Some properties of a simple moving average when applied to forecasting a time series

Simple (equally weighted) moving averages are frequently used to estimate the current level of a time series, with this value being projected as a forecast for future observations. A key measure of the effectiveness of the method is the sampling error of the estimator, which this paper defines in terms of characteristics of the data. This enables the optimal length of the average for any steady state model to be established and the lead time forecast error derived. A comparison of the performance of a simple moving average (SMA) with an exponentially weighted moving average (EWMA) is made. It is shown that, for a steady state model, the variance of the forecast error is typically less than 3% higher than the appropriate EWMA. This relatively small difference may explain the inconclusive results from the empirical studies about the relative predictive performance of the two methods.     

Estimation for regression with infinite variance errors

This paper addresses the problem of modelling time series with nonstationarity from a finite number of observations. Problems encountered with the time varying parameters in regression type models led to the smoothing techniques. The smoothing methods basically rely on the finiteness of the error variance, and thus, when this requirement fails, particularly when the error distribution is heavy tailed, the existing smoothing methods due to [1], are no longer optimal. In this paper, we propose a penalized minimum dispersion method for time varying parameter estimation when a regression model generated by an infinite variance stable process with characteristic exponent α ε (1, 2). Recursive estimates are evaluated and it is shown that these estimates for a nonstationary process with normal errors is a special case.     

Minimizing the variance of a mathematical expectation estimate for a diffusion process functional based on a parametric transformation of a parabolic boundary value problem

This paper deals with finding ways of reducing the variance of a mathematical expectation estimate for the functional of a diffusion process moving in a domain with an absorbing boundary. The estimate of mathematical expectation of the functional is obtained based on a numerical solution of stochastic differential equations (SDEs) by using the Euler method. A formula of the limiting variance is derived with decreasing integration step in the Euler method. A method of reducing the variance value of the estimate based on transformation of the parabolic boundary value problem corresponding to the diffusion process is proposed. Some numerical results are presented.     

A universal approach to estimate the conditional variance in semimartingale limit theorems

The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard task, in particular when the underlying process contains jumps. For this reason, several authors have recently discussed methods to automatically estimate the conditional variance, i.e. they build a consistent estimator from the original statistics, but computed at different time scales. Their methods work in several situations, but are essentially restricted to the case of continuous paths always. The aim of this work is to present a new method to consistently estimate the conditional variance which works regardless of whether the underlying process is continuous or has jumps. We will discuss the case of power variations in detail and give insight to the heuristics behind the approach.

     

Lexicographical dynamic goal programming approach to a robust design optimization within the pharmaceutical environment

The primary objective of this paper is to develop a new robust design (RD) optimization procedure based on a lexicographical dynamic goal programming (LDGP) approach for implementing time-series based multi-responses, while the conventional experimental design formats and frameworks may implement static responses. First, a parameter estimation method for time-dependent pharmaceutical responses (i.e., drug release and gelation kinetics) is proposed using the dual response estimation concept that separately estimates the response functions of the mean and variance, as a part of response surface method. Second, a multi-objective RD optimization model using the estimated response functions of both the process mean and variance is proposed by incorporating a time-series components within a dynamic modeling environment. Finally, a pharmaceutical case study associated with a generic drug development process is conducted for verification purposes. Based on the case study results, we conclude that the proposed LDGP approach effectively provides the optimal drug formulations with significantly small biases and MSE values, compared to other models.     

A variance minimization problem for a Markov decision process

In the steady state of a discrete time Markov decision process, we consider the problem to find an optimal randomized policy that minimizes the variance of the reward in a transition among the policies which give the mean not less than a specified value. The problem is solved by introducing a parametric Markov decision process with average cost criterion. It is shown that there exists an optimal policy which is a mixture of at most two pure policies. As an application, the toymaker's problem is discussed.     

Optimization of a fully adaptive quality and maintenance model in the presence of multiple location and scale quality shifts

This paper presents a new model for the economic-statistical optimization of a Variable-Parameter Shewhart control scheme. The proposed model can be utilized to monitor processes where apart from multiple independent assignable causes, affecting both the mean and variance, failures can also occur. Each time an alarm is issued by the control scheme, preventive maintenance actions are initiated, whereas, corrective maintenance actions are required after a failure. The more realistic assumption of imperfect preventive maintenance actions has been considered. The optimal parameter values are selected through a bi-objective optimization problem formulated by the long-run average cost per time unit minimization, and the long-run expected availability maximization, subject to statistical constraints. A real case example is presented to illustrate the application of the model. An extended numerical investigation is utilized to evaluate the superiority of the proposed model.     

