The likelihood ratio test for non-standard hypotheses near the boundary of the null with application to the assessment of non-inferiority

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1 Statistics & Decisions 7, 75 9 (009) / DOI 0.54/stnd c Oldenbourg Wissenschaftsverlag, München 009 The likelihood ratio test for non-standard hypotheses near the boundary of the null with application to the assessment of non-inferiority Fadoua Balabdaoui, Matthias Mielke, Axel Munk Received: February 6, 009; Accepted: September, 009 Summary: We consider a class of testing problems where the null space is the union of k subgraphs of the form h j (θ j ) θ k, with j =,...,k, (θ,...,θ k ) the unknown parameter, and h j given increasing functions. The data consist of k independent samples, assumed to be drawn from a distribution with parameter θ j, j =,...,k, respectively. An important class of examples covered by this setting is that of non-inferiority hypotheses, which have recently become important in the evaluation of drugs or therapies. When the true parameter approaches the boundary at a / n rate, we give the explicit form of the asymptotic distribution of the log-likelihood ratio statistic. This extends previous work on the distribution of likelihood ratio statistics to local alternatives. We consider the prominent example of binomial data and illustrate the theory for k = and 3 samples. We explain how this can be used for planning a non-inferiority trial. To this end we calculate the optimal sample ratios yielding the maximal power in a binomial non-inferiority trial. Introduction Consider the testing problem H : θ H versus K : θ K where H and K are disjoint subsets of R k, k N \{0}.LetX,...,X n be i.i.d. from an unknown density function f(x,θ), θ. The classical asymptotic theory of the log-likelihood statistic goes back to Wilks (938) who showed that the limiting distribution of the log-likelihood statistic is a central chi-square of k r degrees of freedom when H and K are hyperplanes of dimensions r and k r, respectively. In the case where H and K are of the same dimension and can be approximated respectively by cones C and C at θ 0, a point on the boundary of H and K but in the interior of, Chernoff (954) showed that the log-likelihood ratio AMS 000 subject classification: Primary: 6E0, 6F03, 6H5; Secondary: 6P0 Key words and phrases: Likelihood ratio test, optimal allocation of samples, local asymptotic, non-inferiority trials, cone hypotheses

2 76 Balabdaoui -- Mielke -- Munk converges to inf θ C (Z θ) T J(Z θ) inf θ C (Z θ) T J(Z θ), where J is the Fisher information matrix at θ 0,andZ N (θ 0, J ). The work of Chernoff was generalized by Self and Liang (987) to the case where θ 0 is allowed to be on the boundary of. For further extensions see also Liang and Self (996) and Vu and Zhou (997). Investigations of the distribution of the log-likelihood ratio under local alternatives have a long history, i.e. the true state of nature is a sequence (θ n ) n N converging to θ 0 on the boundary of H.For H a hyperplane this dates back to Wald (943) and Davidson and Lever (970), whereas the latter proved Wald s results under less restrictive assumptions. More recently, van der Vaart (998) shows local asymptotic normality under the general assumption that the model is differentiable in quadratic mean. This assumption is weaker than the originally used conditions on the second or even third derivative of the log likelihood. As an important generalization extending Chernoff s result Feder (968) obtains the asymptotic distribution of the log-likelihood ratio where both the hypothesis and alternative regions are approximable by cones C and C at the boundary point θ 0, where he considers different rates of convergence of (θ n ) n N.The case which is of much interest to us in the testing problems considered here is his Case, where (θ n ) n N converges at the rate of n /.Ifδ jn = n(θ n ξ jn ) and F n is the distribution of the log-likelihood ratio, then it follows from Theorem of Feder (968) that lim n d L (F n, G n ) = 0whereG n is the distribution function of inf (Z + δ n θ) T J(Z + δ n θ) inf (Z + δ n θ) T J(Z + δ n θ) θ C θ C and d L is the Lévy distance. The convergenceis uniform in θ n such that δ jn c, j =, for some c > 0. It should be noted that the asymptotic distribution above corresponds to the following definition of the likelihood ratio λ n = sup θ H nj= f(x j,θ) sup θ K nj= f(x j,θ). In this paper, we consider an equivalent definition of the likelihood ratio where K is replaced by the whole parameter space ; that is, we take the likelihood ratio to be λ n = sup θ H nj= f(x j,θ) sup θ nj= f(x j,θ). As customary the corresponding likelihood ratio test (LRT) for H versus K is given through the rejection region {λ n c}, where the critical value c depends on the distribution (or on an approximation of the distribution) of the likelihood ratio λ n under H and the desired significance level α. In general, the supremum of the likelihood over or H might not exist. However, it can be shown under some regularity assumptions on the model that with probability tending to as n the likelihood attains a local maximum at θ n where

