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Given Xj,...,xn, the proposed QB estimate of 9 is the mean of the approximating normal density (6.3.40),
T_2 + Z w2^.ft-l)
T~2ft)+ Z W,(x,-, ft _ ! )X,
i = 1
which can be expressed in the recursive form
ft,+ i = ft, ~ ft,V(ftn *„ +1),
T ~ + Z w2U|.ft-l)
/(ftp xn + 1 ) — ftiW2(x„ + 1 > ft) ~ *„+ 1 W1 (-xn + 1 > fti)-
Under the assumption that 9 ^ Af, for some specified A/, using Theorem 6.3.1, we can establish that if
?( A) = ? nial
K(/4)= ? TT^2 - ?2(/l),
then the relative asymptotic efficiency of ft, defined by (6.3.44) to (6.3.46) is given
Sequential problems and procedures
V(A) + E2(A) 1(0)
V(A) + E2(A) 1(0)
which reduces in the bipolar signal detection case to
where, as usual, [/(0)]_l denotes the Cramer-Rao lower bound for a single observation from the appropriate mixture density. Comparing this with the efficiency which is obtained in the signal versus noise case (6.2.23), we note that the presence of the factor n, renders the latter less efficient. The reason for this lies in the structure of the underlying mixture density. In the case of signal versus noise, information about 0 is obtained only from a proportion nl of the observations; in the bipolar case all observations contribute information on 0. Clearly, as 7r, —? 1 the efficiencies become identical.
6.3.5 Problems with several unknown parameters
There appear to be few reports in the literature of sequential problems of case B type involving several unknown 0 parameters; see, however, Gregg and Hancock (1968), Patrick and Costello (1968), Patrick, Costello, and Monds (1970), and Farjo and Young (1976). In any case, the multiparameter extensions of the kinds of recursive algorithms we have been considering tend to apply equally well to case C. We therefore present our discussion of such algorithms in the next section.
6.4 APPROXIMATE SOLUTIONS: UNKNOWN MIXING AND COMPONENT PARAMETERS
6.4.1 A review of some pragmatic approaches
As we remarked in Section 6.3.5, there are relatively few detailed studies of case B problems involving several 0 parameters. This remark applies with even more force in case C, where the combination of unknown mixing weights, n, and component density parameters, 0, inevitably leads to a highly parametrized form, which, in the mixture context, is inherently far more complex, mathematically, than the one- or two-parameter cases on which we have tended to concentrate for case A and case B. For this reason, there are only a limited number of published theoretical studies of sequential methods for jointly estimating n and the unknown parameters in 0. From a pragmatic point of view, however, many of the approximate procedures described for cases A and B can be extended in an obvious way to case C. We shall briefly mention a couple of these pragmatic procedures and then turn, in Section 6.4.2, to the theoretical study of a powerful
Statistical analysis offinite mixture distributions
Decision-directed (DD) procedures estimate the unknown parameters by allocating an observation x to one of the underlying densities using a selected decision rule. Thus, if it is decided that x derives from the component density fi(x\0j), the n and 0 parameter estimates are updated on this assumption.
A commonly used decision rule in this context is that based on the maximum a posteriori probability, which acts as if x derives from /,(x|0,) if and only if
j = I
but other decision rules have also been suggested. In Titterington (1976), alternative DD criteria were used in the context of updating a medical databank in the presence of unconfirmed cases. However, as in most other papers dealing with DD procedures, no asymptotic analysis was provided.
We note, however, that properties of various DD procedures were thoroughly investigated in case B by Patrick and Costello (1968) and Patrick, Costello, and Monds (1970). In these papers, a mixture of two normal densities was assumed, with the unknown parameters taken to be the means of the distributions. The DD procedures were shown to be asymptotically biased, with a degree of bias related to the overlap between the two densities. In the light of these results, one should be rather circumspect about extending DD procedures to case C, where their performance is likely to be much worse than for case B, since n is now also assumed unknown.
Katopis and Schwartz (1972) analysed the DD solution to the problem of signal versus noise with unknown signal and noise probabilities (cf. Section 6.3.2). Although no proof of convergence is provided, the paper contains a useful discussion about the conditions under which good asymptotic properties might obtain.
The method of moments (MM) for estimating all the parameters of a mixture of distributions has been examined by a number of authors; see, for example, Rider (1961), Patrick, Carayannopoulos, and Costello (1966), Fu (1968), and our earlier discussion in Section 4.2. In essence, the method consists of solving the r simultaneous equations