Salary cost evaluation by means of non-homogeneous semi-Markov processes

A useful model for forecasting the future development of salary costs in a firm is presented in this paper. This problem is relevant in the field of pension funds and also when a company decides to change the structure of its workforce. In the latter case, it might be necessary to forecast future salary costs in the new organizational hierarchy.

The problem is solved by means of a special kind of stochastic process. To be more precise, this paper presents a generalization of discrete time non-homogeneous semi-Markov processes and of the related reward process. This new stochastic process is able to take in account all the aspects of the problem.     

Point Processes with a Generalized Order Statistic Property

Mixed Poisson processes are characterized by a well-known order statistic property: their occurrence times are distributed like ordinary uniform order statistics, given the state of the process at a certain time. We study a generalization of this property using a generalized model of ordered random variables, including a sequence of real parameters.     

Forecasting Contemporaneously Aggregated Vector ARMA Processes

Given a multiple time series that is generated by a multivariate ARMA process and assuming the objective is to forecast a weighted sum of the individual variables, then under a mean squared error measure of forecasting precision, it is preferable to forecast the disaggregated multiple time series and aggregate the forecasts, rather than forecast the aggregated series directly, if the involved processes are known. This result fails to hold if the processes used for forecasting are estimated from a given set of time series data. The implications of these results for empirical research are investigated using different sets of economic data.     

On Comparison of Stopping Times in Sequential Procedures for Exponential Families of Stochastic Processes

For curved ( k + 1), k -exponential families of stochastic processes a natural and often studied sequential procedure is to stop observation when a linear combination of the coordinates of the canonical process crosses a prescribed level. For such procedures the model is, approximately or exactly, a non-curved exponential family. Subfamilies of these stopping rules defined by having the same Fisher (expected) information are considered. Within a subfamily the Bartlett correction for a point hypothesis is also constant. Methods for comparing the durations of the sampling periods for the stopping rules in such a subfamily are discussed. It turns out that some stopping times tend to be smaller than others. For exponential families of diffusions and of counting processes the probability that one such stopping time is smaller than another can be given explicity. More generally, an Edgeworth expansion of this probability is given     

Output Dead-Time in Point Processes

When events of temporal point processes are too close to each other they can be erased by dead-time effects. Among various possible mechanisms of dead-time, the output dead-time is the most important. Dead-time effects modify the statistical properties of point processes and some of these modifications are analyzed in this article. To do so, we note that a point process is defined by the distance between its successive points called life-time which constitutes a discrete time positive signal. The dead-time mechanism is a system which transforms such a signal into another discrete time positive signal. Except in very specific cases this transformation cannot be expressed in closed form. We show, however, that it can be written in a recursive form analogous to the state representation of systems. By using this recursion, various statistical properties of point processes with dead-time are analyzed in computer experiments. In this study, we focus on the probability distribution of the intervals between points and the coincidence function which describes the second-order properties of the point process. For the rare processes where theoretical calculations are possible there is an excellent agreement between experiment and theory.     

Bayesian Identification of Multivariate Autoregressive Processes

Identification is one of the most important stages of a time series analysis. This paper develops a direct Bayesian technique to identify the order of multivariate autoregressive processes. By employing the conditional likelihood function and a matrix normal-Wishart prior density, or Jeffrey' vague prior, the proposed identification technique is based on deriving the exact posterior probability mass function of the model order in a convenient form. Then one may easily evaluate the posterior probabilities of the model order and choose the order that maximizes the posterior mass function to be the suitable order of the time series data being analyzed. Assuming the bivariate autoregressive processes, a numerical study, with different prior mass functions, is carried out to assess the efficiency of the proposed technique. The analysis of the numerical results supports the adequacy of the proposed technique in identifying the orders of multivariate autoregressive processes.     

On Minimum Distance Estimation in Recurrent Markov Step Processes. I

Consider a Markov step process X=(X_t)_t≥0 whose generator depends on an unknown d -dimensional parameter ϑ. We look at certain empirical measures for recurrent Markov step processes and their a.s. convergence; based on this, we introduce a class of minimum distance estimators. For broad families of sequential observation schemes (at stage n, the trajectory of X is observed up to time S_n, (S_n)_n a sequence of stopping times increasing to ∞), we formulate a stochastic expansion of the suitably rescaled estimation error; for a particular scheme, asymptotic normality is obtained as n →∞. A minimax property under misspecification of the model (in the sense that the true probability law is contiguous to the parametric model but not contained in it) is given.     

