Online control by attributes in the presence of classification errors with variable inspection interval

ABSTRACT

The procedure for online control by attribute consists of inspecting a single item at every m items produced (m ≥ 2). On each inspection, it is determined whether the fraction of the produced conforming items decreased. If the inspected item is classified as non conforming, the productive process is adjusted so that the conforming fraction returns to its original status. A generalization observed in the literature is to consider inspection errors and vary the inspection interval. This study presents an extension of this model by considering that the inspected item can be rated independently r (r ≥ 1) times. The process is adjusted every time the number of conforming classifications is less than a, 1 ≤ a ≤ r. This method uses the properties of an ergodic Markov chain to obtain the expression for the average cost of this control system. The genetic algorithm methodology is used to search for the optimal parameters that minimize the expected cost. The procedure is illustrated by a numerical example.     

On-Line Process Control using Attributes with Misclassification Errors: An Economical Design for Short-Run Production

On-line process control consists of inspecting a single item for every m (integer and m ≥ 2) produced items. Based on the results of the inspection, it is decided whether the process is in-control (the fraction of conforming items is p ₁; State I) or out-of-control (the fraction of conforming items is p ₂ < p ₁; State II). If the inspected item is non conforming, it is determined that the process is out-of-control, and the production process is stopped for an adjustment; otherwise, production continues. As most designs of on-line process control assume a long-run production, this study can be viewed as an extension because it is concerned with short-run production and the decision regarding the process is subject to misclassification errors. The probabilistic model of the control system employs properties of an ergodic Markov chain to obtain the expression of the average cost of the system per unit produced, which can be minimised as a function of the sampling interval, m. The procedure is illustrated by a numerical example.     

Bayesian estimation in Kibble's bivariate gamma distribution

The authors describe Bayesian estimation for the parameters of the bivariate gamma distribution due to Kibble (1941). The density of this distribution can be written as a mixture, which allows for a simple data augmentation scheme. The authors propose a Markov chain Monte Carlo algorithm to facilitate estimation. They show that the resulting chain is geometrically ergodic, and thus a regenerative sampling procedure is applicable, which allows for estimation of the standard errors of the ergodic means. They develop Bayesian hypothesis testing procedures to test both the dependence hypothesis of the two variables and the hypothesis of equal means. They also propose a reversible jump Markov chain Monte Carlo algorithm to carry out the model selection problem. Finally, they use sets of real and simulated data to illustrate their methodology.     

An autoregressive model based on the generalized hyperbolic distribution

We define a nonlinear autoregressive time series model based on the generalized hyperbolic distribution in an attempt to model time series with non-Gaussian features such as skewness and heavy tails. We show that the resulting process has a simple condition for stationarity and it is also ergodic. An empirical example with a forecasting experiment is presented to illustrate the features of the proposed model.     

A Note on Ergodic Processes Prediction via Estimation of the Conditional Mode Function

We establish the uniform almost-sure convergence of a kernel estimate of the conditional density for an ergodic process. A useful application to the prediction of the ergodic process via the conditional mode function is also given.     

Asymptotic approach for a renewal-reward process with a general interference of chance

ABSTRACT

In this study, a renewal-reward process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.     

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.     

On Empirical Processes for Ergodic Diffusions and Rates of Convergence of M-estimators

Abstract. This paper contributes to the development of empirical process theory for ergodic diffusions. We prove an entropy‐type maximal inequality for the increments of the empirical process of an ergodic diffusion. The inequality is used to study the rate of convergence of M‐estimators.     

A new look at Markov processes of G/M/1-type

We present a new method for deriving the stationary distribution of an ergodic Markov process of G/M/1-type in continuous-time, by deriving and making use of a new representation for each element of the rate matrices contained in these distributions. This method can also be modified to derive the Laplace transform of each transition function associated with Markov processes of G/M/1-type.     

Uniform Ergodicity of the Particle Gibbs Sampler

The particle Gibbs sampler is a systematic way of using a particle filter within Markov chain Monte Carlo. This results in an off‐the‐shelf Markov kernel on the space of state trajectories, which can be used to simulate from the full joint smoothing distribution for a state space model in a Markov chain Monte Carlo scheme. We show that the particle Gibbs Markov kernel is uniformly ergodic under rather general assumptions, which we will carefully review and discuss. In particular, we provide an explicit rate of convergence, which reveals that (i) for fixed number of data points, the convergence rate can be made arbitrarily good by increasing the number of particles and (ii) under general mixing assumptions, the convergence rate can be kept constant by increasing the number of particles superlinearly with the number of observations. We illustrate the applicability of our result by studying in detail a common stochastic volatility model with a non‐compact state space.     

