Wavelet-based estimators of partial derivatives of a multivariate density function for discrete stationary and ergodic processes

Sultana Didi; Salim Bouzebda; Sultana Didi; Salim Bouzebda

doi:10.3934/math.2025565

AIMS Mathematics

2025, Volume 10, Issue 5: 12519-12553. doi: 10.3934/math.2025565

Previous Article Next Article

Research article Special Issues

Wavelet-based estimators of partial derivatives of a multivariate density function for discrete stationary and ergodic processes

Sultana Didi ^{1
,},
Salim Bouzebda ^{2
,
,}

1.
Department of Statistics and Operations Research, College of Sciences, Qassim University, P.O. Box 6688, Buraydah 51452, Saudi Arabi; s.biha@qu.edu.sa
2.
Université de Technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne), CS 60 319 - 60 203 Compiègne Cedex, France

Received: 03 February 2025 Revised: 19 April 2025 Accepted: 25 April 2025 Published: 28 May 2025
MSC : 60G42, 60G46, 62G05, 62G07, 62G08, 62G20, 62H05

In this work, we propose a wavelet-based framework for estimating the derivatives of a density function in the setting of discrete, stationary, and ergodic processes. Our primary focus is the derivation of the integrated mean square error (IMSE) over compact subsets of $ \mathbb{R}^d $, which provides a quantitative measure of estimation accuracy. In addition, the uniform convergence with rate and the normality are established. To establish the asymptotic behavior of the proposed estimators, we adopt a martingale approach that accommodates the ergodic nature of the underlying processes. Importantly, beyond ergodicity, our analysis does not require additional assumptions on the data. By demonstrating that the wavelet methodology remains robust under these weaker dependence conditions, we extend earlier results originally developed in the context of independent observations.
- density estimation,
- stationarity,
- ergodicity,
- rates of strong convergence,
- wavelet-based estimators,
- martingale differences,
- discrete time,
- stochastic processes,
- time series
Citation: Sultana Didi, Salim Bouzebda. Wavelet-based estimators of partial derivatives of a multivariate density function for discrete stationary and ergodic processes[J]. AIMS Mathematics, 2025, 10(5): 12519-12553. doi: 10.3934/math.2025565

Related Papers:

Abstract

In this work, we propose a wavelet-based framework for estimating the derivatives of a density function in the setting of discrete, stationary, and ergodic processes. Our primary focus is the derivation of the integrated mean square error (IMSE) over compact subsets of $ \mathbb{R}^d $, which provides a quantitative measure of estimation accuracy. In addition, the uniform convergence with rate and the normality are established. To establish the asymptotic behavior of the proposed estimators, we adopt a martingale approach that accommodates the ergodic nature of the underlying processes. Importantly, beyond ergodicity, our analysis does not require additional assumptions on the data. By demonstrating that the wavelet methodology remains robust under these weaker dependence conditions, we extend earlier results originally developed in the context of independent observations.