Algorithms for Point Processes Analysis

A time point process can be defined either by the statistical properties of the time intervals between successive points or by those of the number of points in arbitrary time intervals. There are mathematical expressions to link up these two points of view, but they are in many cases too complicated to be used in practice. In this article, we present an algorithmic procedure to obtain the number of points of a stationary point process recorded in some time intervals by processing the values of the distances between successive points. We present some results concerning the statistical analysis of these numbers of points and when analytical calculations are possible the experimental results obtained with our algorithms are in excellent agreement with those predicted by the theory. Some properties of point processes in which theoretical calculations are almost impossible are also presented.     

Generalized Laplacian Distributions and Autoregressive Processes

Generalized Laplacian distribution is considered. A new distribution called geometric generalized Laplacian distribution is introduced and its properties are studied. First- and higher-order autoregressive processes with these stationary marginal distributions are developed and studied. Simulation studies are conducted and trajectories of the process are obtained for selected values of the parameters. Various areas of application of these models are discussed.     

基于自相关视角的弱平稳过程之间的伪回归分析

随机干扰项之间的未知形式自相关是导致相互独立的弱平稳过程之间伪回归的主要原因.通过理论分析和一系列的蒙特卡罗模拟,揭示了数据过程本身的持久性、样本容量T和随机干扰项自相关之间的内在联系.研究发现随机干扰项往往呈现出与数据过程阶数相同的自相关.进一步研究表明,运用广义差分法和Cochrane- Orcutt迭代法虽然能大大减少伪回归概率,但在有些情况下,即使当样本容量较大时,较高阶的Cochrane- Orcutt迭代法仍然无法避免伪回归的发生.     

Inversion of the Renewal Density with Dead-time

Stationary renewal point processes are defined by the probability distribution of the distances between successive points (lifetimes) that are independent and identically distributed random variables. For some applications it is also interesting to define the properties of a renewal process by using the renewal density. There are well-known expressions of this density in terms of the probability density of the lifetimes. It is more difficult to solve the inverse problem consisting in the determination of the density of the lifetimes in terms of the renewal density. Theoretical expressions between their Laplace transforms are available but the inversion of these transforms is often very difficult to obtain in closed form. We show that this is possible for renewal processes presenting a dead-time property characterized by the fact that the renewal density is zero in an interval including the origin. We present the principle of a recursive method allowing the solution of this problem and we apply this method to the case of some processes with input dead-time. Computer simulations on Poisson and Erlang (2) processes show quite good agreement between theoretical calculations and experimental measurements on simulated data.     

Generalized confidence interval for the slope in linear measurement error model

A generalized confidence interval for the slope parameter in linear measurement error model is proposed in this article, which is based on the relation between the slope of classical regression model and the measurement error model. The performance of the confidence interval estimation procedure is studied numerically through Monte Carlo simulation in terms of coverage probability and expected length.     

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.     

Estimating the probability of simultaneous rainfall extremes within a region: a spatial approach

In this paper we investigate the impact of model mis-specification, in terms of the dependence structure in the extremes of a spatial process, on the estimation of key quantities that are of interest to hydrologists and engineers. For example, it is often the case that severe flooding occurs as a result of the observation of rainfall extremes at several locations in a region simultaneously. Thus, practitioners might be interested in estimates of the joint exceedance probability of some high levels across these locations. It is likely that there will be spatial dependence present between the extremes, and this should be properly accounted for when estimating such probabilities. We compare the use of standard models from the geostatistics literature with max-stables models from extreme value theory. We find that, in some situations, using an incorrect spatial model for our extremes results in a significant under-estimation of these probabilities which – in flood defence terms – could lead to substantial under-protection.     

