Quantile Scale Curves

This article was downloaded by: [Zhejiang University] On: 24 March 2014, At: 16:06 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Computational and Graphical Statistics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ucgs20 Quantile Scale Curves Kesar Singh a, David E. Tyler a, Jingshan Zhang a & Somnath Mukherjee a a Kesar Singh is Professor, Department of Statistics, Rutgers University, David E. Tyler is Professor, Department of Statistics, Rutgers University, Jingshan Zhang is PhD, Statistician, Forest Laboratories, and Somnath Mukherjee is Doctoral Candidate, Department of Statistics, Rutgers University. Published online: 01 Jan 2012. To cite this article: Kesar Singh, David E. Tyler, Jingshan Zhang & Somnath Mukherjee (2009) Quantile Scale Curves, Journal of Computational and Graphical Statistics, 18:1, 92-105, DOI: 10.1198/jcgs.2009.0006 To link to this article: http://dx.doi.org/10.1198/jcgs.2009.0006 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions

Supplemental materials for this article are available through the JCGS web page at http://www.amstat.org/publications. Quantile Scale Curves Kesar SINGH,David E. TYLER, Jingshan ZHANG, and Somnath MUKHERJEE A concept of scale curve based on data-depth was introduced in Liu, Parelius, and Singh. A scale curve describes the growth of a scalar (positive real-valued) scale measure of a multivariate population/data-cloud, as the distribution/data progresses centeroutwardly. This article proposes a new concept for a scale curve, called the quantile scale curve, which is conceptually and computationally simpler than a data-depth based scale curve. The article focuses on the data-analytic utility of the proposed quantile scale curve, and in particular on a proposed graphical test for detecting linear and nonlinear association between two groups of variables. Other problems addressed using the concept of a quantile scale curve are exploring heavy-tailedness (via a bootscale plot), and multivariate location and scale testing. The article includes a distributional result on simplicial volumes which is of independent interest. Supplemental materials, including a technical appendix, are available online. Key Words: Bootstrap; Data-depth; Heavy-tailedness; Multivariate location; Multivariate scale; Nonlinear relation; Nonparametrics; Scale curve; Simplex volume; Testing for association. 1. INTRODUCTION A notion of scale curve for a multivariate data-cloud was introduced in Liu, Parelius, and Singh (LPS) (1999), based on measures of centrality popularly known as data-depth (see LPS 1999; Zuo and Serfling 2000; Mizera 2002 for a recent account). A scale curve essentially describes the growth of scalar scale measured as the probability distribution progresses center-outwardly in an overall sense, to include more and more probability mass. To be precise, a data-depth scale curve c(t), fort between 0 and 1, is the Lebesgue measure (length in dimension 1, area in dimension 2, etc.) of the 100t% central region of a population/data-cloud as characterized by an appropriate notion of data-depth. The concept is easily visualized and it contains more information than a single scalar scale measure, such as the generalized variance. In particular, the scale curve carries information Kesar Singh is Professor, Department of Statistics, Rutgers University, David E. Tyler is Professor, Department of Statistics, Rutgers University, Jingshan Zhang is PhD, Statistician, Forest Laboratories, and Somnath Mukherjee is Doctoral Candidate, Department of Statistics, Rutgers University. 2009 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America Journal of Computational and Graphical Statistics, Volume 18, Number 1, Pages 92 105 DOI: 10.1198/jcgs.2009.0006 92

