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Functionals of order statistics and their multivariate concomitants with application to semiparametric estimation by nearest neighbours. (English) Zbl 1282.62128
Summary: This paper studies the limiting behavior of general functionals of order statistics and their multivariate concomitants for weakly dependent data. The asymptotic analysis is performed under a conditional moment-based notion of dependence for vector-valued time series. It is argued, through analysis of various examples, that the dependence conditions of this type can be effectively implied by other dependence formations recently proposed in time-series analysis, and thus it may cover many existing linear and nonlinear processes. The utility of this result is then illustrated in deriving the asymptotic properties of a semiparametric estimator that uses the \(k\)-nearest neighbour estimator of the inverse of a multivariate unknown density. This estimator is then used to calculate the consumer surplus for electricity demand in Ontario for the period 1971 to 1994. A Monte Carlo experiment also assesses the efficacy of the derived limiting behavior in finite samples for both these general functionals and the proposed estimator.

MSC:
62G30 Order statistics; empirical distribution functions
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
62G05 Nonparametric estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P20 Applications of statistics to economics
65C05 Monte Carlo methods
Software:
np
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References:
[1] Andrews, D.W.K. (1984). Nonstrong mixing autoregressive processes. J. Appl. Probab., 21, 930–934. · Zbl 0552.60049
[2] Apostol, T.M. (1969). Multi-variable calculus and linear algebra, with applications to differential equations and probability, 2nd edn. Wiley & Sons, New York, USA. · Zbl 0185.11402
[3] Arnold, B.C., Castillo, E. and Sarabia, J.M. (2009). Multivariate order statistics via multivariate concomitants. J. Multivariate Anal., 100, 946–951. · Zbl 1167.62046
[4] Barnett, V., Green, P.J. and Robinson, A. (1976). Concomitants and correlation estimates. Biometrika, 63, 323–328. · Zbl 0329.62042
[5] Bhattacharya, P.K. and Mack, Y.P. (1987). Weak convergence of k-nn density and regression estimators with varying k and applications. Ann. Statist., 15, 976–994. · Zbl 0643.62027
[6] Bickel, P.J. and Bühlmann, P. (1999). A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli, 5, 413–446. · Zbl 0954.62102
[7] Boente, G. and Fraiman, R. (1988). Consistency of a nonparametric estimate of a density function for dependent variables. J. Multivariate Anal., 25, 90–99. · Zbl 0664.62038
[8] Boente, G. and Fraiman, R. (1990). Asymptotic distribution of robust estimators for nonparametric models from mixing processes. Ann. Statist., 18, 891–906. · Zbl 0703.62025
[9] Bradley, R. (1986). Basic Properties of Strong Mixing Conditions. In Dependence in Probability and Statistics (E. Eberlein and M. Taqqu, eds.). Progress in Probability and Statistics, Birkhäuser, Boston, pp. 165–192. · Zbl 0603.60034
[10] Bradley, R.C. (2007). Introduction to strong mixing conditions, vols. I, II, III. Kendrick Press, Utah, USA.
[11] Carroll, R.J., Fan, J., Gijbels, I. and Wand, M.P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc., 92, 477–489. · Zbl 0890.62053
[12] Chow, Y.S. and Teicher, H. (1978). Probability theory: independence, interchangeability, martingales, 1st edn. Springer-Verlag New York, Inc. · Zbl 0399.60001
[13] Chu, B.M. and Jacho-Chávez, D.T. (2012). k-Nearest neighbour estimation of inverse-density-weighted expectations with dependent data. Econometric Theory, 28, 769–803. · Zbl 1419.62072
[14] Csörgö, S. (1981). Empirical Characteristic Functions. In Carleton Mathematical Lecture Notes, vol. 26. Carleton University, Ottawa. · Zbl 0438.60025
[15] David, H.A. and Galambos, J. (1974). The asymptotic theory of concomitants of order statistics. J. Appl. Probab., 11, 762–770. · Zbl 0299.62026
[16] David, H.A. and Nagaraja, H.N. (1998). Concomitants of Order Statistics. In Handbook of Statistics (N. Balakrishnan and C.R. Rao, eds.). Order Statistics: Theory and Methods, Elsevier Science, vol. 16, pp. 487–513. · Zbl 0905.62055
[17] Davidson, J. (1994). Stochastic limit theory. Oxford University Press, Oxford, New York.
[18] Davydov, Y.A. (1968). Convergence of distributions generated by stationary stochastic processes. Theory Probab. Appl., 13, 691–696. · Zbl 0181.44101
[19] Dedecker, J. and Prieur, C. (2005). New dependence coefficients. Examples and applications to statistics. Probab. Theory Related Fields, 132, 203–236. · Zbl 1061.62058
[20] Dedecker, J. and Prieur, C. (2007). An empirical central limit theorem for dependent sequences. Stochastic Process. Appl., 117, 121–142. · Zbl 1117.60035
[21] Devroye, L. and Lugosi, G. (2001). Combinatorial methods in density estimation. Springer-Verlag, New York, USA. · Zbl 0964.62025
[22] Engle, R.F., Granger, C.W.J., Rice, J. and Weiss, A. (1986). Semiparametric estimates of the relation between weather and electricity sales. J. Amer. Statist. Assoc., 81, 310–320.
