17.4 Partial credibility
If it is decided that full credibility is inappropriate, then for competitive reasons (or otherwise) it may be desirable to reflect the past experience in the net premium as well as the externally obtained mean, M. An intuitively appealing method for combining the two quantities is through a weighted average, that is, through the credibility premium
where the credibility factor Z [0, 1] needs to be chosen. There are many formulas for Z that have been suggested in the actuarial literature, usually justified on intuitive rather than theoretical grounds. (We remark that Mowbray  considered full but not partial credibility.) One important choice is
where k needs to be determined. This particular choice will be shown to be theoretically justified on the basis of a statistical model to be presented in Chapter 18. Another choice, based on the same idea as full credibility (and including the full-credibility case Z = 1), is now discussed.
A variety of arguments have been used for developing the value of Z, many of which lead to the same answer. All ...