3 LRT for non-standard hypotheses near the boundary of the null 77 n = (n,...,n k ) and θ H n H, respectively. In this case, the likelihood ratio is nj= f ( X j, θ H ) n λ n = nj=. f ( ). X j, θ n We consider the special case where δ jn = δ are independent of n, and will extend the result of Feder (968) when the observations consist of k independent samples X i,..., X ini, i =,...,k, where the sizes n,...,n k are not assumed to be necessarily equal. We assume that sample sizes are asymptotically of the same order, i.e. lim n n i /n = r i ]0, [. Each sample is assumed to be drawn from the density f i = f(,θ i ), i =,...,k, where f is a common and known density. Our work is motivated by the investigations of local power of the LRT for the null hypothesis H 0 : θ k h (θ )... θ k h k (θ k ), (.) where is used in the sense of or and h,...,h k are given increasing functions. See Röhmel and Mansmann (999), Munk et al. (007) or Skipka et al. (006) for a discussion of several choices of h j. This type of hypotheses occurs also in the context of order restricted inference, see e.g. Robertson, Wright and Dykstra (994), Barmi and Dykstra (997) or the special issue on Statistical Inference and Inequality Constraints (00). Assuming that the functions h j are differentiable, it turns out that the asymptotic distribution of the log-likelihood ratio involves a weighted version of the Fisher information matrix, where the weights depend on the first derivatives of h j at the first k components of the parameter at the boundary, and on the sample ratios r,...,r k.the case k = 3 has been considered by Munk et al. (007) (see their hypothesis H U )for the particular case of binomial samples. They showed that, under the null hypothesis, the log-likelihood ratio converges weakly to a random variable which is stochastically greater than (/) : (/) mixture of a Dirac at zero and a central χ. This can be also obtained by replacing δ by zero in the setting of local alternatives. Hypotheses as in (.) arise in many applications, e.g. where the mean response of a certain group is to be compared to those of the remaining groups. This includes dose finding studies, superiority trials or non-inferiority clinical trials where the goal is to show that a new treatment is not worse up to prespecified threshold than the existing treatments. For a good review on the subject, see e.g. Röhmel (998), Jones et al. (996), Pigeot et al. (003), Kieser and Friede (007), the Special Issue on Therapeutic Equivalence Clinical Issues and Statistical Methodology in Noninferiority Trials (005), D Agostino et al. (003), Senn (997), Skipka et al. (004) and Munk et al. (007). W.l.o.g. we assume that the density f of the endpoints of interest is parametrized in a way that smaller values of θ i are desired, e.g. θ i is the rate of unrecovered patients after a certain treatment period. The number of arms in non-inferiority trials is typically k = or 3, depending on whether or not an additional placebo is taken into account. Our approach here is more general than in the classical setting but also different: (a) It allows for comparing a certain treatment to an arbitrary number of other treatments as it is the case in dose finding studies. (b) It does not restrict the analysis to normal or binary distributed endpoints, but rather provides a theory for general parametric families of distributions, see also Mielke et al. (008) for exponentially distributed endpoints and Lui (005) for Poisson distributed

4 78 Balabdaoui -- Mielke -- Munk endpoints. (c) The investigation of the power of the LRT under local alternatives, to the best of our knowledge, has not been done before. The paper is organized as follows. In Section, we fix the notation and the main assumptions, and we give subsequently the asymptotic distribution of the log-likelihood statistic under the local alternatives. The explicit form of this limiting distribution is established in Section.. The proof is deferred to the Appendix. In Section 3, we consider the example of the binomial model, and the special case where h j (θ) = η j θ, j =,. For k = and 3, we apply the main result of this paper to calculate the optimal sample ratios, r,...,r k, yielding the maximal power. It turns out that for the two sample case (k = ) the optimal allocation is given by r = + θ(0) η θ (0), which is always greater than / for the non-inferiority setting (η >) and less than / for substantial superiority (η <). Hence, as opposed to the classical setting, i.e. η =, optimality is achieved for an unbalanced design. For k = 3 and beyond, unfortunately it is no more possible to come up with such closed form for the optimal allocations. However, the results from Section provide the basis for determining these allocations numerically as presented in Section 3. Asymptotic theory. Model and assumptions We consider a multiple sample with k independentgroups. Fori =,...,k, we observe i.i.d. data X i,...,x ini from a common density f i with respect to a dominating σ-finite measure ν. We assume that the densities f i, i =,...,kdepend on a known density f and unknown parameters θ (0) i, where is an open subset of R. Thus, f i ( ) = f(,θ (0) i ) for i =,...,k.letbeθ (0) = (θ (0),...,θ(0) k ) the true unknown parameter of the whole parameter space k. For local asymptotics, the true parameter is allowed to depend on the sample sizes n i, i =,...,k.ifn = (n,...,n k ), we denote by θ n (0) = (θ (0),n,...,θ (0) k,n k ) this parameter. Finally, let n = k i= n i be the total sample size. We denote by L n (θ) = k n i f(x ij,θ i ) i= j= the likelihood function, and the log-likelihood function by l n (θ). For given strictly increasing functions h,...,h k defined on respectively, we will focus in this paper on testing problems where the null space H is given by H ={(θ,...,θ k ) : θ k h (θ ) θ k h k (θ k )}. (.)