QUANTILE SCALE CURVES 93 on peakedness and tailedness of the distribution/data (see LPS 1999; Wang and Serfling 2005). Mid to lower parts of the curve avoid extreme data and thus are somewhat insensitive to outliers. As seen in LPS, scale curve yields interesting applications such as comparing vector-valued estimators using their scale curves. The scope of the scale curve goes beyond descriptive statistics. Different distributional changes can be shown to have predictable effect on scale curve, which can be easily exploited to construct tests for various hypotheses. The notion of data-depth is quite challenging to work with theoretically as well as computationally (see Serfling 2002 for asymptotics). We propose in this article an alternative notion, namely the quantile scale curve (qsc), which is conceptually and computationally simpler. From this proposed notion, we derive several applications mentioned later in this section. The proposed notion has its roots in Oja s (see Oja 1983) notion of scalar scale of multivariate data. Oja defined the mean volume of all possible simplices formed by a dataset on R d as a scalar measure of multivariate scale. We simply define the quantile curve of these volumes as our quantile scale curve. To be more precise, consider a dataset X 1,X 2,...,X n, with each X i being a d- dimensional column vector. There are N = ( n d+1) choices of subsets of size (d + 1). Each of these choices forms a simplex, with its vertices at the chosen data points. The volume of the simplex formed by {X 1,X 2,...,X d+1 } has a handy formula worth mentioning here: (X 1,...,X d+1 ) = 1 [ ] 1 1 1 d!. X 1 X 2 X d+1 The proposed quantile scale curve, s(t), is then defined as s(t) = ([Nt]), 0 t 1, where (i), i = 1, 2,...,N are the ordered values of the volumes. Under random sampling the qsc, s(t), is a consistent estimator of the quantile curve of (X 1,...,X d+1 ) if X 1,...,X d+1 are independent replicas of the population random vector. For our purposes, however, we are interested in the qsc as an exploratory and inferential tool. If the task of going through all possible simplices for a given n and d is found infeasible, the set of all possible simplices could be replaced by a random selection of a large number of such simplices. We recommend randomly selecting a minimum of 1000 to 2000 simplices. The rationale behind this recommendation lies in the observation that, in essence, one is approximating the univariate empirical distribution of simplex-volumes (for a given sample), using a random selection of a sizable number of simplices. The SE in such an 1 approximation is bounded by 2, attained at the median, where M is the number of M random simplices selected. This bound is 1/60 for M = 900 and 1/100 for M = 2500. In all our plots, except in Example 3.2 where a complete enumeration is done, we have taken M = 1000. Our figures are quite reproducible in terms of the essence, in repeated runs. If the situation allows, one may want to draw 5,000 to 10,000 random simplices, or just consider all possible simplices. A close approximation of probability distribution generally entails (via Bahadur Kiefer type asymptotics) a close approximation of quantiles, especially in high density zone. The curve s(t) tends to be smooth and reflects the scale of

94 K. SINGH ET AL. the data quite accurately. A real data example is presented in the next section. The computing simply involves evaluating determinants for which software is readily available. On the theoretical side, s(t) is simply a U-quantile, a topic which is well studied (see Serfling 1980; Arcones 1996). Turning to the interpretation of qsc, in the unimodal cases one would argue that the lower side of qsc mostly reflects the scale of the central zone of the distribution, as the low volume simplices would typically (though not always) arise from the choices of vertices from the central high concentration zone. The upper part of the qsc would focus on the outskirts of the distribution. With a multimodal distribution the interpretation is not as clear for the lower side of qsc. In the next section some theoretical properties are presented. In particular, it is shown in the case of a bivariate normal population that qsc is proportional to a data-depth based scale curve (LPS 1999). The claim for the bivariate normal case is deduced from a general distributional result on the simplex volumes when the vertices are iid draws from a multivariate normal population. Our result also points to an oversight in Efron (1965), namely that the formula for expected volume given in Efron (1965) is inaccurate by a factor of 2. The main item of the application (Section 3) which inspired this article concerns testing for a relationship between two groups of variables. Graphs of qsc help indicate if there is any significant evidence of a relationship between the two sets of variables under consideration and furthermore if the relationship appears to be a nonlinear one. Such a graphical test seems to be quite novel. In another application (Section 4) we propose a plot, called a bootscale plot, which points out if the underlying population has infinite variance. Such a plot would be handy for assessing heavy-tailedness and may be useful in extreme value studies. In Section 5, we briefly discuss derived tests for multivariate location and scale. The test results are presentable in graphical form. The graphs are intuitive and easy to comprehend. A condensed summary of a study on simulated powers in the case of correlation is also presented. The graphical approach proposed here bears the distinction of presenting simultaneously a robust (lower end of the plot) and a fairly powerful (upper end of the plot) testing outcome on the given testing problem. 2. SOME THEORY AND AN ILLUSTRATIVE EXAMPLE Let us first note an affine equivariance property of the quantile scale curve. For a nonsingular matrix of size d d and a d 1 column vector b, define Y i = AX i + b. Letthe qsc for Y i s be denoted by s (t). Recall that s(t) denotes the qsc of X i s. Proposition 2.1. s (t) = A s(t),0 t 1. Proof: We use the determinant-based formula for a simplex volume mentioned earlier. The claim follows from the following: [ ] [ ][ ] [ ] 1 1 1 1 0 1 1 1 0 0 0 = +. Y 1 Y 2 Y d+1 0 A X 1 X 2 X d+1 b b b

