Abstract
This paper discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives in the presence of long memory in the common covariate and errors. The proposed test is based on a marked empirical process of the differences between the response variables. We discuss asymptotic null distribution of this process and consistency of the test for a class of general alternatives. We also conduct a Monte Carlo simulation study to evaluate the finite sample level and power behavior of the test at some alternatives.
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References
Beran J (1994) Statistics for long-memory processes, vol 61. Monographs on statistics and applied probability. Chapman and Hall, London
Beran J, Feng Y, Ghosh S, Kulik R (2013) Long-memory processes. Probabilistic properties and statistical methods. Springer, Heidelberg
Cox D, Koh E, Wahba G, Yandell BS (1988) Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann Stat 16:113–119
Dahlhaus R (1989) Efficient parameter estimation for self-similar processes. Ann Stat 17:1749–1766
Dalla V, Giraitis L, Hidalgo J (2006) Consistent estimation of the memory parameter for nonlinear time series. J Time Ser Anal 27:211–251
Davies RB, Harte DS (1987) Tests for hurst effect. Biometrika 74:95–101
Davydov YA (1970) The invariance principle for stationary processes. Theory Probab Appl 15:487–498
Delgado MA (1993) Testing the equality of nonparametric regression curves. Stat Probab Lett 17:199–204
Eubank RL, Spigelman CH (1990) Testing the goodness-of-fit of a linear model via nonparametric regression techniques. J Am Stat Assoc 85:387–392
Ferreira E, Stute W (2004) Testing for difference between conditional means in a time series context. J Am Stat Assoc 99:169–174
Fox R, Taqqu MS (1986) Large-sample properties of parameter estimates for strongly dependent stationary gaussian time series. Ann Stat 14:517–532
Giraitis L, Surgailis D (1999) Central limit theorem for the empirical process of a linear sequence with long memory. J Stat Plan Inference 80:81–93
Giraitis L, Koul HL, Surgailis D (2012) Large sample inference for long memory processes. Imperial College Press, London
Gonzalez-Manteiga W, Crujeiras RM (2013) An updated review of goodness-of-fit tests for regression models. TEST 22(3):361–411
Guo H, Koul HL (2008) Asymptotic inference in some heteroscedastic regression models with long memory design and errors. Ann Stat 36(1):458–487
Hall P, Hart JD (1990) Bootstrap test for difference between means in nonparametric regression. J Am Stat Assoc 85:1039–1049
Hall P, Huber C, Speckman PL (1997) Covariate-matched one-sided tests for the difference between functional means. J Am Stat Assoc 92:1074–1083
Härdle W, Mammen E (1993) Comparing nonparametric versus parametric regression fits. Ann Stat 21:1926–1947
Härdle W, Marron JS (1990) Semiparametric comparisons of regression curves. Ann Stat 18:63–89
Hart JD (1997) Nonparametric smoothing and lack-of-fit tests. Springer series in statistics. Springer, New York
Hidalgo J, Robinson PM (2002) Adapting to unknown disturbance autocorrelation in regression with long memory. Econom Econom Soc 70(4):1545–1581
Ho HC, Hsing T (1996) On the asymptotic expansion of the empirical process of long memory moving averages. Ann Stat 24:992–1024
Ho HC, Hsing T (1997) Limit theorems for functions of moving averages. Ann Probab 25:1636–1669
Koul HL, Ni P (2004) Minimum distance regression model checking. J Stat Plan Inference 119(1):109–142
Koul HL, Schick A (1997) Testing the equality of two nonparametric regression curves. J Stat Plan Inference 65:293–314
Koul HL, Schick A (2003) Testing for superiority among two regression curves. J Stat Plan Inference 117:15–33
Koul HL, Stute W (1998) Regression model fitting with long memory errors. J Stat Plan Inference 71:35–56
Koul HL, Surgailis D (1997) Asymptotic expansion of M-estimators with long memory errors. Ann Stat 25:818–850
Koul HL, Surgailis D (2002) Asymptotic expansion of the empirical process of long memory moving averages. In: Dehling H, Mikosch T, Sörensen M (eds) Empirical process techniques for dependent data. Birkhäuser, Boston, pp 213–239
Koul HL, Baillie RT, Surgailis D (2004) Regression Model fitting with a long memory covariate process. Econom Theory 20:485–512
Koul HL, Surgailis D, Mimoto N (2015) Minimum distance lack-of-fit tests under long memory errors. Metrika 78(2):119–143
Kulasekera KB (1995) Comparison of regression curves using quasi-residuals. J Am Stat Assoc 90:1085–1093
Li F (2006) Testing for the equality of two nonparametric regression curves with long memory errors. Commun Stat Simul Comput 35(3):621–643
Mielniczuk J, Wu WB (2004) On random-design model with dependent errors. Stat Sin 14(4):1105–1126
Raz J (1990) Testing for no effect when estimating a smooth function by nonparametric regression: a randomization approach. J Am Stat Assoc 85:132–138
Robinson PM (1995a) Log-periodogram regression of time series with long range dependence. Ann Stat 23:1048–1072
Robinson PM (1995b) Gaussian semiparametric estimation of long range dependence. Ann Stat 23:1630–1661
Robinson PM (1997) Large-sample inference for nonparametric regression with dependent errors. Ann Stat 25:2054–2083
Scheike TH (2000) Comparison of non-parametric regression functions through their cumulatives. Stat Probab Lett 46:21–32
Zygmund A (1968) Trigonometric series. Cambridge University Press, Cambridge
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Koul, H.L., Li, F. Comparing two nonparametric regression curves in the presence of long memory in covariates and errors. Metrika 83, 499–517 (2020). https://doi.org/10.1007/s00184-019-00735-4
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DOI: https://doi.org/10.1007/s00184-019-00735-4