Testing nonparametric shape restrictions. (English) Zbl 1539.62113
Summary: We describe and examine a test for a general class of shape constraints, such as signs of derivatives, U-shape, quasi-convexity, log-convexity, among others, in a nonparametric framework using partial sums empirical processes. We show that, after a suitable transformation, its asymptotic distribution is a functional of a Brownian motion index by the c.d.f. of the regressor. As a result, the test is distribution-free and critical values are readily available. However, due to the possible poor approximation of the asymptotic critical values to the finite sample ones, we also describe a valid bootstrap algorithm.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 62H15 | Hypothesis testing in multivariate analysis |
Keywords:
monotonicity; convexity; concavity; U-shape; quasi-convexity; log-convexity; convexity in means; B-splines; CUSUM transformation; distribution-free estimationSoftware:
FITPACKReferences:
| [1] | ABREVAYA, J. and JIANG, W. (2005). A nonparametric approach to measuring and testing curvature. J. Bus. Econom. Statist. 23 1-19. Digital Object Identifier: 10.1198/073500104000000316 Google Scholar: Lookup Link MathSciNet: MR2108688 · doi:10.1198/073500104000000316 |
| [2] | AGARWAL, G. G. and STUDDEN, W. J. (1980). Asymptotic integrated mean square error using least squares and bias minimizing splines. Ann. Statist. 8 1307-1325. MathSciNet: MR0594647 · Zbl 0522.62032 |
| [3] | Akakpo, N., Balabdaoui, F. and Durot, C. (2014). Testing monotonicity via local least concave majorants. Bernoulli 20 514-544. Digital Object Identifier: 10.3150/12-BEJ496 Google Scholar: Lookup Link zbMATH: 1400.62090 MathSciNet: MR3178508 · Zbl 1400.62090 · doi:10.3150/12-BEJ496 |
| [4] | ANDREWS, D. W. K. (1997). A conditional Kolmogorov test. Econometrica 65 1097-1128. Digital Object Identifier: 10.2307/2171880 Google Scholar: Lookup Link MathSciNet: MR1475076 · doi:10.2307/2171880 |
| [5] | Baraud, Y., Huet, S. and Laurent, B. (2005). Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function. Ann. Statist. 33 214-257. Digital Object Identifier: 10.1214/009053604000000896 Google Scholar: Lookup Link MathSciNet: MR2157802 · Zbl 1065.62109 · doi:10.1214/009053604000000896 |
| [6] | BIRKE, M. and DETTE, H. (2007). Estimating a convex function in nonparametric regression. Scand. J. Stat. 34 384-404. Digital Object Identifier: 10.1111/j.1467-9469.2006.00534.x Google Scholar: Lookup Link MathSciNet: MR2346646 · Zbl 1142.62019 · doi:10.1111/j.1467-9469.2006.00534.x |
| [7] | BOLLAERTS, K., EILERS, P. H. C. and VAN MECHELEN, I. (2006). Simple and multiple P-splines regression with shape constraints. Br. J. Math. Stat. Psychol. 59 451-469. Digital Object Identifier: 10.1348/000711005X84293 Google Scholar: Lookup Link MathSciNet: MR2282224 · doi:10.1348/000711005X84293 |
| [8] | Bowman, A. W., Jones, M. C. and Gijbels, I. (1998). Testing monotonicity of regression. J. Comput. Graph. Statist. 7 489-500. |
| [9] | BROWN, R. L., DURBIN, J. and EVANS, J. M. (1975). Techniques for testing the constancy of regression relationships over time. J. Roy. Statist. Soc. Ser. B 37 149-192. MathSciNet: MR0378310 · Zbl 0321.62063 |