5 LRT for non-standard hypotheses near the boundary of the null 79 Here, we focus on local alternatives; i.e., on the case where the true parameter lies almost on the boundary of the null and alternative spaces in the following sense ( (0) ) (0) h j θ j,n j = θ k,n k + δ ( ) j + o n (.) n for j =,...,k whereδ,...,δ k > 0. The o(/ n ) term will be suppressed in the sequel for simplicity and because the theoretical results will not be affected. To be able to establish the asymptotic theory of the log-likelihood ratio, we need to make first some general regularity assumptions on the functions h j and the parametric model itself. In the following, X k denotes the common sample space, and is any arbitrary norm on or k. Finally, for any real vectors x,...,x k R s,...,r s k we denote by diag(x,...,x k ) the diagonal matrix with all elements of x,...,x k on its diagonal and dimension s + +s k. Assumption G. For j =,...,k, the function h j is and continuously differentiable on. Assumption G. We assume that for j =,...,k n j lim n n = r j, ]0, [. This assumption ensures that no group vanishes asymptotically. Let R denote diag(r,...,r k ). Assumption G3. The family f(x, ) is identifiable; i.e. f(x,ϑ)= f(x,ϑ )ν-a.e. x implies ϑ = ϑ for all ϑ, ϑ. Assumption L. ϑ log f(x,ϑ)is twice continuously differentiable on for ν-a.e. x, and its third derivative exists for ν-a.e. x. Let J ϑ denote the Fisher information matrix corresponding to f(,ϑ),and J θ (0) = diag ( ) J θ (0), J θ (0),...,J θ (0) k the Fisher information corresponding to the density of the whole model; that is, ki= f(,θ (0) i ). To be able to derive the limiting distribution of the log-likelihood ratio, we will assume that there exist neighborhoods N i of θ (0) i, and a function F such that for all ϑ N i, i =,...,k: Assumption L. log f(x,ϑ) F(x), T log f(x,ϑ) F(x), and T log f(x,ϑ) F(x), where E (0) θ F(X) <. i Assumption L3. J (0) θ is finite and strictly positive definite, i =,...,k. i

6 80 Balabdaoui -- Mielke -- Munk Remark. (a) L L3 are in the spirit of the classical conditions to obtain asymptotic normality of maximum likelihood estimators. As already mentioned in the introduction they could be replaced by the weaker conditions of differentiability in quadratic mean and that the Fisher information matrices J θ (0) i, i =,...,k are nonsingular (confer the LAN property of van der Vaart (998, Theorem 7.0)). (b) We mention that our analysis can be extended to the case when the densities f i (,θ i ) do not come from the same parametric family f(, ) or that the parameter spaces i, i =,...,k for each sample may differ. However, in order to simplify notation, we restrict to this case.. Asymptotic distribution of the log-likelihood ratio under local alternatives Our goal in this subsection is to give the asymptotic distribution of the log-likelihood ratio under the local alternatives for the hypothesis H 0 in (.). Let and put ( θ (0) = h ( θ (0)) k,...,h k ( θ (0) k ),θ (0) k ( D θ (0) = diag / [ h ( (0)) θ ],...,/[h (0) )] ), k ( θ k. Then we obtain for the asymptotic distribution of the likelihood ratio logλ n D inf ψ C (Z ψ + δ)t RD θ (0) J θ (0)(Z ψ + δ) (.3) where Z = (Z,...,Z k ) N (0, [RD θ (0) J θ (0)] ) and C = {ψ R k : ψ k min{ψ,...,ψ k }}. This result can be easily derived as a direct extension of Feder (968) to the multi-sample case allowing for differentsample sizes in each groupor by the LAN property from van der Vaart (998). In the following, we use the notation r j J (0) θ j α j = [ ( h (0))], (.4) j θ j for j =,...,k andα k = r k J (0) θ. Note that with the above notation we have that k Z i N (0, /α i ), i =,...,k. The next result is our main theorem and gives an explicit representation of the limiting distribution in (.3). Theorem. Let Z m + δ m = min(z + δ,...,z k + δ k ),i.e. m = arg min {Z j + δ j }. j=,...,k Under the assumptions above, we have { 0 if Zk Z m + δ m logλ n D = α m α k α m +α k (Z k Z m δ m ) otherwise. )

