Asymptotic properties of the likelihood ratio test statistics test ...

The Canadian Journal of StatisticsVol. 34, No. 4, 2007, Pages ???-???La revue canadienne de statistiqueAsymptotic properties of the likelihood ratiotest statisticstest statistics in affected-sib-pair analysisZeny Z. Feng ∗ , Jiahua Chen and Mary E. ThompsonKey words and phrases: ASP method, identical by descent (IBD), likelihood ratio test (LRT),linkage analysis, possible triangle constraint.MSC 2000 : Primary 92D10; secondary 92D30.Abstract: In an Affected-Sib-Pair (ASP) genetic linkage analysis, IBD data for affected sib pairs are routinelycollected at a large number of markers along chromosomes. Under very general genetic assumptions,the IBD distribution at each marker satisfies the possible triangle constraint. Statistical analysis of IBDdata hence should utilize this information to improve efficiency. At the same time, this constraint rendersthe usual regularity conditions for likelihood based statistical methods unsatisfied. In this paper, authorsstudy the asymptotic properties of the likelihood ratio test under the possible triangle constraint. Thelimiting distribution of the LRT statistic based on data from a single locus is derived. The precision of thelimiting distribution and the power of the test are investigated by simulation. Further, authors study thetest based on the supremum of the LRT statistics over the markers distributed throughout a chromosome.Instead of deriving a limiting distribution theoretically, a mixture of chisquare distributions to is used toapproximate the true distribution. The simulation results show that this approach has desirable simplicityand satisfactory precision.Title in French: we can supply thisRésumé : Insérer votre résumé ici. In an Affected-Sib-Pair (ASP) genetic linkage analysis, IBD data foraffected sib pairs are routinely collected at a large number of markers along chromosomes. Under verygeneral genetic assumptions, the IBD distribution at each marker satisfies the possible triangle constraint.Statistical analysis of IBD data hence should utilize this information to improve efficiency. At the sametime, this constraint renders the usual regularity conditions for likelihood based statistical methods unsatisfied.In this paper, authors study the asymptotic properties of the likelihood ratio test under thepossible triangle constraint. The limiting distribution of the LRT statistic based on data from a singlelocus is derived. The precision of the limiting distribution and the power of the test are investigated bysimulation. Further, authors study the test based on the supremum of the LRT statistics over the markersdistributed throughout a chromosome. Instead of deriving a limiting distribution theoretically, a mixtureof chisquare distributions to is used to approximate the true distribution. The simulation results showthat this approach has desirable simplicity and satisfactory precision.1. INTRODUCTIONThe sib-pair method is a non-parametric statistical method for linkage analysis introduced inPenrose (1935). The analysis is based on genetic information on sib pairs and their parents. In1

