiality is proved. Special cases of our results lead to new theorems for the NBAFR class. Finally weak convergence and related issues are established for this class. [ABSTRACT FROM AUTHOR]

In this paper, we introduce lacunary weak statistical convergence of order α and weak N θ α -convergence for 0 < α ≤ 1 . We prove some properties of these modes of convergence and give some inclusion relations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the exponential-type inequalities for maximal partial sums of negatively superadditive dependent (NSD) random variables, which extends the corresponding results for independent and negatively associated (NA) random variables. Using these inequalities, we further investigate the weak convergence of the M-estimators in the generalized linear model with NSD errors, which generalize and improve the corresponding results of the independent random errors to that of NSD random errors. [ABSTRACT FROM AUTHOR]

The extremal ratio has been used in several fields, most notably in industrial quality control, life testing, small-area variation analysis, and the classical heterogeneity of variance situation. In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be an random variable (rv). Generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered rvs. The concept of dual generalized order statistics (DGOS) is introduced to enable a common approach to descendingly ordered rvs. In this article, the limit dfs are obtained for the extremal ratio and product with random indices under non random normalization based on GOS and DGOS. Moreover, this article considers the conditions under which the cases of random and non random indices give the same asymptotic results. Some illustrative examples are obtained, which lend further support to our theoretical results. [ABSTRACT FROM AUTHOR]

We prove an approximation in law of the real-valued mild solution for a class of semilinear stochastic heat equation driven by an additive fractional noise using martingale differences under suitable assumptions. [ABSTRACT FROM PUBLISHER]

In this paper we show an approximation in law to the subfractional Brownian motion within the Skorohod topology. The construction of these approximations is based on random walks. [ABSTRACT FROM AUTHOR]

In this article, we first introduce multifrational Riemann-Liouville Brownian sheets. Then, we show a result of approximation in law of the multifractional Riemann-Liouville Brownian sheet. The construction of these approximations is based on a sequence of I.I.D random variables. [ABSTRACT FROM PUBLISHER]

Necessary and sufficient conditions for the weak convergence of the generalized range, midrange, extremal quotient, and extremal product are obtained. The classes of possible non degenerate limit distribution functions of these simple statistics are characterized. Comparison study between these statistics with some examples for the most important distribution functions are given. [ABSTRACT FROM PUBLISHER]

In this article, we discuss nonparametric estimation of a mean residual life function from length-biased data. Precisely, we prove strong uniform consistency and weak converge of the nonparametric mean residual life estimator in length-biased setting. [ABSTRACT FROM AUTHOR]

The mean residual life of a life distribution,X, with a finite mean is defined byM(t) =E[X−tX>t] fort⩾ 0. Kochar et al. (2000) provided an estimator ofMwhen it is assumed to be decreasing. They showed that its asymptotic distribution was the same as that of the empirical estimate, but only under very stringent analytic and distributional assumptions. We provide a more general asymptotic theory, and under much weaker conditions. We also provide improved asymptotic confidence bands. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of stochastic Liénard equations with random switching. The motivation of our study stems from modeling of complex systems in which both continuous dynamics and discrete events are present. The continuous component is a solution of a stochastic Liénard equation and the discrete component is a Markov chain with a finite state space that is large. A distinct feature is that the processes under consideration are time inhomogeneous. Based on the idea of nearly decomposability and aggregation, the state space of the switching process can be viewed as “nearly decomposable” into l subspaces that are connected with weak interactions among the subspaces. Using the idea of aggregation, we lump the states in each subspace into a single state. Considering the pair of process (continuous state, discrete state), under suitable conditions, we derive a weak convergence result by means of martingale problem formulation. The significance of the limit process is that it is substantially simpler than that of the original system. Thus, it can be used in the approximation and computation work to reduce the computational complexity. [ABSTRACT FROM PUBLISHER]

In survival analysis, the classical Koziol-Green random censorship model is commonly used to describe informative censoring. Hereby, it is assumed that the distribution of the censoring time is a power of the distribution of the survival time. In this article, we extend this model by assuming a general function between these distributions. We determine this function from a relationship between the observable random variables which is described by a copula family that depends on an unknown parameter θ. For this setting, we develop a semi-parametric estimator for the distribution of the survival time in which we propose a pseudo-likelihood estimator for the copula parameter θ. As results, we show first the consistency and asymptotic normality of the estimator for θ. Afterwards, we prove the weak convergence of the process associated to the semi-parametric distribution estimator. Furthermore, we investigate the finite sample performance of these estimators through a simulation study and finally apply it to a practical data set on survival with malignant melanoma. [ABSTRACT FROM AUTHOR]

The Marshall and Olkin (1997) parametrization operation is given by [image omitted] where F1 is a given distribution function and α > 0, [image omitted] = 1 - α. In this article, we show that both Fα = Pα(F1) and the generating distribution F1 belong to the same domain of maximal (minimal) (upper record value) (lower record value) attraction. Moreover, it is shown that the weak convergence of any non extreme order statistic (order statistic with variable rank) based on F1, under given normalizing constants, to a non degenerate limit type implies that the same non extreme order statistic based on the family Fα = Pα(F1), α≠1, under the same normalizing constants, does not converge to any non degenerate distribution and vice versa. [ABSTRACT FROM AUTHOR]