7 LRT for non-standard hypotheses near the boundary of the null 8 In other words, the limiting distribution of logλ n is the mixture distribution where and p [0, ) + ( p)f, p = P(Z k < Z + δ,...,z k < Z k + δ k ) ( ) αm α k F (λ) = P (Z k Z m δ m ) λ Z k < Z m + δ m. α m + α k λ Figure. Plot of the conditional densities of the limiting distribution of the LR for binomial samples under the local alternatives for k =, θ (0) = /. Left plot: η =. and different values of the parameter δ: δ = for the solid line, δ = for the dashed line and δ = 4forthe dotted line. Right plot: δ = 4 and and different values of the parameter η: η =. for the solid line, η =.3 for the dashed line, η =.5 for the dotted line. For all plots, the sample ratio r is taken to be optimal (see Section 3..). The asymptotic distribution of the log-likelihood ratio takes the form of a mixture of a Dirac at the origin and random variables which could be thought to follow a mixture of non-central chi-squared distributions. However, this is not the case. If k = andδ = δ, the unconditional distribution of α α (Z Z δ) α + α is indeed that of a non-central chi-squared distribution but this does not hold true when we condition on the event {Z < Z + δ}. We show that, conditionally on that event, the c.d.f. of is given by ( δ /α ) λ +/α ( ) λ 0. (.5) δ /α +/α λ

8 8 Balabdaoui -- Mielke -- Munk The proof is deferred to the Appendix. In addition, the explicit limiting distribution is derived for k = 3. While we have omitted giving its expression here as it is not of a compact form, it is an essential and necessary result for the considerations of optimal allocation of the samples in Section 3. The non-degenerated components of the explicit limiting distributions derived in the Appendix are illustrated in Figure. and. for binomial samples where h j (θ j ) = η j θ j, j =,...,k is considered. The conditional densities are displayed for different choices of η(=.,.3,.5), δ(=,, 4), k = andk = 3, respectively. This setting covers the assessment of non-inferiority based on the relative risk, which will be again considered in Section 3. There, we use Theorem. to obtain the optimal sample allocations yielding the maximal the power of the LRT λ Figure. Plot of the conditional densities of the limiting distribution of the LR for binomial samples under the local alternatives for k = 3, θ (0) = θ (0) = /. Left plot: η =. and different values of the parameter δ: δ = for the solid line, δ = for the dashed line and δ = 4 for the dotted line. Right plot: δ = 4 and and different values of the parameter η: η =. for the solid line, η =.3 for the dashed line, η =.5 for the dotted line. For all plots, the sample ratio r = r is taken to be optimal (see Section 3..). Remark.3 The result of Theorem. can be used to determine the critical region of the likelihood ratio test by simply setting δ j = 0, j =,...,k. For k = e.g., we reject the null hypothesis at the asymptotic level α if logλ n c α where c α is the ( α)-th quantile of the mixture of a Dirac at 0 and a central χ with the mixing probability P (Z Z ) = ; this means that c α is the ( α)-quantile of χ. However, for an arbitrary k, itis less obvious to find a simple relationship between c α and some large quantile of χ. λ

