Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics

In this article we propose three distribution-free (or nonparametric) statistical quality control charts for monitoring a process center when an in-control target center is not specified. These charts are of the Shewhart-type, the exponentially moving average-type, and the cumulative sum-type. The constructions of the proposed charts require the availability of an initial reference sample taken when the process was operating in-control to calculate an estimator for the unknown in-control target process center. This estimated center is then used in the calculation of signed-rank-like statistics based on grouped observations taken periodically from the process output. As long as the in-control process underlying distribution is continuous and symmetric, the proposed charts have a constant in-control average run length and a constant false alarm rate irrespective of the process underlying distribution. Other advantages of the proposed distribution-free charts include their robustness against outliers and their superior efficiency over the traditional normal-based control charts when applied to processes with moderate- or heavy-tailed underlying distributions, such as the double exponential or the Cauchy distributions.     

A subset selection procedure for multinomial distributions

A subset selection procedure is developed for selecting a subset containing the multinomial population that has the highest value of a certain linear combination of the multinomial cell probabilities; such population is called the ‘best’. The multivariate normal large sample approximation to the multinomial distribution is used to derive expressions for the probability of a correct selection, and for the threshold constant involved in the procedure. The procedure guarantees that the probability of a correct selection is at least at a pre-assigned level. The proposed procedure is an extension of Gupta and Sobel's [14] selection procedure for binomials and of Bakir's [2] restrictive selection procedure for multinomials. One illustration of the procedure concerns population income mobility in four countries: Peru, Russia, South Africa and the USA. Analysis indicates that Russia and Peru fall in the selected subset containing the best population with respect to income mobility from poverty to a higher-income status. The procedure is also applied to data concerning grade distribution for students in a certain freshman class.     

Estimation efficiency of grouped and censored observation schemes for a generalised regression model

A multiple regression model is considered in which the density of the response variable is a member of a very wide family which includes many well-known distributions. Schemes of observation in which the response observations are grouped or type 1 right censored are examined. Results on the asymptotic variance efficiencies of the maximum likelihood estimators of the regression coefficients and standard deviation of the error distribution are presented for the two schemes.     

Analysis of Means Using Ranks

A nonparametric procedure, called the analysis of means using ranks (ANOMR), is proposed for testing the equality of several population means. The ANOMR procedure may be used graphically in the form of a Shewhart control chart and so has the advantage of pinpointing which population mean, if any, is significantly different from the others. Exact and asymptotic critical values are given for the implementation of ANOMR. Results from a Monte Carlo power study are presented which indicate that for light-tailed distributions such as the uniform and the normal, ANOMR is only slightly less powerful than the parametric competitive procedures based on analysis of variance and analysis of means. For heavy-tailed distributions such as the Cauchy, ANOMR is shown to provide greater power than the parametric procedures. The results also indicate that for both light and heavy-tailed distributions the use of the ANOMR test instead of the Kruskal-Wallis test leads to only a small loss of power for a range of alternatives.     

Nonparametric quality control charts based on the sign statistic

Nonparametric control chart are presented for the problem of detecting changes in the process median (or mean), or changes in the process variability when samples are taken at regular time intervals. The proposed procedures are based on sign-test statistics computed for each sample, and are used in Shewhart and cumulative sum control charts. When the process is in control the run length distributions for the proposed nonparametric control charts do not depend on the distribution of the observations. An additional advantage of the non-parametric control charts is that the variance of the process does not need to be established in order to set up a control chart for the mean. Comparisons with the corresponding parametric control charts are presented. It is also shown that curtailed sampling plans can considerably reduce the expected number of observations used in the Shewhart control schemes based on the sign statistic.