[23] Francq, C. and Zakoïan, J.-M. (2010). GARCH models: structure, statistical inference and financial applications. John Wiley & Sons, West Sussex, UK.
[24] Gordin, M.I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR, 188, 739–741. · Zbl 0212.50005
[25] Hall, P. and Yatchew, A. (2005). Unified approach to testing functional hypotheses in semiparametric contexts. J. Econometrics, 127, 225–252. · Zbl 1334.62080
[26] Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models. Ann. Statist., 21, 157–178. · Zbl 0770.62049
[27] Härdle, W. and Stoker, T.M. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc., 84, 986–995. · Zbl 0703.62052
[28] Hart, J.D. and Vieu, P. (1990). Data-driven bandwidth choice for density estimation based on dependent data. Ann. Statist., 18, 873–890. · Zbl 0703.62045
[29] Hausman, J.A. and Newey, W.K. (1995). Nonparametric estimation of exact consumers surplus and deadweight loss. Econometrica, 63, 1445–1476. · Zbl 0840.90044
[30] Hayfield, T. and Racine, J.S. (2008). Nonparametric econometrics: the np package. J. Stat. Soft, 27, 1–32.
[31] Hong, Y. and White, H. (2005). Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica, 73, 837–901. · Zbl 1152.91729
[32] Huynh, K.P. and Jacho-Chávez, D.T. (2009). Growth and governance: a nonparametric analysis. J. Comp. Econ., 37, 121–143. · Zbl 1190.91120
[33] Ibragimov, I.A. (1962). Some limit theorems for stationary processes. Theory Probab. Appl., 7, 349–382. · Zbl 0119.14204
[34] Ichimura, H. (1993). Semiparametric Least Squares (SLS) and weighted SLS estimation of single index models. J. Econometrics, 58, 71–120. · Zbl 0816.62079
[35] Jacho-Chávez, D.T. (2008). k nearest-neighbor estimation of inverse density weighted expectations. Econ. Bull., 3, 1–6. · Zbl 1419.62072
[36] Khaledi, B.-E. and Kochar, S. (2000). Stochastic comparisons and dependence among concomitants of order statistics. J. Multivariate Anal., 73, 262–281. · Zbl 0953.62048
[37] Koroljuk, V.S. and Borovskich, Y.V. (1994). Theory of U -statistics. Kluwer Academic Publishers, Dordrecht/Boston/London.
[38] Lewbel, A. (1998). Semiparametric latent variable model estimation with endogenous or mismeasured regressors. Econometrica, 66, 105–121. · Zbl 1055.62574
[39] Lewbel, A. and Schennach, S.M. (2007). A simple ordered data estimator for inverse density weighted expectations. J. Econometrics, 136, 189–211. · Zbl 1418.62138
[40] Li, J. and Tran, L.T. (2009). Nonparametric estimation of conditional expectation. J. Statist. Plann. Inference, 139, 164–175. · Zbl 1149.62078
[41] Loftsgaarden, D.O. and Quesenberry, C.P. (1965). A nonparametric estimate of a multivariate density function. Ann. Math. Statist., 36, 1049–1051. · Zbl 0132.38905
[42] Lu, X., Lian, H. and Liu, W. (2012). Semiparametric estimation for inverse density weighted expectations when responses are missing at random. J. Nonparametr. Stat., 24, 139–159. · Zbl 1241.62044
[43] Mack, Y.P. and Rosenblatt, M. (1979). Multivariate k-nearest neighbor density estimates. J. Multivariate Anal., 9, 1–15. · Zbl 0406.62023
[44] Moore, D.S. and Yackel, J.W. (1977a). Consistency properties of nearest neighbor density function estimators. Ann. Statist., 5, 143–154. · Zbl 0358.60053
[45] Moore, D.S. and Yackel, J.W. (1977b). Large Sample Properties of Nearest Neighbor Density Estimators. In Statistical Decision Theory and Related Topics II (S.S. Gupta and D.S. Moore, eds.). Academic Press, New York. · Zbl 0419.62036
[46] Nagaraja, H.N. and David, H.A. (1994). Distribution of the maximum of concomitants of selected order statistics. Ann. Statist., 22, 478–494. · Zbl 0795.62010
[47] Peligrad, M., Utev, S. and Wu, W.B. (2007). A maximal $\(\backslash\)mathbb{L}_{p}$ -inequality for stationary sequences and its applications. Proc. Amer. Math. Soc., 135, 541–550. · Zbl 1107.60011
[48] Pham, T.D. (1986). The mixing property of bilinear and generalized random coefficient autoregressive models. Stochastic Process. Appl., 23, 291–300. · Zbl 0614.60062
[49] Pham, T.D. and Tran, L.T. (1985). Some mixing properties of time series models. Stochastic Process. Appl., 13, 297–303. · Zbl 0564.62068
[50] Philipp, W. and Stout, W.F. (1975). Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc., 2. · Zbl 0361.60007
[51] Priestley, M. (1988). Nonlinear and Nonstationary Time Series Analysis. Academic Press. · Zbl 0667.62068
[52] Puri, M.L. and Tran, L.T. (1980). Empirical distribution functions and functions of order statistics for mixing random variables. J. Multivariate Anal., 10, 405–425. · Zbl 0466.62043
[53] Rao, B.L.S.P. (2009). Conditional independence, conditional mixing and conditional association. Ann. Inst. Statist. Math., 61, 441–460. · Zbl 1314.60054
[54] Rinott, Y. and Rotar, V. (1999). Some bounds on the rate of convergence in the CLT for martingales. I. Theory Probab. Appl., 43, 604–619. · Zbl 0963.60017
[55] Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. In Collection Mathématiques & Applications, vol. 31. Springer, Berlin.
