962 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



sequence, conditional on starting at n. We use the notation : 



Vn 



A definition of likelihood as Lc has been used by Moran. Clearly 



\nLc = A\na + H\ny - yZ - aT. 



Appendix B 

 unconditional maximum likelihood estimates 



The definition of likelihood as L leads to complicated results which 

 are of theoretical rather than practical interest. For this reason these 

 results have been relegated to an appendix. 



The results of setting d/dy In L and d/da In L equal to zero lead, re- 

 spectively, to the likelihood equations 



a - y{n - H) - y'Z = 0, 



yn — a + 7-i — ayT = 0. 



Considered as a system of equations for y and a, this pair has the non- 

 negative roots 



H - n - M + {{H - n - Mf + 4MK}''' 



7 = 



a = ^ - W. 



2Z 



These are the unconditional maximum likelihood estimators for y and a. 

 Although dc depended only on A and T, and 7c only on H and Z, the 

 unconditional estimators depend on all of n, A, H, Z, and T. We may 

 obtain a maximum unconditional likelihood estimator for b as well, 

 either by considering L to be a function of b and 7, or from general 

 properties of maximiun likelihood estimators. Since b = a/y, we expect 



that b = d/y, as can be verified by an argument similar to that used 

 above for a and 7 . 



The estimators d, b, and 7 obtained in this Appendix may turn out to 

 be useful in practice, but their complicated dependence on the sufficient 

 statistics n, A, H, and Z makes a study of their statistical properties 

 difficult. As a first step along such a study, we have derived the gen- 

 erating function of the joint distribution of the sufficient statistics in 

 Appendix C. The greater simplicity of the conditional estimators of 

 Section VI makes it possible to study their statistical properties. This 