References

[1]	C. R. Genovese, M. Perone-Pacifico, I. Verdinelli, L. Wasserman, Non-parametric inference for density modes, J. R. Stat. Soc. Ser. B Stat. Methodol., 78 (2016), 99–126. https://doi.org/10.1111/rssb.12111 doi: 10.1111/rssb.12111
[2]	H. Sasaki, Y. K. Noh, G. Niu, M. Sugiyama, Direct density derivative estimation, Neural Comput., 28 (2016), 1101–1140. https://doi.org/10.1162/NECO_a_00835 doi: 10.1162/NECO_a_00835
[3]	Y. K. Noh, M. Sugiyama, S. Liu, M. C. du Plessis, F. C. Park, D. D. Lee, Bias reduction and metric learning for nearest-neighbor estimation of Kullback-Leibler divergence, Neural Comput., 30 (2018), 1930–1960. https://doi.org/10.1162/neco_a_01092 doi: 10.1162/neco_a_01092
[4]	V. A. Vasiliev, A. V. Dobrovidov, G. M. Koshkin, Neparametricheskoe otsenivanie funktsionalov ot raspredeleniĭ statsionarnykh posledovatel' nosteĭ, Moscow: Nauka, 2004.
[5]	A. V. Dobrovidov, I. M. Ruds'ko, Bandwidth selection in nonparametric estimator of density derivative by smoothed cross-validation method, Autom. Remote Control, 71 (2010), 209–224. https://doi.org/10.1134/S0005117910020050 doi: 10.1134/S0005117910020050
[6]	C. R. Genovese, M. Perone-Pacifico, I. Verdinelli, L. Wasserman, On the path density of a gradient field, Ann. Statist., 37 (2009), 3236–3271. https://doi.org/10.1214/08-AOS671 doi: 10.1214/08-AOS671
[7]	R. S. Singh, Applications of estimators of a density and its derivatives to certain statistical problems, J. R. Stat. Soc. Ser. B Stat. Methodol., 39 (1977), 357–363. https://doi.org/10.1111/j.2517-6161.1977.tb01635.x doi: 10.1111/j.2517-6161.1977.tb01635.x
[8]	T. G. Meyer, Bounds for estimation of density functions and their derivatives, Ann. Statist., 5 (1977), 136–142.
[9]	B. W. Silverman, Weak and strong uniform consistency of the kernel estimate of a density and its derivatives, Ann. Statist., 6 (1978), 177–184. https://doi.org/10.1214/aos/1176344076 doi: 10.1214/aos/1176344076
[10]	R. S. Singh, Mean squared errors of estimates of a density and its derivatives, Biometrika, 66 (1979), 177–180. https://doi.org/10.2307/2335262 doi: 10.2307/2335262
[11]	W. Härdle, G. Kerkyacharian, D. Picard, A. Tsybakov, Wavelets, approximation, and statistical applications, In: Lecture notes in statistics, New York: Springer, 1998. https://doi.org/10.1007/978-1-4612-2222-4
[12]	B. L. S. P. Rao, Nonparametric estimation of the derivatives of a density by the method of wavelets, Bull. Inform. Cybernet., 28 (1996), 91–100. https://doi.org/10.5109/13457 doi: 10.5109/13457
[13]	Y. P. Chaubey, H. Doosti, B. L. S. P. Rao, Wavelet based estimation of the derivatives of a density with associated variables, Int. J. Pure Appl. Math., 27 (2006), 97–106.
[14]	B. L. S. P. Rao, Nonparametric estimation of partial derivatives of a multivariate probability density by the method of wavelets, In: Asymptotics in statistics and probability, 2000,321–330. https://doi.org/10.1515/9783110942002-022
[15]	N. Hosseinioun, H. Doosti, H. A. Niroumand, Nonparametric estimation of a multivariate probability density for mixing sequences by the method of wavelets, Ital. J. Pure Appl. Math., 28 (2011), 31–40.
[16]	G. Koshkin, V. Vasil'iev, An estimation of a multivariate density and its derivatives by weakly dependent observations, In: Statistics and control of stochastic processes, Singapore: World Scientific, 1997,229–241.
[17]	B. L. S. P. Rao, Wavelet estimation for derivative of a density in the presence of additive noise, Braz. J. Probab. Stat., 32 (2018), 834–850.
[18]	Y. Meyer, Wavelets and operators, Cambridge University Press, 1992.
[19]	I. Daubechies, Ten lectures on wavelets, In: CBMS-NSF Regional conference series in applied mathematics, Philadelphia: Society for Industrial and Applied Mathematics (SIAM), 1992.