QUANTILE SCALE CURVES 95 The determinant of the right side equals A 1 1 1 X 1 X 2 X d+1. In the rest of this section s( ) will stand for the population version of qsc. Let us recall that the data-depth based scale curve (LPS 1999) c(t) = the volume of the 100t% central region of the population as given by an (affine equivariant) notion of data-depth. For normal populations, we have the following theorem: Theorem 2.1. For normal distribution of dimensions d = 1 and 2, c(t) s(t) = constant, 0 <t<1. For d 3, the proportionality between c(t) and s(t) does not hold. Proof: For d = 1, the claim is nearly obvious. In fact, in the univariate case, it follows from the definitions of c(t) and s(t) that c(t) = 2σz 0.5(1 t), s(t) = 2σz 0.5(1 t). Turning to the bivariate case, we first note that by Proposition 2.1 we can assume wlg that = I. For an affine equivariant data-depth and an elliptical population, data-depth contours coincide with the density contours. This is seen by making the standard linear transformation to make the population spherical. Using this fact, it follows that c(t) is the tth quantile of π(x 2 + Y 2 ), where (X, Y ) have the standard bivariate normal distribution. Thus, c(t) = 2π log(1 t). For the qsc, one obtains the expression 3 s(t) = log(1 t) 2 from a general distributional result given in Lemma 2.1 below for the simplex volume when the vertices are random draws from a multivariate normal population. Lemma 2.1. Let X 1,X 2,...,X d+1 be iid draws from N(μ, ) on R d, and let denote the volume of the simplex with its vertices at X 1,X 2,...,X d+1. Then has the same distribution as 1/2 (d + 1)1/2 (ξ 1 ξ 2 ξ d ) 1/2, d! L where ξ i s are independent random variables, ξ i=χ 2 i. To derive the expression for s(t), for d = 2, one uses the well-known result (ξ 1 ξ 2 ) 1/2 = L 1 2 χ 2 2 (see Srivastava and Khatri 1979). The lemma is proved in the Appendix, where it is shown using the lemma that the proportionality between c(t) and s(t) breaks

96 K. SINGH ET AL. Figure 1. Illustrative quantile scale curves. down for d 3. The Appendix also includes a correction to Efron s formula for expected simplex volumes. See supplemental materials online. We conclude this section with the quantile scale curves for pairs of test score data given in Kent, Bibby, and Mardia (1979). The data-depth based scale curve for the same data is given in LPS (1999). The two types of scale curves seem to have identical features in this illustrative example, perhaps indicating the bivariate distributions are approximately normal. (See Figure 1.) 3. EXPLORING FOR ASSOCIATION Let the measurement X i,ad 1 column vector, consist of two sets of measurements T i and Q i, both column vectors of length p and q, respectively. Thus d = p + q and X i = (T i,q i ). Given the data X 1,X 2,...,X n, suppose the interest lies in detecting a possible linear or a nonlinear association between the T and Q measurements. TESTING FOR A LINEAR TREND Suppose there is some degree of linearity between T i and Q i. We propose to permute the n columns of Q i s randomly and then realign with the original columns of T i.this process essentially gets rid of any existing trend between the two sets of measurements. The resulting dataset can be represented as Y i = (T i,q j i ) where j 1,j 2,...,j n is a random permutation of (1, 2,...,n). Consider the scale of the Y -data. The destruction of linearity via the above randomization will inflate the scale. Romanazzi (2004) has already noted this phenomenon with reference to the data-depth based scale curve. The degree of scale inflation will depend upon the strength of the original linear trend. A trend of linearity compresses the (p + q)-dimensional data along a lower dimension hyperplane, which in turn deflates the overall scale. Once the linear trend is taken out, the volume inflates. The test procedure would consist of plotting the qsc of the original data and a large number of copies of qsc of the postpermutation data (as described previously), in the same graph. If the qsc of the original data lies at the bottom (among 5% lowest by some criterion) of this qsc band, then the existence of a linearity is concluded. We present here two illustrations one based on a simulation and the other on a real dataset.

