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**Chapter 2 Generalised multivariate beta type II distributions**

**Introduction**

In this chapter the generalised multivariate beta type II distribution is derived; this distribution stems from monitoring the variance of a normal random variable by taking successive, independent samples of measurements over time. These measurements are independent and identically distributed collected from a N (µ_{0}, σ^{2}) distribution. The mean (µ_{0}) is known and the unknown variance change from σ^{2} to σ^{2}_{1} = λσ^{2} (also unknown) where λ = 1and λ > 0 somewhere between samples κ−1 and κ. Therefore, from sample onwards the process is considered out-of-control.

Section 2.2 gives an overview of the process monitoring problem that leads to the random variables of interest. The joint distribution of these random variables gives rise to a new distribution that can be regarded as a generalised multivariate beta type II distribution. Sections 2.3 and 2.4 focus on the joint and marginal distributions, respectively. Since the moments are needed to investigate the correlation structure between the random variables, this is the only statistical property of the distribution that is focused on in Section 2.5. A shape analysis of the univariate and bivariate distributions as well as the correlation of the charting statistics for the bivariate case is considered in Section 2.6. An example of using the generalised multivariate beta type II distribution to calculate some run-length probabilities is deferred to Chapter 5.

**Problem statement**

The example in Section 1.2 illustrated the use of a Q-chart and it highlighted the im-portance of the joint distribution of the charting statistics once a shift in the process parameter occurred. In this section the random variables of interest are derived when monitoring the variance of normal random variables using a Q-chart.

Let (Y_{i1}, Y_{i2}, . . . , Y_{in}_{i} ), i = 1, 2, . . . represent successive, independent samples of size n_{i} ≥ 1 measurements made on a sequence of items produced in time. Assume that these values are independent and identically distributed having been collected from a N (µ_{0}, σ^{2}) dis-tribution where the parameters µ_{0} and σ^{2} denotes the known process mean and unknown process variance, respectively. Suppose that the unknown process variance has encoun-tered a permanent upward or downward step shift between samples (time periods) κ − 1 and κ with κ > 1 from σ^{2} to σ^{2}_{1} = λσ^{2} (also unknown) where λ =1 and λ > 0. In practice κ and λ would be unknown (but deterministic) values. Take note that other types of shifts, for example a trend or cycle, falls outside the scope of this thesis.

Since the process variance σ^{2} is unknown, the first sample is used to obtain an initial estimate of σ^{2}. It is assumed that the process starts in-control. This is an important assumption because the data is used to compute estimates of the unknown process para-meter which will in turn be incorporated in the charting statistic that is used to determine if the process is in-control. This initial estimate is continuously updated using the new incoming samples as they are being collected, as long as the estimated value of σ^{2} does not change, i.e.

where Sr^{2}^{pooled} denotes the pooled sample variance of all the measurements up to and including sample r; and S_{i}^{2} denote the variance of the i^{th} sample with µ_{0} representing the known population mean. According to (2.1), i denotes time and r is a specific point in time. Take note that a sample can even consist of an individual observation because the process mean is assumed to be known and the variance of the sample can still be calculated as S_{i}^{2} = (Y_{i1} − µ_{0})^{2} for i = 1, 2, . . .. The sequential sample quantity in (2.1) is computed as each new sample becomes available. Therefore, the variance of the first sample, i.e. S_{1}^{2}, estimates σ^{2} at sample one. At sample two S_{2}^{2} is compared to S_{1}^{2} to determine whether the value of σ^{2} is still the same. If the hypothesis of equal variances cannot be rejected, i.e. the charting statistic plotted inside the control limits, a new updated estimate of σ^{2} is obtained. The updated estimate is S_{2}^{2}^{pooled} and includes information from samples one and two; this point estimate is then used to check if the value of σ^{2} is still the same at sample three by comparing S_{3}^{2} to S_{2}^{2}^{pooled} . This sequential updating-and-testing procedure continues until a change is detected in the value of σ^{2}. A change in the variance will be detected when a charting statistic plots on or outside the control limits.

The control chart and the charting statistic are based on the in-control distribution of the process, in other words they are derived under the null hypothesis of no change in the process variance. The two sample test statistic for testing the hypothesis at time i = r that the two independent samples (the measurements of the r^{th} sample alone and the measurements of the first r −1 samples combined) are from normal distributions with the same unknown variance σ^{2} (see Bain and Engelhardt, 1992 [3], p.402), is based on the statistic

The focus will be on the part where the process is out-of-control, i.e. encountered a shift. The exact distribution of the charting statistic is then unknown because after a change in the process parameter occurred, the charting statistics are no longer independent. Essentially this means that the distribution of the statistic (2.2) is investigated under the alternative hypothesis that the process is out-of-control. To simplify the notation used in expression (2.2), following a change in the process variance between samples κ − 1 and κ, define the random variable

Take note that this is an on-going process and even though j could theoretically go up to infinity, it will be restricted to p in this thesis when defining the random variables and determining the joint distribution. This will serve as a cut-oﬀ point for the monitoring of the process; therefore only the random variables from immediately after the change in the variance took place up to p samples afterwards will be considered. However, there is not a restriction placed on the value of p. The properties of the multivariate and marginal distributions as p tends to infinity fall outside the scope of this thesis.

