On sequential versions of the generalized likelihood ratio test

I t is shown that the Wilks large sample likelihood ratio statistic An, for testing between composite hypotheses 0O <= Q1 on the basis of a sample of size n, behaves as n varies like a diffusion process related to an equilibrium OrnsteinUhlenbeck process, whenever the null hypothesis is true. This fact is used to construct large sample sequential tests based on An, which are the same whatever the underlying distributions. In particular, the underlying distributions need not belong to an exponential family. The classical weak convergence theory of partial sums of independent identically distributed random variables concentrates on the convergence of Slnt]/o-*Jn to the Wiener process. In some respects, this is a rather unnatural way of formulating the result, since one is more usually interested in the behaviour of the sequence Sn/o~Jn than that of Slnt]/a,Jn as t varies. A functional limit theorem for Sn/^n is, however, easy to deduce from the classical theorem, and can be expressed in many ways, one of which is as follows. Let (Xn)n>1 be a sequence of independent identically distributed random variables with mean zero and variance a, and set Sn = 2TM=i -̂ yLet Z)[0, oo) be the space of all right continuous functions with left limits on [0, oo), and D' the subspace consisting of those functions x which also satisfy supt>0 \x(t)\/Jlog(t v 3) < oo. Define a metric m' on D' by taking m'(x, y) to be the infimum of those e > 0 for which there exists a continuous and strictly increasing real function A with A(0) = 0 such that supt>0 MO -y(A(<))|/Vlog (t V 3) < e and suPf+s>0 |log[(A(*)-A(«))/(*-«)]| < e. Finally, set YN(u) = SlNeu]/cr^j(Ne ) for each u > 0. THEOREM 1. As N -> co, YN=> Y in (D',m'), where Y is an equilibrium OrnsteinUhlenbeck diffusion process with drift coefficient a(x) = — x/2 and infinitesimal variance fi(x) = 1. 86 A N D R E W D. B A E B O U R Proof. The theorem follows from the analogue in D[0, oo) of Theorem 2 of Mtiller (6), using the fact that the map 0: D2 -» D': (x) (u) = e~"' x(e) is continuous. Here, D2 is the subspace of D[0, oo) consisting of those functions for which l i m s u p ^ \x(t)\/*J(tloglogt) < oo, endowed with the appropriate metric. | The description of (Sn/a*Jn)n>1 given by Theorem 1 is rather interesting. Y is a strictly stationary process, with Y(t) distributed as a standard normal random variable. However, because YN(u) corresponds to Sn/a^jn with n = Ne , the sequence {Sn/o-yjn)n>N behaves like [Y(log(n/N))]n>N, so that, as measured in 7i-time, the fluctuations get progressively slower as n increases. Qualitatively, the picture is attractive: quantitatively, however, the Ornstein-Uhlenbeck process is somewhat more difficult to work with than the Wiener process. As an illustration of the use of such a description of (Sn/crjn), consider the generalized likelihood ratio statistic An = suY>ge&iLn;(XM; 0)/sup,60o£„(*<»>; d), where X^ is a sample of n independent identically distributed random vectors (Xi)f=1 from a distribution on R with density f(x;6), and where 0O <= Qv Wilks' theorem states that, under suitable regularity conditions, 2 log An is approximately distributed as x% a s n gets large, where d = t1 —10 = dim 0X — dim 0O, whenever 0*, the true value of 6, belongs to 0O; and is stochastically larger when 6* e @i\@0. This result is used to provide a widely useful fixed sample test of Ho: 6*e 0O against H±: d*e 0X\ 0O. What about sequential analogues ? The main line of development stems from a paper by Schwarz(7). The densities f(x; 6) are assumed to come from an exponential family, so that the log-likelihood depends on X only through Sn = STM=1 X;-. Sequential tests are then derived, by analogy with the usual sequential probability ratio test, by stopping when Sn first reaches an appropriate boundary. There are, however, difficulties with this procedure when testing contiguous or nested hypotheses, since, for certain values of the alternative, the average sample number can become large, as observed, for example, by Bechhofer (2). A more natural approach is to consider tests of the form ' stop when 2 log An gets large'. This has been suggested, for instance, by Armitage (l), and some large deviation results associated with such a test, again in the context of exponential families, have been derived by Woodroofe(9). Here, we start by showing that, under local regularity conditions on the family of densities of the sort given in Cramer(4), Chapter 33.3, there is a process Uid) which approximates the sequence (2 log An)n>N as N gets large whenever 0* e 0O, and which does not depend on / . Large sample tests can then be constructed, and significance levels tabulated, by using the properties of £7(d). In proving convergence to U(d), we essentially follow the asymptotic theory developed in detail for fixed sample tests (albeit in the more general context of Markov chains) in Billingsley (3), noting such modifications as are required. The main assumptions are that, for any i, j and k, (1) For almost all x, the derivatives ft(x; 6), /^(x; 6) and fijk(x; 6) exist and are continuous in 6 throughout Qv where f{ denotes df/8dit etc. On the generalized likelihood ratio test 87 (2) For any 6 e Qlt there exists a neighbourhood N of 6 such that I = N \fi(> 9')\dx < CO,