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In many practical situations the variance of a set of measurements can be attributed to several known sources of variability. For example, if several measurements of each item of a lot are taken, then one may need to deal not only with the within-item variability, but also with item-to-item within-lot and lot-to-lot components of variability. In such cases conventional control charts tend to produce an unacceptably high rate of false alarms and in general represent a rather weak diagnostic tool. This article shows how to build a control system based on likelihood ratio tests capable of monitoring the mean and variance components of a nested random effect model. The strong points and weaknesses of this approach are compared to those of competing methods, and some examples related to manufacturing of integrated circuits are discussed.

We consider the efficient estimation of the semiparametric additive transformation model with current status data. A wide range of survival models and econometric models can be incorporated into this general transformation framework. We apply the B-spline approach to simultaneously estimate the linear regression vector, the nondecreasing transformation function, and a set of nonparametric regression functions. We show that the parametric estimate is semiparametric efficient in the presence of multiple nonparametric nuisance functions. An explicit consistent B-spline estimate of the asymptotic variance is also provided. All nonparametric estimates are smooth, and shown to be uniformly consistent and have faster than cubic rate of convergence. Interestingly, we observe the convergence rate interfere phenomenon, i.e., the convergence rates of B-spline estimators are all slowed down to equal the slowest one. The constrained optimization is not required in our implementation. Numerical results are used to illustrate the finite sample performance of the proposed estimators., 32 pages, 5 figures

Suppose a process yields independent observations whose distributions belong to a family parameterized by \theta\in\Theta. When the process is in control, the observations are i.i.d. with a known parameter value \theta_0. When the process is out of control, the parameter changes. We apply an idea of Robbins and Siegmund [Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 (1972) 37-41] to construct a class of sequential tests and detection schemes whereby the unknown post-change parameters are estimated. This approach is especially useful in situations where the parametric space is intricate and mixture-type rules are operationally or conceptually difficult to formulate. We exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a Gamma distribution, in both a univariate and a multivariate setting., Comment: Published at http://dx.doi.org/10.1214/009053605000000183 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

It is known that both the optimal exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts are based on a given reference value $\delta$, which, for the CUSUM chart, is the magnitude of a shift in the mean to be detected quickly. In this paper a generalized EWMA control chart (GEWMA) which does not depend on $\delta$ is proposed for detecting the mean shift. We compare theoretically the GEWMA control chart with the optimal EWMA, CUSUM and the generalized likelihood ratio (GLR) control charts. The results of the comparison in which the in-control average run length approaches infinity show that the GEWMA control chart is better than the optimal EWMA control chart in detecting a mean shift of any size and is also better than the CUSUM control chart in detecting the mean shift which is not in the interval $(0.7842\delta ,1.3798\delta )$. Moreover, the GLR control chart has the best performance in detecting mean shift among the four control charts except when detecting a particular mean shift $\delta,$ when the in-control average run length approaches infinity.

Observations are generated according to a regression with normal error as a function of time,when the process is in control. The process potentially changes at some unknown point oftime and then the ensuing observations are normal with the same mean function plus an arbitrary function under suitable regularity conditions. The problem is to obtain a stopping rule that is optimal in the sense that the rule minimizes the expected delay in detecting a change subject to a constraint on the average run length to a false alarm. A bound on the expected delay is first obtained. It is then shown that the cusum and Shiryayev–Roberts procedures achieve this bound to first order.

Surveillance can be based, in some change-point detection problems, on a sequence of invariant statistics. Gordon and Pollak prove that, under certain conditions, the average run length (ARL) to false alarm of invariance- based Shiryayev–Roberts detection schemes is asymptotically the same as that of the dual classical scheme that is based on the original sequence of observations. In this paper we give alternative conditions under which the two ARLs coincide and demonstrate that these conditions are satisfied in cases where Gordon and Pollak’s conditions are difficult to check.

Suppose $X_1, X_2, \dots, X_{\nu - 1}$ are iid random variables with distribution $F_0$, and $X_{\nu}, X_{\nu + 1}, \dots$ are are iid with distributed $F_1$. The change point $\nu$ is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from $F_0$ to $F_1$ (detect the change), but to avoid false alarms. ¶ Pollak found a version of the Shiryayev-Roberts procedure to be asymptotically optimal for the problem of minimizing the average run length to detection over all stopping times which satisfy a given constraint on the rate of false alarms. Here we find that this procedure is strictly optimal for a slight reformulation of the problem he considered. ¶ Explicit formulas are developed for the calculation of the average run length (both before and after the change) for the optimal stopping time.