Joint economic design of EWMA control charts for mean and variance

Control charts with exponentially weighted moving average (EWMA) statistics (mean and variance) are used to jointly monitor the mean and variance of a process. An EWMA cost minimization model is presented to design the joint control scheme based on pure economic or both economic and statistical performance criteria. The pure economic model is extended to the economic-statistical design by adding constraints associated with in-control and out-of-control average run lengths. The quality related production costs are calculated using Taguchi’s quadratic loss function. The optimal values of smoothing constants, sampling interval, sample size, and control chart limits are determined by using a numerical search method. The average run length of the control scheme is computed by using the Markov chain approach. Computational study indicates that optimal sample sizes decrease as the magnitudes of shifts in mean and/or variance increase, and higher values of quality loss coefficient lead to shorter sampling intervals. The sensitivity analysis results regarding the effects of various inputs on the chart parameters provide useful guidelines for designing an EWMA-based process control scheme when there exists an assignable cause generating concurrent changes in process mean and variance.     

Variation analysis of uncertain stationary independent increment processes

A stationary independent increment process is an uncertain process with stationary and independent increments. This paper aims to calculate the variance of stationary independent increment processes, and gains that, for each fixed time, the variance is a constant multiplying the square of time. Based on this result, it is proved that the total variation of stationary independent increment process with finite variance is bounded almost surely. Besides, the quadratic variation of stationary independent increment process with finite variance is 0 almost surely and in mean.     

Markov decision processes with a minimum-variance criterion

We consider the optimization of the variance of the sum of costs as well as that of an average expected cost in Markov decision processes with unbounded cost. In case of general state and action space, we find the stationary policy which makes the average variance as small as possible in the class of policies which are ε-optimal in an average expected cost.     

Travel time estimation and order batching in a 2-block warehouse

Order batching problem (OBP) is the problem of determining the number of orders to be picked together in one picking tour. Although various objectives may arise in practice, minimizing the average throughput time of a random order is a common concern. In this paper, we consider the OBP for a 2-block rectangular warehouse with the assumptions that orders arrive according to a Poisson process and the method used for routing the order-pickers is the well-known S-shape heuristic. We first elaborate on the first and second moment of the order-picker’s travel time. Then we use these moments to estimate the average throughput time of a random order. This enables us to estimate the optimal picking batch size. Results from simulation show that the method provides a high accuracy level. Furthermore, the method is rather simple and can be easily applied in practice.     

An Algorithm for a Letter-Based Representation of All-Pairwise Comparisons

All-pairwise comparisons among a set of t treatments or groups are one of the most frequent tasks in applied statistics. Users of statistical software are accustomed to the familiar lines display, in which treatments that do not differ significantly, are connected by a common line or letter. Availability of the lines display is restricted mainly to the balanced analysis of variance setup. This limited availability is at stark variance with the diversity of statistical methods and models, which call for multiple comparisons. This article describes a general method for graphically representing any set of t(t?1)/2 all-pairwise significance statements (p values) for t treatments by a familiar letter display, which is applicable regardless of the underlying data structure or the statistical method used for comparisons. The method reproduces the familiar lines display in case of the balanced analysis of variance. Its broad applicability is demonstrated using data from an international multienvironment wheat yield trial and from a fish catching survey.     

Spectral Representation of Multivariate Regularly Varying Lévy and CARMA Processes

A spectral representation for regularly varying Lévy processes with index between one and two is established and the properties of the resulting random noise are discussed in detail, giving also new insight in the L ²-case where the noise is a random orthogonal measure. This allows a spectral definition of multivariate regularly varying Lévy-driven continuous time autoregressive moving average (CARMA) processes. It is shown that they extend the well-studied case with finite second moments and coincide with definitions previously used in the infinite variance case when they apply.     

Separation of random noise models from seasonal time series data

The autocorrelation function of seasonal time series data is shown to have peaks which occur at the correlation lags equal to the integer multiples of the fundamental period that is present in the series. This property is shown to be valid even if some of the harmonics including the fundamental are removed from the time series data. Using this property, an analytical procedure is presented for estimating the variance of the white noise generating the low frequency random walk model present in the data. The procedure is similarly extended to estimate the variance of white noise generating the autoregressive (AR) and moving average (MA) noise models. The method is validated on several seasonal time series data whose components are known a priori.     

