bluuebunny/crossref_metadata_2025 huggingface dataset:
General Minimal Repair Models
Authors: Aven Terje
Published: February 2008

Minimal repairs have been given considerable attention in the reliability literature. A minimal repair restores the system to the state it had just before failure. But the state just before failure depends on the information that is available about the system. This article presents general models for analyzing such repairs, using a point process framework and taking into account the information available about the state of the system. Several examples demonstrate the wide range of applications.
Global and Dynamic Information Measures for Reliability
Authors: Ebrahimi Nader , Soofi Ehsan S.
Published: February 2008

In modeling a duration variable, such as the lifetime of an object as it ages, consideration of the age partitions the support of the lifetime distribution progressively and induces truncated distributions for the past and residual lifetimes. Global information measures of the distribution of the entire lifetime do not change with the age, and thus are static. Information measures of the past and residual lifetime distributions are functions of the age, and hence are dynamic. This article presents an overview of the global and dynamic Shannon entropy, Kullback–Leibler information, and mutual information. Examples include information measures for parallel and series systems, proportional hazard, exponential, and Pareto models.
Cumulative Damage Models Based on Gamma Processes
Authors: Park Chanseok , Padgett William J.
Published: February 2008

A statistical model for cumulative damage based on gamma processes, incorporating an accelerating variable, is derived. A simpler approximation to the resulting failure distribution is also given which is a Birnbaum–Saunders‐type model. The proposed model is illustrated with a real‐data example which shows that it is a strong competitor to the widely used Weibull accelerated test model.
Multistate Reliability Theory
Authors: Natvig Bent
Published: February 2008

An inherent weakness of traditional reliability theory (
see
Coherent Systems
) is that the system and the components are always described either as functioning or failed. Early attempts to replace this by a theory of multistate systems for multistate components were made in the late 1970s. By the mid‐1980s the basic theory in this area was established. The objective of this article is to introduce concepts of multistate systems and present some upper and lower bounds for the availabilities and unavailabilities, to any level, in a fixed time interval. A series of applications of multistate reliability theory have been suggested during the last years. For example, the theory enables one to consider applications in electrical power generation systems, where the system state is the amount of power generated by the system or in offshore gas pipeline networks, where the system state is the amount of gas delivered.
Masked Failure Data: Competing Risks
Authors: Basu Sanjib
Published: February 2008

The competing‐risks model considers a setting when a system is exposed to multiple risks and a system failure can be caused by one of the multiple risks. The case of masked causes arises when the causes of failure for some of the systems are not completely identified but are narrowed down to a subset of the potential risks. This article provides a selective review of statistical models and methods for analysis of competing‐risks data with or without masking in the context of reliability applications. Latent failure times and cause‐specific hazards provide two parallel approaches to competing‐risks models; identifiability issues in these models in both nonparametric and parametric settings are discussed. For the case of masked data, recent developments in both likelihood‐based and Bayesian approaches are reviewed. A special emphasis is given to the treatment and models for the masking probabilities. The article concludes with a review of different reliability applications.
Reliability Growth Modeling
Authors: Sen Ananda
Published: February 2008

At the initial developmental phase of any production involving complex systems, a test‐analyze‐and‐fix (TAAF) process is often undertaken to promote the detection of system faults and incorporate appropriate redesigns or other corrective actions, and the modified system is retested. As this test–redesign–retest sequence contributes to an improvement in the system performance, failure data become increasingly sparse at the later stages of testing, making it more difficult to assess the current reliability. A reliability growth model provides a structure through which failure data from the current as well as previous stages of testing could be analyzed in an integrated way in order to make efficient inference on reliability, and other process parameters. Reliability growth models have been applied extensively in analyzing failure data arising from hardware as well as a piece of software. Further, for systems with single‐shot missions that undergo a developmental program, reliability growth models arise quite naturally while studying the effectiveness of design fixes. In this article, we describe some popular reliability growth models that have been developed in the context of observing either time to failure or sequence of polytomous outcomes.
Multicomponent Maintenance
Authors: Nicolai Robin P. , Dekker Rommert
Published: February 2008

In the past few decades the maintenance of multicomponent systems has become more and more complex and, as a consequence, it has got the attention of many scholars. Multicomponent maintenance is the problem of finding optimal maintenance policies for a system consisting of several units of machines or many pieces of equipment, which may or may not depend on each other. Generally, three types of interactions or dependencies can be distinguished: economic dependence, stochastic dependence, and structural dependence. Economic dependence implies that grouping maintenance actions either saves costs (e.g. because to economies of scale) or yields higher costs (e.g. because of manpower restrictions) compared to multiple individual maintenance actions. Stochastic dependence occurs if the condition of the components influences the lifetime distribution of other components. Structural dependence applies if components structurally form a part, so that maintenance of a failed component implies maintenance of working components as well.