The paper is organized as follows. In Section 2, we derive the limiting distribution of the LRTstatistic under the constraint at a single locus. Simulation results for assessing the precision andthe power follow. In Section 3, we study the empirical distribution of the supremum of the LRTstatistics.2. Limiting distribution of the LRT statisticIn this section, we derive the limiting distribution of the LRT statistic. The precision of thelimiting distribution is examined by a simulation study. The power of the LRT with possibletriangle constraint is also investigated via simulation.2.1 The limiting distribution of the LRT statistic under the possible triangle constraintSuppose N independent sib pairs affected with a certain disease are recruited and the IBD valuesat a given marker are collected. Let N 0 , N 1 and N 2 be observed frequencies of IBD values equaling0, 1 and 2. The random variables N 0 and N 1 are jointly multinomially distributed with parametersπ 0 , π 1 and π 2 = 1 − π 0 − π 1 . The log-likelihood function of π 0 and π 1 isl(π 0 , π 1 ) = N 0 log π 0 + N 1 log π 1 + (N − N 0 − N 1 ) log(1 − π 0 − π 1 ).It is well known that ˆπ 0 = N 0N and ˆπ 1 = N 1Nare the unique maximum likelihood estimates (MLE) ofπ 0 and π 1 with no constraints imposed. To test the null hypothesis of no linkage, the log-likelihoodratio statistic is given byΛ N = 2N[ˆπ 0 log(4ˆπ 0 ) + ˆπ 1 log(2ˆπ 1 ) + (1 − ˆπ 0 − ˆπ 1 ) log(4 − 4ˆπ 0 − 4ˆπ 1 )]. (1)By a classical result in Wilks (1938), Λ N in (1) converges to χ 2 2 in distribution as N → ∞ underthe null hypothesis of no linkage.As we pointed out earlier, the IBD distribution of an affected sib pair satisfies the possibletriangle constraint. It is advantageous to reject the null hypothesis only if the deviation of the IBDdistribution is in the direction of the triangle region. Thus, the test for linkage becomes a test ofH 0 : (π 0 , π 1 , π 2 ) = (1/4, 1/2, 1/4)vsH A : 2π 0 ≤ π 1 ≤ 1/2.The problem of the computation of the restricted MLE, denoted as ˜π, under the possible triangleconstraint was studied in Holmans (1993). The numerical solution can be easily obtained followinga simple procedure. The likelihood ratio statistic under the possible triangle constraint is given by˜Λ N = 2N[ˆπ 0 log(4˜π 0 ) + ˆπ 1 log(2˜π 1 ) + (1 − ˆπ 0 − ˆπ 1 ) log(4 − 4˜π 0 − 4˜π 1 )]. (2)Due to the violation of regularity conditions, the limiting distribution of ˜Λ N differs from the usualchisquare. We derive its limiting distribution as follows.For simplicity, we orthogonalize parameters by transforming from the (π 0 , π 1 ) to the (w 0 , w 1 )space, wherew 0 = √ 2(2π 0 + π 1 − 1), w 1 = 2π 1 − 1.See Figure 1 for illustration. By this transformation, the MLEs ˆπ and the restricted MLEs ˜π aretransformed to ŵ and ˜w respectively. Under the null hypothesis, it is easy to show thatvar(ˆπ 0 ) = 316N , var(ˆπ 1) = 14N , cov(ˆπ 0, ˆπ 1 ) = − 18N .3

w 1−1.414II000000001111111100000000111111110000000011111111 φ000000001111111100000000111111110000000011111111 ρ000000001111111100000000111111110000000011111111Ι0000000011111111 .0000000011111111(−1.414,−1) (w 0 , w 1 )( w 0 , w 1)IIIw 1 = -0.707w0w 0w0ΙΙΙw 1. Iρφ000000000111111111000000000111111111000000000111111111000000000111111111000000000111111111ΙΙFigure 2: Rotate the (w 0 , w 1 ) plane such that Cone I is the first quadrant.For x > 0, we work for an expression of P (˜Λ N ≥ x) under the null hypothesis. Let θ bethe angle between the positive w 0 axis and the vector (ŵ 0 , ŵ 1 ). Note that under the normalityassumption on ( √ Nŵ 0 , √ Nŵ 1 ), θ has a uniform distribution in [0, 2π] and is independent of thenorm ρ = √ N(ŵ0 2 + ŵ2 1 ). Decomposing the probability by conditioning on the position of θ, wemay writeP (˜Λ N ≥ x) = P (θ ∈ Cone I) P (˜Λ N ≥ x|θ ∈ Cone I)+ P (θ ∈ Cone II) P (˜Λ N ≥ x|θ ∈ Cone II)+ P (θ ∈ Cone III) P (˜Λ N ≥ x|θ ∈ Cone III)+ P (θ ∈ Cone IV) P (˜Λ N ≥ x|θ ∈ Cone IV)Suppose (ŵ 0 , ŵ 1 ) falls