Bernoulli

  • Bernoulli
  • Volume 20, Number 3 (2014), 1484-1506.

Comparison of multivariate distributions using quantile–quantile plots and related tests

Subhra Sankar Dhar, Biman Chakraborty, and Probal Chaudhuri

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Abstract

The univariate quantile–quantile (Q–Q) plot is a well-known graphical tool for examining whether two data sets are generated from the same distribution or not. It is also used to determine how well a specified probability distribution fits a given sample. In this article, we develop and study a multivariate version of the Q–Q plot based on the spatial quantile. The usefulness of the proposed graphical device is illustrated on different real and simulated data, some of which have fairly large dimensions. We also develop certain statistical tests that are related to the proposed multivariate Q–Q plot and study their asymptotic properties. The performance of those tests are compared with that of some other well-known tests for multivariate distributions available in the literature.

Article information

Source
Bernoulli, Volume 20, Number 3 (2014), 1484-1506.

Dates
First available in Project Euclid: 11 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1402488947

Digital Object Identifier
doi:10.3150/13-BEJ530

Mathematical Reviews number (MathSciNet)
MR3217451

Zentralblatt MATH identifier
06327916

Keywords
characterization of distributions contiguous alternatives Gaussian process Pitman efficacy spatial quantiles tests for distributions the level and the power of test

Citation

Dhar, Subhra Sankar; Chakraborty, Biman; Chaudhuri, Probal. Comparison of multivariate distributions using quantile–quantile plots and related tests. Bernoulli 20 (2014), no. 3, 1484--1506. doi:10.3150/13-BEJ530. https://projecteuclid.org/euclid.bj/1402488947


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References

  • [1] Ahmad, I.A. (1993). Modification of some goodness-of-fit statistics to yield asymptotically normal null distributions. Biometrika 80 466–472.
  • [2] Ahmad, I.A. (1996). Modification of some goodness of fit statistics. II. Two-sample and symmetry testing. Sankhyā Ser. A 58 464–472.
  • [3] Anderson, T.W. and Darling, D.A. (1954). A test of goodness of fit. J. Amer. Statist. Assoc. 49 765–769.
  • [4] Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika 83 715–726.
  • [5] Bickel, P.J. (1967). Some contributions to the theory of order statistics. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calf., 1965/66), Vol. I: Statistics 575–591. Berkeley, CA: Univ. California Press.
  • [6] Bickel, P.J. and Wichura, M.J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 1656–1670.
  • [7] Breckling, J. and Chambers, R. (1988). $M$-quantiles. Biometrika 75 761–771.
  • [8] Burke, M.D. (1977). On the multivariate two-sample problem using strong approximations of the EDF. J. Multivariate Anal. 7 491–511.
  • [9] Chambers, J., Cleveland, W., Kleiner, B. and Tukey, P. (1983). Graphical Methods for Data Analysis. Belmont: Wadsworth.
  • [10] Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91 862–872.
  • [11] Dhar, S.S., Chakraborty, B. and Chaudhuri, P. (2013). Supplement to “Comparison of multivariate distributions using quantile–quantile plots and related tests.” DOI:10.3150/13-BEJ530SUPP.
  • [12] Doksum, K. (1974). Empirical probability plots and statistical inference for nonlinear models in the two-sample case. Ann. Statist. 2 267–277.
  • [13] Doksum, K.A. and Sievers, G.L. (1976). Plotting with confidence: Graphical comparisons of two populations. Biometrika 63 421–434.
  • [14] Easton, G.S. and McCulloch, R.E. (1990). A multivariate generalization of quantile-quantile plots. J. Amer. Statist. Assoc. 85 376–386.
  • [15] Friedman, J.H. and Rafsky, L.C. (1979). Multivariate generalizations of the Wald–Wolfowitz and Smirnov two-sample tests. Ann. Statist. 7 697–717.
  • [16] Friedman, J.H. and Rafsky, L.C. (1981). Graphics for the multivariate two-sample problem. J. Amer. Statist. Assoc. 76 277–287.
  • [17] Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of Multivariate Observations. New York: Wiley.
  • [18] Gnanadesikan, R. and Wilk, M.B. (1968). Probability plotting methods for the analysis of data. Biometrika 55 1–17.
  • [19] Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. New York: Academic Press.
  • [20] Justel, A., Peña, D. and Zamar, R. (1997). A multivariate Kolmogorov–Smirnov test of goodness of fit. Statist. Probab. Lett. 35 251–259.
  • [21] Kiefer, J. (1959). $K$-sample analogues of the Kolmogorov–Smirnov and Cramér–V. Mises tests. Ann. Math. Statist. 30 420–447.
  • [22] Kiefer, J. and Wolfowitz, J. (1958). On the deviations of the empiric distribution function of vector chance variables. Trans. Amer. Math. Soc. 87 173–186.
  • [23] Koenker, R. (2005). Quantile Regression. Econometric Society Monographs 38. Cambridge: Cambridge Univ. Press.
  • [24] Koltchinskii, V.I. (1997). $M$-estimation, convexity and quantiles. Ann. Statist. 25 435–477.
  • [25] Lehmann, E.L. and Romano, J.P. (2005). Testing Statistical Hypotheses. New Delhi: Springer.
  • [26] Liu, R.Y., Parelius, J.M. and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference. Ann. Statist. 27 783–840.
  • [27] Marden, J.I. (1998). Bivariate qq-plots and spider web plots. Statist. Sinica 8 813–826.
  • [28] Marden, J.I. (2004). Positions and QQ plots. Statist. Sci. 19 606–614.
  • [29] Möttönen, J. and Oja, H. (1995). Multivariate spatial sign and rank methods. J. Nonparametr. Stat. 5 201–213.
  • [30] Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
  • [31] Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley.
  • [32] Serfling, R.J. (2004). Nonparametric multivariate descriptive measures based on spatial quantiles. J. Statist. Plann. Inference 123 259–278.
  • [33] Shapiro, S.S. and Wilk, M.B. (1965). An analysis of variance test for normality: Complete samples. Biometrika 52 591–611.
  • [34] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
  • [35] Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. Monographs on Statistics and Applied Probability. London: Chapman & Hall.

Supplemental materials

  • Supplementary material: Supplement to “Comparison of multivariate distributions using quantile–quantile plots and related tests”. In the supplement, we provide additional multivariate Q–Q plots and discuss the performance of various tests for univariate data.