QUANTILE SCALE CURVES 97 Figure 2. Test plots for linearity: (a) σ 2 = 1, (b) σ 2 = 2, (c) σ 2 = 4, (d) σ 2 = 8. Example 3.1: We simulated 40 observations from the following linear model: [ ] Z i1 [ ] 1 1 1 X i = Z i2 ɛi1 +. 1 1 1 ɛ i2 Z i3 Here the Z i = (Z i1,z i2,z i3 ) are standard normal; ɛ i = (ɛ i1,ɛ i2 ) are bivariate normal with mean 0 and dispersion σ 2 I, and Z i,ɛ i are independent. The proposed test plots for four dimensional data are presented in Figure 2, with σ 2 = 1, 2, 4, 8. For all the four plots, the solid curve represents the qsc of the prepermutation data and the band of dotted curves belongs to the postpermutation data. The figures support the theory. Example 3.2: The article Sensory and Physical Properties of Inherently Flame Retardant Fabrics (Fabric Research (1984), 61 68) reports measurements on stiffness and thickness on certain types of fabric samples. The scatterplot is presented in Figure 3. The Pearsonian sample correlation is 0.773. In spite of this seemingly high correlation the p-value for testing H 0 : ρ = 0versusH 1 : ρ 0 under the bivariate normality assumption turns out to be 0.0715. Thus the parametric test fails to reject H 0 : ρ = 0. Our method Figure 3. Scatterplot of fabric data.

98 K. SINGH ET AL. Table 1. Simulated power. Rho t-test AU qsc qsc 0.75 qsc 0.50 qsc 0.25 0.0 0.057 0.052 0.047 0.055 0.062 0.2 0.182 0.178 0.144 0.112 0.085 0.4 0.609 0.566 0.484 0.353 0.255 0.6 0.943 0.941 0.894 0.794 0.648 0.8 1.000 1.000 1.000 0.999 0.993 at s(0.5) yields p-value = 0.0333, that is, of all the 720 qsc s arising out of postpermuation data, only 24 have s(0.5) lower than that of the original data. One pair of data may be an outlier, or perhaps the relationship is not strictly linear. This may explain the failure of the parametric test in rejecting the absence of a linear trend. We carried out a simulation study to compare the power of the proposed testing scheme for linearity with that of the standard t-test under bivariate normality. Table 1 summarizes the finding. Here the sample size n = 50. AU qsc refers to the test based on area under the qsc; qsc 0.75 refers to the test based on qsc at t = 0.75, etc. The findings are quite encouraging. NONLINEAR RELATIONSHIP AND A CROSSOVER PHENOMENON Assume that there is a nonlinear relationship between T and Q. Even then, typically, there is a fair amount of linearity on local subregions of the domain of T. In such a case, the volume of a simplex whose vertices are close in the coordinate(s) associated with the T variable(s) would tend to be relatively small due to local linearity. In the postpermutation data, the local linearity is destroyed. Consequently, the permutation would raise the lower to middle part of the scale curve. In other words, the simplex volume based scale curve of postpermutation data would tend to lie above that of the original data for t away from the upper end. For t near 1, because of the nonlinearity between T and Q, this phenomenon should be much weaker or absent altogether. In view of the observations above, we would expect the random permutation based band of qsc to lie above the original qsc of Y near the beginning to the middle part, then possibly cross over toward the upper end. We simulated 30 sample points from the following nonlinear model: W = 2T 2 + ɛ, T N(0, 1), ɛ N(0,σ 2 ), where T and ɛ are independent and with σ 2 = 0.5. The scale curves from the original samples and the permuted samples are shown in Figure 4, represented by a solid curve and dashed curves, respectively. We note that this plot confirms our conjecture. We also applied this proposed method to some real dataset. The dataset we used is from a central composite design conducted in a chemical process (Myers and Montgomery 2002, p. 274). There are two process variables: time and temperature; and three responses of interest: yield, viscosity, and number-average molecular weight. First, we are interested in the association between the two process variables and the individual response variable. Figure 5, (a) (c), shows the scale curve plots for detecting