Schmid extended the classical EWMA control chart to autocorrelated processes. Here, we consider the tail probability of the run length in the in-control state. The in-control process is assumed to be a stationary Gaussian process. It is proved that the tails for the autocorrelated process are larger than in the case of independent variables if all autocovariances are greater than or equal to zero. The inequality is strict. Moreover, this result is still valid for stationary processes having elliptically contoured marginal distributions.

We show that if dynamic sampling is feasible, then there exist surveillance schemes that satisfy a probability constraint on false alarm. Procedures are suggested for detecting a change of a normal mean from 0 to a (unknown) positive value. These procedures are optimal (up to a constant term) when the post-change mean is known, and almost optimal [up to an $o(\log(1/\alpha))$ term when the post-change mean is unknown.

We consider a sequential procedure for comparing three treatments with the goal of ultimately selecting the best treatment. This procedure starts with a sequential test to detect an overall treatment difference and eliminates the apparently inferior treatment if this test rejects the equality of the treatments. It then proceeds with a sequential test of the remaining two treatments. We base these sequential tests on the stopping boundaries popularized by O'Brien and Fleming. Our procedure is similar in structure to that used by Siegmund in conjunction with modified repeated significance tests. We compare the performances of the two procedures via a simulation experiment. We derive analytic approximations for an error probability, the power and the expected sample size of our procedure, which we compare to simulated values. Furthermore, we propose a modification of the procedure for the comparison of a standard treatment with experimental treatments.

An inequality that relates the probability of false alarm of a change-point detection policy to its average run length to detection is proved. By means of this inequality, a lower bound on the rate of detection, when the change occurs after a long delay, is derived.

A new proof is given to a known result on the average run length to false alarm of the Shiryayev-Roberts change-point detection policy when the observations are nonlattice. Via the approach of this new proof, the average run length to false alarm can be calculated in the lattice case and for the mixed-type Shiryayev-Roberts scheme.

In this paper, a dynamic sampling plan in the Shiryayev-Roberts procedure is considered. It is shown that a two-rate dynamic sampling plan is optimal in the sense that it minimizes the stationary average delay time (SADT). Analytical results as well as numerical comparisons show that it is significantly superior to the fixed sampling plan. The comparison also shows that it is as powerful as the dynamic sampling procedure of Assaf and Ritov. The generalizations to the fast initial response and to the CUSUM procedure are also briefly discussed.

In this article, a model of single variable sampling plan with type I censoring is studied. Under the assumption that the variable is exponentially distributed and the loss function is a polynomial, an explicit expression of the Bayes risk is derived. Then, a simple and finite algorithm to determine an optimal sampling plan for minimizing the Bayes risk is suggested. Furthermore, a discretization method is proposed so that one can easily obtain an approximately optimal sampling plan.

Suppose that a system in its standard state produces i.i.d. observations whose distribution is symmetric about zero. At an unknown time the system may leave its standard state, and the observations would subsequently be stochastically larger. Subject to a bound on the rate of false alarms, one wants to detect quickly such a departure from the standard state. We present a robust method of detection which is computationally feasible and remarkably efficient. The method is based on the sequential vectors of signs and ranks of the observations. The methodology is one of likelihood ratio; a sequence of likelihood ratios for these vectors is computed, and the Shiryayev-Roberts approach to changepoint detection is then applied to yield a class of statistics and associated stopping rules. Inequalities and asymptotic approximations for the operating characteristics of these rules are developed. These are found to be valid also for small average run lengths and early changepoints as well. The relative efficiency of these schemes (with respect to a normal parametric shift detection policy) is very high, making them a robust alternative to parametric methods.