in the triangle region. In this case, we have ˜Λ N = N(ŵ 2 0 + ŵ 2 1) = ρ 2 .Since ρ and θ are independent, we haveP (˜Λ N ≥ x|θ ∈ Cone IV) = P (χ 2 2 ≥ x),where χ 2 2 stands for a random variable with a χ 2 2 distribution. This takes care of the fourth termin (4).For (ŵ 0 , ŵ 1 ) falling in Cone I, ( ˜w 0 , ˜w 1 ) is the projection of (ŵ 0 , ŵ 1 ) on the line w 1 = √ 22 w 0.See Figure 2. Let φ be the angle between the line w 1 = √ 22 w 0 and the vector (ŵ 0 , ŵ 1 ). Then˜Λ N = ρ 2 cos 2 φ,and ρ 2 cos 2 φ has a χ 2 1 distribution.Rotate the (w 0 , w 1 ) plane as we show in Figure 2, such that Cone I is the first quadrant (Q 1 )of the rotated plane. By circular symmetry of the distribution of (ŵ 0 , ŵ 1 ) and the periodicityproperties of cos φ,P (˜Λ N ≥ x|θ ∈ Cone I) = P (ρ 2 cos 2 φ ≥ x|φ ∈ Q 1 ) = P (χ 2 1 ≥ x).Similarly, P (˜Λ N ≥ x|θ ∈ Cone II) = P (χ 2 1 ≥ x). For (ŵ 0 , ŵ 1 ) falling in Cone III, the corresponding( ˜w 0 , ˜w 1 ) = 0. Thus we have P (˜Λ N ≥ x|θ ∈ Cone III) = 0.In summary, the expression of ˜Λ N can be written as:⎧ √ √21N( ⎪⎨ 3ŵ0 + 3ŵ1) 2 + o p (1), when (ŵ 0 , ŵ 1 ) ∈ Cone I.˜Λ N = Nŵ0 2 + o p (1),when (ŵ 0 , ŵ 1 ) ∈ Cone II.(5)o ⎪⎩ p (1),when (ŵ 0 , ŵ 1 ) ∈ Cone III.N(ŵ0 2 + ŵ1) 2 + o p (1), when (ŵ 0 , ŵ 1 ) ∈ Cone IV,(4)5

Table 3: The power of LRT of restricted and unrestricted model.N=100 N=200Size of Test Restricted Unrestricted Restricted Unrestricted0.05 0.860 0.680 0.988 0.9510.01 0.669 0.486 0.949 0.8590.005 0.576 0.399 0.912 0.8160.001 0.396 0.217 0.816 0.6800.0001 0.190 0.098 0.621 0.449statistics Λ N without constraint and ˜Λ N with constraint. The powers at a number of significancelevels are computed based on the outcomes of 10,000 sets of sample. The powers of the test withpossible triangle constraint (restricted) and without possible triangle constraint (unrestricted) aresummarized in Table 3. The simulation result reveals the expected power gain by the constrainedlikelihood ratio test, which is particularly striking when N = 100.3. Approximating the sample distribution of the supremum of the LRT statistics.With the advance of modern technology, IBD data of ASPs are usually collected at a largenumber of markers over stretches of chromosomes. The IBD values are related to each otherthrough a crossover process along the chromosomes. Thus, ˜Λ N values computed at these markersform a stochastic process indexed by the location of these markers on the chromosome. We maywrite the process as {˜Λ N (t), 0 ≤ t ≤ T } with t being the marker locus in cM and T being thetotal genetic length of a chromosome segment under investigation. Here,“cM” is an abbreviationfor centiMorgan, a unit of genetic length. During meiosis (the sex cell division process), a pair ofhomologous (same type, same length) chromosomes will duplicate and exchange genetic materialby crossing-over. The number of crossovers between two loci depends on the distance betweenthem. Thus the expected number of crossovers occurring in a chromosomal interval serves as ameasure of chromosomal length. In a genetic map, if two loci are 1cM apart, that means, among100 gametes (eggs or sperms) that pass from randomly selected parents to offspring, it is expectedto observe 1 crossover between the two loci. If a chromosome segment under investigation containsa disease gene, an unusually large value of ˜Λ N (t) is expected at t near the disease locus. Thus, themaximum value of ˜Λ N (t) over t ∈ [0, T ] is a natural test statistic for linkage. The next problemis to determine the sample distribution of the supremum