On the basis of the planning aspect, we may distinguish between two types of models for multicomponent maintenance: stationary models and dynamic models. In stationary models, a long‐term stable situation is assumed and mostly these models assume an infinite planning horizon. In dynamic models, short‐term information can be taken into account. These models generate dynamic decisions that may change over the planning horizon.
M
arkov Renewal Processes in Reliability Modeling
Authors: Çekyay Bora , Özekici Süleyman
Published: February 2008

Renewal theory is one of the most useful tools used in reliability modeling and analysis. The stochastic structure of most models involve an embedded renewal process where the state of the system regenerates itself over these renewal times. This allows one to analyze various issues related to reliability, availability, maintenance, and repair using renewal theory. In this setting, we aim to explore this idea further by considering Markov renewal models. However, we propose to use a Markov renewal process to model a mission‐based system where the device is designed to perform a random sequence of phases or stages of the mission with random durations. The mission process is modeled as a Markov renewal process and different reliability measures are analyzed using Markov renewal theory. First, it is assumed that the system is replaced with a brand‐new one (maximal repair model) at the beginning of each stage. Under this hypothesis, the probability of survival until a predetermined time (system reliability) and the probability of completing a predetermined number of phases of the mission (mission reliability) are determined. Then, we consider cases without replacement (no repair model) and use intrinsic aging concepts to analyze mission reliability of a series system.
Warranty Cost Analysis
Authors: Murthy D. N. Prabhakar , Blischke Wallace R.
Published: February 2008

For products sold with warranty, the manufacturer incurs costs resulting from the servicing of failures occurring under warranty. These costs depend on several factors, the most important being the reliability of the product. For planning purposes, the manufacturer must estimate (a) the expected cost of warranty per unit sold and (b) the expected cost per unit time over the product life cycle. This must be done in the product development stage. This article deals with models that may be used to obtain these cost estimates for products sold with free‐replacement and
pro rata
warranties.
Age‐Dependent Minimal Repair and Maintenance
Authors: Sheu Shey‐Huei , Chien Yu‐Hung
Published: February 2008

This article presents and discusses two generalized maintenance replacement policies (policy 1 and 2) for a system with two types of failures, when it fails. Type I failure is assumed to be minor and can thus be rectified through minimal repair. Type II failure is catastrophic and can only be removed by replacement. Under policy 1, a system is replaced at the
n
th type I failure or first type II failure or at age
T
, whichever occurs first. Under policy 2, if an operating system fails before time
t
, it is either replaced by a new system (type II failure) or it undergoes minimal repair (type I failure); otherwise, the system is replaced when the first failure after
t
occurs or the total operating time reaches age
T
(0 ≤ 
t
 ≤ 
T
), whichever occurs first. Type I and type II failures are age dependent. The cost of the
i
th minimal repair of a system at age
y
depends on the random part
C
(
y
) and the deterministic part
c

i

(
y
). The expected cost rates for operating a system under policy 1 and 2 in the long run are obtained, respectively. The optimal (
n
*,
T
*) for policy 1 and optimal (
t
*,
T
*) for policy 2, which minimize the corresponding cost rate are derived and discussed. Various special cases are included and numerical examples are finally demonstrated. Because the models and its analysis are general, several existing results are shown to be subsumed by these two models.
Warranty Servicing
Authors: Murthy D.N. Prabhakar , Nat Jack
Published: February 2008

Warranty costs are the additional expenses that a manufacturer incurs to service all the claims made by customers. These costs can be reduced by performing effective waranty servicing. In this article, a framework is given to link all the issues involved in warranty servicing and models that a manufacturer can use to make optimal servicing decisions are received.
Sampling Inspection of Products
Authors: Göb Rainer
Published: February 2008

Product assessment by inspection is a traditional tool of industrial quality control. Statistical product inspection (SPI), also called
acceptance sampling
, is product assessment by sampling inspection. Appropriately conceived, SPI continues to be a valid tool of quality control in modern industrial environments. The article analyzes the basics of SPI in several aspects: history, position of SPI in modern quality control, quality modeling, sampling and its statistical analysis, design of SPI schemes.
Multiple Sampling Plans
Authors: Suresh Kodakanallur Krishnaswamy , Göb Rainer
Published: February 2008

