Curved Chaotic Map Time Series Models and Their Stochastic Reversals

This paper considers two types of chaotic map time series models, including the well-known tent, logistic and binary-shift maps as special cases; these are called curved tent and curved binary families. Deterministic behaviour is investigated by invariant distributions, Lyapunov exponents, and by serial dependency. Stochastic time reversal of the families is shown to produce models which have a broader range of stochastic and chaotic properties than their deterministic counterparts. The marginal distributions may have concentrations and restricted supports and are shown to be a non-standard class of invariant distribution. Dependenc y is generally weaker with the reversed stochastic models. The work gives a broad statistical account of deterministic and stochastically reversed map models, such as are emerging in random number generation, communica tion systems and cryptography     

Ordering results for series and parallel systems comprising heterogeneous exponentiated Weibull components

In this paper, we discuss the usual stochastic and reversed hazard rate orders between the series and parallel systems from two sets of independent heterogeneous exponentiated Weibull components. We also obtain the results concerning the convex transform orders between parallel systems and obtain necessary and sufficient conditions under which the dispersive and usual stochastic orders, and the right spread and increasing convex orders between the lifetimes of the two systems are equivalent. Finally, in the multiple-outlier exponentiated Weibull models, based on weak majorization and p-larger orders between the vectors of scale and shape parameters, some characterization results for comparing the lifetimes of parallel and series systems are also established, respectively. The results of this paper can be used in practical situations to find various bounds for the important aging characteristics of these systems.     

Exponentiated models preserve stochastic orderings of parallel and series systems

Abstract

In this paper, we establish that the usual stochastic, hazard rate, reversed hazard rate, likelihood ratio, dispersive and star orders are all preserved for parallel systems under exponentiated models for lifetimes of components. We then use the multiple-outlier exponentiated gamma models to illustrate this result. Finally, we consider the dual family with exponentiated survival function and establish similar results for series systems. The results established here extend some well-known results for series and parallel systems arising from different exponentiated distributions such as generalized exponential and exponentiated Weibull, established previously in the literature.     

A class of regression models for parallel and series systems with a random number of components

We extend the Weibull power series (WPS) class of distributions to the new class of extended Weibull power series (EWPS) class of distributions. The EWPS distributions are related to series and parallel systems with a random number of components, whereas the WPS distributions [Morais AL, Barreto-Souza W. A compound class of Weibull and power series distributions. Computational Statistics and Data Analysis. 2011;55:1410–1425] are related to series systems only. Unlike the WPS distributions, for which the Weibull is a limiting special case, the Weibull law is a particular case of the EWPS distributions. We prove that the distributions in this class are identifiable under a simple assumption. We also prove stochastic and hazard rate order results and highlight that the shapes of the EWPS distributions are markedly more flexible than the shapes of the WPS distributions. We define a regression model for the EWPS response random variable to model a scale parameter and its quantiles. We present the maximum likelihood estimator and prove its consistency and asymptotic normal distribution. Although series and parallel systems motivated the construction of this class, the EWPS distributions are suitable for modelling a wide range of positive data sets. To illustrate potential uses of this model, we apply it to a real data set on the tensile strength of coconut fibres and present a simple device for diagnostic purposes.     

Bathtub distributions: a review

Bathtub distributions are characterized by bathtub failure rate functions . These are possibly more realisitic models than the monotone failure rate models . A systematic account of such distributions is not available and this review aims to give such an account . We give some easily verifiable conditions to check the bathtub property of a distribution along with methods to construct such distributions . We also discuss some stochastic and reliablity mechanisms which lead to bathtub distributions. These include mixtures ( stochastic failure rate models ) , series system , stochastic differential equation models and so on. We also review inference on bathtub distributions. The paper concludes with a rather exhaustive list of bathtub distributions.     

Graphical Techniques for Selecting Explanatory Variables for Time Series Data

Bayesian model building techniques are developed for data with a strong time series structure and possibly exogenous explanatory variables that have strong explanatory and predictive power. The emphasis is on finding whether there are any explanatory variables that might be used for modelling if the data have a strong time series structure that should also be included. We use a time series model that is linear in past observations and that can capture both stochastic and deterministic trend, seasonality and serial correlation. We propose the plotting of absolute predictive error against predictive standard deviation. A series of such plots is utilized to determine which of several nested and non-nested models is optimal in terms of minimizing the dispersion of the predictive distribution and restricting predictive outliers. We apply the techniques to modelling monthly counts of fatal road crashes in Australia where economic, consumption and weather variables are available and we find that three such variables should be included in addition to the time series filter. The approach leads to graphical techniques to determine strengths of relationships between the dependent variable and covariates and to detect model inadequacy as well as determining useful numerical summaries.     