The valuation of convertible bonds with numeraire changes

The changes of numeraire can be used as a very powerful mean in pricing contingent claims in the context of a complete market. We apply the method of nurmeraire changes to evaluate convertible bonds when the instantaneous growth and variance of the value of issuer and those of zero-coupon bonds follow a general adapted stochastic process in this paper. A closed-form solution is derived when the instantaneous growth and variance of the value of issuer and those of zero-coupon bonds are deterministic function of time. We also consider a special case when the asset price follows GBM （Geometric Brownian Motion） and interest rate follows Vasicek＇s model.     

Efficiency measure,modelling and estimation in combined array designs

In off‐line quality control, the settings that minimize the variance of a quality characteristic are unknown and must be determined based on an estimated dual response model of mean and variance. The present paper proposes a direct measure of the efficiency of any given design‐estimation procedure for variance minimization. This not only facilitates the comparison of different design‐estimation procedures, but may also provide a guideline for choosing a better solution when the estimated dual response model suggests multiple solutions. Motivated by the analysis of an industrial experiment on spray painting, the present paper also applies a class of link functions to model process variances in off‐line quality control. For model fitting, a parametric distribution is employed in updating the variance estimates used in an iteratively weighted least squares procedure for mean estimation. In analysing combined array experiments, Engel and Huele (Technometrics, 1996; 39:365) used log‐link to model process variances and considered an iteratively weighted least squares leading to the pseudo‐likelihood estimates of variances as discussed in Carroll and Ruppert (Transformation and Weighting in Regression, Chapman & Hall: New York). Their method is a special case of the approach considered in this paper. It is seen for the spray paint data that the log‐link may not be satisfactory and the class of link functions considered here improves substantially the fit to process variances. This conclusion is reached with a suggested method of comparing ‘empirical variances’ with the ‘theoretical variances’ based on the assumed model. Copyright © 2003 John Wiley & Sons, Ltd.     

关于机器随机故障完工时间方差最小化单机调度问题

讨论了机器随机故障时,工件完工时间方差的期望最小化单机调度问题,其中描述机器故障的计数过程为广义泊松过程.推导出了目标函数等价的确定形式,而后进一步给出了工件加工时间相同时问题的最优解.     

基于平衡样本的校准估计研究--以规模以下工业企业测算为例

为解决规模以下工业企业调查中存在的样本代表性不足的问题,提出基于平衡样本的校准估计方法,并得出相应的估计量和估计量方差。该方法在抽样设计阶段采用了平衡抽样设计,在估计阶段采用了校准估计方法,较大限度地使用了辅助信息;通过数据分析得出基于平衡样本的校准估计方法要优于基于平衡抽样的HT估计方法。同时,为满足平衡变量间线性无关的假定,提出使用主成分分析、切片逆回归和切片平均方差估计三种方法对相关的平衡变量进行处理的思路。该方法对我国规模以下工业企业调查的完善具有理论与实践的双重意义,可适当的推广至我国政府统计的其他调查中。     

最小方差准则下一个未定权益的近似对冲

本文研究了不完备的离散时间股票市场下未定权益的定价的对冲问题.利用在最小方差准则下选择概率测度Q或权重函数LN来求最优投资组合的方法,给出了离散时间情况下的鞅表示定理,在最小方差准则下提供一个简单的方法来近似对冲一个未定权益或一个欧氏期权.     

A simple inequality for the variance of the number of zeros of a differentiable Gaussian stationary process

The variance of the number of zeros of a Gaussian differentiable stationary process in a finite time interval can be represented by a single integral of a sophisticated function having singularities in the vicinity of zero, which complicates computer calculations. In this paper, for a wide class of correlation functions, an inequality estimating this variance in simpler terms is proved. Two of three considered examples demonstrate the limits of the effectiveness of the obtained inequality by comparison with special processes earlier established by the author for which the variance is calculated by formulas without integrals. In the two subsequent cases, the inequality is used for the asymptotic estimation of the variance of the number of zeros in a small time interval and, in the last one, in addition to this asymptotics, the upper and lower bounds for the most widely used analytic process in all time intervals.