Estimation for Continuous Branching Processes

The maximum-likelihood estimator for the curved exponential family given by continuous branching processes with immigration is investigated. These processes originated from population biology but also model the dynamics of interest rates and development of the state of technology in economics. It is proved that in contrast to branching processes with discrete space and/or time the MLE gives a unified approach to the inference. In order to include singular subdomains of the parameter space we modify the MLE slightly. Consistency and asymptotic normality for the MLE are considered. Concerning the asymptotic theory of the experiments, all three properties LAQ, LAN, and LAMN occur for different submodels     

EXPONENTIAL SMOOTHING AND NON‐NEGATIVE DATA

The most common forecasting methods in business are based on exponential smoothing, and the most common time series in business are inherently non‐negative. Therefore it is of interest to consider the properties of the potential stochastic models underlying exponential smoothing when applied to non‐negative data. We explore exponential smoothing state space models for non‐negative data under various assumptions about the innovations, or error, process. We first demonstrate that prediction distributions from some commonly used state space models may have an infinite variance beyond a certain forecasting horizon. For multiplicative error models that do not have this flaw, we show that sample paths will converge almost surely to zero even when the error distribution is non‐Gaussian. We propose a new model with similar properties to exponential smoothing, but which does not have these problems, and we develop some distributional properties for our new model. We then explore the implications of our results for inference, and compare the short‐term forecasting performance of the various models using data on the weekly sales of over 300 items of costume jewelry. The main findings of the research are that the Gaussian approximation is adequate for estimation and one‐step‐ahead forecasting. However, as the forecasting horizon increases, the approximate prediction intervals become increasingly problematic. When the model is to be used for simulation purposes, a suitably specified scheme must be employed.     

Optimal Alarm Systems for Count Processes

In many phenomena described by stochastic processes, the implementation of an alarm system becomes fundamental to predict the occurrence of future events. In this work we develop an alarm system to predict whether a count process will upcross a certain level and give an alarm whenever the upcrossing level is predicted. We consider count models with parameters being functions of covariates of interest and varying on time. This article presents classical and Bayesian methodology for producing optimal alarm systems. Both methodologies are illustrated and their performance compared through a simulation study. The work finishes with an empirical application to a set of data concerning the number of sunspot on the surface of the sun.     

Reverse Regression and Employment Discrimination

The importance of interval forecasts is reviewed. Several general approaches to calculating such forecasts are described and compared. They include the use of theoretical formulas based on a fitted probability model (with or without a correction for parameter uncertainty), various “approximate” formulas (which should be avoided), and empirically based, simulation, and resampling procedures. The latter are useful when theoretical formulas are not available or there are doubts about some model assumptions. The distinction between a forecasting method and a forecasting model is expounded. For large groups of series, a forecasting method may be chosen in a fairly ad hoc way. With appropriate checks, it may be possible to base interval forecasts on the model for which the method is optimal. It is certainly unsound to use a model for which the method is not optimal, but, strangely, this is sometimes done. Some general comments are made as to why prediction intervals tend to be too narrow in practice to encompass the required proportion of future observations. An example demonstrates the overriding importance of careful model specification. In particular, when data are “nearly nonstationary,” the difference between fitting a stationary and a nonstationary model is critical.     

Modeling Two-state Disease Processes with Random Effects

Many chronic medical conditions are manifested by alternating sojourns in symptom-free and symptomatic states. In many cases, in addition to their relapsing and remitting nature, these conditions lead to worsening disease patterns over time and may exhibit seasonal trends. We develop a mixed-effect two-state model for such disease processes in which covariate effects are modeled multiplicatively on transition intensities. The transition intensities, in turn, are functions of three time scales: the semi-Markov scale involving the backward recurrence time for the cyclical component, the Markov scale for the time trend component, and a seasonal time scale. Multiplicative bivariate log-normal random effects are introduced to accommodate heterogeneity in disease activity between subjects and to admit a possible negative correlation between the transition intensities. Maximum likelihood estimation is carried out using Gauss-Hermite integration and a standard Newton-Raphson procedure. Tests of homogeneity are presented based on score statistics. An application of the methodology to data from a multi-center clinical trial of chronic bronchitis is provided for illustrative purposes.     

Generalized modified slash distribution with applications

Abstract

In this article a generalization of the modified slash distribution is introduced. This model is based on the quotient of two independent random variables, whose distributions are a normal and a one-parameter gamma, respectively. The resulting distribution is a new model whose kurtosis is greater than other slash distributions. The probability density function, its properties, moments, and kurtosis coefficient are obtained. Inference based on moment and maximum likelihood methods is carried out. The multivariate version is also introduced. Two real data sets are considered in which it is shown that the new model fits better to symmetric data with heavy tails than other slash extensions previously introduced in literature.