Asymptotics of an empirical bridge of regression on induced order statistics

A. P. Kovalevskii

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 954-963.

Résumé

We develop a class of statistical tests for analysis of multivariate data. These statistical tests verify the hypothesis of a linear regression model. To solve the question of the applicability of the regression model, one needs a statistical test to determine whether the actual multivariate data corresponds to this model. If the data does not correspond to the model, then the latter should be corrected. The developed statistical tests are based on an ordering of data array by some null variable. With this ordering, all observed variables become concomitants (induced order statistics). Statistical tests are based on functionals of the process of sequential (under the introduced ordering) sums of regression residuals. We prove a theorem on weak convergence of this process to a centered Gaussian process with continuous trajectories. This theorem is the basis of an algorithm for analysis of multivariate data for matching a linear regression model. The proposed statistical tests have several advantages compared to the commonly used statistical tests based on recursive regression residuals. So, unlike the latter, the statistics of the new tests are invariant to a change in ordering from direct to reverse. The proof of the theorem is based on the Central Limit Theorem for induced order statistics by Davydov and Egorov (2000).

Détail

Keywords: weak convergence, regression residuals, empirical bridge.
Mots-clés : concomitants

Bibliographie

[1] A. Aue, L. Horvath, M. Huskova, P. Kokoszka, “Testing for changes in polynomial regression”, Bernoulli, 14 (2008), 637–660 | DOI | MR | Zbl

[2] A. Aue, L. Horvath, “Structural breaks in time series”, Journal of Time Series Analysis, 34:1 (2013), 1–16 | DOI | MR | Zbl

[3] A. Aue, G. Rice, O. Sonmez, “Detecting and dating structural breaks in functional data without dimension reduction”, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80:3 (2018), 509–529 | DOI | MR | Zbl

[4] N. Balakrishnan, A.C. Cohen, Order Statistics and Inference: Estimation Methods, Academic Press, Boston, MA, 1991 | MR | Zbl

[5] P.K. Bhattacharya, “Convergence of sample paths of normalized sums of induced order statistics”, The Annals of Statist., 2 (1974), 1034–1039 | DOI | MR | Zbl

[6] P.K. Bhattacharya, “An invariance principle in regression analysis”, The Annals of Statist., 4 (1976), 621–624 | DOI | MR | Zbl

[7] P.K. Bhattacharya, “Induced order statistics: Theory and applications”, Handbook of Statistics, v. 4, eds. P.R. Krishnaiah, P. K. Sen, North-Holland, Amsterdam, 1984, 383–403 | DOI | MR | Zbl

[8] W. Bischoff, “A functional central limit theorem for regression models”, Ann. Stat., 26 (1998), 1398–1410 | DOI | MR | Zbl

[9] R.L. Brown, J. Durbin, J.M. Evans, “Techniques for testing the constancy of regression relationships over time”, J. R. Statist. Soc., 37 (1975), 149–192 | MR | Zbl

[10] H.A. David, “Concomitants of order statistics”, Bull. Internat. Statist. Inst., 45 (1973), 295–300 | MR | Zbl

[11] H.A. David, J. Galambos, “The asymptotic theory of concomitants of order statistics”, J. Appl. Probab., 11 (1974), 762–770 | DOI | MR | Zbl

[12] H.A. David, H N. Nagaraja, “Concomitants of order statistics”, Order Statistics: Theory and Methods, 1998, 487–513 | MR | Zbl

[13] H.A. David, H N. Nagaraja, Order Statistics, John Wiley Sons, New Jersey, 2003 | MR | Zbl

[14] H.A. David, M.J. O'Connell, S.S. Yang, “Distribution and expected value of the rank of a concomitant of an order statistics”, The Annals of Statist., 5 (1977), 216–223 | DOI | MR | Zbl

[15] Y. Davydov, V. Egorov, “Functional limit theorems for induced order statistics”, Mathematical Methods of Statistics, 9:3 (2000), 297–313 | MR | Zbl

[16] Y. Davydov, R. Zitikis, “Convex rearrangements of random elements”, Fields Institute Communications, 44 (2004), 141–171 | MR | Zbl

[17] P. Deheuvels, G.V. Martynov, “Cramer-von mises-type tests with applications to tests of independence for multivariate extreme-value distributions”, Communications in Statistics — Theory and Methods, 25:4 (1996), 871–908 | DOI | MR | Zbl

[18] F. Domma, S. Giordano, “Concomitants of m-generalized order statistics from generalized Farlie-Gumbel-Morgenstern distribution family”, Journal of Computational and Applied Mathematics, 294 (2016), 413–435 | MR | Zbl

