Sum Of Processes Research Articles

Pfeifer records process

In this article, we study the limit distribution of sums of Pfeifer records. Motivated by the results obtained by Arnold and Villaseñor [Generalized order statistics process and Pfeifer records, Statistics 46(3) (2012), pp. 373–385], we show that the partial sum process of Pfeifer records converge to a function of the Brownian motion. The normalization is either a sequence of appropriate constants or a sequence of functions, depending on the tail behaviour of the underlying variables. These results, in particular, prove stronger version of results obtained in Villaseñor and Arnold [On limit laws for sums of Pfeifer records, Extremes 10 (2007), pp. 235–248] and Bose and Gangopadhyay [Convergence of linear functions of Pfeifer records, Extremes 13 (2010), pp. 89–106] and extends results of Bose et al. [Partial sum process for records, Extremes 8 (2005), pp. 43–56] from classical records to Pfeifer records.

A functional limit theorem for dependent sequences with infinite variance stable limits

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable L\'{e}vy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable L\'{e}vy process. Due to clustering, the L\'{e}vy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of c\`{a}dl\`{a}g functions endowed with Skorohod's $M_1$ topology, the more usual $J_1$ topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared GARCH(1,1) processes and stochastic volatility models.

The cumulative quantile regression function with censored and truncated covariate

In this paper, we propose a nonparametric estimator of the cumulative quantile regression (CQR) function when the response is subjected to random truncation and censorship. The observed responses are weighted by the increments of the product-limit estimator for the underlying response distribution. Strong Gaussian approximations of the associated weighted partial sum process and the CQR process are established under appropriate assumptions. A functional law of the iterated logarithm for the CQR process is also derived. The construct provides a foundation for the asymptotic theory of functional statistics based on these processes.

The influence of sequential extremal processes on the partial sum process

In this paper we derive general upper bounds for the total variation distance between the distributions of a partial sum process in row-wise independent, non-negative triangular arrays and the sum of a fixed number of corresponding extremal processes. As a special case we receive bounds for the supremum distance between the distribution functions of a partial sum and the sum of corresponding upper extremes which improve upon existing results. The outcome may be interpreted as the influence of large insurance claims on the total loss. Moreover, under an additional infinitesimal condition we also prove explicit bounds for limits of the above quantities. Thereby we give a didactic and elementary proof of the Ferguson–Klass representation of Levy processes on ℝ ≥ 0 which reflects the influence of extremal processes in insurance.

Removing logarithms in Poisson approximation to the partial sum process of a Markov chain

He and Xia (1997, Stochastic Processes Appl. 68, pp. 101–111) gave some error bounds for a Wasserstein distance between the distributions of the partial sum process of a Markov chain and a Poisson point process on the positive half-line. However, all these bounds increase logarithmically with the mean of the Poisson point process. In this paper, using the coupling method and a general deep result for estimating the errors of Poisson process approximation in Brown and Xia (2001, Ann. Probab. 29, pp. 1373–1403), we give a new error bound for the above Wasserstein distance. In contrast to the previous results of He and Xia (1997), our new error bound has no logarithm anymore and it is bounded and asymptotically remains constant as the mean increases.

The Pólya Sum Process: Limit Theorems for Conditioned Random Fields

Zessin (J. Contemp. Math. Anal. 44(1):36–44, 2009) constructed the so-called Pólya sum process via partial integration technique. This process shares some important properties with the Poisson process such as complete randomness and infinite divisibility. This work discusses H-sufficient statistics for the Pólya sum process as was done for the Poisson process by Nguyen and Zessin (Z. Wahrscheinlichkeitstheor. Verw. Geb. 37(3):191–200, 1976/77).

Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance

The random walk is used as a model expressing equitableness and the effectiveness of various finance phenomena. Random walk is included in unit root process which is a class of nonstationary processes. Due to its nonstationarity, the least squares estimator (LSE) of random walk does not satisfy asymptotic normality. However, it is well known that the sequence of partial sum processes of random walk weakly converges to standard Brownian motion. This result is so-called functional central limit theorem (FCLT). We can derive the limiting distribution of LSE of unit root process from the FCLT result. The FCLT result has been extended to unit root process with locally stationary process (LSP) innovation. This model includes different two types of nonstationarity. Since the LSP innovation has time-varying spectral structure, it is suitable for describing the empirical financial time series data. Here we will derive the limiting distributions of LSE of unit root, near unit root and general integrated processes with LSP innovation. Testing problem between unit root and near unit root will be also discussed. Furthermore, we will suggest two kind of extensions for LSE, which include various famous estimators as special cases.