| [10] | Brunk, H. D. (1955). Maximum likelihood estimates of monotone parameters. Ann. Math. Stat. 26 607-616. Digital Object Identifier: 10.1214/aoms/1177728420 Google Scholar: Lookup Link MathSciNet: MR0073894 · Zbl 0066.38503 · doi:10.1214/aoms/1177728420 |
| [11] | CHEN, X. (2007). Large sample sieve estimation of semi-nonparametric models. In Handbook of Econometrics 6B. Elsevier, Amsterdam. |
| [12] | CHEN, X. and CHRISTENSEN, T. M. (2015). Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions. J. Econometrics 188 447-465. Digital Object Identifier: 10.1016/j.jeconom.2015.03.010 Google Scholar: Lookup Link MathSciNet: MR3383220 · Zbl 1337.62101 · doi:10.1016/j.jeconom.2015.03.010 |
| [13] | CHETVERIKOV, D. (2019). Testing regression monotonicity in econometric models. Econometric Theory 35 729-776. Digital Object Identifier: 10.1017/s0266466618000282 Google Scholar: Lookup Link MathSciNet: MR3974977 · Zbl 1420.62480 · doi:10.1017/s0266466618000282 |
| [14] | Claeskens, G., Krivobokova, T. and Opsomer, J. D. (2009). Asymptotic properties of penalized spline estimators. Biometrika 96 529-544. Digital Object Identifier: 10.1093/biomet/asp035 Google Scholar: Lookup Link MathSciNet: MR2538755 · Zbl 1170.62031 · doi:10.1093/biomet/asp035 |
| [15] | DE BOOR, C. (1978). A Practical Guide to Splines. Applied Mathematical Sciences 27. Springer, New York-Berlin. MathSciNet: MR0507062 · Zbl 0406.41003 |
| [16] | DELGADO, M. A., HIDALGO, J. and VELASCO, C. (2005). Distribution free goodness-of-fit tests for linear processes. Ann. Statist. 33 2568-2609. Digital Object Identifier: 10.1214/009053605000000606 Google Scholar: Lookup Link MathSciNet: MR2253096 · Zbl 1084.62038 · doi:10.1214/009053605000000606 |
| [17] | DETTE, H., NEUMEYER, N. and PILZ, K. F. (2006). A simple nonparametric estimator of a strictly monotone regression function. Bernoulli 12 469-490. Digital Object Identifier: 10.3150/bj/1151525131 Google Scholar: Lookup Link MathSciNet: MR2232727 · Zbl 1100.62045 · doi:10.3150/bj/1151525131 |
| [18] | DIACK, C. A. T. and THOMAS-AGNAN, C. (1998). A nonparametric test of the non-convexity of regression. J. Nonparametr. Stat. 9 335-362. Digital Object Identifier: 10.1080/10485259808832749 Google Scholar: Lookup Link MathSciNet: MR1646904 · Zbl 0954.62058 · doi:10.1080/10485259808832749 |
| [19] | DIERCKX, P. (1993). Curve and Surface Fitting with Splines. Monographs on Numerical Analysis. The Clarendon Press, Oxford University Press, New York. MathSciNet: MR1218172 · Zbl 0782.41016 |
| [20] | Dümbgen, L. and Spokoiny, V. G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124-152. Digital Object Identifier: 10.1214/aos/996986504 Google Scholar: Lookup Link MathSciNet: MR1833961 · doi:10.1214/aos/996986504 |
| [21] | DURBIN, J. (1973). Distribution Theory for Tests Based on the Sample Distribution Function. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 9. SIAM, Philadelphia, PA. MathSciNet: MR0305507 · Zbl 0267.62002 |
| [22] | Eilers, P. H. C. and Marx, B. D. (1996). Flexible smoothing with \(B\)-splines and penalties. Statist. Sci. 11 89-121. Digital Object Identifier: 10.1214/ss/1038425655 Google Scholar: Lookup Link MathSciNet: MR1435485 · Zbl 0955.62562 · doi:10.1214/ss/1038425655 |
| [23] | Ghosal, S., Sen, A. and van der Vaart, A. W. (2000). Testing monotonicity of regression. Ann. Statist. 28 1054-1082. Digital Object Identifier: 10.1214/aos/1015956707 Google Scholar: Lookup Link MathSciNet: MR1810919 · Zbl 1105.62337 · doi:10.1214/aos/1015956707 |