Under a single sampling plan, the decision of rejection or acceptance is based on a unique sample. If product quality is extreme, either very good or very bad, the inspection amount can be reduced by
iterative sampling plans
: smaller sample portions are taken and evaluated successively so that extreme product quality is detected rapidly. Two kinds of iterative sampling procedures are distinguished: Under
multiple m‐stage sampling
a maximum number of
m
sample portions are selected, a definite decision is reached after evaluating the
m
th sample.
Sequential sampling plans
impose no upper limit on the number of samples taken. The procedure may continue until 100% inspection of a lot. The article describes the operation of iterative sampling plans, provides their essential performance measures, namely the operating characteristic function and the average cumulative sample size, and compares iterative against single sampling.
Sampling in Industrial Standards
Authors: Palcat Frank
Published: February 2008

This article reviews the history leading up to the development of the first sampling‐related standards from research in the 1920s following through to the very formative years of World War II and beyond. The transition in subsequent years from military‐specific standards to national standards and then to international standards is also covered. Finally, the array of sampling inspection standards currently available from the International Organization for Standardization (ISO) is briefly reviewed.
Attributes Sampling Schemes in International Standards
Authors: Balamurali Saminathan , Göb Rainer , Jun Chi‐Hyuck
Published: February 2008

Sampling tables for the inspection of attribute quality characteristics were developed during World War II by the United States Department of Defense. In 1950, these tables led to the first military standard MIL‐STD‐105A which evolved into the latest version MIL‐STD‐105E issued in 1989. Further national and international standards emerged as derivatives of MIL‐STD. On international level, the International Standards Organization (ISO) introduced the attributes sampling standard ISO 2859 which has been adopted as a national standard in many countries. The article analyzes the purpose, structure, and operation of MIL‐STD‐105E and of ISO 2859.
Variables Sampling Schemes in International Standards
Authors: Balamurali Saminathan , Göb Rainer , Jun Chi‐Hyuck
Published: February 2008

Variables sampling schemes are based on the specification range quality model: a specification range
R
is prescribed for a continuous measurement characteristic like weight, length, volume, and an item is considered as conforming, if the associated measurement
X
is in
R
. Issued in 1957, MIL‐STD‐414 was the first industrial standard on variables sampling. On the international level, the International Standards Organization (ISO) introduced in 1989, the first variables sampling standard ISO 3951, compatible with the attributes sampling standard ISO 2859, which is a civilian counterpart of MIL‐STD‐105E. ISO 3951 has been adopted as a national standard in many countries. The article analyzes the purpose, structure, and operation of the latest version of ISO 3951 which was issued in three parts in 2005 and 2006.
Reliability Sampling
Authors: Hryniewicz Olgierd
Published: February 2008

Reliability sampling is a part of acceptance sampling dedicated to the control of those quality characteristics of a product, which are related to its functioning in time. Therefore, in reliability sampling we are interested in such quality characteristics of a product as: mean time to failure (MTTF), mean time between failures (MTBF), probability of good performance during a specified mission time, and so on. The aim of the reliability sampling procedures, known as
sampling plans
, is to provide means for testing requirements related to these characteristics. In the article we present reliability sampling plans by attributes and variables, which are useful in reliability practice. For sampling plans by variables, we present procedures valid in cases of the exponential and the Weibull distributions of lifetimes. We consider different types of sampling plans with different types of censoring. Special attention is devoted to procedures described in international standards related to reliability.
Database Systems for Acceptance Sampling
Authors: Lenz Hans‐J.
Published: February 2008

Data collected in on‐line or off‐line quality control are no longer read into files owned by single departments but are stored in a database system. The data can be accessed according to the user's view (profile) and his read/write rights. The use of such operative data is called
on‐line transaction processing
(OLTP). It is of interest to low ranked officers responsible for administrative tasks and concerned with workflows like incoming batch inspection. Middle managers need data for planning and decisions, i.e., for “on‐line analytical processing” (OLAP), where grouped and aggregated data are mandatory. Such databases are called
data warehouses
and the managers' views are known as
data marts
. The main design ideas of such systems are presented.
Acceptance Sampling in Modern Industrial Environments
Authors: Von Collani Elart , Göb Rainer
Published: February 2008

In the industrial mass production environments of the first half of the twentieth century, statistical product inspection (SPI) was a central tool of statistical quality control (SQC). Contemporary SQC emphasizes methods of provision and prevention such as experimental design or process control rather than inspection. To remain a valid tool of modern SQC, SPI has to be redefined in its purpose and in its procedures. This article analyzes important challenges for SPI in modern industrial environments: the adjunct role of SPI in product control, adaptiveness and flexibility of SPI, the role of prior information on product quality, the economic concern of SPI, and quality measurement as a purpose of SPI.
Sampling in Pharmaceutical and Chemical Industries
Authors: Melgaard Henrik
Published: February 2008