The Generation of Correlated Gamma Distributed Random Fields for the Simulation of Synthetic Aperture Radar Images

Simulation plays an important role in both the development and analysis of new radar imaging and processing systems and in the analysis of their data. We have developed a synthetic aperture radar (SAR) simulation system, cSAR, that simulates the raw signal data. An important component of this system is the simulation of clutter, or the random spatial fluctuations of backscatter. The prevailing statistical model for clutter is the K distribution. This model is founded on a gamma distributed scattering coefficient. In this paper we present a system for simulating correlated gamma distributed fields of scattering coefficients. Our system starts with the requirement that in radar and other coherent imaging scenarios the locations of elemental scene scatterers must be random to achieve fully developed speckle in the image. We start with a set of scatterers whose spatial location follows a uniform distribution. To generate a correlated random field with this random distribution of scatterers we use the turning bands method. We show that the autocorrelation function (ACF) of the simulated random fields matches the analytical ACF. To transform the correlated Gaussian random field into a correlated gamma random field we use the memoryless nonlinear transform (MNLT). It has been shown that the MNLT skews the ACF of the simulated field. We present a method to preserve the form of the ACF of the field using a mapping function developed by Tough and Ward [J. Phys. D: Appl. Phys., 32 (1999), pp. 3075--3084]. We demonstrate that the ACF of the simulated field is preserved. Once the random field of scatterers is created, the scattering coefficient $s$ can be approximated with a local summing process, which emulates the incoherent imaging process of a SAR sensor. There is evidence in the literature that the sum of correlated gamma variates on a regular grid is also gamma distributed. We develop a new theory for the first and second order linear sum process (LSP) statistics for a random sum of random variables. We use this theory to derive two estimators for the gamma order parameter. Through a Monte Carlo experiment, it is shown that $s$, as represented by the LSP, does generally follow a gamma distribution for scenarios with at least nine scatterers per resolution cell. The order parameter and correlation properties are modified from the original random field according to patterns predicted by our theory, though some parameters must be calculated through simulation.

Convergence of clock process in random environments and aging in Bouchaud's asymmetric trap model on the complete graph

In this paper the celebrated arcsine aging scheme of Ben Arous and Černý is taken up. Using a brand new approach based on point processes and weak convergence techniques, this scheme is implemented in a broad class of Markov jump processes in random environments that includes Glauber dynamics of discrete disordered systems. More specifically, conditions are given for the underlying clock process (a partial sum process that measures the total time elapsed along paths of a given length) to converge to a subordinator, and consequences for certain time correlation functions are drawn. This approach is applied to Bouchaud's asymmetric trap model on the complete graph for which aging is for the first time proved, and the full, optimal picture, obtained. Application to spin glasses are carried out in follow up papers.

Asymptotic Normality for Weighted Sums of Linear Processes

We establish asymptotic normality of weighted sums of stationary linear processes with general triangular array weights and when the innovations in the linear process are martingale differences. The results are obtained under minimal conditions on the weights and as long as the process of conditional variances of innovations is covariance stationary with lag k auto-covariances tending to zero, as k tends to infinity. We also obtain weak convergence of weighted partial sum processes. The results are applicable to linear processes that have short or long memory or exhibit seasonal long memory behavior. In particular they are applicable to GARCH and ARCH(∞) models. They are also useful in deriving asymptotic normality of kernel type estimators of a nonparametric regression function when errors may have long memory.

ON SET-INDEXED RESIDUAL PARTIAL SUM LIMIT PROCESS OF SPATIAL LINEAR REGRESSION MODELS

In this paper we derive the limit process of the sequence of set-indexedleast-squares residual partial sum processes of observations obtained form a spatiallinear regression model. For the proof of the result we apply the uniform central limittheorem of Alexander and Pyke (1986) and generalize the geometrical approach ofBischo (2002) and Bischo and Somayasa (2009). It is shown that the limit processis a projection of the set-indexed Brownian sheet onto the reproducing kernel Hilbertspace of this process. For that we dene the projection via Choquet integral of theregression function with respect to the set-indexed Brownian sheet.DOI : http://dx.doi.org/10.22342/jims.17.2.4.73-83

The Pólya sum process: a Cox representation

In [1], Zessin constructed the so-called Polya sum process via partial integration. Here we use the technique of integration by parts to the Polya sum process to derive representations of the Polya sum process as an infinitely divisible point process and a Cox process directed by an infinitely divisible random measure. This result is related to the question of the infinite divisibilty of a Cox process and the infinite divisibility of its directing measure. Finally we consider a scaling limit of the Polya sum process and show that the limit satisfies an integration by parts formula, which we use to determine basic properties of this limit.