| [24] | Giacomini, R., Politis, D. N. and White, H. (2013). A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometric Theory 29 567-589. Digital Object Identifier: 10.1017/S0266466612000655 Google Scholar: Lookup Link MathSciNet: MR3064050 · Zbl 1272.62033 · doi:10.1017/S0266466612000655 |
| [25] | Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653-1698. Digital Object Identifier: 10.1214/aos/1015345958 Google Scholar: Lookup Link MathSciNet: MR1891742 · doi:10.1214/aos/1015345958 |
| [26] | Guntuboyina, A. and Sen, B. (2015). Global risk bounds and adaptation in univariate convex regression. Probab. Theory Related Fields 163 379-411. Digital Object Identifier: 10.1007/s00440-014-0595-3 Google Scholar: Lookup Link MathSciNet: MR3405621 · Zbl 1327.62255 · doi:10.1007/s00440-014-0595-3 |
| [27] | Hall, P. and Heckman, N. E. (2000). Testing for monotonicity of a regression mean by calibrating for linear functions. Ann. Statist. 28 20-39. Digital Object Identifier: 10.1214/aos/1016120363 Google Scholar: Lookup Link MathSciNet: MR1762902 · Zbl 1106.62324 · doi:10.1214/aos/1016120363 |
| [28] | HALL, P. and HUANG, L.-S. (2001). Nonparametric kernel regression subject to monotonicity constraints. Ann. Statist. 29 624-647. Digital Object Identifier: 10.1214/aos/1009210683 Google Scholar: Lookup Link MathSciNet: MR1865334 · Zbl 1012.62030 · doi:10.1214/aos/1009210683 |
| [29] | Hanson, D. L. and Pledger, G. (1976). Consistency in concave regression. Ann. Statist. 4 1038-1050. MathSciNet: MR0426273 · Zbl 0341.62034 |
| [30] | Harville, D. A. (1997). Matrix Algebra from a Statistician’s Perspective. Springer, New York. Digital Object Identifier: 10.1007/b98818 Google Scholar: Lookup Link MathSciNet: MR1467237 · Zbl 0881.15001 · doi:10.1007/b98818 |
| [31] | Hildreth, C. (1954). Point estimates of ordinates of concave functions. J. Amer. Statist. Assoc. 49 598-619. MathSciNet: MR0065093 · Zbl 0056.38301 |
| [32] | JUDITSKY, A. and NEMIROVSKI, A. (2002). On nonparametric tests of positivity/monotonicity/convexity. Ann. Statist. 30 498-527. Digital Object Identifier: 10.1214/aos/1021379863 Google Scholar: Lookup Link MathSciNet: MR1902897 · Zbl 1012.62048 · doi:10.1214/aos/1021379863 |
| [33] | KHMALADZE, È. V. (1981). A martingale approach in the theory of goodness-of-fit tests. Theory Probab. Appl. 26 240-257. · Zbl 0481.60055 |
| [34] | KOMAROVA, T. and HIDALGO, J. (2023). Supplement to “Testing nonparametric shape restrictions.” https://doi.org/10.1214/23-AOS2311SUPP |
| [35] | KOSTYSHAK, S. (2017). Non-parametric testing of U-shaped relationships. Working paper, University of Florida. |
| [36] | KOUL, H. L. and STUTE, W. (1999). Nonparametric model checks for time series. Ann. Statist. 27 204-236. Digital Object Identifier: 10.1214/aos/1018031108 Google Scholar: Lookup Link MathSciNet: MR1701108 · doi:10.1214/aos/1018031108 |
| [37] | LEE, J. and ROBINSON, P. M. (2016). Series estimation under cross-sectional dependence. J. Econometrics 190 1-17. Digital Object Identifier: 10.1016/j.jeconom.2015.08.001 Google Scholar: Lookup Link MathSciNet: MR3425269 · Zbl 1419.62515 · doi:10.1016/j.jeconom.2015.08.001 |
| [38] | LI, W., NAIK, D. and SWETITS, J. (1996). A data smoothing technique for piecewise convex/concave curves. SIAM J. Sci. Comput. 17 517-537. Digital Object Identifier: 10.1137/S1064827592239822 Google Scholar: Lookup Link MathSciNet: MR1374294 · Zbl 0841.62031 · doi:10.1137/S1064827592239822 |