This article discusses the specific issues related to sampling in the pharmaceutical and chemical industry. The effect of measurement uncertainty in case of acceptance sampling by variables is one issue. The largest deterioration on the operating characteristics is seen from the low‐frequency error type. Sampling in bulk materials and retesting is also discussed. The trend in this industry is in the direction of faster measurement methods replacing off‐line by on‐line measuring and release.
Sampling in Software Development
Authors: Runeson Per
Published: February 2008

In software development, sampling inspection is applied during the development process, as the production process for software is trivial. Insoftware inspection, which is a manual scrutiny of software code or specifications, sampling is applied to estimate the number of remaining faults and to guide which parts of the artifact to inspect. Capture–recapture techniques are used in the fault content estimation. Insoftware testing, which is the dynamic execution of code, sampling is implicitly or explicitly applied to select test cases. Since the number of states is extremely high in software products, a complete coverage of all cases is practically impossible. In addition to scoping the tests to a reasonable magnitude of size, sampling is used to estimate the number of remaining faults and to predict the operational failure behavior of the software, e.g. through software reliability growth models. The theory is quite well established in the field, while the industrial practice is underdeveloped.
Sampling in Implementation of Statistical Process Control
Authors: Nishina Ken
Published: February 2008

In this article, we explain how to obtain data by sampling and implement statistical process control (SPC) over its life cycle. The SPC life cycle consists of three steps: mass production preparation, early‐stage mass production, and routine mass production. In mass production preparation, machine performance is ensured by sampling according to an experimental design. In mass production stage, by sampling in control charts, process variation is reduced and process capability is ensured. Sampling to prevent the escape of a nonconforming product in routine mass production is also explained. In addition, considering semiconductor wafer process, the inherent process variation related to sampling is introduced.
Sampling from Virtual Populations
Authors: Polansky Alan M.
Published: February 2008

The increasing complexity of the problems addressed in modern statistical inference has created many situations where a closed‐form analytical solution is not available. For example, when constructing a rejection region for a hypothesis‐testing procedure, the sampling distribution of the test statistic under the null hypothesis may not have a closed form, even when the exact form of the population is known. In nonparametric situations where the population may not be known, estimates of the sampling distribution of a test statistic or parameter estimate computed using the bootstrap often do not have a closed form. The use of empirical approximations based on simulated realizations of the random process that produces the observed data have been implemented for many years. Almost universal access to computers has made such simulations easy to perform. The purpose of this article is to review the most common methodologies for performing these simulations.
Sampling Designs in Surveys
Authors: Gabler Siegfried
Published: February 2008

Survey sampling basically consists of the selection of a part of a population and the subsequent inference from this selected sample. To derive statistical conclusions from the actually collected data, a stochastic background based on a random selection procedure is necessary. Depending on the availability of frames, surveys will use simple or complex samples.
Adaptive and Spatial Sampling Designs
Authors: Siegfried Gabler
Published: February 2008

On the one hand, spatial and adaptive sampling designs are informative since they depend on the characteristic of interest—in contrast to the classical sampling designs such as simple random sampling. On the other hand, better results can be obtained using adaptive designs for populations with special patterns. Dealing with spatial inferences, model‐based approaches are often used. The fixed‐population approach is replaced by a stochastic one. The characteristic of interest is defined for a continuous region of space and time that can be described by a spatial stochastic process.
Selection and Validation of Response Scales
Authors: Ograjenšek Irena
Published: February 2008

Measurement is defined as a set of rules for assigning either numbers or numerals to empirical properties of a concept under consideration. Concepts can be operationalized as variables with different properties of measurement scale: nominal, ordinal, interval, and ratio. The level of measurement determines which mathematical operations are possible on a variable, which, in turn, define the tools of statistical analysis at researchers' disposal.

Specific measurement tools applicable in the survey setting include list questions, category questions, ranking questions, and rating or scale questions. We want these measures to be reliable and valid, capturing what they are supposed to capture.

In the process of measurement, data can be obtained either in formal or informal settings and can represent verbal (oral and written) as well as nonverbal acts and/or responses. In order to minimize the possible effect of the chosen data collection method on the validity and/or scope of the research findings, researchers can use two or more methods of data collection to test hypotheses and measure variables; this is the essence of the so‐called triangulation.
Data Management in Survey Sampling
Authors: Heerwegh Dirk
Published: February 2008

An overview of new technology applications in data collection and data management is given. Attention is paid to advantages and disadvantages, both in methodological and statistical terms.
Processing of Survey Data
Authors: Biemer Paul P.
Published: February 2008

Upon completion of data collection, the survey data must be converted from its raw and unedited state to a verified and corrected state that is ready for analysis and/or dissemination to the users. These steps may include questionnaire inspection, data entry, editing, coding, data preparation, documentation, and data analysis. This chapter discusses these steps as well as their potential effect on data quality.
Statistical Analysis of Survey Data
Authors: Ograjenšek Irena
Published: February 2008