[20]	R. A. DeVore, G. G. Lorentz, Constructive approximation, In: Grundlehren der mathematischen Wissenschaften, Springer Science & Business Media, 303 (1993).
[21]	D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, D. Picard, Density estimation by wavelet thresholding, Ann. Statist., 24 (1996), 508–539.
[22]	C. Schneider, Beyond Sobolev and Besov: Regularity of solutions of PDEs and their traces in function spaces, In: Lecture notes in mathematics, Cham: Springer, 2291 (2021). https://doi.org/10.1007/978-3-030-75139-5
[23]	Y. Sawano, Theory of Besov spaces, In: Developments in mathematics, Singapore: Springer, 56 (2018). https://doi.org/10.1007/978-981-13-0836-9
[24]	J. Peetre, New thoughts on Besov spaces, Duke University, 1976.
[25]	J. Bergh, J. Löfström, Interpolation spaces: An introduction, Springer Science & Business Media, 2012.
[26]	U. Krengel, Ergodic theorems, In: De Gruyter studies in mathematics, Walter de Gruyter, 6 (2011).
[27]	E. Masry, Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series, Stochastic Process. Appl., 67 (1997), 177–193. https://doi.org/10.1016/S0304-4149(96)00005-1 doi: 10.1016/S0304-4149(96)00005-1
[28]	S. Bouzebda, I. Elhattab, A strong consistency of a nonparametric estimate of entropy under random censorship, C. R. Math. Acad. Sci. Paris, 347 (2009), 821–826. https://doi.org/10.1016/j.crma.2009.04.021 doi: 10.1016/j.crma.2009.04.021
[29]	R. E. Bellman, Adaptive control processes: A guided tour, Princeton University Press, 1961.
[30]	D. W. Scott, M. P. Wand, Feasibility of multivariate density estimates, Biometrika, 78 (1991), 197–205.
[31]	S. Bouzebda, Weak convergence of the conditional single index $U$-statistics for locally stationary functional time series, AIMS Mathematics, 9 (2024), 14807–14898. https://doi.org/10.3934/math.2024720 doi: 10.3934/math.2024720
[32]	S. Attaoui, B. Bentata, S. Bouzebda, A. Laksaci, The strong consistency and asymptotic normality of the kernel estimator type in functional single index model in presence of censored data, AIMS Mathematics, 9 (2024), 7340–7371. https://doi.org/10.3934/math.2024356 doi: 10.3934/math.2024356
[33]	Z. C. Elmezouar, F. Alshahrani, I. M. Almanjahie, S. Bouzebda, Z. Kaid, A. Laksaci, Strong consistency rate in functional single index expectile model for spatial data, AIMS Mathematics, 9 (2024), 5550–5581. https://doi.org/10.3934/math.2024269 doi: 10.3934/math.2024269
[34]	S. Bouzebda, Limit theorems in the nonparametric conditional single-index u-processes for locally stationary functional random fields under stochastic sampling design, Mathematics, 12 (2024), 1996. https://doi.org/10.3390/math12131996 doi: 10.3390/math12131996
[35]	S. Bouzebda, S. Didi, Multivariate wavelet density and regression estimators for stationary and ergodic discrete time processes: asymptotic results, Comm. Statist. Theory Methods, 46 (2017), 1367–1406. https://doi.org/10.1080/03610926.2015.1019144 doi: 10.1080/03610926.2015.1019144
[36]	A. Mokkadem, M. Pelletier, The law of the iterated logarithm for the multivariable kernel mode estimator, ESAIM Probab. Stat., 7 (2003), 1–21. https://doi.org/10.1051/ps:2003004 doi: 10.1051/ps:2003004
[37]	S. Allaoui, S. Bouzebda, C. Chesneau, J. Liu, Uniform almost sure convergence and asymptotic distribution of the wavelet-based estimators of partial derivatives of multivariate density function under weak dependence, J. Nonparametr. Stat., 33 (2021), 170–196. https://doi.org/10.1080/10485252.2021.1925668 doi: 10.1080/10485252.2021.1925668
[38]	M. Rosenblatt, Uniform ergodicity and strong mixing, Z. Wahrscheinlichkeitstheorie verw Gebiete, 24 (1972), 79–84. https://doi.org/10.1007/BF00532465 doi: 10.1007/BF00532465
