Test for Parameter Change in Linear Processes Based on Whittle's Estimator

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.     

Asymptotic inference in poverty indices: an empirical processes approach

We derive an empirical poverty index containing most of those proposed in the literature. Then, we study its asymptotic behavior by using empirical processes theory. From the results obtained, we derive a Wald-type test for comparing a vector of theoretical poverty indices to a vector of reference indices. We study the test statistic under the null hypothesis of equality, fixed alternatives, and a sequence of local alternatives. A simulation experiment conducted shows that our test performs well for exponential and Pareto data.     

Simulation of the typical Poisson–Voronoi–Cox–Voronoi cell

We consider stationary Poisson–Voronoi tessellations (PVT) in the Euclidean plane and study the properties of Voronoi tessellations induced by linear Poisson processes on the edges of the PVT. We are especially interested in simulation algorithms for the typical cell. Two different simulation algorithms are introduced. The first algorithm directly simulates the typical cell, whereas the second algorithm simulates cells from which distributional properties of the typical cell can be obtained. This second algorithm can also be used for simulating the typical cell of other Cox–Voronoi tessellations. The implementation of both algorithms is tested for their correctness using random software tests. Then different cell characteristics are studied by simulation and compared with the typical cell of PVT and Cox–Voronoi tessellations based on linear Poisson processes on the lines of Poisson line processes. Our results can be applied, for example, in the analysis of telecommunication networks and vesicle paths on cytoskeletal networks.     

Darling–Erdős limit results for change-point detection in panel data

We wish to test the null hypothesis if the means of N panels remain the same during the observation period of length T. A quasi-likelihood argument leads to self-normalized statistics whose limit distribution under the null hypothesis is double exponential. The main results are derived assuming that the each panel is based on independent observations and then extended to linear processes. The proofs are based on an approximation of the sum of squared CUSUM processes using the Skorokhod embedding scheme. A simulation study illustrates that our results can be used in case of small and moderate N and T. We apply our results to detect change in the “corruption index”.     

A NOTE ON SAMPLE QUANTILE PROCESSES OF FINITE POPULATIONS

A representation is established for sample quantile processes of simple random samples drawn without replacement from finite populations. The result is then used to obtain weak limits of properly normalized quantile processes and their linear functions.     

The Relative Deprivation Curve and Its Applications

This article introduces the relative deprivation curve to represent the size distribution of income and wealth. The curve has many useful applications in the measurement of poverty and inequality, which are explored. The methodology developed is then applied to the data obtained from the Australian Household Expenditure Survey, 1975–1976.     

Central Limit Theorems of Local Polynomial Threshold Estimator for Diffusion Processes with Jumps

Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the non‐parametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the σ ‐field generated by diffusion processes is destroyed, so the classical central limit theorem for martingale difference sequences cannot work. In high‐frequency data, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps by Jacod's stable convergence theorem. We believe that our proof procedure for local polynomial threshold estimators provides a new method in this field, especially in the local linear case.     

Bayesian estimation of renewal function for inverse Gaussian renewal process

Two approximation methods are used to obtain the Bayes estimate for the renewal function of inverse Gaussian renewal process. Both approximations use a gamma-type conditional prior for the location parameter, a non-informative marginal prior for the shape parameter, and a squared error loss function. Simulations compare the accuracy of the estimators and indicate that the Tieney and Kadane (T–K)-based estimator out performs Maximum Likelihood (ML)- and Lindley (L)-based estimator. Computations for the T–K-based Bayes estimate employ the generalized Newton's method as well as a recent modified Newton's method with cubic convergence to maximize modified likelihood functions. The program is available from the author.     

Approximation to Multifractional Riemann-Liouville Brownian Sheet

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.     

Bounds on Bivariate Distribution Functions with Given Margins and Known Values at Several Points

Let H(x, y) be a continuous bivariate distribution function with known marginal distribution functions F(x) and G(y). Suppose the values of H are given at several points, H(x _i , y _i ) = θ _i , i = 1, 2,…, n. We first discuss conditions for the existence of a distribution satisfying these conditions, and present a procedure for checking if such a distribution exists. We then consider finding lower and upper bounds for such distributions. These bounds may be used to establish bounds on the values of Spearman's ρ and Kendall's τ. For n = 2, we present necessary and sufficient conditions for existence of such a distribution function and derive best-possible upper and lower bounds for H(x, y). As shown by a counter-example, these bounds need not be proper distribution functions, and we find conditions for these bounds to be (proper) distribution functions. We also present some results for the general case, where the values of H(x, y) are known at more than two points. In view of the simplification in notation, our results are presented in terms of copulas, but they may easily be expressed in terms of distribution functions.     