[19] N.R. Draper, H. Smith, Applied Regression Analysis, Wiley Series in Probability and Statistics, 1998 | DOI | MR | Zbl

[20] V.A. Egorov, V.B. Nevzorov, “Some theorems on induced order statistics”, Theory Prob. Appl., 27:3 (1983), 633–639 | MR | Zbl

[21] V.A. Egorov, V.B. Nevzorov, “Rate of convergence to the normal law of sums of induced order statistics”, J. of Soviet Math., 25:3 (1984), 1139–1146 | MR | Zbl

[22] J. Einmah, D. Mason, “Strong limit theorems for weighted quantile processes”, Annals. Probab., 16:4 (1988), 1623–1643 | DOI | MR | Zbl

[23] S. EryIlmaz, I.G. Bairamov, “On a new sample rank of an order statistics and its concomitant”, Statistics $\$ Probability Letters, 63:2 (2003), 123–131 | DOI | MR | Zbl

[24] F. Galton, Natural Inheritance, Macmillan, London, 1889 http://galton.org/books/natural-inheritance/index.html

[25] J.L. Gastwirth, “A general definition of the Lorenz curve”, Econometrica, 39 (1971), 1037–1039 | DOI | Zbl

[26] C.M. Goldie, “Convergence theorems for empirical Lorenz curves and their inverses”, Adv. Appl. Prob., 9 (1977), 765–791 | DOI | MR | Zbl

[27] H.L. Koul, Weighted Empirical Processes in Dynamic Nonlinear Models, Springer Verlag, New York, 2002 | MR | Zbl

[28] A. Kovalevskii, “A regression model for prices of second-hand cars. Applied methods of statistical analysis”, Applications in survival analysis, reliability and quality control, 2013, 124–128

[29] A.P. Kovalevskii, E.V. Shatalin, “Asymptotics of Sums of Residuals of One-Parameter Linear Regression on Order Statistics”, Theory of probability and its applications, 59:3 (2015), 375–387 | DOI | MR | Zbl

[30] A. Kovalevskii, E. Shatalin, “A limit process for a sequence of partial sums of residuals of a simple regression on order statistics”, Probability and Mathematical Statistics, 36:1 (2016), 113–120 | MR | Zbl

[31] I.B. MacNeill, “Limit processes for sequences of partial sums of regression residuals”, Ann. Prob., 6 (1978), 695–698 | DOI | MR | Zbl

[32] G.V. Martynov, Omega-square tests, Nauka, M., 1978 (in Russian) | MR | Zbl

[33] A.I. Sakhanenko, O.A. Sukhovershina, “On accuracy of approximation in Koul's theorem for weighted empirical processes”, Sib. Elektron. Mat. Izv., 12 (2015), 784–794 | MR | Zbl

[34] P.K. Sen, “A note on invariance principles for induced order statistics”, The Annals of Probab., 4 (1976), 474–479 | DOI | MR | Zbl

[35] M.Q. Shahbaz, S. Shahbaz, M. Mohsin, A. Rafiq, “On distribution of bivariate concomitants of records”, Applied Mathematics Letters, 23:5 (2010), 567–570 | DOI | MR | Zbl

[36] G. Shorack, J. Wellner, Empirical processes with applications to statistics, Wiley, N. Y., 1986 | MR | Zbl

[37] A. Stepanov, A. Berred, V. B. Nevzorov, “Concomitants of records: Limit results, generation techniques, correlation”, Statistics $\$ Probability Letters, 109 (2016), 184–188 | DOI | MR | Zbl

[38] W. Stute, “Nonparametric model checks for regression”, Ann. Statist., 25 (1997), 613–641 | DOI | MR | Zbl

[39] N. Wirth, Algorithms and Data Structures, Prentice-Hall, NJ, 1986 | MR | Zbl

[40] S.S. Yang, “General distribution theory of the concomitants of order statistics”, The Annals of Statist., 5 (1977), 996–1002 | DOI | MR | Zbl

[41] E. Zamanzade, M. Vock, “Variance estimation in ranked set sampling using a concomitant variable”, Statistics $\$ Probability Letters, 105 (2015), 1–5 | DOI | MR | Zbl

[42] A. Zeileis, F. Leisch, K. Hornik, Ch. Kleiber, “Strucchange: An R Package for Testing for Structural Change in Linear Regression Models”, Journal of Statistical Software, 7:2 (2002), 1–38 | DOI