[39]	S. Bouzebda, S. Didi, L. El Hajj, Multivariate wavelet density and regression estimators for stationary and ergodic continuous time processes: Asymptotic results, Math. Meth. Stat., 24 (2015), 163–199. https://doi.org/10.3103/S1066530715030011 doi: 10.3103/S1066530715030011
[40]	S. Bouzebda, S. Didi, Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes, Rev. Mat. Complut., 34 (2021), 811–852. https://doi.org/10.1007/s13163-020-00368-6 doi: 10.1007/s13163-020-00368-6
[41]	R. C. Bradley, Introduction to strong mixing conditions, Kendrick Press, 2007.
[42]	A. Leucht, M. H. Neumann, Degenerate $U$- and $V$-statistics under ergodicity: Asymptotics, bootstrap and applications in statistics, Ann. Inst. Statist. Math., 65 (2013), 349–386. https://doi.org/10.1007/s10463-012-0374-9 doi: 10.1007/s10463-012-0374-9
[43]	M. H. Neumann, Absolute regularity and ergodicity of Poisson count processes, Bernoulli, 17 (2011), 1268–1284. https://doi.org/10.3150/10-BEJ313 doi: 10.3150/10-BEJ313
[44]	S. Bouzebda, M. Chaouch, Uniform limit theorems for a class of conditional $Z$-estimators when covariates are functions, J. Multivariate Anal., 189 (2022), 104872. https://doi.org/10.1016/j.jmva.2021.104872 doi: 10.1016/j.jmva.2021.104872
[45]	S. Bouzebda, A. Nezzal, T. Zari, Uniform consistency for functional conditional $U$-statistics using delta-sequences, Mathematics, 11 (2023), 161. https://doi.org/10.3390/math11010161 doi: 10.3390/math11010161
[46]	S. Didi, S. Bouzebda, Wavelet density and regression estimators for continuous time functional stationary and ergodic processes, Mathematics, 10 (2022), 4356. https://doi.org/10.3390/math10224356 doi: 10.3390/math10224356
[47]	P. Hall, J. S. Marron, Estimation of integrated squared density derivatives, Statist. Probab. Lett., 6 (1978), 109–115. https://doi.org/10.1016/0167-7152(87)90083-6 doi: 10.1016/0167-7152(87)90083-6
[48]	J. Jurečková, Asymptotic linearity of a rank statistic in regression parameter, Ann. Math. Statist., 40 (1969), 1889–1900. https://doi.org/10.1214/aoms/1177697273 doi: 10.1214/aoms/1177697273
[49]	E. Giné, D. M. Mason, Uniform in bandwidth estimation of integral functionals of the density function, Scand. J. Statist., 35 (2008), 739–761. https://doi.org/10.1111/j.1467-9469.2008.00600.x doi: 10.1111/j.1467-9469.2008.00600.x
[50]	B. Ya. Levit, Asymptotically efficient estimation of nonlinear functionals, Probl. Peredachi Inf., 14 (1978), 65–72.
[51]	E. Masry, Wavelet-based estimation of multivariate regression functions in Besov spaces, J. Nonparametr. Statist., 12 (2000), 283–308. https://doi.org/10.1080/10485250008832809 doi: 10.1080/10485250008832809
[52]	S. Allaoui, S. Bouzebda, J. Liu, Multivariate wavelet estimators for weakly dependent processes: Strong consistency rate, Comm. Statist. Theory Methods, 52 (2023), 8317–8350. https://doi.org/10.1080/03610926.2022.2061715 doi: 10.1080/03610926.2022.2061715
[53]	S. Bouzebda, Limit theorems for wavelet conditional $U$-statistics for time series models, Math. Methods Statist., 34 (2025), 181–225.
[54]	P. Hall, S. Penev, Cross-validation for choosing resolution level for nonlinear wavelet curve estimators, Bernoulli, 7 (2001), 317–341.
[55]	M. Chaouch, N. Laïb, Regression estimation for continuous-time functional data processes with missing at random response, J. Nonparametr. Stat., 37 (2024), 1–32. https://doi.org/10.1080/10485252.2024.2332686 doi: 10.1080/10485252.2024.2332686
[56]	L. Giraitis, R. Leipus, A generalized fractionally differencing approach in long-memory modeling, Lith. Math. J., 35 (1995), 53–65. https://doi.org/10.1007/BF02337754 doi: 10.1007/BF02337754