of ˜Λ N (t) under the null hypothesis ofno linkage. Given the threshold value, say x, the type I error of the test for each individualchromosome isP 0 { max ˜Λ N (t) ≥ x},0≤t≤Twhere P 0 represents the probability under the null hypothesis.At each locus t, ˜Λ N (t) is determined by the value of unrestricted processed ŵ 0 (t) and ŵ 1 (t)defined in (5). The limiting distribution of ˜Λ N (t) is a mixture of χ 2 1, χ 2 2 and a point mass at 0. Itis noted that, for large N, √ Nŵ 0 (t) and √ Nŵ 1 (t) are asymptotically two independent stationaryGaussian processes. Thus, the process ˜Λ N is built up from the underlying stationary Gaussianprocesses. The limiting distribution of max 0≤t≤T ˜ΛN (t) is difficult to obtain theoretically. Thus, wedo not have a limiting distribution to be used to compute approximate sample quantiles. Instead,we use simulation to find a simple method to approximate the sample distribution.We placed markers at points of a 1cM grid, 3cM grid, 5cM grid, and 10cM grid along ahypothetical chromosome of some length. We let ∆t denote the marker density, where, for example,∆t = 1cM means that markers are placed at points of a 1cM grid. A crossover process was thensimulated as a pure Poisson process according to the Haldane mapping function (Haldane, 1919).7

Figure 3: Proportions of zeroes for different chromosomal length (T ) and marker density (∆t).markers are 1cM apartmarkers are 3cM apartproportion of 00.0 0.2 0.4proportion of 00.0 0.2 0.420 60 100 140chromosomal length20 60 100 140chromosomal lengthmarkers are 5cM apartmarkers are 10cM apartproportion of 00.0 0.2 0.4proportion of 00.0 0.2 0.420 60 100 140chromosomal length20 60 100 140chromosomal lengthMarker data were consequently determined along the chromosome. We set the overall chromosomelength equal to 150cM, as the longest mean genetic length (mean of male and female) of a humanchromosome arm is about 150cM (Morton, 1991). The IBD data at each marker for each sibpair were then obtained, with markers assumed fully informative. We set the number of sib pairsat 500 and generated 10,000 samples for each setting of marker density. In applications, variouslengths of chromosome segments might be investigated. We aim at providing precise approximatesample quantiles of max 0≤t≤T ˜ΛN (t) for T = 10cM, 20cM, . . . , 150cM, along with different markerdensities. The sample quantile approximation will be discussed later for T values in between. Forconvenience, we denote X(T, ∆t) = max 0≤t≤T ˜ΛN (t, ∆t). Notation of ∆t only comes in the casewhere multiple markers on the same chromosomal arm are tested.Let X i (T, ∆t) be the observed X(T, ∆t) based on the ith simulated sample. For each givenvalue of T and ∆t , we may write the cumulative distribution function of X(T, ∆t) as:G T,∆t (x) = P {X(T, ∆t) ≤ x} = wF T,∆t (x) + (1 − w), for x > 0, (7)where 1 − w is the point mass of X i (T, ∆t) at 0. As the chromosomal length T increases, 1 − wdecreases. The empty dots in Figure 3 show the empirical proportions of zeroes as a function of Tand ∆t.Using the simulated X i (T, ∆t) for each given T and ∆t, the corresponding quantiles of thedistribution of X(T, ∆t) can be approximated by the sample quantiles. In applications, we mayprovide a table of simulated quantiles for various combinations of T and ∆t and a number of prechosensignificance levels. This is not very convenient whenever a new significance level is required.A more useful solution is to find a simple family of parametric distributions that provide accurateapproximations to the distributions of X(T, ∆t) with parameter values being smooth functions of8

