Inferences for the Parameters

It would be possible to estimate 6 and cr jointly via Eq. (6.6-1) however, this is not the usual procedure. The standard procedure is to integrate Eq. (6.1-13) over the permitted range of cr, thus obtaining the marginal posterior density function 

Introduction of a new variable. (j = / 2a ), into the latter integral gives 

The marginal posterior density for a Ua-parameter subset 6a of 6e is obtained by integrating Eq. (6.6-4) over any other estimated parameters, as described in Problem 6.E. When the ranges of the parameters are unrestricted. one obtains 

Each of these density functions has its maximum at the least-squares values of its included parameters. The resulting distributions of parameters are known as t-distributions. Such distributions were first investigated by Gossett (1908) for singleresponse problems of quality control at the Guinness brewery in Dublin. The covariance matrix of the estimated parameter vector 6e in Eq. (6.6-4) is 

GREGPLUS for all members of the estimated parameter set de- Several examples of these are given in Appendix C. 

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