QUANTILE SCALE CURVES 99 Figure 4. Detecting nonlinear trend using qsc. these three associations, time and temperature versus one of the response variables, that is, yield, viscosity, and molecular weight. From the plots, we can see that the first two suggest nonlinear associations and the third one suggests linear association. This agrees with the models fitted to the data which are given in Myers and Montgomery (2002, pp. 276 278). One can also use the scale curve to detect the overall association between the two process variables and the three response variables. Toward this task, Figure 5(d) gives the scale curve plot of the original samples and the permuted samples. The plot suggests an overall (somewhat) nonlinear association. Geiser and Randles (1997) comes close to this subject matter. DETECTING LACK OF FIT Another application of our proposed method is to detect a possible lack of fit in the regression analysis, even though there is convincing evidence according to a test for an overall linear trend. The residuals in the regression analysis are supposed to be free of Figure 5. Test plots for chemical process data: (a) time and temperature versus yield, (b) time and temperature versus viscosity, (c) time and temperature versus molecular weight, (d) time and temperature versus yield, viscosity, and molecular weight.

100 K. SINGH ET AL. Figure 6. Test plots for lack-of-fit example: (a) scatterplot and the fitted line, (b) scale curve plot between X and Y, (c) scale curve plot between residuals and covariates. any association with the covariates if the model is adequate in describing the data. Our methodology can be called upon to investigate if that is the case. An example is given above. The dataset is taken from Myers and Montgomery (2002, p. 55). The scatterplot of the data and the fitted line are given in Figure 6(a). The scale curve plot between X and Y is shown in Figure 6(b). Parametric tests as well as our method for testing linearity confirm the existence of overall linearity. We apply our scale curve method on the covariates and the residuals. Figure 6(c) shows the resulting scale curve plot, which suggests that there exists a nonlinear pattern in the residuals. This means that the original linear model is inadequate to describe the data and some nonlinear model should be considered. 4. EXPLORING TAILEDNESS The concept of quantile scale curves can be a very handy tool for exploring the heavy/light-tailedness of the underlying population. We present here two qsc-based graphical tools toward this end and then conclude the section with a remark on the appearance of the qsc when the dataset contains some outlier(s). STANDARDIZED QSC To assess how heavy-tailed an underlying population is, one would need to study how fast the sample scale curve grows toward its upper end. For this purpose, we suggest to normalize s(t) as follows: d(t) = s(t)/s(0.5), 0.5 t 1, and plot d( ) over the range t 0.5. The choice of s(0.5) with which to normalize seems to be a rather natural one, as s(0.5) is the median value of all possible simplex volumes.