After data collection and entry, survey data need to be prepared for analysis and analyzed. The data preparation stage involves three important components: data formatting, data editing, and computation of survey weights. The data analysis stage involves the choice of relevant statistical tools according to the measurement scale properties of survey variables.
Survey Quality and Survey Ethics
Authors: Djerf Kari
Published: February 2008

Producers of statistics have devoted considerable efforts to improve the quality of their products. Traditionally, surveys are annotated with quality descriptions providing the clients with support to interpret the results. Furthermore, there can be studies on the quality of the entire sample survey, called
quality reports
. The clients often need more profound information on the organization, applied procedures and processes as well as the products. Thus, the approach is changing from a mere focus on product quality to total quality management (TQM).

This article describes various perspectives of survey quality, starting from traditional reporting and ending up with recent more comprehensive quality approaches.
Dependence
Authors: Shore Haim
Published: February 2008

Dependence and independence are addressed from three perspectives: definition, relationship between dependence and correlation, and the expected value of the product of univariate functions of independent random variables.
Descriptive Statistics
Authors: Shore Haim
Published: February 2008

Descriptive statistics, calculated from the data, provide summary information about properties of a data set. In this article some of the most widely used descriptive statistics are detailed with the allied formulae. These include the sample average, median and other sample quantiles, mode, standard deviation, coefficient of variation, range, interquartile range, skewness, and kurtosis.
Measures of Scale
Authors: Di Bucchianico Alessandro
Published: February 2008

Example of sample statistics that describe spread are given, together with some properties.
M
onte
C
arlo Methods
Authors: Shore Haim
Published: February 2008

Monte Carlo methods are widely used in all branches of science and technology in general, and in quality and reliability engineering, in particular. In this article, Monte Carlo methods are first defined and major areas of application are outlined. Commonly used techniques that are helpful in the implementation of Monte Carlo methods are briefly addressed, and applications in quality and reliability engineering are described.
Moving Averages
Authors: Shore Haim
Published: February 2008

The moving average is defined and its common uses in quality engineering are described. Two versions of the moving average, the standard moving average and the exponentially weighted moving average, are briefly introduced
Multi‐Vari Chart
Authors: De Mast Jeroen
Published: February 2008

The multi‐vari chart is used to identify and display sources of variation in processes. It is a cousin of the control chart.
Normality Tests
Authors: Koning Alex J.
Published: February 2008

For testing normality with unknown parameters μ and σ
2
, extensive simulation studies have shown the superiority of the Shapiro–Wilk test and variants. The Shapiro–Francia test is a simplified version of the Shapiro–Wilk test, which uses the squared normal probability plot correlation as a test statistic. The Cramér–von Mises and Anderson–Darling tests with plug‐in estimators for μ and σ
2
perform slightly worse than the Shapiro–Wilk test.
Pooled Variance, Pooled Estimate
Authors: Di Bucchianico Alessandro
Published: February 2008

Pooled estimators are estimators that combine the statistical information from different samples.
Statistical Methods for Counts, Rates, and Proportions
Authors: Thyregod Poul
Published: February 2008

Poisson distribution and multinomial distribution are natural distribution models for count and frequency data for categorical variables. When there are only two categories, the multinomial reduces to the binomial distribution.

The distributions are skewed with a variance that changes with the mean value, and therefore standard methods for normally distributed variables such as ANOVA (analysis of variance) and the general linear model cannot reasonably be applied.

When events are counted and classified by two or more categorical variables or when sample units are classified, the counts or frequencies are often presented in tabular form in so‐called contingency tables.

A general framework for analyzing association in multidimensional contingency tables is provided by the log‐linear models. When a set of nested models is formulated, higher‐order interaction terms may be removed successively in analogy with ANOVA models for normally distributed variables. A subset of these models, the logit models, may be used for assessing the effect of explanatory variables on a categorical response variable.

Another important subset of the log‐linear models is constituted by the graphical models. These models allow for a factorization of the joint probability into a product of conditional distributions.

The log‐linear and logit models are special cases of a wider class of models, the generalized linear models. In this framework, the effect of continuous explanatory variables may be modeled in regression‐like models. An example is the logistic regression where the effect of explanatory variables on a response probability is modeled.