The Papangelou process. A concept for Gibbs, Fermi and Bose processes

This note is a revised and enlarged version of the german article [16] in a slightly different framework. We here correct a serious mistake in the first version and generalize the class of Polya sum processes considered there. (A corrected version of the same results can be found already in the thesis of Mathias Rafler [12].) Moreover, the class of Polya difference processes is constructed here for the first time. In analogy to classical statistical mechanics we propose a theory of interacting Bosons and Fermions. We consider Papangelou processes. These are point processes specified by some kernel which represents the conditional intensity of the process. The main result is a general construction of a large class of such processes which contains Cox, Gibbs processes of classical statistical mechanics, but also interacting Bose and Fermi processes.

A lack-of-fit test in Tobit errors-in-variables regression models

The problem of fitting a parametric model in Tobit errors-in-variables regression models is discussed in this paper. The proposed test is based on the supremum of the Khmaladze type transformation of a certain partial sum process of calibrated residuals. This framework covers the usual error-free Tobit model as a special case. The asymptotic null distribution of this transformed process is shown to be the same as that of a time transformed standard Brownian motion. Consistency against some fixed alternatives and asymptotic power under some local nonparametric alternatives of this test are also discussed. Simulation studies are conducted to assess the finite sample performance of the proposed test.

Checking semiparametric transformation models with censored data

Semiparametric transformation models provide a very general framework for studying the effects of (possibly time-dependent) covariates on survival time and recurrent event times. Assessing the adequacy of these models is an important task because model misspecification affects the validity of inference and the accuracy of prediction. In this paper, we introduce appropriate time-dependent residuals for these models and consider the cumulative sums of the residuals. Under the assumed model, the cumulative sum processes converge weakly to zero-mean Gaussian processes whose distributions can be approximated through Monte Carlo simulation. These results enable one to assess, both graphically and numerically, how unusual the observed residual patterns are in reference to their null distributions. The residual patterns can also be used to determine the nature of model misspecification. Extensive simulation studies demonstrate that the proposed methods perform well in practical situations. Three medical studies are provided for illustrations.

Distribution-free test in Tobit mean regression model

This paper discusses the problem of fitting a parametric model in Tobit mean regression models. The proposed test is based on the supremum of the Khamaladze-type transformation of a partial sum process of calibrated residuals. The asymptotic null distribution of this transformed process is shown to be the same as that of a time-transformed standard Brownian motion. Consistency of this sequence of tests against some fixed alternatives and asymptotic power under some local nonparametric alternatives are also discussed. Simulation studies are conducted to assess the finite sample performance of the proposed test. The power comparison with some existing tests shows some superiority of the proposed test at the chosen alternatives.

Log-likelihood ratio test for detecting transient change

The limit behaviour of scan statistics in a form of maxima of moving sum processes with changing window size for detecting a transient change is studied. Approximate critical values obtained from tail behaviour of the limit variables that are maxima of some locally homogeneous Gaussian fields are compared with critical values obtained by simulation.

The invariance principle for fractionally integrated processes with strong near-epoch dependent innovations

In this paper, we show the invariance principle for the partial sum processes of fractionally integrated processes, otherwise known as I(d + m) processes, where |d| < 1/2 and m is a nonnegative integer, with strong near-epoch dependent innovations. The results are applied to the test of unit root. The conditions given improve previous results in the literature concerning fractionally integrated processes.

Functional Convergence to Stable Lévy Motions for Iterated Random Lipschitz Mappings

It is known that, in the dependent case, partial sums processes which are elements of $D([0,1])$ (the space of right-continuous functions on $[0,1]$ with left limits) do not always converge weakly in the $J_1$-topology sense. The purpose of our paper is to study this convergence in $D([0,1])$ equipped with the $M_1$-topology, which is weaker than the $J_1$ one. We prove that if the jumps of the partial sum process are associated then a functional limit theorem holds in $D([0,1])$ equipped with the $M_1$-topology, as soon as the convergence of the finite-dimensional distributions holds. We apply our result to some stochastically monotone Markov chains arising from the family of iterated Lipschitz models.

Composing the cumulative quantile regression function and the Goldie concentration curve

The model we discuss in this paper deals with inequality in distribution in the presence of a covariate. To elucidate that dependence, we propose to consider the composition of the cumulative quantile regression (CQR) function and the Goldie concentration curve, the standardized counterpart of which gives a fraction to fraction plot of the response and the covariate. It has the merit of enhancing the visibility of inequality in distribution when the latter is present. We shall examine the asymptotic properties of the corresponding empirical estimator. The associated empirical process involves a randomly stopped partial sum process of induced order statistics. Strong Gaussian approximations of the processes are constructed. The result forms the basis for the asymptotic theory of functional statistics based on these processes.