On a class exhibiting trend change in average failure rate

Abstract

Weak convergence and moment convergence issues are investigated for the New Better than Average Failure Rate (NBAFR) family (introduced by Loh (1984 Loh, W. Y. 1984. A new generalization of the class of NBU distributions. IEEE Transactions on Reliability R-33 :97–113[Crossref], [Web of Science ®] , [Google Scholar])). 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.     

Max-stable processes for modeling extremes observed in space and time

Max-stable processes have proved to be useful for the statistical modeling of spatial extremes. For statistical inference it is often assumed that there is no temporal dependence; i.e., that the observations at spatial locations are independent in time. In a first approach we construct max-stable space–time processes as limits of rescaled pointwise maxima of independent Gaussian processes, where the space–time covariance functions satisfy weak regularity conditions. This leads to so-called Brown–Resnick processes. In a second approach, we extend Smith’s storm profile model to a space–time setting. We provide explicit expressions for the bivariate distribution functions, which are equal under appropriate choice of the parameters. We also show how the space–time covariance function of the underlying Gaussian process can be interpreted in terms of the tail dependence function in the limiting max-stable space–time process.     

Improved Asymptotics of a Decreasing Mean Residual Life Estimator

The mean residual life of a life distribution, X, with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. Kochar et al. (2000 Kochar, S.C., Mukerjee, H., Samaniego, F.J. (2000). Estimation of a monotone mean residual life. Ann. Stat. 28: 905–921.[Crossref], [Web of Science ®] , [Google Scholar]) provided an estimator of M when 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.     

Limit theory of bivariate dual generalized order statistics with random index

The class of limit distribution functions of bivariate extreme, intermediate and central dual generalized order statistics from independent and identically distributed random variables with random sample size is fully characterized. Two cases are considered. The first case is when the random sample size is assumed to be independent of all basic random variables. The second case is when the interrelation of the random size and the basic random variables is not restricted.     

Time series analysis of categorical data using auto-odds ratio function

In this paper, we consider the auto-odds ratio function (AORF) as a measure of serial association for a stationary time series process of categorical data at two different time points. Numerical measures such as the autocorrelation function (ACF) have no meaningful interpretation, unless the time series data are numerical. Instead, we use the AORF as a measure of association to study the serial dependency of the categorical time series for both ordinal and nominal categories. Biswas and Song [Discrete-valued ARMA processes. Stat Probab Lett. 2009;79(17):1884–1889] provided some results on this measure for Pegram's operator-based AR(1) process with binary responses. Here, we extend this measure to more general set-ups, i.e. for AR(p) and MA(q) processes and for a general number of categories. We discuss how this method can effectively be used in parameter estimation and model selection. Following Weiß [Empirical measures of signed serial dependence in categorical time series. J Stat Comput Simul. 2011;81(4):411–429], we derive the large sample distribution of the estimator of the AORF under independent and identically distributed (iid) set-up. Some simulation results and two categorical data examples (one is ordinal and other nominal) are presented to illustrate the proposed method.     

On Infinitely Divisible Exponential Dispersion Model Related to Poisson-Exponential Distribution

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.     

ALGORITHMS FOR RETURN PROBABILITIES FOR STOCHASTIC FLUID FLOWS

Abstract

We consider several known algorithms and introduce some new algorithms that can be used to calculate the probability of return to the initial level in the Markov stochastic fluid flow model. We give the physical interpretations of these algorithms within the fluid flow environment. The rates of convergence are explained in terms of the physical properties of the fluid flow processes. We compare these algorithms with respect to the number of iterations required and their complexity. The performance of the algorithms depends on the nature of the process considered in the analysis. We illustrate this with examples and give appropriate recommendations.     

Asymptotic Distributions of the Generalized Range,Midrange, Extremal Quotient,and Extremal Product,with a Comparison Study

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.     

New acceptance sampling plans based on life tests for Birnbaum–Saunders distributions

In this paper, we consider the Birnbaum–Saunders distribution as a life model to develop various acceptance sampling schemes based on the truncated life tests. We develop the double sampling plan and determine the design parameters satisfying both the producer's and consumer's risks simultaneously for the specified reliability levels in terms of the mean ratio to the specified life. We also propose a group sampling plan and determine the parameters by the above-mentioned two-point method. Tables are constructed for the proposed sampling plans and results are explained with examples.