When the assumption of independence is violated, overdispersion might result. A method for modeling overdispersion is also presented.
Sample‐Size Determination
Authors: Shore Haim
Published: February 2008

Sample data may be collected with different objectives in different scenarios. In some cases, one may wish to collect enough observations that would guarantee minimal prespecified reliability for the conclusions derived from the data. On other occasions, in addition to specified constraints regarding reliability, an optimal value is desired of some objective function that would guarantee that the sampling procedure is optimal in some sense. In all these cases, a major decision variable is the required sample size. In this article, sample‐size determination is addressed, relating, in particular, to four often‐encountered scenarios: hypothesis testing, parameter estimation, relational modeling, and optimal sampling.
Sum of Squares
Authors: Di Bucchianico Alessandro
Published: February 2008

We describe the notion of sums of squares, which it the leading idea behind analysis of variance. We special attention to the case of unbalanced designs, where different types of sums of squares can be used. We outline the appropriate use of these sums of squares.
Measures of Association
Authors: Hershberger Scott L. , Fisher Dennis G.
Published: February 2008

Measures of association are numerical estimates of the magnitude of the statistical dependence between two variables, especially if these variables are categorical. A multitude of such measures have been proposed; we review some of the most commonly used ones, dividing them into two types. The first is suitable for nominal scale data, and includes measures of predictive association. The second type is suitable for ordinal data, and includes measures of concordance‐discordance.
Probability Density Functions
Authors: Everitt Brian S.
Published: February 2008

Probability density functions are the foundations of almost all statistical methodology. Here we give a brief account of the density functions most often encountered in practice along with a summary of their most important properties.
Confidence Intervals
Authors: Dracup Chris
Published: February 2008

In estimating a population parameter from sample data, it is more informative to provide a range of “plausible” parameter values rather than a one‐point estimate. A 95% confidence interval contains those values of the population parameter that would not be rejected at the 0.05 significance level by a statistical test applied to the sample data. In the long run, such intervals should provide 95% “coverage”, with 95% of them including their true population value (assuming the data meet any distribution assumptions). Topics covered include the construction of confidence intervals for means, proportions, and mean differences; the effects of discrete and nonsymmetrical sampling distributions; central and noncentral intervals; bootstrapped intervals; and the Bayesian approach.
Least‐Squares Estimation
Authors: Van De Geer Sara A.
Published: February 2008

The method of least‐squares is about estimating parameters by minimizing the squared discrepancies between observed data, on the one hand, and their expected values on the other. It is commonly used in the context of a regression model, where one observes pairs of variables (
X
,
Y
), with
X
, a covariable and
Y
, a response variable. The expectation of the response variable
Y
given the covariable
X
is modeled as a function
f
β
(
X
) depending on parameters β.
Quantiles
Authors: Lovie Pat
Published: February 2008

A quantile is defined as a value that cuts off a certain proportion
p
of the area under a probability distribution function (a theoretical quantile) or of the distribution of a set of data (empirical quantile). Quantile is a general name for a class of statistics, which includes familiar measures such as the median, quartiles, and percentiles.
Time Series Analysis
Authors: Brockwell P. J.
Published: February 2008

The time‐domain approach to the modeling and forecasting of time series is outlined. Weakly stationary processes are introduced, and an introduction to autoregressive moving average processes and their properties is given. Estimation, model selection, and prediction with these models are considered, and extensions to multivariate time series, state‐space models, and nonlinear models are indicated.
Hypothesis Testing
Authors: Watson G. S.
Published: February 2008

The sections in this article are

Introduction

Early Ideas

The Neyman–Pearson Lemma

Several‐Parameter Problems

Miscellaneous Remarks
Maximum Likelihood
Authors: Tosteson Tor D. , Demidenko Eugene
Published: February 2008

Maximum likelihood is a general method of parameter estimation of great practical and theoretical significance. The method is defined and the optimal properties of maximum‐likelihood estimates are explained along with the underlying assumptions. An example is given and computational methods are discussed.
Exponentially Weighted Moving Average (
EWMA
) Control Chart
Authors: Crowder Stephen V. , Vander Wiel Scott A.
Published: February 2008

An exponentially weighted moving average (EWMA) is a filter of sequential process data, giving the most recent observation the most weight and older observations exponentially less weight. EWMA control charts are especially useful in detecting small shifts in the mean or variance of a process. For processes that tend to wander over time, EWMAs are also a basic technique for predicting future observations, for making automatic adjustments to a process and for monitoring a sequence of forecast or regulation errors.
Control Charts for Attributes
Authors: Jones‐Farmer L. Allison
Published: February 2008

Control charts can be used to monitor characteristics of a process that are measured using categorical or attribute data. The most common attribute control charts are the
p
‐chart and
c
‐chart. This section discusses the scenarios in which the
p
‐ and
c
‐charts are useful. A detailed example of the use of a
p
‐chart is given. Strengths and weaknesses of attribute charts are discussed.
Multivariate Exponentially Weighted Moving Average (
MEWMA
) Control Chart
Authors: Conerly Michael D.
Published: February 2008