QUANTILE SCALE CURVES 101 Figure 7. Standardized scale curves for bivariate Cauchy, double exponential, and normal. Thus s(0.5) is a quite decent measure of overall scale. In Figure 7, we present the plot of d( ) for samples from a number of bivariate distributions. The graphs perfectly agree with our expectation. The sample size in the plot in Figure 7 is 50. The two coordinates are taken to be independent in each case. BOOTSCALE PLOT FOR DETECTING INFINITE DISPERSION Described below is a proposed plot which holds promise to diagnose the heavytailedness of the underlying population in a univariate or multivariate sample, especially the possibility of infinite dispersion ( x 2 df = ). Consider a figure which plots the scale curves s 1 (t), s 2 (t), s 4 (t), etc., where s 1 (t) = s(t), the scale curve of the original data, and s k (t) is the scale curve of the data cloud D(k) defined as follows: { Y1 + +Y k D k =, with Y 1,...,Y k as random draws from X 1,X 2,...,X n }. k Suppose now that x 2 df <. In view of the Central Limit Theorem one would not expect any noticeable inflation in the curves as one goes from s 1 (t) to s 2 (t) to s 4 (t), etc. However, in the infinite variance case, there should be a rise in the scale curves s k (t) as k rises. In our simulated trials, we generated 100 random draws from bivariate normal, t with degree of freedom 6, double exponential, and Cauchy. Figure 8 shows the plots of s 1 (t), s 2 (t), s 4 (t), s 8 (t), and s 16 (t) for these distributions. As we expected, in all the finite variance cases, the curves s 1 (t), s 2 (t), s 4 (t), s 8 (t), and s 16 (t) were close to each other. In a case of infinite variance (Cauchy distribution) there was an obvious rise, that is, s 2 (t) was substantially higher than s 1 (t),aswass 4 (t) compared to s 2 (t),etc. QSC IN PRESENCE OF OUTLIER(S) In the presence of outlier(s), the scale curve approximates a horizontal line for the most part and then ends with a sharp rise. This phenomenon is illustrated in Figure 9. One expects such a qsc due to the fact that a small portion of simplex volumes would dominate the rest of them, when the sample consists of some very extreme data. The sample here is bivariate standard normal with n = 20, plus an outlier (5, 5).

102 K. SINGH ET AL. Figure 8. Bootscale plots: (a) bivariate normal, (b) t 6, (c) double exponential, (d) Cauchy. 5. QSC-BASED MULTIVARIATE LOCATION AND SCALE TESTING In this section we briefly describe quantile scale curve based testing plans for the classical location shift problem in a symmetric population setting and also the multivariate scale comparison problem for two given multivariate samples. LOCATION TESTING Suppose X 1,X 2,...,X n are coming from a symmetric population, with θ as the center L of symmetry, that is, X i = 2θ Xi. Suppose the null hypothesis specifies a value of θ, that is, we are testing H 0 : θ = θ 0 versus H 1 : θ θ 0. Often this problem arises from a paired data setting, where X i = X i1 X i2, with the pair (X i1,x i2 ) as a measurement of interest on related subjects or the same subject under two different circumstances. The inferential problem is to determine if there is a location difference among the paired populations, which of course leads to H 0 : θ = 0, for the population associated with the differences. Figure 9. qsc in presence of outliers.

QUANTILE SCALE CURVES 103 Figure 10. Location test plot under H 0 and H 1. The testing idea stems from a reflection principle. If θ 0 is the true center of symmetry, the qsc of the combined data {X 1,X 2,...,X n } {2θ 0 X 1,...,2θ 0 X n } will nearly agree with that of {X 1,...,X n }. However, if the point of reflection moves away from θ 0, the scale of the combined data will inflate, because the reflected data will cover new grounds beyond the region covered by the original data. This scale inflation could be utilized to construct tests for θ. We modify the idea somewhat for technical convenience in the following specific proposal: reflect each X i randomly across θ 0 with probability 0.5 and leave it as it is with the remaining 0.5 probability and thus create the dataset ξ ={ξ 1,ξ 2,...,ξ n } of the same size. There are 2 n copies of ξ in all, but one could use a random subset of it in the testing scheme. Plot the qsc of the original data and a band of qsc s of a large number of copies of ξ. If the qsc of {X 1,...,X n } lies within the bottom 5% of the band (in a precise sense) or below the band altogether, the null hypothesis is rejected. For illustration, we present plots of the above testing scheme in Figure 10, one for the case when H 0 holds and the other when H 1 holds. We have carried out simulation-based power comparison of this test with the benchmark Hotelling s T 2 -test under normality. With s(0.75) and auc (area under qsc) as the criterion, our tests possess quite respectable power (often 80). Detailed study will appear elsewhere. TESTING FOR MULTIVARIATE SCALE DIFFERENCE The concept of qsc combines nicely with that of bootstrap for constructing an asymptotic test for a (scaler) scale difference. Suppose we have two independent samples at hand and we want to assess if the second underlying population has a higher (or lower) scale in an overall sense. Let s 1 ( ), s 2 ( ) be the qsc s of the two samples. At a fixed t, 0<t<1, one could form a two-sample asymptotic t-statistic: T n =[s 1 (t) s 2 (t)] /[ a 2 1 n 1 + a2 2 n 2 ] 1/2, where a1 2 and a2 2 are the estimated variances of s 1(t) and s 2 (t), respectively, using bootstrap. Given the U-quantile structure of qsc, it easily follows that such a T n converges to N(0, 1) in distribution under H 0. The following is a graphical test for scale which uses the whole qsc: Plot a band of d ( ), with d (t) = s1 (t) s 2 (t) where s 1 ( ), s 2 ( ) are the qsc s of bootstrap samples drawn from the first and the second samples, respectively. Under H 0, this band will lie on both sides of the X-axis whereas under H 1, nearly all of it