A Note on Consistent Estimation of Multivariate Parameters in Ergodic Diffusion Models

Certain aspects of maximum likelihood estimation for ergodic diffusions are studied via recently developed empirical process theory for martingales. This approach enables us to remove some undesirable regularity conditions that usually appear in the statistical literature on ergodic diffusions. In particular, dimension dependent conditions for the existence of a continuous likelihood and for consistency of the maximum likelihood estimator turn out to be unnecessary.     

Bootstrapping an Econometric Model: Some Empirical Results

The bootstrap, like the jackknife, is a technique for estimating standard errors. The idea is to use Monte Carlo simulation, based on a nonparametric estimate of the underlying error distribution. The bootstrap will be applied to an econometric model describing the demand for capital, labor, energy, and materials. The model is fitted by three-stage least squares. In sharp contrast with previous results, the coefficient estimates and the estimated standard errors perform very well. However, the model's forecasts show serious bias and large random errors, significantly understated by the conventional standard error of forecast.     

On the efficiency of regression analysis with AR(p) errors

In this paper we will consider a linear regression model with the sequence of error terms following an autoregressive stationary process. The statistical properties of the maximum likelihood and least squares estimators of the regression parameters will be summarized. Then, it will be proved that, for some typical cases of the design matrix, both methods produce asymptotically equivalent estimators. These estimators are also asymptotically efficient. Such cases include the most commonly used models to describe trend and seasonality like polynomial trends, dummy variables and trigonometric polynomials. Further, a very convenient asymptotic formula for the covariance matrix will be derived. It will be illustrated through a brief simulation study that, for the simple linear trend model, the result applies even for sample sizes as small as 20.     

Ergodic decomposition of a stationary channel operator

Pointwise weak* topology is defined for the set of all channel operators and the latter is shown to be compact with this topology. Since the set of all ergodic stationary channel operators coincides with the set of all extreme points of the set of stationary channel operators, each stationary channel operator is represented as an integral of ergodic stationary channel operators with respect to a suitable probability measure on the set of all ergodic stationary channel operators.     

Convergence Analysis of MCMC Algorithms for Bayesian Multivariate Linear Regression with Non‐Gaussian Errors

When Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals. Combining this robust regression model with a default prior on the unknown parameters results in a highly intractable posterior density. Fortunately, there is a simple data augmentation (DA) algorithm and a corresponding Haar PX‐DA algorithm that can be used to explore this posterior. This paper provides conditions (on the mixing density) for geometric ergodicity of the Markov chains underlying these Markov chain Monte Carlo algorithms. Letting d denote the dimension of the response, the main result shows that the DA and Haar PX‐DA Markov chains are geometrically ergodic whenever the mixing density is generalized inverse Gaussian, log‐normal, inverted Gamma (with shape parameter larger than d /2) or Fréchet (with shape parameter larger than d /2). The results also apply to certain subsets of the Gamma, F and Weibull families.     

The Co-Integrated Vector Autoregression with Errors–in–Variables

The co-integrated vector autoregression is extended to allow variables to be observed with classical measurement errors (ME). For estimation, the model is parametrized as a time invariant state-space form, and an accelerated expectation-maximization algorithm is derived. A simulation study shows that (i) the finite-sample properties of the maximum likelihood (ML) estimates and reduced rank test statistics are excellent (ii) neglected measurement errors will generally distort unit root inference due to a moving average component in the residuals, and (iii) the moving average component may–in principle–be approximated by a long autoregression, but a pure autoregression cannot identify the autoregressive structure of the latent process, and the adjustment coefficients are estimated with a substantial asymptotic bias. An application to the zero-coupon yield-curve is given.     

Bootstrap-based ARMA order selection

Modelling the underlying stochastic process is one of the main goals in the study of many dynamic phenomena, such as signal processing, system identification and time series. The issue is often addressed within the framework of ARMA (Autoregressive Moving Average) paradigm, so that the related task of identification of the ‘true’ order is crucial. As it is well known, the effectiveness of such an approach may be seriously compromised by misspecification errors since they may affect model capabilities in capturing dynamic structures of the process. As a result, inference and empirical outcomes may be heavily misleading. Despite the big number of available approaches aimed at determining the order of an ARMA model, the issue is still open. In this paper, we bring the problem in the framework of bootstrap theory in conjunction with the information-based criterion of Akaike (AIC), and a new method for ARMA model selection will be presented. A theoretical justification for the proposed approach as well as an evaluation of its small sample performances, via simulation study, are given.     

LAN property for an ergodic Ornstein–Uhlenbeck process with Poisson jumps

In this article, we consider an ergodic Ornstein–Uhlenbeck process with jumps driven by a Brownian motion and a compensated Poisson process, whose drift and diffusion coefficients as well as its jump intensity depend on unknown parameters. Considering the process discretely observed at high frequency, we derive the local asymptotic normality property. To obtain this result, Malliavin calculus and Girsanov’s theorem are applied to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied.     

LAN property for an ergodic diffusion with jumps

In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a compound Poisson process, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.