The multivariate exponentially weighted moving average (MEWMA) control chart is used to simultaneously monitor several correlated process variables. The primary focus of this article is monitoring the process mean by constructing an MEWMA control chart. An overview of the application and use of MEWMA control charts is discussed. The plotting statistic is clearly defined and an illustrative example is presented.
Multivariate Control Charts, Interpretation of
Authors: Mason Robert L. , Young John C.
Published: February 2008

Signal interpretation is straightforward when only one variable is being monitored in a process. This is because separate charts can be constructed to monitor changes in the location and/or changes in the variation of the variable of interest. For a multivariate process consisting of
p
variables, signal interpretation is more complex because the signaling statistic can potentially be a function of all
p
variables. Not only do we have to detect location and variation changes in the
p
variables, but we also have to detect changes in the relationships (i.e., correlations) among the many variables. Further, procedures that identify the particular variable or group of variables that cause or contribute to the signal are limited. In this presentation, we discuss two procedures that are useful in signal interpretation when using Hotelling's
T
2
statistic in multivariate process control and briefly describe several alternative methods.
Regression Control Charts
Authors: Shu Lianjie , Tsui Kwok‐Leung , Tsung Fugee
Published: February 2008

The regression control chart, which integrates regression analysis and control chart theory, has proved useful in various fields of application in the last few years. The basic idea of the regression control chart is to monitor the process output after it has been adjusted for the effect of the external covariate(s). This paper provides an overall review of regression control techniques. The formulation of the regression control chart is presented, and its computation is illustrated. Some remarks on future issues are also made.
Control Charts, Nonparametric
Authors: Chakraborti Subhabrata , Graham Marien A.
Published: February 2008

Standard control charts are often based on the assumption that the observations follow a specific parametric distribution, such as the normal. In many applications we do not have enough information to make this assumption and in such situations, development and application of control charts that do not depend on a particular distributional assumption is desirable. Nonparametric or distribution‐free control charts can serve this broader purpose. We discuss some ideas and recent developments in the area of univariate nonparametric control charts. Most of these charts are for monitoring the location; few charts are available for scale. A key advantage of nonparametric charts is that the in‐control run length distribution is the same for all continuous process distributions. Nonparametric charts are more robust and efficient than their parametric counterparts under heavy‐tailed and/or asymmetric distributions.
Variable Sampling Rate Control Charts
Authors: Reynolds Marion R. , Stoumbos Zachary G.
Published: February 2008

When control charts are used to monitor a process, the traditional practice is to take samples from the process using a fixed sampling rate (FSR). Variable sampling rate (VSR) control charts vary the sampling rate as a function of the data obtained from the process. With a VSR control chart the sampling rate is increased when there is some indication of a problem with the process, and decreased when there is no indication of a problem with the process. A VSR control chart signals in the same way as a standard FSR control chart when the indication of a problem with process is strong enough. There are several approaches to implementing the VSR concept. One approach is to allow the sampling interval to vary as a function of the data. Another approach is to allow the sample size for the current sample to vary as a function of the data in previous samples. A third approach is to allow the sample size for the current sample to depend on the data in this sample. Compared to traditional FSR control charts, all of the approaches to implementing the VSR concept will produce much faster detection of most process changes.
Neural Networks: Construction and Evaluation
Authors: Saithanu Kidakan
Published: February 2008

The use of neural networks (NNs) for process monitoring is discussed. An example illustrating the use of NNs applied to multivariate quality control is provided. The strengths and weaknesses of the NN approach relative to more traditional methods in multivariate quality control are outlined.
Average Run Lengths and Operating Characteristic Curves
Authors: Human Schalk W. , Graham Marien A.
Published: February 2008

Control charts are typically used in establishing whether a process is in a state of statistical control or not. The performance of any control chart is an important consideration and characteristics of the run‐length distribution are generally used to assess its performance. Two popular measures of a control chart's performance are the expected value of the run‐length distribution, called the
average run length
(
ARL
), and the operating characteristic (
OC
). Both are measures of how quickly a chart will detect or react to a change in the process and can be used to compare control charting plans. Both these measures provide useful information about the operational performance of a control chart. We discuss the
ARL
and the
OC
of the well‐known Shewhart

X

chart when both the process mean and variance are known. The application of
OC
curves in acceptance sampling is also considered.
Autoregressive Integrated Moving Average (ARIMA) Modeling
Authors: Luceño Alberto , Peña Daniel
Published: February 2008