104 K. SINGH ET AL. Figure 11. Scale test plot under H 0 and H 1. will lie above (or below) the X-axis. Precise p-values can be easily defined. We provide an illustration in Figure 11. In a simulation study we found members of the above two sample t-tests to be as powerful as the test based on ratio of generalized variances under multivariate normality. See Pesarin (2001) for discussions on related subject matter. CONCLUDING REMARKS The rather simple quantile scale curve, which is just the quantile curve of the volumes of random simplices arising from a multivariate sample, promises many applications. The output is presentable in a graphical form, which should make the proposed methodology attractive to the user. The original idea of scale curve based on data-depth proposed in LPS (1999) may be somewhat superior in terms of its interpretability, but the quantile scale curve of the present article is much more friendly in terms of computing and it is uniquely suited for some of the applications presented in this article. APPENDIX: SUPPLEMENTAL MATERIALS The following supplemental materials can be found on the JCGS web page. 1. Appendix: The proof of Lemma 2.1 and a note on Efron s formula for computing expected volumes of random simplices (.pdf). 2. Data Files: List of published data sources (.txt). 3. Computer Code (S-Plus and FORTRAN): Codes used to generate the graphs and the power table (Table 1) (.ssc). ACKNOWLEDGMENTS We thank the referee and the associate editor for careful reading of the manuscript. Research supported in part by NSF grants. [Received June 2007. Revised January 2009.]

QUANTILE SCALE CURVES 105 REFERENCES Arcones, M. (1996), The Bahadur Kiefer Representation of U-Quantiles, Annals of Statistics, 24, 1400 1422. Efron, B. (1965), The Convex Hull of a Random Set of Points, Biometrika, 52, 331 343. Gieser, P., and Randles, R. (1997), A Nonparametric Test of Independence Between Two Vectors, Journal of the American Statistical Association, 92, 561 567. Liu, R., Parelius, J., and Singh, K. (1999), Multivariate Analysis by Data-Depth: Descriptive Statistics, Graphics and Inferences, Annals of Statistics, 27, 783 840. Mardia, K. V., Kent, J. T., and Bibby, J. M. (1979), Multivariate Analysis (1st ed.), New York: Academic Press. Mizera, I. (2002), On Depth and Deep Points: A Calculus, Annals of Statistics, 30, 1681 1736. Myers, R., and Montgomery, D. (2002), Response Surface Methodology (2nd ed.), New York: Wiley. Oja, H. (1983), Descriptive Statistics for Multivariate Distributions, Statistics & Probability Letters, 1, 327 332. Pesarin, F. (2001), Multivariate Permutation Tests: With Applications in Biostatistics, New York: Wiley. Romanazzi, M. (2004), Data Depth and Correlation, Allgemeines Statistisches Archiv, 88 (2), 191 214. Serfling, R. (1980), Approximation Theorems of Mathematical Statistics, New York: Wiley. (2002), Generalized Quantile Process Based on Multivariate Depth Function, With Application in Nonparametric Multivariate Analysis, Journal of Multivariate Analysis, 83 (1), 232 247. Srivastava, M. S., and Khatri, C. G. (1979), An Introduction to Multivariate Statistics, New York: Elsevier/North- Holland. Wang, J., and Serfling, R. (2005), Nonparametric Multivariate Kurtosis and Tailweight Measures, Journal of Nonparametric Statistics, 17, 441 456. Zuo, Y., and Serfling, R. (2000), General Notions of Statistical Data-Depth, Annals of Statistics, 28, 461 482.