A rank-sign analogue of the Page-CUSUM procedure is introduced here for detecting location change in a sequentially observed series of data following a symmetric distribution. This Rank-CUSUM procedure is asymptotically equivalent to the Page-CUSUM procedure if the score functions used in both procedures are appropriate for the underlying density; but even otherwise, it maintains its prescribed false alarm rate and has good detection property. Results of simulation studies comparing the Rank-CUSUM and the Page-CUSUM procedures are reported. 1. Introduction. For the problem of sequential detection of change in distribution, the CUSUM chart proposed by Page (1954) is widely accepted for its simplicity. The stopping rule for the procedure also has a minimax property as shown by Lorden (1971) and Moustakides (1986). However, one has to recognize the fact that the method is based on strict distributional assumptions and may perform poorly when these assumptions fail. In this paper, we consider a rank-sign analogue of the Page-CUSUM procedure for the situation where one wants to detect a small change in location of a symmetric density without knowing the actual form of the density. If the score function used to calculate the rank-CUSUM's is appropriately chosen for the underlying density, then this procedure is asymptotically equivalent to the Page-CUSUM procedure (based on the true density); but even otherwise, the rank-CUSUM procedure maintains its prescribed false alarm rate and also has good detection property. In Section 2, the rank-CUSUM 's are introduced and the weak convergence properties of both types of CUSUM are discussed. Some simulation results comparing the performance of the rank-CUSUM procedure with that of the Page-CUSUM procedure are given in Section 3. Technical details of

Pollak and Siegmund compared the Shiryayev-Roberts procedure with the CUSUM procedure for detecting a change in the drift of a Brownian motion based on the conditional average delay time. In this paper, the exponentially weighted moving average (EWMA) procedure proposed by Roberts is compared with the Shiryayev-Roberts and CUSUM procedures. The comparison is based on the stationary average delay time as advocated by Shiryayev. The optimal design for the EWMA procedure and its asymptotic properties are studied when the average in-control run length is large. The results show that the EWMA procedure is less efficient than the other two procedures.

A model for sequential clinical trials involving three treatments is proposed and various error probabilities are defined. A specific procedure involving a concatenation of repeated significance tests is studied in detail. The first phase of testing proceeds until at least one treatment is eliminated; and then a second phase is used to compare the remaining treatments. Analytic approximations to the operating characteristics of the test are derived and compared with the results of a Monte Carlo experiment.

The origins of statistical quality control are first reviewed relative to the concept of statistical control. A recent Bayesian approach developed at ATT namely i) variation due to chance causes (called common causes by Deming, 1986); ii) variation due to assignable causes (called special causes by Deming, 1986). Chance causes are inherent in the system of production while assignable causes, if they exist, can be traced to a particular machine, a particular worker, a particular material, etc. According to both Shewart and Deming, if variation in product is only due to chance causes, then the process is said to be in statistical control. Nelson (1982) describes a process in statistical control as follows: "A process is said to have reached a state of statistical control when changes in Now at George Washington University, Washington, D.C. 20006. This research was partially supported by the U.S. Air Force Office of Scientific Research (AFOSR-90-0087) to the University of California at Berkeley. This content downloaded from 157.55.39.111 on Wed, 03 Aug 2016 04:54:04 UTC All use subject to http://about.jstor.org/terms 100 R.E. Barlow k T.Z. Irony measures of variability and location from one sampling period to the next are no greater than statistical theory would predict. That is, assignable causes of variation have been detected, identified, and eliminated." Duncan (1974) describes chance variations: "If chance variations are ordered in time or possibly on some other basis, they will behave in a random manner. They will show no cycles or runs or any other defined pattern. No specific variation to come can be predicted from knowledge of past variations." Duncan, in the last sentence, is implying statistical independence and not statistical control. Neither Shewhart nor Duncan have given us a mathematical definition of what it means for a process to be in statistical control. The following example shows that statistical independence depends on the knowledge of the observer and, therefore, we think it should not be a part of the definition of statistical control. Example The idea of chance causes apparently comes from or can be associated with Monte Carlo experiments. Suppose I go to a computer and generate ? random quantities normally distributed with mean 0 and variance 1. Since I know the distribution used to generate the observed quantities, I would use a iV(0,l) distribution to predict the (n+l)st quantity yet to be generated by the computer. For me, the process is random and the generated ? random quantities provide no predictive information. However, suppose I show a plot of these ? numbers to my friend and I tell her how the numbers were generated except that I neglect to tell her that the variance was 1. Then for her, zn+1 is not independent of the first ? random quantities because she can use these ? quantities to estimate the process variance and, therefore, better predict xn+iWhat is interesting from this example is that for one of us the observations are from an independent process while for the other the observations are from a dependent process. But of course (objectively) the plot looks exactly the same to both of us. The probability distribution used depends on the state of knowledge of the analyst. I think we both would agree however that the process is in statistical control. All authors seem to indicate that the concept of statistical control is somehow connected with probability theory although not with any specific probability model. We think de Finetti (1937, 1979) has given us the concept which provides the correct mathematical definition of statistical control. Definition: Statistical control We say that a product process is in statistical control with respect to some measurement variable, x, on units 1, 2,...,? if and only if in our judgement p(xv 22,..,i?) = ?(???, xi2,...,xin) for all permutations {ix, ?2,...,in} of units {1, 2,...,n}. That is, the units are This content downloaded from 157.55.39.111 on Wed, 03 Aug 2016 04:54:04 UTC All use subject to http://about.jstor.org/terms