A Resampling Algorithm for Chaotic Time Series

In the field of chaotic time series analysis, there is a lack of a distributional theory for the main quantities used to characterize the underlying data generating process (DGP). In this paper a method for resampling time series generated by a chaotic dynamical system is proposed. The basic idea is to develop an algorithm for building trajectories which lie on the same attractor of the true DGP, that is with the same dynamical and geometrical properties of the original data. We performed some numerical experiments on some short noise-free and high-noise series confirming that we are able to correctly reproduce the distribution of the largest finite-time Lyapunov exponent and of the correlation dimension.     

Hazard Rate Properties of Systems with Two Components

In this article, we study the hazard rate ordering of lifetimes of two-component systems (series and parallel) by considering some bivariate distributions for the joint distribution of component lifetimes. Such that, we aim to investigate the lifetimes of systems with stochastically dependent and with stochastically independent components, the lifetimes of the components, and a stochastic ordering relation between these lifetimes. In addition to these, the mononotonicity of the hazard rates of the parallel and the series systems for the bivariate Farlie–Gumbel–Morgenstern (FGM) family is studied.     

Stochastic comparisons of parallel and series systems with heterogeneous resilience-scaled components

By adding a resilience parameter to the scale model, a general distribution family called resilience-scale model is introduced including exponential, Weibull, generalized exponential, exponentiated Weibull and exponentiated Lomax distributions as special cases. This paper carries out stochastic comparisons on parallel and series systems with heterogeneous resilience-scaled components. On the one hand, it is shown that more heterogeneity among the resilience-scaled components of a parallel [series] system with an Archimedean [survival] copula leads to better [worse] performance in the sense of the usual stochastic order. On the other hand, the [reversed hazard] hazard rate order is established for two series [parallel] systems consisting of independent heterogeneous resilience-scaled components. The skewness and dispersiveness are also investigated for the lifetimes of two parallel systems consisting of independent heterogeneous and homogeneous [multiple-outlier] resilience-scaled components. Numerical examples are provided to illustrate the effectiveness of our theoretical findings. These results not only generalize and extend some known ones in the literature, but also provide guidance for engineers to assemble systems with higher reliability in practical situations.     

Volatility estimation from short time series of stock prices

We consider estimation of the historical volatility of stock prices. It is assumed that the stock prices are represented as time series formed as samples of the solution of a stochastic differential equation with random and time-varying parameters; these parameters are not observable directly and have unknown evolution law. The price samples are available with limited frequency only. In this setting, the estimation has to be based on short time series, and the estimation error can be significant. We suggest some supplements to the existing nonparametric methods of volatility estimation. Two modifications of the standard summation formula for the volatility are derived. In addition, a linear transformation eliminating the appreciation rate and preserving the volatility is suggested.     

Analysis of Stochastic Online Bin Packing Processes

Abstract

A stochastic online version of the classical bin packing problem, where a bin corresponds to the capacity of a resource allocated among streams of requests at discrete time units, is a fundamental problem that arises in a wide variety of application areas including bandwidth allocation in networks, memory management in computers, and message transmission in slotted network channels. We derive a mathematical analysis of the corresponding multi-dimensional stochastic process, potentially infinite in each dimension, under a general class of scheduling policies based on a combination of a Lyapunov function technique and matrix-analytic methods. Our analysis yields stability conditions and stationary distributions for this stochastic bin packing process under general probability distributions. We further provide some algorithmic techniques for the numerical computation of these measures.     

Simple diagnostic tools for inverstigating linear trends in time series

This paper presents a simple diagnostic tool for time series. Based on a coefficient α that veries between 1 and 0, the tool measures the approximation of a time series to an arithmetic progression (i.e., a linear function of time). The proposed α is based on the ratio of the average squared second difference to the average squared first difference of the ginven series. As such, α reduces to the Von Neumann ratio η of the series of first differences, namely, α = 1-η/4. For an arithmetic progression α = 1, and deviations therefrom cause it to decrease. Unlike the correlation coefficient (between the entries and the indics), α is sensitive to local, or piecewise, linearity. Here α is evaluated for an assortment of simple time series models such as random walk, AR(1) and MA(1). Large-sample distribution yields a number of commonly used stochastic models including non-normal process. For most standard deterministic and stochastic models, α stabilizes as n approaches infinity, and provides a statistic that is capable of distinguishing between many different standard random and deterministic models. A further measure τ, which together with α distinguisches between random walks and deterministic trend plus i.i.d., is also suggested. Some examples based on empirical data are also studied.     