9 LRT for non-standard hypotheses near the boundary of the null 83 If F (0) denotes the c.d.f. of the conditional distribution of given Z k m when δ j = 0, j =,...,k ; i.e., ( ) F (0) (λ) = P αm α k (Z k Z m ) λ Z k < Z m α m + α k and if p (0) = P(Z k > Z m ),thenc α is the quantile determined by the equality F (0) (c α) = α p(0) p (0). (.6) 3 Power calculations Application to the binomial model In Munk et al. (007) the power of three sample non-inferiority likelihood ratio test for binary outcomes has been investigated numerically. The question of optimal allocation of treatments, when the total number of samples is prespecified has not been addressed for this problem. More generally, let r,...,r k be the optimal sample ratios yielding the asymptotically maximal power under local alternatives of the LRT for an arbitrary k. Explicit expressions of the r j, j =,...,k and the corresponding power are not available in general, and hence one has to resort to find numerically good approximations for the relevant quantities, based on Theorem.. For simplicity, we take δ = =δ k = δ. (3.) The critical region of the LRT corresponding to the asymptotic level α is given by logλ n c α, where c α is the ( α)-th quantile of the mixture p (0) [0, ) + ( p (0) )F (0),wherethe expressions of p (0) and F (0) are given above. Now, the asymptotic power is equal to where P = ( p)( F (c α )) (3.) p = P (Z k Z + δ,...,z k Z k + δ) = P (Z k Z m + δ) in view of the assumption (3.). The mixing probability p as well as the distribution F and the quantile c α are functions of r,...,r k as they all depend on /α i, i =,...,k, the variances of Z,...,Z k which involve the sample ratios. Thus, the sample allocations affect the power in (3.), and the aim is to determine the optimal sample allocation at which this power is maximal; i.e., determine (r,...,r k ) = argmax { (r,...,r k ):r i 0, k i= r i = ( p)( F (c α )) }. Binomial model: One important application is the binomial model. In clinical trials, this model is widely used for measuring the efficacy of a certain treatment (see e.g. Hesketh et al. (996) for a application). In the following examples, we consider the case of two

10 84 Balabdaoui -- Mielke -- Munk (k = ) and three (k = 3) groups, and we shall determine the optimal sample ratios. We consider the case where the functions h j are given by h j (θ j ) = η j θ j, j =,...,k, with η j > fixed(seee.g.röhmel and Mansmann (999) or Munk et al. (007) for a discussion of this hypothesis). That is non-inferiority expressed in terms of the relative risk θ k /θ j. In this case, we have For i {,...,k},let In this model, we have that θ (0) j,n j = ( θ (0) k + δn /) /η j, j k. α j = r r j η j σ j σ i = θ (0) i ( (0)) θ. for j =,...,k andα k = r k σk θ (0) Figure 3. Plots of the optimal sample ratio r for k = as a function of θ (0), and for different values of η: η =. for the solid line, η =.3 for the dashed line, and η =.5 for the dotted line. 3. The asymptotic optimal allocations 3.. Two groups (k = ) Put r = r and η = η. The asymptotic power in (3.) specializes to ( δ /α ) c +/α α P = ( p) ( ) δ /α +/α with p = P(Z Z + δ) i

11 LRT for non-standard hypotheses near the boundary of the null 85 and c α is the ( α)-quantile of a central χ. Now, note that if X N (0, ), wehave that ( [ P(Z Z + δ) = P X δ + ] ) /, α α and hence the expression of P takes the simpler form ( δ P = ) c α. (3.3) /α + /α Note that the power is an increasing function of δ. This is expected since the alternative hypothesis is further away from the null for larger values of δ. The power in (3.3) is maximal if is minimal; i.e., if r δ = θ (0) 3 + = η σ α α r δ = + σ r θ (0) δ = Figure 3. Plots of the optimal sample ratio r = r = r as a function of θ(0) 3 for k = 3, θ (0) = θ (0), different values of η: η =. for the solid line, η =.3 for the dashed line, η =.5 for the dotted line and δ = (,, 4). r = σ η σ η + σ = θ (0) θ (0) ( θ (0) ( (0)) θ η ) η + θ (0) ( θ (0) Substituting θ (0) = ηθ (0) yields r = + θ(0) η θ (0). For η =, which corresponds to the classical setting (H : θ θ ) the balanced design (r = /) is optimal. In contrast, the non-inferiority setting with η>yields an unbalanced optimal design with r > /; that is more patients in the first arm have to be enrolled. Note that the aim is to show that the first group is non-inferior to the second one θ (0) 3 ).