Inspection and repair policies for $(n - r + 1)$-out-of-$n$ systems are compared stochastically with respect to two partial orderings $\geq^{b_1}$ and $\geq^{b_2}$ on the set of permutations of $\{1, 2, \ldots, n\}$. The partial ordering $\geq^{b_1}$ is finer than the partial ordering $\geq^{b_2}$. A given permutation $\pi$ of $\{1,2, \ldots, n\}$ determines the order in which components are visited and inspected. We assume that the reliability of the $i$th independent component is given by $P_i$, where $P_1 \leq P_2 \leq \cdots \leq P_n$. If $\pi$ and $\pi'$ are two permutations such that $\pi \geq^{b_1} \pi'$, then we show that the number of inspections necessary to achieve minimal or complete repair is stochastically smaller with $\pi$ than with $\pi'$. We also consider three policies for minimal repair when the components are each made up of $t$ "parts" assembled in parallel. It is shown that if $\pi \geq^{b_2} \pi'$, then the number of repairs necessary under $\pi$ is stochastically smaller than the number necessary under $\pi'$, but that in general this is not true for the finer ordering $\geq^{b_1}$. The results enable one to make interesting comparisons between various inspection and repair policies, as well as to understand better the relationship between the orderings $\geq^{b_1}$ and $\geq^{b_2}$ on the set of permutations.

Three stopping rules are proposed to detect a change in a normal mean, when the initial value of the mean is unknown but an estimate obtained from a training sample is available. Asymptotic approximations are given for the average run length when there is no change. Under certain hypotheses about the length of time before the change occurs and the magnitude of the change, we obtain asymptotic approximations for the expected delay in detection in terms of the corresponding expected delay in the much simpler case of a known initial value. The results of a Monte Carlo experiment supplement our asymptotic theory to yield some general conclusions about the relative merits of the three stopping rules and guidelines for choosing the size of the training sample.

The average run length (ARL) is a key variable for assessing the properties of process control procedures. For continuous sampling procedures that are based on finite weighted sums (such as the moving sum technique) closed form expressions of the ARL are not available in the literature. For normally distributed random variables, Lai gives bounds for the ARL. In this paper we derive a lower bound of the ARL that (1) does not depend on normality and (2) in many situations is much sharper than the one obtained by Lai. Our results also imply that Lai's upper bound deviates from the true value less than the number of terms in the sum. Furthermore, we show that the applicability of Lai's bounds is not restricted to normally distributed control variables.

Suppose one accumulates independent observations from a certain process. Initially, the process is at State 0. At some unknown point in time something occurs (e.g., a breakdown ) which puts the process in State 1, and consequently the stochastic behavior of the observations changes. It is of interest to declare that a change took place (to raise an alarm) as soon as possible after its occurrence, subject to a restriction on the rate of false detections. It is assumed that the aforementioned observations are the only information one has about the process, and the problem is to construct a good detection scheme. Practical examples of this problem arise in areas such as health, quality control, ecological monitoring, etc. For instance, consider surveillance for congenital malformations in newborn infants. Under normal circumstances, the percentage of babies born with a certain type of malformation has a known value. Should something occur (such as an environmental change, the introduction of a new drug to the market, etc.) the percentage may increase. One would want to raise an alarm as quickly as possible after a change would have taken place, subject to an acceptable rate of false alarms.

The Quality Measurement Plan (QMP) is a new kind of control chart based on a hierarchical Bayes model? QMP has been successfully implemented at ATT but with probability P, the defect changes to a random Primal State. We present a recursive adaptive filter for this model and make comparisons to QMP. A fascinating result is that for a very large set of real control chart data, QMP forecasts as well as the Primal State model. This indicates that QMP is good enough for practical purposes .