Let [image omitted] be a triangular array of independent elliptical random vectors in d,d ≥ 2. In a recent article, we show for the case d = 2 that the bivariate sample maxima of this triangular array is attracted by a max-infinitely divisible bivariate distribution function, provided that the components of the triangular array become asymptotically dependent and further the random radius associated to the elliptical random vectors has distribution function in the Weibull max-domain of attraction. [ABSTRACT FROM AUTHOR]

Self-consistency equations are established for the distribution functions of right- and left-censored one- and two-dimensional variables and sojourn times of a Markov renewal process. They have a unique solution that equals the product-limit estimator if a hazard function may be defined. Under right-censoring, the results presented here provide new formulations of known estimators. If the left- and right-censoring times are dependent, no estimators were available and simple algorithms are defined. All the estimators are [image omitted]-consistent and converge weakly to centered Gaussian processes. [ABSTRACT FROM AUTHOR]

In this article, we develop a cusum test for testing for parameter changes in linear processes based on Whittle's estimator. It is shown that under regularity conditions, the test statistic converges to the sup of a Brownian bridge. The result is particularly useful in handling the change point test in stationary ARMA processes. A simulation result is provided for illustration. [ABSTRACT FROM AUTHOR]

We construct a univariate exponential dispersion model comprised of discrete infinitely divisible distributions. This model emerges in the theory of branching processes. We obtain a representation for the Lévy measure of relevant distributions and characterize their laws as Poisson mixtures and/or compound Poisson distributions. The regularity of the unit variance function of this model is employed for the derivation of approximations by the Poisson-exponential model. We emphasize the role of the latter class. We construct local approximations relating them to properties of special functions and branching diffusions. [ABSTRACT FROM AUTHOR]

We derive new properties for the power-variance family of probability distributions, which are also referred to as Tweedie laws. They emerge in the theory of generalized linear models and have a wide range of applications. We present decomposition criteria for Tweedie distributions which generalize Cochran's and Raikov's theorems. These criteria are interpreted in terms of the additivity of a shape parameter. The intersection of the power-variance family with the class of distributions which have no indecomposable components is determined. We reveal continuity of this family and obtain convenient common formulas for the cumulants. [ABSTRACT FROM AUTHOR]

We consider one dimensional probability distributions μ having a continuous and positive probability density function. We find the asymptotic of the size and the mass of the Voronoi cells and we prove that the local distortion associated with stationary or optimal quantizers is asymptotically uniform. Numerical simulations and computations illustrate the theoretical results and lead to the design of some good-fit test for the stationary equilibria. [ABSTRACT FROM AUTHOR]

Let X be a nonnegative random variable with cumulative distribution function F. In this article, we introduce a concept of deviation from exponentiality for F, based on the Mean Residual Life function. Our aim is to estimate this deviation under the random censorship model. For this purpose, we propose two slightly different estimators. For each one, we determine the limit distribution of an appropriately normalized version. Then, this work leads to the construction of a test of H0 : F is exponential vs. H1 : F is not exponential. It is applied to the Koziol and Green data set (Koziol, S. A., Green, S. B. (1976). A cramer-von mises statistic for randomly censored data. Biometrika 23:465–474). [ABSTRACT FROM AUTHOR]

We consider the issue of bias correction when the kernel method is used in constructing confidence intervals for wildlife abundance based on transect data. Our method is based on some linear model which is the limit of the weak convergence of the kernel estimate process (after suitable centering and scaling) (See Bhattacharya and Mack, also Mack and Müller). The implementation of this method is demonstrated on a well-known example in transect sampling, alongside with two other bias-reduction devices: one based on the jackknife and the other one involving the estimation of the second derivative term in the Taylor expansion of the asymptotic bias. Some comments are made in comparison with the Fourier method. [ABSTRACT FROM AUTHOR]

The smooth nonparametric estimator &Qtilde;[sub n](p) of a quantile function Q(p) is defined as the solution of &Ftilde;[sub n] (&Qtilde;[sub n] (p)) = p, where &Ftilde;[sub n] is the distribution function corresponding to a kernel estimator of a density function. The asymptotic properties of the smooth quantile process, √n(&Qtilde;[sub n] (p) - Q(p)), based on randomly right censored lifetime data are studied. The bootstrap approaches to approximate the distributions of the smooth quantile processes are investigated and are used to construct simultaneous confidence bands for quantile functions. Data-based selection of the bandwidth required for computing &Ftilde;[sub n] is also investigated using bootstrap methods. A Monte Carlo simulation is carried out to assess small sample performance of the proposed confidence bands. An application to construct confidence bands for the quantile function of the time between a manuscript's submission and its first review is provided using a JASA data set. The developed results can be applied to construct simultaneous confidence bands for the difference of two quantile functions and to check whether there is a location shift or scale change for two distributions under study. [ABSTRACT FROM AUTHOR]

Weak convergence of random variables is characterized by pointwise convergence of the Fourier transform of the respective distributions, and in some cases can also be characterized through the Laplace transform. For some distributions, the Laplace transform is easier to compute and provides an alternative approach to the method of characteristic functions that facilitates proving weak convergence. We show that, for a bivariate distribution, a joint Fourier-Laplace transform always characterizes the distribution when the second variate is positive almost surely. [ABSTRACT FROM AUTHOR]