[57]	D. Guégan, S. Ladoucette, Non-mixing properties of long memory processes, C. R. Acad. Sci., 333 (2001), 373–376. https://doi.org/10.1016/S0764-4442(01)02052-3 doi: 10.1016/S0764-4442(01)02052-3
[58]	D. W. K. Andrews, Non-strong mixing autoregressive processes, J. Appl. Probab., 21 (1984), 930–934. https://doi.org/10.2307/3213710 doi: 10.2307/3213710
[59]	C. Francq, J. M. Zakoïan, GARCH models: Structure, statistical inference and financial applications, John Wiley & Sons, Ltd., 2010. https://doi.org/10.1002/9780470670057
[60]	S. Bouzebda, N. Limnios, On general bootstrap of empirical estimator of a semi-Markov kernel with applications, J. Multivariate Anal., 116 (2013), 52–62. https://doi.org/10.1016/j.jmva.2012.11.008 doi: 10.1016/j.jmva.2012.11.008
[61]	S. Bouzebda, C. Papamichail, N. Limnios, On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications, J. Nonparametr. Stat., 30 (2018), 49–86. https://doi.org/10.1080/10485252.2017.1404059 doi: 10.1080/10485252.2017.1404059
[62]	S. Bouzebda, On the weak convergence and the uniform-in-bandwidth consistency of the general conditional $U$-processes based on the copula representation: Multivariate setting, Hacet. J. Math. Stat., 52 (2023), 1303–1348. https://doi.org/10.15672/hujms.1134334 doi: 10.15672/hujms.1134334
[63]	S. Bouzebda, General tests of conditional independence based on empirical processes indexed by functions, Jpn. J. Stat. Data Sci., 6 (2023), 115–177. https://doi.org/10.1007/s42081-023-00193-3 doi: 10.1007/s42081-023-00193-3
[64]	D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab., 1 (1973), 19–42.
[65]	V. H. Peña, E. Giné, Decoupling, In: Probability and its applications, New York: Springer, 2012. https://doi.org/10.1007/978-1-4612-0537-1
[66]	M. A. Kon, L. A. Raphael, Characterizing convergence rates for multiresolution approximations, Wavelet Anal. Appl., 7 (1998), 415–437. https://doi.org/10.1016/S1874-608X(98)80016-2 doi: 10.1016/S1874-608X(98)80016-2
[67]	P. Hall, C. C. Heyde, Martingale limit theory and its application, In: probability and mathematical statistics: A series of monographs and textbooks, Academic Press, 1980. https://doi.org/10.1016/C2013-0-10818-5
[68]	Y. S. Chow, H. Teicher, Probability theory: Independence, interchangeability, Martingales, Springer Science & Business Media, 2012.
[69]	M. Schipper, Optimal rates and constants in $L_2$-minimax estimation of probability density functions, Math. Methods Statist., 5 (1996), 253–274.
[70]	S. Efromovich, Adaptive estimation of and oracle inequalities for probability densities and characteristic functions, Ann. Statist., 36 (2008), 1127–1155. https://doi.org/10.1214/009053607000000965 doi: 10.1214/009053607000000965
[71]	S. Efromovich, Lower bound for estimation of Sobolev densities of order less ${1\over2}$, J. Statist. Plann. Inference, 139 (2009), 2261–2268. https://doi.org/10.1016/j.jspi.2008.10.017 doi: 10.1016/j.jspi.2008.10.017
[72]	H. Triebel, Theory of function spaces, In: Modern birkhäuser classics, Birkhäuser Basel, 2010. https://doi.org/10.1007/978-3-0346-0416-1
[73]	G. Bourdaud, M. L. de Cristoforis, W. Sickel, Superposition operators and functions of bounded $p$-variation, Rev. Mat. Iberoam., 22 (2006), 455–487. https://doi.org/10.4171/rmi/463 doi: 10.4171/rmi/463
[74]	R. M. Dudley, R. Norvaiša, Differentiability of six operators on nonsmooth functions and $p$-variation, In: Lecture notes in mathematics, Heidelberg: Springer Berlin, 1703 (1999). https://doi.org/10.1007/BFb0100744
[75]	D. Geller, I. Z. Pesenson, Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds, J. Geom. Anal., 21 (2011), 334–371. https://doi.org/10.1007/s12220-010-9150-3 doi: 10.1007/s12220-010-9150-3

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)