To simplify matters going forward and for notational purposes the factors, ^{κ−1 }ni/nκ i=1 κ+j−1 and _{i=1} n_{i}/n_{κ+j} in (2.4) and (2.6) are omitted and the (∗) superscript dropped, since the factors are deterministic and can therefore be incorporated in the control limits when they are calculated (see Chapter 5, Section 5.1.1).

the shift in the variance occurred. The random variable W_{j} for j = 0, 1, 2, . . . , p relate to the samples after the change in the variance occurred, with degrees of freedom denoted by v_{j} = n_{κ+j} which represents the sample size of sample κ + j. Take note that j = 0 corresponds to sample κ, j = 1 to sample κ + 1, and so forth. The parameter λ is deterministic and represent the ratio of the new variance (after the change) with respect σ^{2 }to the previous variance (before the change), i.e. λ = _{σ}1_{2} and indicates the unknown size of the shift in the variance. The information regarding the time from which the shift occurred (κ), is contained in the degrees of freedom of the random variable X; if samples of equal size (n) are taken at equally spaced time periods the degrees of freedom of X reduces to (κ − 1) × n.

Remark 2.3 Consider the case where independent samples each of size n ≥ 1 measure-ments are made on a sequence of items where these values are independent and identi-cally distributed having been collected from an exponential distribution with a parameter θ as the unknown process mean. Suppose that from sample κ the process parameter has changed from θ to θ_{1} = λθ where λ = 1and λ > 0. The two sample test statistic for the Q-chart for comparing the means follows from the likelihood ratio test and is based on the ratio of the mean for a specific sample to the overall mean of all the preceding samples (see Bain and Engelhardt, 1992 [3], p.418). The resulting random variables

**The generalised multivariate beta type II distribution**

In this section the joint probability density function (pdf) of U_{0}, U_{1}, . . . , U_{p} (see (2.7)) is derived. This distribution is unknown and is important for studying the probabilistic properties and performance of the control chart, for example determining the run-length probabilities.

Remark 2.4 The joint distribution of U_{0},U_{1}, … ,U_{p} (see (2.7)) derived in Theorem 2.1 gives rise to a new distribution that can be regarded as the generalised multivariate beta type II distribution. The random variables in (2.7) are constructed from independent chi-squared random variables using the variables-in-common (or trivariate reduction) tech-nique and the resulting distribution is defined on the positive domain. A literature overview of relevant multivariate beta type II distributions is briefly discussed in order to contextualise the new distribution (2.8). Tiao and Guttman (1965) [45] obtained a multivariate analogue of the beta type II distribution (see (B.29)) by performing the appro-priate transformation on the Dirichlet distribution. Note that the beta type II distribution is also referred to as the inverted beta distribution or the betaprime distribution. They also considered an alternative development through stochastic representation. Suppose that X, W_{j} with j = 0, 1, 2, … ,p are independent chi-squared random variables as mentioned in Theorem 2.1, but with degrees of freedom 2a and respectively.

1 INTRODUCTION

1.1 Background

1.2 Example

1.3 Objectives and scope

2 GENERALISED MULTIVARIATE BETA TYPE II DISTRIBUTIONS

2.1 Introduction

2.2 Problem statement

2.3 The generalised multivariate beta type II distribution

2.4 Marginal generalised beta type II distributions

2.4.1 Generalised univariate beta type II distribution

2.4.2 Generalised bivariate beta type II distribution

2.4.3 Distribution of a subset

2.5 Product moments of the random variables

2.6 Shape analysis and correlation

2.7 Conclusion

3 NONCENTRAL GENERALISED MULTIVARIATE BETA TYPE II DISTRIBUTIONS

3.1 Introduction

3.2 Problem statement

3.3 The noncentral generalised trivariate beta type II distribution

3.4 The noncentral generalised multivariate beta type II distribution

3.5 Shape analysis

3.6 Conclusion

4 GENERALISED BIMATRIX VARIATE BETA TYPE II DISTRIBUTIONS

4.1 Introduction

4.2 Problem statement

4.3 The generalised bimatrix variate beta type II distributions

4.3.1 The covariance structure change from to

4.3.1.1 The probability density function

4.3.1.2 Marginal probability density functions

4.3.1.3 Product moment of the determinants .

4.3.1.4 Distributions of |U0|, |U1| and |U0U1|

4.3.2 The covariance structure change from to 1

4.3.2.2 Marginal probability density function

4.3.2.3 Product moment of the determinants .

4.3.2.4 Distributions of |U0|, |U1| and |U0U1|

4.3.2.5 Distribution of (|U0| , |U1|)

4.4 Conclusion

5 ILLUSTRATIVE EXAMPLES

5.1 Run-length probabilities if the unknown process variance has changed

5.1.1 The generalised distribution approach

5.1.2 The simulation approach

5.1.3 Assuming independence

5.1.4 A conditioning approach

5.2 Run-length probabilities if the unknown process variance and known process mean have changed

5.3 Run-length probabilities in the matrix environment

5.4 Conclusion

6 CONCLUSIONS

APPENDICES

BIBLIOGRAPHY

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