A functional central limit theorem for Markov additive processes with an application to the closed Lu–Kumar network

A functional central limit theorem for a class of time-homogeneous continuous-time Markov processes (X,Y) is proved. The process X is a positive recurrent Markov process on a countable-state space and the process Y has conditionally independent increments given X. The pair (X,Y) is called a Markov additive process. This paper unifies and generalizes several functional central limit theorems for Markov additive processes. An explicit expression for the variance parameter of the limit process is calculated using the local characteristics of the X process. The functional central limit theorem is then used to prove a heavy traffic limit theorem for the closed Lu–Kumar network.     

Elliptical safety region plots for C pk

The process capability index C _pk is widely used when measuring the capability of a manufacturing process. A process is defined to be capable if the capability index exceeds a stated threshold value, e.g. C _pk >4/3. This inequality can be expressed graphically using a process capability plot, which is a plot in the plane defined by the process mean and the process standard deviation, showing the region for a capable process. In the process capability plot, a safety region can be plotted to obtain a simple graphical decision rule to assess process capability at a given significance level. We consider safety regions to be used for the index C _pk . Under the assumption of normality, we derive elliptical safety regions so that, using a random sample, conclusions about the process capability can be drawn at a given significance level. This simple graphical tool is helpful when trying to understand whether it is the variability, the deviation from target, or both that need to be reduced to improve the capability. Furthermore, using safety regions, several characteristics with different specification limits and different sample sizes can be monitored in the same plot. The proposed graphical decision rule is also investigated with respect to power.     

The double sampling S2 chart with estimated process variance

This paper proposes useful exact bounds for the parameters of the double sampling S² chart with known process variance and it also investigates the properties of the double sampling S² chart with estimated process variance, in terms of the average run length, the standard deviation of the run length and the average sample size, providing a numerical comparison with the known process variance case. It also provides guidelines to systematically design the double sampling S² chart both with known and estimated process variance and proposes two optimal design procedures with estimated process variance, for (a) minimizing the out-of-control average run length and (b) minimizing the out-of-control average sample size.     

Non-homogeneous Pólya-Aeppli process

Abstract

In this paper we suppose that the intensity parameter of the Pólya-Aeppli process is a function of time t and call the resulting process a non-homogeneous Pólya-Aeppli process (NHPAP). The NHPAP can be represented as a compound non-homogeneous Poisson process with geometric compounding distribution as well as a pure birth process. For this process we give two definitions and show their equivalence. Also, we derive some interesting properties of NHPAP and use simulation the illustrate the process for particular intensity functions. In addition, we introduce the standard risk model based on NHPAP, analyze the ruin probability for this model and include an example of the process under exponentially distributed claims.     

Efficient estimation in a nonlinear counting-process regression model

Suppose we observe i.i.d. copies of X, C, Y, where X is a counting process, C is a censoring process talcing only values 0 and 1, and Y is a covariate process. Assume that the intensity process of X is of the form C(s)a(s, Y(s)) with a unknown, but that the distribution of X, C, Y is unspecified otherwise. McKeague and Utikal proposed an estimator for the doubly cumulative hazard f f a(s, y) ds dy and determined its asymptotic distribution. We show that the estimator is regular and efficient in the sense of a Hájek-Inagaki convolution theorem for partially specified models.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

A General Class of Test Procedures for Left-Truncated and Right-Censored Data

When the distribution of one of the characteristics of a process is non normal, methods based on empirical percentiles suggest the use of several process capability indices (PCIs) which are similar to the usual C _p , C _pk , C _pm , and C _pmk indices. However most of these PCIs apply only to the case of symmetrical tolerances. To take into account the asymmetry of the tolerances as well as the asymmetry of the process distribution, new PCIs which improve the previous ones are proposed. In the end and in order to validate the method proposed here, we apply it to a real production case.     

Bayes Estimation of Shift Point in Normal Sequence and Its Application to Statistical Process Control

A sequence of independent observations X ₁, X ₂, …, X _m , X _m+1, …, X _n was observed on some measurable characteristic X in statistical process control. The shift in process mean is reflected in the sequence after X _m . The Bayes estimators of shift point m, and past and future process means, μ₁ and μ₂, are derived using various priors and loss functions. An application in statistical process control is given and a simulation study of the estimators is carried out.     

Bivariate control charts for testing x data patterns

A new control chart, called the θ chart, for monitoring the mean of a process with bivariate quality characteristics is proposed. It can identify a rotation, shift or alternation between the subgroups of the process mean. The conventional application of X² chart to identify a sudden shift of the process mean is also expanded to identify a change of the process mean or a change of the process dispersion. Furthermore, when used together, the θ and X² charts could provide further insight into the process.     

On improvement of prediction in arma processes

Necessary and sufficient conditions are derived in the paper that enable to decide whether an additional multivariate process will improve the prediction in a given multivariate discrete stationary process. The both processes are assumed to form together a process ARMAm n Further it was investigated wnen one can asser t that the both processes are uncorrelated provided the additional process did not improve the prediction in the original process, Some hints for the actual construction of predictors in a multivariate ARMA. (m n) process can be found in the paper.     