12 86 Balabdaoui -- Mielke -- Munk by rejecting the hypothesis H : θ ηθ, whereas smaller values of θ i are desired. The optimal allocation r is displayed in Figure 3. for different choices of η(=.,.3,.5). It is important to note that for k = the optimal allocation of the samples is independent of δ. 3.. Three groups (k = 3) We recall that the asymptotic power is given by where P = ( p)( F (c α )), (3.4) p = P(Z 3 Z + δ, Z 3 Z + δ) and c α is the ( α)-th quantile of the mixing distribution p (0) [0, ) + ( p (0) )F (0). As mentioned above, the probability distribution F (0) is not exactly that of a central χ (c.f. derivation in the Appendix). Although these distributions are not very different, it is not straightforward to find a closed form for the optimal ratios r and r as presented for k =. Nevertheless, the explicit limiting distribution derived in the Appendix based on Theorem. allows to compute the power in (3.4) for a given allocation (r, r, r 3 ) and fixed θ (0), η, δ and therefore the optimal allocation can be determined by numerical as a function of θ (0) 3 for different choices of η(=.,.3,.5) and δ(=,, 4). The plots are based investigations. Assuming that θ (0) = θ (0), Figure 3. shows r = r = r on smoothing out the obtained values of r for equally spaced values of θ (0) 3 using cubic spline interpolation. It is important to note that the numerical investigations turn out that, in contrast to the case of k =, the optimal allocation is not independent of δ. 4 Conclusion In this paper, the problem of testing whether the true parameter belongs to the union of k smooth subgraphs based on the log-likelihood ratio statistic and k independent data sets with different sizes is considered. Under local alternatives of order / n,the distribution of the log-likelihood ratio and hence the power are explicitly determined. It turns out that, in this non-standard case, the distribution involves a weighted Fisher information matrix, where the weights depend on the sample ratios. In order to maximize the power of the test, these sample ratios should be optimized. While a closed form of the optimal ratios cannot be given in general, one can always simulate this distributions using Theorem.. This is not feasible for the general form (.3) because this involves a numerically difficult minimization step. Even though the theory of Section is applied to binary endpoints and k = and 3, it also does provide the basis for investigating the optimal allocations for samples with arbitrary sizes and drawn from an arbitrary (common) parametric distribution. Also, the theory should not be confined to the noninferiority setting, even if the latter remains still the most prominent application and has been served as a major motivation for this work. In particular, it would be interesting to extend our findings to the so called retention of effect hypothesis, where the effect of two treatments is investigated relatively to a third one, e.g. placebo as in most of clinical trials.

13 LRT for non-standard hypotheses near the boundary of the null 87 A Appendix Proof of Theorem.: The asymptotic distribution of the likelihood ratio is given by inf (Z + δ ψ)t RD θ (0) J θ (0)(Z + δ ψ). ψ C Putting ν j = ψ k ψ j for j =,...,k in (.3) and ν k = ψ k, we can write (Z + δ ψ) T RD θ (0) J θ (0)(Z + δ ψ) = α (Z + δ (ν k ν )) α k (Z k + δ k (ν k ν k )) + α k (Z k ν k ) where the α j s are defined in (.4). Thus, finding the asymptotic distribution of the likelihood ratio under the local alternatives amounts to solving the minimization problem: where { inf α (Z + δ ν k + ν ) +... η E 0 +α k (Z k + δ k ν k + ν k ) + α k (Z k ν k ) } (A.) E 0 = {η = (ν,...,ν k ) : ν 0 ν k 0,ν k R}. To solve (A.), we first minimize the target function first with respect to ν k for fixed ν,...,ν k. To do this, let us write Z m + δ m = min{z + δ,...,z k + δ k }. Note first that if Z k m, then the infimum is equal to 0 and (Z k Z δ,...,z k Z k δ k, Z k ) is the unique solution to the problem. In the following, we assume that Z k < Z m + δ m. We will start by minimizing the target function in (A.) over the set E + 0,m = {η = (ν,...,ν k ) : ν 0 ν k 0andν k Z m + δ m }. Then, we have inf (ν,...,ν k ) E + 0,m { α (Z + δ ν k + ν ) +... = inf ν k Z m +δ m α k (Z k ν k ) + α k (Z k + δ k ν k + ν k ) + α k (Z k ν k ) } = α k (Z k Z m δ m ) since we assumed that Z k < Z m + δ m. Now, let us consider the complementary set E 0,m = {η = (ν,...,ν k ) : ν 0 ν k 0andν k < Z m + δ m }.