Bivariate Birnbaum-Saunders accelerated lifetime model: estimation and diagnostic analysis

In this paper, we discuss the bivariate Birnbaum-Saunders accelerated lifetime model, in which we have modeled the dependence structure of bivariate survival data through the use of frailty models. Specifically, we propose the bivariate model Birnbaum-Saunders with the following frailty distributions: gamma, positive stable and logarithmic series. We present a study of inference and diagnostic analysis for the proposed model, more concisely, are proposed a diagnostic analysis based in local influence and residual analysis to assess the fit model, as well as, to detect influential observations. In this regard, we derived the normal curvatures of local influence under different perturbation schemes and we performed some simulation studies for assessing the potential of residuals to detect misspecification in the systematic component, the presence in the stochastic component of the model and to detect outliers. Finally, we apply the methodology studied to real data set from recurrence in times of infections of 38 kidney patients using a portable dialysis machine, we analyzed these data considering independence within the pairs and using the bivariate Birnbaum-Saunders accelerated lifetime model, so that we could make a comparison and verify the importance of modeling dependence within the times of infection associated with the same patient.     

Volatility and GMM — Monte Carlo studies and empirical estimations

In this article we examine small sample properties of a generalized method of moments (GMM) estimation using Monte Carlo simulations. We assume that the generated time series describe the stochastic variance rate of a stock index. we use a mean reverting square-root process to simulate the dynamics of this instantaneous variance rate. The time series obtained are used to estimate the parameters of the assumed variance rate process by applying GMM. Our results are described and compared to estimates from empirical data which consist of volatility as well as daily volume data of the German stock market. One of our main findings is that estimates of the mean reverting parameter that are not significantly different from zero do not necessarily imply a rejection of the hypothesis of a mean reverting behavior of the underlying stochastic process.     

Some reliability characteristics and stochastic ordering of series and parallel systems of bivariate generalized exponential distribution

In this paper, the reliability properties of two-component parallel and series systems are considered for bivariate generalized exponential (BVGE) distribution introduced by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivar Anal. 2009;100:581–593]. For this model, the moments and mean residual life functions of these systems and the regression mean residual life function are derived. Stochastic comparisons of series and parallel systems of BVGE distribution are investigated. Moreover, various ordering results for the comparisons of series and parallel systems arising from BVGE random vectors are obtained with respect to the usual stochastic, reversed hazard rate and likelihood ratio orderings.     

Outlier Detection in Time Series Models Using Local Influence Method

We propose a new procedure for detecting a patch of outliers or influential observations for autoregressive integrated moving average (ARIMA) model using local influence analysis. It is shown that the dependency aspects of time series data gives rise to masking or smearing effects when the local influence analysis is performed using current perturbation schemes. We suggest a new perturbation scheme to take into account the dependent structure of time series data, and employ the stepwise local influence method to give a diagnostic procedure. We show that the new perturbation scheme can avoid the smearing effects, and the stepwise technique of local influence can successfully deal with masking effects. Various simulation studies are performed to show the efficiency of proposed methodology and a real example is used for illustrations.     

Curve of Correlation for Time Series

The curve of correlation is a measure of local correlation between two random variables X and Y at the point X = x of the support of this variable. This article studies this local measure using the theory of time series for bivariate and univariate stationary stochastic process. We suggest local polynomial estimators for time series observing their consistency both theoretically and through simulations. For this, different sizes of series, bandwidths, and kernels, besides lags and models’ configurations were used. Applications have also been made using the daily returns of two financial series.     

Stochastic properties and parameter estimation for a general load-sharing system

We study here a general load-sharing parallel system in which the lifetimes of the components of the system are arbitrary continuous random variables. The system functions if at least one component in the system functions and the surviving unit shares the whole load. Some sufficient conditions are obtained for the usual stochastic order between two different load-sharing systems. We then consider the optimal allocation problem of one load standby in a series system with two independent components. Finally, the maximum likelihood estimation of the parameters for some specific systems is discussed.     

Tail exponent estimation via broadband log density-quantile regression

Heavy tail probability distributions are important in many scientific disciplines such as hydrology, geology, and physics and therefore feature heavily in statistical practice. Rather than specifying a family of heavy-tailed distributions for a given application, it is more common to use a nonparametric approach, where the distributions are classified according to the tail behavior. Through the use of the logarithm of Parzen's density-quantile function, this work proposes a consistent, flexible estimator of the tail exponent. The approach we develop is based on a Fourier series estimator and allows for separate estimates of the left and right tail exponents. The theoretical properties for the tail exponent estimator are determined, and we also provide some results of independent interest that may be used to establish weak convergence of stochastic processes. We assess the practical performance of the method by exploring its finite sample properties in simulation studies. The overall performance is competitive with classical tail index estimators, and, in contrast, with these our method obtains somewhat better results in the case of lighter heavy-tailed distributions.