Tests of Separate Families of Hypotheses

So far we have considered testing or choosing hypotheses on parameters within one family of models or likelihood functions. The procedures dis­cussed in the preceding sections cannot be used to choose between two entirely different likelihood functions, say Ly(0) and Lg(y). For example, this case arises when we must decide whether a particular sample comes from a lognor­mal or a gamma population. Such models, which do not belong to a single parametric family of models, are called nonnested models.

Suppose we want to test the null hypothesis Lf against the alternative Lt. Cox (1961, 1962) proposed the test statistic

Rf= log Lf{Q) – [Ee log Ьу{в)]в – log Le(y) (4.5.31)

+ [Eg log Lg(ye)]g,

where ye = plime у (meaning the probability limit is taken assuming L/0) is the true model) and “ indicates maximum likelihood estimates. We are to accept Lf if Rf is larger than a critical value determined by the asymptotic distribution of Rf. Cox proved that Rf is asymptotically normal with zero mean and variance equal to E(vj) — E(vfWf)(EwfWf)~lE(VfWf), where vf= log Lf{ в) – log Lg(ye) – Eg [log Lf{ в) – log Lg(ye)] and wf=d log Lf{ в)/ дв. Amemiya (1973b) has presented a more rigorous derivation of the asymp­totic distribution of Rf.

A weakness of Cox’s test is its inherent asymmetry; the test of Zy-against Lg based on Rf may contradict the test of Lg against Ly based on the analogous test statistic Rg. For example, Lf may be rejected by Ry-and at the same time Lg may be rejected by Rg.

Pesaran (1982) compared the power of the Cox test and other related tests. For other recent references on the subject of nonnested models in general, see White (1983).

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