Analysis of means based on Rayleigh variate

Introduction

In classical statistical inference confidence for unknown parameters of a statistical population is an inferential procedure. An application of confidence interval usually the 99.73% confidence interval when the variate follows normal distribution is the origin of the well knownshewart control charts. Construction of control charts using the theory of confidence intervals when the variate follows inverse Gaussian distribution is considered by Edgemen (1989) [1] . The justification Edgemen (1989) [1] is that the appeal to the central limit theorem for non-normal process control charts is not possible as sample size in the control chart analysis is usually 10 or less. Moreover quality variate such as product life is often better modeled by the probability distribution of a normal distribution. If normal distribution is considered as a central model for any classical inferential procedure similar place can be attributed to exponential distribution in life testing and reliability studies. It is the only model exemplifying constant failure rate of a product which has life and eventual failure or death. On the other hand we know that ageing of the product is a primary criterion that contributes to its failure. Here models that represent aging phenomenon of as product are also equally desirable in problems of quality control and reliability. Among many such models Rayleigh distribution is an ageing model also called an increasing failure rate model. From a different version is generalization of weibull distribution with shape parameter 2 also. Kantam and SriRam (2001) [4] developed control charts to be used when the process variate follows Rayleigh distribution. Similarly study is made in Kantam et al. (2006). Kantam et al. (2012) [16] developed ANOM procedure for exponential and gamma variate. In this paper we made an attempt to study the ANOM procedure representing a quality characteristic modeled by Rayleigh distribution. The procedure of ANOM is presented in section 2 and respective illustrations are given in section 3 and are compared.

Analysis of Means

The Shewart control chart is as common tool of statistical quality control for many practioners. When these charts indicate the presences of an assignable cause (of non random variability) are adjustment of the process is made if the remedy is known. Otherwise the suspected presence of assignable cause is regarded to be an indication of heterogeneity of the sub group statistic for which the control chart is developed. For instance the statistic is sample mean, this leads to heterogeneity of process mean indicating departure from target mean. Such an analysis is generally carried out with the help of the well-known analysis of variance. Ott (1967) [8] developed a procedure called Analysis of Means (ANOM) to divide a collection of a given number of sub group means into categories such that means within a category are homogeneous and those between categories are heterogeneous, under the assumption that the probability model of the of the variate is normal. In this section we make an attempt to develop the ANOM procedure of Ott (1967) [8] when the data variate is supposed to follow Rayleigh distribution. Suppose 1

x , 3 x ,…….., k x are arithmetic means of k sub groups of size 'n' each drawn from an Rayleigh distribution. If these subgroup means are used to develop control charts to assess whether the population from which these subgroups are drawn is operating admissible quality variations depending on the basic population model we may use the control chart constants developed by us or the popular Shewart constants given in any SQC text book. Generally we say that the process is in control if all the subgroup means fall within the control limits. Otherwise we say the process lacks control. If α is the level of significance of the above decisions we can have the following probability statements.

Using the notation of independent subgroups equation (2.1) becomes

With equitailed probability for each subgroup mean we can find two constants say L * and U * such that

Accordingly, these depend on the subgroup size 'n' and the number of subgroups 'k', We can make use of equation s (2.3) and (2.4)for specified values of 'n' and 'k' in order to get the percentile values called L * ,U * for various values of'α'.

In the case of normal population L * and U * satisfy U * = -L * .For the skewed populations like Rayleigh distribution we have calculated L * , , U * separately from the sampling distribution of i x , these L * and U * calculated for Rayleigh distribution are

A control chart for averages giving "In control" indicates that all the subgroup means though vary among themselves are homogeneous in some sense. This is exactly the null hypothesis in an analysis of variance technique. Hence the constants tables of (2.1), (2.2), (2.3),(2.4), (2.5), (2.6). can be used as an alternative to analysis of variance technique Which are presented at the end.

For a normal population one can use the tables of Ott (1967). For Rayleigh population our Tables can be used. We therefore present below some examples for which the 'goodness of fit' of an appropriate model is assessed with Q-Q plot technique (strength of linearity between observed and theoretical quintiles of a model) and tested homogeneity of means involved in each case are given in table 3.1

Illustrations

Example 1: Wadsworth (1986) [11]: Consider the following data of 25 observations on manufacture of metal products that suspect variations in iron content of raw material supplied by five suppliers. Five ingots were randomly selected from each of the five suppliers. The following table contains the data for the iron determinations on each ingots in percent by weight. The goodness of the of data in these three examples as revealed by Q-Q plot (correlation coefficient) are summarized in the following table, which shows that Rayleigh is a better model, exhibiting significant linear relation between sample and population quantiles. Treating these observations in the data as a single sample, we have calculated the decision limits for the Normal distribution, Rayleigh distribution and HLD and have given them in the following tables respectively.

References

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