| [39] | MAMMEN, E. (1991). Estimating a smooth monotone regression function. Ann. Statist. 19 724-740. Digital Object Identifier: 10.1214/aos/1176348117 Google Scholar: Lookup Link MathSciNet: MR1105841 · Zbl 0737.62038 · doi:10.1214/aos/1176348117 |
| [40] | Mammen, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741-759. Digital Object Identifier: 10.1214/aos/1176348118 Google Scholar: Lookup Link MathSciNet: MR1105842 · doi:10.1214/aos/1176348118 |
| [41] | MAMMEN, E. and THOMAS-AGNAN, C. (1999). Smoothing splines and shape restrictions. Scand. J. Stat. 26 239-252. Digital Object Identifier: 10.1111/1467-9469.00147 Google Scholar: Lookup Link MathSciNet: MR1707587 · doi:10.1111/1467-9469.00147 |
| [42] | Meyer, M. C. (2008). Inference using shape-restricted regression splines. Ann. Appl. Stat. 2 1013-1033. Digital Object Identifier: 10.1214/08-AOAS167 Google Scholar: Lookup Link MathSciNet: MR2516802 · Zbl 1149.62033 · doi:10.1214/08-AOAS167 |
| [43] | MUKERJEE, H. (1988). Monotone nonparameteric regression. Ann. Statist. 16 741-750. Digital Object Identifier: 10.1214/aos/1176350832 Google Scholar: Lookup Link MathSciNet: MR0947574 · doi:10.1214/aos/1176350832 |
| [44] | NICULESCU, C. P. (2003). Convexity according to means. Math. Inequal. Appl. 6 571-579. Digital Object Identifier: 10.7153/mia-06-53 Google Scholar: Lookup Link MathSciNet: MR2013520 · Zbl 1047.26010 · doi:10.7153/mia-06-53 |
| [45] | Ramsay, J. O. (1988). Monotone regression splines in action. Statist. Sci. 3 425-461. |
| [46] | SCHLEE, W. (1982). Nonparametric tests of the monotony and convexity of regression. In Nonparametric Statistical Inference, Vol. I, II (Budapest, 1980). Colloquia Mathematica Societatis János Bolyai 32 823-836. North-Holland, Amsterdam. MathSciNet: MR0719743 · Zbl 0525.62049 |
| [47] | SEN, P. K. (1982). Invariance principles for recursive residuals. Ann. Statist. 10 307-312. MathSciNet: MR0642743 · Zbl 0484.62098 |
| [48] | Stute, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613-641. Digital Object Identifier: 10.1214/aos/1031833666 Google Scholar: Lookup Link MathSciNet: MR1439316 · Zbl 0926.62035 · doi:10.1214/aos/1031833666 |
| [49] | STUTE, W., THIES, S. and ZHU, L.-X. (1998). Model checks for regression: An innovation process approach. Ann. Statist. 26 1916-1934. Digital Object Identifier: 10.1214/aos/1024691363 Google Scholar: Lookup Link MathSciNet: MR1673284 · doi:10.1214/aos/1024691363 |
| [50] | WANG, J. C. and MEYER, M. C. (2011). Testing the monotonicity or convexity of a function using regression splines. Canad. J. Statist. 39 89-107. Digital Object Identifier: 10.1002/cjs.10094 Google Scholar: Lookup Link MathSciNet: MR2815340 · Zbl 1349.62141 · doi:10.1002/cjs.10094 |
| [51] | Wright, F. T. (1981). The asymptotic behavior of monotone regression estimates. Ann. Statist. 9 443-448. MathSciNet: MR0606630 · Zbl 0471.62062 |
| [52] | Zhou, S., Shen, X. and Wolfe, D. A. (1998). Local asymptotics for regression splines and confidence regions. Ann. Statist. 26 1760-1782. Digital Object Identifier: 10.1214/aos/1024691356 Google Scholar: Lookup Link MathSciNet: MR1673277 · doi:10.1214/aos/1024691356 |
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