[56] Robinson, P.M. (1987). Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica, 55, 875–891. · Zbl 0651.62107
[57] Robinson, P.M. (1995). Nearest-neighbour estimation of semiparametric regression models. J. Nonparametr. Stat., 5, 33–41. · Zbl 0873.62043
[58] Rosenblatt, M. (1956a). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA, 42, 43–47. · Zbl 0070.13804
[59] Rosenblatt, M. (1956b). Remarks on some non-parametric estimates of a density function. Ann. Math. Statist., 27, 832–837. · Zbl 0073.14602
[60] Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York. · Zbl 0538.62002
[61] Shiryaev, A.N. (1996). Probability, 2nd edn. Springer, New York, USA. · Zbl 0909.01009
[62] Stokes, S.L. (1977). Ranked set sampling with concomitant variables. Comm. Statist. Theory Methods, 6, 1207–1211.
[63] Stute, W. (1984). Asymptotic normality of nearest neighbor regression function estimates. Ann. Statist., 12, 917–926. · Zbl 0539.62026
[64] Stute, W. (1993). U-functions of concomitants of order statistics. Probab. Math. Statist., 14, 143–155. · Zbl 0806.60020
[65] Tong, H. (1993). Non-linear time series: a dynamical system approach. Oxford University Press, Oxford, UK.
[66] Tran, L.T. and Wu, B. (1993). Order statistics for nonstationary time series. Ann. Inst. Statist. Math., 45, 665–686. · Zbl 0801.62047
[67] Tran, L.T. and Yakowitz, S. (1993). Nearest neighbor estimators for random fields. J. Multivariate Anal., 44, 23–46. · Zbl 0764.62076
[68] Truong, Y.K. and Stone, C.J. (1992). Nonparametric function estimation involving time series. Ann. Statist., 20, 77–97. · Zbl 0764.62038
[69] Varadhan, S.R.S. (2001). Probability theory. American Mathematical Society, Courant Institute of Mathematical Sciences, New York, USA. · Zbl 0980.60002
[70] Watterson, G.A. (1958). Linear estimation in censored samples from multivariate normal populations. Ann. Math. Statist., 30, 814–824. · Zbl 0090.11401
[71] Wheeden, R.L. and Zygmund, A. (1977). Measure and integral. Dekker, New York. · Zbl 0362.26004
[72] White, H. and Domowitz, I. (1984). Nonlinear regression with dependent observations. Econometrica, 52, 143–161. · Zbl 0533.62055
[73] Wu, B. (1988). On Order Statistics in Time Series Analysis. Ph.D. Thesis, Indiana University.
[74] Wu, W.B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA, 102, 14150–14154. · Zbl 1135.62075
[75] Wu, W.B. (2007). Strong invariance principles for dependent random variables. Ann. Probab., 35, 2294–2320. · Zbl 1166.60307
[76] Wu, W.B. and Woodroofe, M. (2004). Martingale approximations for sums of stationary processes. Ann. Probab., 32, 1674–1690. · Zbl 1057.60022
[77] Yang, S.S. (1977). General distribution theory of the concomitants of order statistics. Ann. Statist., 5, 996–1002. · Zbl 0367.62017
[78] Yang, S.S. (1981a). Linear combinations of concomitants of order statistics with application to testing and estimation. Ann. Inst. Statist. Math., 33, 463–470. · Zbl 0478.62036
[79] Yang, S.S. (1981b). Linear functions of concomitants of order statistics with application to nonparametric estimation of a regression function. J. Amer. Statist. Assoc., 76, 658–662. · Zbl 0475.62031
[80] Yatchew, A. (2003). Semiparametric Regression for the Applied Econometrician. In Themes in Modern Econometrics, 1st edn. Cambridge University Press, Cambridge, UK. · Zbl 1067.62041
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