Existence and Uniqueness of Q-Processes with a Given Finite μ-Invariant Measure

Let Q be a stable and conservative Q‐matrix over a countable state space S consisting of an irreducible class C and a single absorbing state 0 that is accessible from C. Suppose that Q admits a finite μ‐subinvariant measure m on C. We derive necessary and sufficient conditions for there to exist a Q‐process for which m is μ‐invariant on C, as well as a necessary condition for the uniqueness of such a process.     

Three New Multivariate Process Capability Indices

Different multivariate process capability indices are developed by researchers to evaluate process capability when vectors of quality characteristics are considered in a study. This article presents three indices referred to as NCpM, MCpM, and NMC _PM in order to evaluate process capability in multivariate environment. The performance of the proposed indices is investigated numerically. Simulation results indicate that the proposed indices have descended estimation error and improved performance compared to the existing ones. These results can be important to researchers and practitioners who are interested in evaluating process capability in multivariate domain.     

Switching nonparametric regression models

We propose a methodology to analyse data arising from a curve that, over its domain, switches among J states. We consider a sequence of response variables, where each response y depends on a covariate x according to an unobserved state z. The states form a stochastic process and their possible values are j=1,?…?, J. If z equals j the expected response of y is one of J unknown smooth functions evaluated at x. We call this model a switching nonparametric regression model. We develop an Expectation–Maximisation algorithm to estimate the parameters of the latent state process and the functions corresponding to the J states. We also obtain standard errors for the parameter estimates of the state process. We conduct simulation studies to analyse the frequentist properties of our estimates. We also apply the proposed methodology to the well-known motorcycle dataset treating the data as coming from more than one simulated accident run with unobserved run labels.     

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.     

Defective Galton–Watson processes

The Galton–Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability p_k, k ? 0. In this paper, we consider defective Galton–Watson processes having defective reproduction laws, so that ∑_{k ? 0}p_k = 1 ? ? for some ? ∈ (0, 1). In this setting, each particle may send the process to a graveyard state Δ with probability ?. Such a Markov chain, having an enhanced state space {0, 1, …}∪{Δ}, gets eventually absorbed either at 0 or at Δ. Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t → ∞ and ? → 0.     

Particle swarm optimization based Liu-type estimator

In this study, a new method for the estimation of the shrinkage and biasing parameters of Liu-type estimator is proposed. Because k is kept constant and d is optimized in Liu’s method, a (k, d) pair is not guaranteed to be the optimal point in terms of the mean square error of the parameters. The optimum (k, d) pair that minimizes the mean square error, which is a function of the parameters k and d, should be estimated through a simultaneous optimization process rather than through a two-stage process. In this study, by utilizing a different objective function, the parameters k and d are optimized simultaneously with the particle swarm optimization technique.     

Nonparametric empirical Bayes for the Dirichlet process mixture model

The Dirichlet process prior allows flexible nonparametric mixture modeling. The number of mixture components is not specified in advance and can grow as new data arrive. However, analyses based on the Dirichlet process prior are sensitive to the choice of the parameters, including an infinite-dimensional distributional parameter G ₀. Most previous applications have either fixed G ₀ as a member of a parametric family or treated G ₀ in a Bayesian fashion, using parametric prior specifications. In contrast, we have developed an adaptive nonparametric method for constructing smooth estimates of G ₀. We combine this method with a technique for estimating α, the other Dirichlet process parameter, that is inspired by an existing characterization of its maximum-likelihood estimator. Together, these estimation procedures yield a flexible empirical Bayes treatment of Dirichlet process mixtures. Such a treatment is useful in situations where smooth point estimates of G ₀ are of intrinsic interest, or where the structure of G ₀ cannot be conveniently modeled with the usual parametric prior families. Analysis of simulated and real-world datasets illustrates the robustness of this approach.     

An Improved Approach for Estimating Product Performance Based on the Capability Index C pmk

In this article, we construct an improved procedure for estimating the process capability index C _pmk . We propose a new C _pmk lower-bound approach based on the GCI concept, and compare it with other existing methods. Based on the comparison results, we conclude with a recommendation, and construct a step-by-step procedure for the recommended approach to estimate the actual process capability C _pmk for various sample sizes. The lower bound attended by our recommended approach, indeed, improves other existing lower bound methods. We also investigate a real-world application to illustrate how we could apply the recommended approach to the actual manufacturing processes.     

Probabilistic Properties of Parametric Dual and Inverse Time Series Models Generated by ARMA Models

For the class of autoregressive-moving average (ARMA) processes, we examine the relationship between the dual and the inverse processes. It is demonstrated that the inverse process generated by a causal and invertible ARMA (p, q) process is a causal and invertible ARMA (q, p) model. Moreover, it is established that this representation is strong if and only if the generating process is Gaussian. More precisely, it is derived that the linear innovation process of the inverse process is an all-pass model. Some examples and applications to time reversibility are given to illustrate the obtained results.