CUSUM Procedure Research Articles

Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues

For a sequence ξ1, ξ2, · ·· of i.i.d. random variables let X0 = 0 and Xk = max(Xk–1 + ξ k, 0) for k = 1, 2, ···. Let . These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties of Nx are derived.

Multivariate CUSUM Quality-Control Procedures

It is a common practice to use, simultaneously, several one-sided or two-sided CUSUM procedures of the type proposed by Page (1954). In this article, this method of control is considered to be a single multivariate CUSUM (MCUSUM) procedure. Methods are given for approximating parameters of the distribution of the minimum of the run lengths of the univariate CUSUM charts. Using a new method of comparing multivariate control charts, it is shown that an MCUSUM procedure is often preferable to Hotelling's TZ procedure for the case in which the quality characteristics are bivariate normal random variables.

Comments on zacks' recent paper on one-sided cusum procedures

Comments on zacks' recent paper on one-sided cusum procedures

The Distribution of the Run Length of One-Sided CUSUM Procedures for Continuous Random Variables

A method is given for approximating the moments and percentage points of the run length distribution of one-sided CUSUM procedures for continuous random variables. The run length probabilities are calculated recursively using numerical quadrature until the ratio of successive probabilities stabilizes. This ratio and the probabilities of low run length values are then used to approximate parameters of the run length distribution. The accuracy of this method is compared with that of previous methods in examples involving normally distributed random variables.

The Distribution of the Run Length of One-Sided CUSUM Procedures for Continuous Random Variables

A method is given for approximating the moments and percentage points of the run length distribution of one-sided CUSUM procedures for continuous random variables. The run length probabilities are calculated recursively using numerical quadrature until the ratio of successive probabilities stabilizes. This ratio and the probabilities of low run length values are then used to approximate parameters of the run length distribution. The accuracy of this method is compared with that of previous methods in examples involving normally distributed random variables.

On Sequential Detection of a Shift in the Probability of a Rare Event

Abstract Suppose one is monitoring a sequence of observations for a possible increase in the probability of a rare event, and that it is not possible to immediately stop the process under observation or influence it to return to its normal state. One would then desire a scheme which takes advantage of observations occurring after a detection of a change is proclaimed. A modification of Page's CUSUM Procedure is developed, taking account of these additional observations. A table is given enabling one to select a modified Page procedure with a specified rate of false alarms. A comparison is made between the modified Page procedure and a procedure proposed by Chen (1978).

On Sequential Detection of a Shift in the Probability of a Rare Event

Abstract Suppose one is monitoring a sequence of observations for a possible increase in the probability of a rare event, and that it is not possible to immediately stop the process under observation or influence it to return to its normal state. One would then desire a scheme which takes advantage of observations occurring after a detection of a change is proclaimed. A modification of Page's CUSUM Procedure is developed, taking account of these additional observations. A table is given enabling one to select a modified Page procedure with a specified rate of false alarms. A comparison is made between the modified Page procedure and a procedure proposed by Chen (1978).

A Cusum for a Scale Parameter

Standard cusum procedures are available for controlling the mean of a process. However, in many industrial applications it is important to control the process variability as well. This article presents a technique for employing the same cusum procedure used for the mean for controlling the variance. The technique is based on cusumming |Xi/σ|1/2, which is very nearly normally distributed if X is N(O, σ2). If the process has a nonzero unknown mean, then the same technique may be applied with Xi replaced by the difference between successive observations.

The probability distribution and the expected value of a stopping variable associated with one-sided cusum procedures for non-negative integer valued random variables

The structure of a stopping variable N based on one-sided CUSUM procedures is analyzed. Stopping occurs when a Markovian sequence of maxima of partial sums {M } crosses a certain boundary. On the basis of a recursive relationship between the Mn+1 and Mn a recursive equation is derived for the determination of the defective distributions Kn(x) = P{M ≤ x, N ≤n} . This recursive equation yields a recursive algorithm for the determination of P {N > n} . The paper studies the case when the basic random variables are non-negative integers-valued. In these cases the values of P{N > n} and E{N} can be determined by solving proper systems of linear equations.

United States Department of Agriculture Cusum Acceptance Sampling Procedures

A set of USDA acceptance sampling plans based on cumulative sum (cusum) procedures is described. The plans, for inspection by attributes, are indexed by AQL's and apply to continuous production grouped in lots or batches. They are simple to apply and provide better discrimination than single sampling plans. The plans differ from published cusum procedures in that starting values are used and the cusum value is not reset to zero after failure.

Some first passage problems related to cusum procedures

Let X 1, X 2,… be independent random variables, and set W n = max(0, W n-1 + X n ), W 0 = 0, n ⩾ 1. The so-called cusum (cumulative sum) procedure uses the first passage time T( h) = inf{ n ⩾ 1: W n ⩾ h}for detecting changes in the mean μ of the process. It is shown that lim h →∞ μET( h)/ h = 1 if μ > 0. Also, a cusum procedure for detecting changes in the normal mean is derived when the variance is unknown. An asymptotic approximation to the average run length is given.

Wald's approximations to the average run length in cusum procedures

Wald's approximation to the ARL(average run length in cusum) (cumulative sum) procedures are given for an exponential family of densities. From these approximations it is shown that Page's (1954) cusum procedure is (in a sense) identical with a cusum procedure defined in terms of likelihood ratios. Moreover, these approximations are improved by estimating the excess over the boundary and their closeness is examined by numerical comparisons with some exact results. Some examples are also given.