14 88 Balabdaoui -- Mielke -- Munk It is not difficult to see that { α (Z + δ ν k + ν ) +... inf (ν,...,ν k ) E 0,m where + α k (Z k + δ k ν k + ν k ) + α k (Z k ν k ) } = inf ν k <Z m +δ m { αm (Z m + δ m ν k ) + α k (Z k ν k ) } = inf ν k <Z m +δ m { (αm + α k )(ν k ν k ) + α k (ν k Z m δ m ) + α k (Z k ν k )} ν k = α m(z m + δ m ) + α k Z k α m + α k. Depending on the value of νk, there are two cases to consider.. ν k < Z m + δ m : In this case, we have { inf (αm + α k )(ν k νk ν k <Z m +δ ) + α m (νk Z m δ m ) + α k (Z k νk )} m = α m (ν k Z m δ m ) + α k (Z k ν k ) = α k α m (α m + α k ) (Z k Z m + δ m ) + α kα m (α m + α k ) (Z k Z m δ m ) = α mα k α m + α k (Z k Z m δ m ).. ν k m: Note that this is equivalent to Z k Z m + δ m, and this case was ruled out. To summarize the results obtained above, we can write { inf α (Z + δ ν k + ν ) +... ν E 0 + α k (Z k + δ k ν k ν k ) + α k (Z k ν k ) } { 0 if Zk > Z m + δ m = α m α k α m +α k (Z k Z m δ m ) otherwise. Note that in the second case, this follows because α k /(α m + α k )<which implies that α m α k (Z k Z m δ m ) /(α m + α k )<α m (Z k Z m δ m ). Now the limiting distribution of the log-likelihood ratio under the local alternative follows easily from the previous result. Distribution of. In the following, we derive the explicit expression of the probability distribution of for k = and 3 as claimed in (.5). We start with k =. Let X N (0,σ ) where σ = /α + /α. Then, d = (X δ) σ conditionally on {X <δ}

15 LRT for non-standard hypotheses near the boundary of the null 89 Now, ( (X δ) ) P σ λ X <δ = ( ) P (X δ) λ, δ X > 0 σ, σ (δ/σ) and we can write ( (X δ) ) ( P σ λ, δ X > 0 = P 0 δ X ) λ ( σ δ = P σ λ X σ δ ) σ = (δ/σ) (δ/σ λ). Hence, the conditional distribution function of and its p.d.f. are given by [ F(λ) = (δ/σ ] λ) λ>0 (δ/σ) and f(λ) = [ λ φ(δ/σ ] λ) λ>0. (δ/σ) Now, we turn to k = 3. Let X = Z 3 Z and X = Z 3 Z,andσ j = Var(X j ) = /α j + /α 3. Note that Z 3 < min(z + δ, Z + δ) is equivalent to max(x, X )<δ. For λ>0wehavethat ( (max(x, X ) δ) ) F(λ) = P λ max(x, X )<δ and Now, σ m ( (max(x, X ) δ) / = P λ, max(x, X )<δ) P(X <δ,x <δ). σ m ( ) (max(x, X ) δ) P λ, max(x, X )<δ σ M ( ) (X δ) = P λ, X <δ,x <δ,x X + P σ ( (X δ) σ = p I (λ) + p II (λ). λ, X <δ,x <δ,x X ) ( δ p I (λ) = P λ X δ ), X X σ σ σ ( δ = P λ U δ ), 0 V σ σ

16 90 Balabdaoui -- Mielke -- Munk where U = X /σ and V = X X, with Var(U ) =, Var(V ) = /α + /α, Cov(U, V ) = /α / /α + /α 3. Hence, ( P δ σ ) ( λ U σ δ, 0 V + P δ σ ) λ U σ δ, V < 0 F(λ) = P(X <δ,x <δ) where U = X /σ and V = V, with Var(U ) =, and Cov(U, V ) = /α / /α + /α 3. To establish the expression of the density, put v = Var(U ), v = Var(V ), ρ = Corr(U, V ) and a (λ) = δ/σ λ.wehave p () I (λ) = λ πv v ρ { exp v a (λ) + } v x ρ v v a (λ)x 0 v v ( ρ ) dx = exp{ a (λ)/v } { exp λ πv v (x ρ } a (λ)v /v ) ρ 0 v ( ρ ) dx = exp{ a (λ)/v } P(W 0) λ πv where W N (a (λ)ρ v /v,v ( ρ )). Similarly, we can show that p () II (λ) = exp {a (λ)/v } P(W < 0) λ πv where W N (a (λ)ρ v /v,v ( ρ )), a (λ) = δ/σ λ, v = Var(U ), v = Var(V ), ρ = Corr(U, V ). The density f U is then given by [ p () I (λ) + p () II f(λ) = (λ) ] λ>0. P(X <δ,x <δ) Acknowledgments. The third author would like to acknowledge the support by the Deutsche Forschungsgemeinschaft, FOR96. The authors thank a referee and the editor for helpful comments which led to a better presentation of results. References Chernoff, H. (954). On the distribution of the true log likelihood ratio test statistic when the true parameter is near the boundaries of the hypothesis regions. The Annals of Mathematical Statistics, 6, D Agostino, R. B.; Massaro, J. M.; Sullivan, L. M. (003). Non-inferiority trials: design concepts and issues-the encounters of academic consultants in statistics. Statistics in Medicine,,