Figure 4: Proportions of zeroes for different chromosomal length (T ) and marker density ∆t = 2cM.Marksers are 2cM apartproportion of 00.0 0.1 0.2 0.3 0.420 40 60 80 100 120 140chromosomal lengthT and ∆t. Thus, for any given values of T and ∆t in an application and the observed value ofX(T, ∆t), one can use this distribution family to compute an approximate p-value conveniently.Recall that when T = 0, the marginal limiting distribution of the LRT statistic is a mixture ofχ 2 1 and χ 2 2 distributions, and a distribution concentrated at 0. Based on simulation results at theselected T and ∆t, we propose to use the following distribution family:whereF T,∆t (x) =∫ x0f(u; p, d 1 , d 2 , T, ∆t)duf(x; p, d 1 , d 2 , T, ∆t) = pf(x; d 1 , T, ∆t) + (1 − p)f(x; d 2 , T, ∆t),with f(x; d, T, ∆t) being a χ 2 ddensity, and p being the mixing proportion, to approximate thesample distribution of the non-zero portion of X(T, ∆t). When d is not an integer, we interpretf(x; d, T, ∆t) as a Gamma distribution with parameters (d/2, 2). Our task now is to find smoothfunctions of T and ∆t for parameters p, d 1 and d 2 . Based on the simulated data, with theconsideration of simplicity, we propose the following function to be used:logit(p) = a 1 + a 2 T + a 3√∆t,d 1 = b 1 + b 2 T + b 3√T + b4√∆t,d 2 = c 1 + c 2 T + c 3√T + c4√∆t,(8)where logit(p) = log(p/(1 − p)). The values of a = (a 1 , a 2 , a 3 ), b = (b 1 , b 2 , b 3 , b 4 ) and c =(c 1 , c 2 , c 3 , c 4 ) will be chosen to best fit the simulated sample distributions of X(T, ∆t). The probabilityof w = 1 − P {X(T, ∆t) = 0} will be modeled later.We now regard the simulated nonzero values of X i (T, ∆t), i = 1, 2, . . . , n = 10, 000 for T =0, 10, . . . , 150 and ∆t = 1, 3, 5, 10 as a random sample from density f(x; p, d 1 , d 2 , T, ∆t) and considerestimating a, b and c by maximum likelihood. At each given T and ∆t, the log-likelihood function9

is given by:l(a, b, c; T, ∆t) =n∑log{pf(x i ; d 1 , T, ∆t) + (1 − p)f(x i ; d 2 , T, ∆t)}.i=1Note that p, d 1 and d 2 are functions of a, b and c for each given T and ∆t. Due to the verycomplex relationship between X(T 1 , ∆t 1 ) and X(T 2 , ∆t 2 ) for any T 1 ≠ T 2 or ∆t 1 ≠ ∆t 2 , wemaximize, instead of the joint log-likelihood function, the following “pseudo” log-likelihood:l(a, b, c) = ∑ ∑l(a, b, c; T, ∆t)T ∆twith summation over T = 0, 10, · · · , 150 and ∆t = 1, 3, 5, 10. Since when T = 0 and ∆t = 0, thelimiting distribution of X(0, 0) is known to have p = 0.836, d 1 = 1 and d 2 = 2, our maximizationwill be done under the constraints: a 1 = 1.629, b 1 = 1 and c 1 = 2.We now use the EM algorithm (Dempster, Laird and Rubin, 1977) for the numerical computation.Let Z i be a latent variable representing the mixture component of the ith observation. Thecomplete log-likelihood function is:l c (a, b, c) =n∑ ∑ ∑n∑z i log f(x i ; d 1 ; T, ∆t) + − z i )i=1 T ∆ti=1(1 ∑ Tn∑n∑+{ z i } log(p) + {n − z i } log(1 − p).i=1i=1∑log f(x i ; d 2 ; T, ∆t)∆tGiven initial values of parameters a, b and c, we compute the conditional expected values of Z i inthe above set up in the E-step. In the M-step, we update the parameter values a, b and c as themaximizer of the complete likelihood. The E-step and M-step are iterated until convergence.The E-M algorithm converges at â 2 = −0.048, â 3 = 0.590, ˆb 2 = −0.016, ˆb 3 = 0.241, ˆb 4 =−0.115, ĉ 2 = −0.016, ĉ 3 = 0.442 and ĉ 2 = −0.519. With (8), we find a set of parameters of p, d 1and d 2 , for any given value of T and ∆t. Naturally, the corresponding distribution may not be agood approximation for the distribution of X(T, ∆t) when T is much larger than 150cM.We approximate the proportion of 0 observations 1 − w by a function of the form1 − w(T, ∆t) = 0.402 exp{α 1 T + α 2 ∆t + α 3 T · ∆t}.This function is motivated by the requirement of w(0, 0) = 0.598 (the total weight of the twochisquare components for T = 0 and ∆t = 0) and the general trend of the observed proportions.By fitting this model to the observed zero proportions, we getw(T, ∆t) = 1 − 0.402 exp{−0.052T + 0.018∆t + 0.002T · ∆t}.The solid line in Figure 3 shows the fitted values. Here, we fixed the coefficient of the exponentialterm to be 0.402, which is the portion of zeroes when T = 0 and ∆t = 0 given by our derivedmarginal limiting distribution of the LRT statistic (6). Figure 3 shows that, although the fit isnot quite as good when T = 10cM, this simple model gives a very good fit when T ≥ 20cM forvarious ∆t. In Figure 4, we plot the empirical proportion of 0 (empty dots) when ∆t = 2cM foreach T = 10, . . . , 150cM. The solid line is the predicted proportion of 0 for each each T based onour fitted model. We see that, our model approximates the proportion of 0 very well.In summary, we use the following probability function to approximate the distribution of thesupremum statistic for any given T and ∆t: for any x ≥ 0,P{X(T, ∆t) < x} = w[pP(χ 2 d 1< x) + (1 − p)P(χ 2 d 2< x)] + (1 − w), (9)10

withw(T, ∆t) = 1 − 0.402 exp{−0.052T + 0.018∆t + 0.002T · ∆t};p(T, ∆t) = exp{1.629 − 0.049T + 0.590 √ ∆t}/[1 + exp{1.629 − 0.048T + 0.590 √ ∆t}];d 1 (T, ∆t) = 1 − 0.016T + 0.241 √ T − 0.115 √ ∆t;d 2 (T, ∆t) = 2 − 0.016T + 0.442 √ T − 0.519 √ ∆t.For example, when T = 10 and ∆t = 1, we have w = 0.752, p = 0.849, d 1 = 1.487 and d 2 = 2.719.To assess the precision of the approximations, we use Q-Q plots to compare the empiricalquantiles with the quantiles from our mixed χ 2 for each size of T and each marker density of ∆t.In Figure 5, 6, 7 and 8, we give the Q-Q plots for T =20cM, 40cM, 60cM, 80cM, 100cM and 120cMand ∆t = 1cM, 3cM, 5cM and 10cM. It is seen that our mixed χ 2 distribution approximates thedistribution of the supremum statistics very well for various values of T and ∆t. Our approximationprovides an approximation conveniently for any values of T and ∆t. To see this, we generated10,000 sample sets. In each set, we have the supremum statistics for T = 20cM, 40cM, 60cM,80cM, 100cM and 120cM with ∆t = 2cM. Values of w’s, p’s, d 1 ’s and d 2 ’s are computed by theequation (9). Figure 9 contains the Q-Q plots between the empirical quantiles and the quantilesfrom the mixture of χ 2 distributions for each T . It is seen that the quantiles computed fromour mixture of χ 2 distributions match the quantiles of the test data very well. Our method isspecifically for the problem of testing on a large number of markers. In human genome, thereare 22 pairs of autosomes. In a genome-wide scan, since we obtain a supremum LRT statisticsfor each chromosome, there are 22 supremum statistics to be tested. Alternatively, if we considercrossover processes to be independent from chromosome arm to chromosome arm (usually, thereare two arms for each chromosome: a longer arm and a shorter arm), there are total 44 supremumstatistics to be tested. If we want to control the family-wise error rate at a level of 0.05, the fitof the distribution for p-value=0.001 is sufficient for each chromosome arm. It can be shown fromthe simulation results that the distribution approximations