Tables for wald tests for the mean of a normal distribution
About: This article is published in Biometrika.The article was published on 1959-06-01. It has received 19 citations till now. The article focuses on the topics: Binomial proportion confidence interval & Wald test.
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TL;DR: In this paper, the operation of a cumulative sum control scheme is regarded as forming a Markov chain and the transition probability matrix for this chain is obtained and then the properties of this matrix used to determine not only the average run lengths for the scheme, but also moments and percentage points of the run-length distribution and exact probabilities of run length.
Abstract: The classical method of studying a cumulative sum control scheme of the decision interval type has been to regard the scheme as a sequence of sequential tests, to determine the average sample number for these component tests and hence to study the average run length for the scheme. A different approach in which the operation of the scheme is regarded as forming a Markov chain is set out. The transition probability matrix for this chain is obtained and then the properties of this matrix used to determine not only the average run lengths for the scheme, but also moments and percentage points of the run-length distribution and exact probabilities of run length. The method may be used with any discrete distribution and also, as ani accurate approximation, with any continuous distribution for the random variable which is to be controlled. Examples are given for the cases of a Poisson random variable and a normal random variable.
851 citations
TL;DR: In this article, the average run length and the distribution of run length for CUSUM schemes with the fast initial response (FIR) feature were compared to standard CUSUME schemes.
Abstract: The fast initial response (FIR) feature for cumulative sum (CUSUM) quality-control schemes permits a more rapid response to an initial out-of-control situation than does a standard CUSUM quality-control scheme. This feature is especially valuable at start-up or after a CUSUM has given an out-of-control signal. This article presents the average run length and the distribution of run length for CUSUM schemes with the FIR feature and compares FIR CUSUM schemes to standard CUSUM schemes. The comparisons show that if the process starts out in control, the fast initial response feature has little effect; however, if the process mean is not at the desired level, an out-of-control signal will be given faster when the FIR feature is used.
333 citations
TL;DR: In this article, the authors consider the problem of controlling the quality of the output of a continuous manufacturing process where quality will normally be maintained at an acceptable level for long periods of time, but where a change can take place which will result in the process manufacturing material to a quality level which is unacceptable.
Abstract: We consider here the problem of controlling the quality of the output of a continuous manufacturing process where quality will normally be maintained at an acceptable level for long periods of time, but where a change can take place which will result in the process manufacturing material to a quality level which is unacceptable. Our first attempts in this field were to take standard sequential and multiple sampling schemes derived for batch inspection problems and to use theae schemes retrospectively, the first sample being the current sample the second sample being that taken immediately previous to the current sample and so oni. An example of such a scheme is quoted by Barnard (1954). Page (1954) introduced the concept of cumulative sum charts based on Wald sequential schemes and since this original paper we have adopted this principle to give a wide range of inispection schemes with known properties. Barnard (1959) has recently drawn attention again to the cumulative sum chart and has prepared certain empirical methods for deciding when changes have occurred and for estimating the magnitude of these. We describe, in this paper, the methods which we have developed for our own use and which we have now been applying in practice since 1955. The method is entirely systematic, the parameters of a scheme with required properties can be determined easily, and in our experience the technique is very simple to apply either in graphical or tabular form.
221 citations
TL;DR: The average run length of a Cusum chart for controlling a normal mean is calculated by solving the systems of linear equations which approximate the integral equations for the required quantities.
Abstract: The Average Run Length of a Cusum chart for controlling a normal mean is calculated by solving the systems of linear equations which approximate the integral equations for the required quantities. The accuracy of approximation by this method is numerically evaluated and the results are compared with those obtained by other approximate methods. The construction and use of a new nomogram based on the contours of Average Run Lengths La . and Lr drawn in the h√n/σ—|μ – k|√n/σ plane is discussed. Numerical examples are given to illustrate the flexibility and convenience provided by this nomogram in the design of Cusum charts.
115 citations
Cites methods from "Tables for wald tests for the mean ..."
...Some methods of obtaining the Average Run Length by either finding approximate expressions or by numerical techniques have been reported in the literature [1, 3, 7, 12, 13] during the last fifteen years....
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TL;DR: In this paper, the authors describe from first principles the direct calculation of the operating characteristic function, O.C., the probability of accepting the hypothesis θ = θ 0, and the average sample size, A.S.N., required to terminate the test, for any truncated sequential test once the acceptance, rejection, and continuation regions are specified at each stage.
Abstract: This paper describes from first principles the direct calculation of the operating characteristic function, O.C., the probability of accepting the hypothesis θ = θ0, and the average sample size, A.S.N., required to terminate the test, for any truncated sequential test once the acceptance, rejection, and the continuation regions are specified at each stage. What is needed is to regard a sequential test as a step by step random walk, which is a Markov chain. The method is contrasted with Wald's and two examples are included.
81 citations