17 LRT for non-standard hypotheses near the boundary of the null 9 Davidson, R. R.; Lever, W. E. (970). The limiting distribution of the likelihood ratio statistic under a class of local alternatives. Sankhya Ser. A, 3, El Barmi, H.; Dykstra, R. L. (997). Likelihood ratio test against a set of inequality constraints. Nonparametric Statistics,, Feder, P. I. (968). On the distribution of the true log likelihood ratio test statistic when the true parameter is near the boundaries of the hypothesis regions. The Annals of Mathematical Statistics, 6, Hesketh, P.; Navari, R.; Grote, T.; Gralla, R.; Hainsworth, J.; Kris, M.; Anthony, L.; Khojasteh, A.; Tapazoglou, E.; Benedict, C.; Hahne, W. (996). Double-blind, randomized comparison of the antiemetic efficacy of intravenous dolasetron mesylate and intravenous ondansetron in the prevention of acute cisplatin-induced emesis in patients with cancer. Dolasetron comparative chemotherapy-induced emesis prevention group. Journal of Clinical Oncology, 4, Jones, B.; Jarvis, P.; Lewis, J. A.; Ebbutt, A. F. (996). Trials to assess equivalence: the importance of rigorous methods. British Medical Journal, 33, Kieser, M.; Friede, T. (007). Planning, analysis of three-arm non-inferiority trial with binary endpoints.statistics in Medicine, 6, Liang, K.-Y.; Self, S. G. (996). On the asymptotic behaviour of the pseudolikelihood ratio test statistic. J. R. Statist. Soc. B, 58, Lui, K.-J. (005). Sample size calculation for testing non-inferiority and equivalence under Poisson distribution. Statistical Methodology,, Mielke, M.; Munk, A.; Schacht, A. (008). The assessment of non-inferiority in a gold standard design with censored, exponentially distributed endpoints. Statistics in Medicine, 7, Munk, A.; Mielke, M.; Skipka, G.; Freitag, G. (007). Testing noninferiority in threearmed clinical trials based on the likelihood ratio statistics. The Canadian Journal of Statistics, 35, Pigeot, I.; Schäfer, J.; Röhmel, J.; Hauschke, D. (003). Assessing non-inferiority of a new treatment in a three-arm clinical trial including a placebo. Statistics in Medicine,, Robertson, T.; Wright, F. T.; Dykstra, R. L. (988). Order Restricted Statistical Inference. John Wiley & Sons. Röhmel, J. (998). Therapeutic equivalence investigations: statistical considerations. Statistics in Medicine, 7, Röhmel, J.; Mansmann, U. (999). Unconditional non-asymptotic one-sided tests for independent binomial proportions when the interest lies in showing non-inferiority and/or superiority. Biometrical Journal, 4, Self, S. G.; Liang, K.-Y. (987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 8,

18 9 Balabdaoui -- Mielke -- Munk Senn, S. (997). Statistical Issues in Drug Development. Wiley: New York. Skipka, G.; Munk, A.; Freitag, G. (004). Unconditional exact tests for the assessment of non-inferiority for the difference of binomial probabilities contrasted and compared. Computational Statistics and Data Analysis, 47, Special Issue on Statistical Inference under Inequality Constraints. (00). Edited by Dykstra, R. L.; Robertson, T.; Silvapulle, M. J. Journal of Statistical Planning and Inference, 07, 80. Special Issue on Therapeutic Equivalence Clinical Issues and Statistical Methodology in Noninferiority Trials. (005). Edited by Munk, A.; Trampisch, H.-J. Biometrical Journal, 47, 07. Van der Vaart, A. W. (998). Asymptotic Statistics. Cambridge University Press. Vu, H. T. V.; Zhou, S. (997). Generalization of likelihood ratio tests under nonstandard conditions. The Annals of Mathematical Statistics, 5, Wald, A. (943). Tests of statistical hypothesis concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society, 54, Wilks, S. S. (938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. The Annals of Mathematical Statistics, 9, Fadoua Balabdaoui Institute for Mathematical Stochastics University of Göttingen Goldschmidtstraße Göttingen Germany and CEREMADE Université de Paris-Dauphine fadoua@ceremade.dauphine.fr Axel Munk Institute for Mathematical Stochastics University of Göttingen Goldschmidtstraße Göttingen Germany munk@math.uni-goettingen.de Matthias Mielke Institute for Mathematical Stochastics University of Göttingen Goldschmidtstraße Göttingen Germany mmielke@math.uni-goettingen.de