give p-values which are accurate whenthey are near 0.05, 0.01 and 0.001 regardless of the values of T and ∆t. See Table 4.In summary, our mixture χ 2 distribution of the formP{X(T, ∆t) < x} = w[pP(χ 2 d 1< x) + (1 − p)P(χ 2 d 2< x)] + (1 − w),is an adaptable approximation to the limiting distribution of the supremum statistics max 0≤t≤T ˜ΛN (t).With any given size of T within our range of simulation, the parameters p, d 1 and d 2 can be obtainedsuch that the threshold value for a given size of test can be easily determined by our probabilityfunction.4. DiscussionsIn this paper, the limiting distribution of the LRT statistic at any given marker under the possibletriangle constraint is obtained via a technique similar to that used in Chernoff (1954) and Self& Liang (1987) for deriving the limiting distribution of the LRT statistics under nonstandardconditions. Our approach is straightforward and simple and consistent with Holmans’ result.Holmans (1993) showed that with a given genetic model, e.g. allele frequencies, the weight ofχ 2 2 can be calculated through a complicated method. He did not appear to notice that, whenmarker is fully informative, the weight of χ 2 2 is fixed. When the marker is not fully informative,the number of alleles shared IBD by a sib pair may not be determined unambiguously. Generallyin this case a conditional IBD distribution can be obtained given observed genotype data. AnEM algorithm can be implemented in order to compute the correct log likelihood ratio statistic.Under the null hypothesis, twice the correct log likelihood ratio statistic should be asymptoticallychi-squared with 2 degrees of freedom. Thus, for a sufficiently large sample size, we expect theempirical distribution of the unrestricted LRT statistic using an incomplete informative marker11

Figure 5: Q-Q plots of approximating mixed χ 2 distributions and empirical distributions of supremumLRT statistics (max 0≤t≤T ˜λN ) for T =20cM, 40cM, 60cM, 80cM, 100cM and 120cM and∆t = 1cM.Fitted mixed chi−square dist0 5 15T=20cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=40cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=60cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=80cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=100cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=120cM0 5 10 15 20empirical supremum LRT statistics12

Figure 6: Q-Q plots of approximating mixed χ 2 distributions and empirical distributions of supremumLRT statistics (max 0≤t≤T ˜λN ) for T =20cM, 40cM, 60cM, 80cM, 100cM and 120cM and∆t = 3cM.Fitted mixed chi−square dist0 5 15T=20cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 25T=40cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=60cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=80cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=100cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=120cM0 5 10 15 20empirical supremum LRT statistics13

Figure 7: Q-Q plots of approximating mixed χ 2 distributions and empirical distributions of supremumLRT statistics (max 0≤t≤T ˜λN ) for T =20cM, 40cM, 60cM, 80cM, 100cM and 120cM and∆t = 5cM.Fitted mixed chi−square dist0 5 15T=20cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=40cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=60cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 20T=80cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 25T=100cM0 5 10 15 20empirical supremum LRT statisticsFitted mixed chi−square dist0 10 25T=120cM0 5 10 15 20empirical supremum LRT statistics14