ARIMA (autoregressive integrated moving average) modeling is a method to approach the dynamic behavior of a process that generates a time series of successive valuesY′ = (y1, …,yn) of a magnitudeYobserved at equispaced periods {S, …,nS}, whereSis a sampling interval. ARIMA is composed of three parts, namely, AR, I, and MA, that can stand alone or be combined at will. Hence there are white noise models (without any part) as well as AR, I (e.g., random walk), MA, ARI, ARMA, IMA, and ARIMA models. When the I part is included, the process is nonstationary. Differentiation of a nonstationary ARIMA process leads to a stationary ARMA process. ARIMA models belong to the class of linear time series models and often assume that vectorY′ is jointly normally distributed, possibly after transformation. ARIMA modeling is usually implemented by identifying the order of the AR, I, and MA parts, estimating the AR and MA parameters, and diagnosing the adequacy of the hypotheses. Short‐term forecasting with ARIMA models is often difficult to beat in the quality context by using nonlinear and/or nonnormal models.
Process Capability, Error in Estimation of
Authors: Pearn Wen Lea , Wu Chien‐Wei
Published: February 2008

Process capability indices (PCIs) are widely used to provide numerical measures of process performance, which have received considerable research attention and increased usage in process assessments and purchasing decisions during the last decade. A substantial majority of capability research works that appeared in the literature do not take into account gauge measurement errors. However, the gauge measurement errors have a significant effect on estimating and assessing process capability since measurement systems are not 100% precise. An inaccurate measurement system can thwart all the benefits of improvement endeavors resulting in poor quality. Analyzing process capability without considering gauge capability may often lead to unreliable decisions. It could result in a serious loss to producers if gauge capability is not being considered in process capability estimation and testing. Since measurement errors unfortunately cannot be avoided, using appropriate confidence coefficients and power becomes an essential task. In this article, we summarize the issue on estimating and testing the process capability is the presence of measurement errors, provide a numerical example discussing the influence of measurement errors, and cite additional references for more discussions on gauge repeatability and reproducibility (GR&R) studies.
Process Capability Indices, Nonnormal
Authors: Chou Youn‐Min , Polansky Alan M.
Published: February 2008

Data from manufacturing processes may often come from nonnormal distributions. Process capability indices for nonnormal distributions have attracted much research in the last 20 years. This paper presents process capability indices for nonnormal distributions and reviews various approaches to deal with the problem of nonnormality in evaluating process capability.
Process Capability Indices, Nonparametric
Authors: Polansky Alan M.
Published: February 2008

The capability of a manufacturing process is a measure of the ability of the process to consistently produce items within engineering specifications. Most early attempts to assess the capability of a process were based on the assumption that the quality characteristic of interest follows a parametric distribution, usually normal. Several studies of process data indicate that many quality characteristics follow nonnormal distributions.Further, theoretical and empirical studies of measures of process capability indicate that violations of the assumption relating to the distribution of the quality characteristic can produce serious errors in conclusions about the capability of the process. This article reviews methods that have been developed to assess process capability within a nonparametric framework. Such methods require few specific assumptions about the distribution of the quality characteristic. The methods reviewed in this article include those based on nonparametric tolerance limits, nonparametric estimates of the process yield, and indices based on the percentiles of the process distribution.
Statistical Process Control, Multivariate
Authors: Fuchs Camil , Kenett Ron S.
Published: February 2008

Owing to pervasive capabilities of modern data collection, analysis, and graphical displays, the multivariate statistical process control is playing an increasingly central role in industry and services. Its application requires careful consideration of the data context and the statistical tools involved. In this section we present various situations in which such techniques are used and provide details on their practical implementation.
Process Capability Indices,
B
ayesian Estimation of
Authors: Spiring Fred , Cheng Smiley
Published: February 2008

Practitioners often base inferences regarding the capability of a process on a point estimate of one of the common indices without examining the distributional properties of the estimator used. The practice is due, in part, to (a) the complexity of the indices developed and, at least initially, (b) the lack of stochastic development in the general area of process capability indices. Since the mid‐1980s the body of work addressing (a) estimation of and (b) drawing inference from process capability indices has grown steadily. Investigating process capability from a Bayesian perspective has mirrored this development. Much like the indices themselves, the Bayesian approach has seen two distinct patterns develop. One focus has been on examining parts nonconforming while a second examines the ability of a process to cluster around a target.
Process Capability Plots
Authors: Vännman Kerstin
Published: February 2008

A process is usually defined to be capable if a process capability index exceeds a stated threshold value. This definition can be expressed graphically as a region in the plane, defined by the process parameters, to obtain a process capability plot. Under the assumption of normality, safety regions can be plotted in the process capability plot to draw conclusions about process capability at a given significance level. Alternatively, an estimated process capability plot defined by the estimators of process parameters can be used. This article discusses these different kinds of plots and presents examples and interpretations.