Abstract
A solution to the problem of useful random signal extraction from a mixture with a noise in the multiplicative observation model is proposed. Unlike conventional filtering tasks, in the problem under consideration it is supposed that the distribution (and the model) of the useful signal is unknown. Therefore, in this case one cannot apply such well-known techniques like Kalman filter or posterior Stratonovich-Kushner evolution equation. The new paper is a continuation and development of the author’s article, reported at the First ISNPS Conference (Halkidiki’2012), where the filtering problem of positive signal with the unknown distribution had been solved using the generalized filtering equation and nonparametric kernel techniques. In the present study, new findings are added concerning the construction of stable procedures for filtering, the search for optimal smoothing parameter in the multidimensional case and some of the convergence results of the proposed techniques. The main feature of the problem is the positive distribution support. In this case, the classical methods of nonparametric estimation with symmetric kernels are not applicable because of large estimator bias at the support boundary. To overcome this drawback, we use asymmetric gamma kernel functions. To have stable estimators, we propose a regularization procedure with a data-driven optimal regularization parameter. Similar filtering algorithms can be used, for instance, in the problems of volatility estimation in statistical models of financial and actuarial mathematics.
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- 1.
For brevity, in the future the arguments of some functions are omitted.
References
Dobrovidov Alexander V., Koshkin Gennady M., Vasiliev Vyacheslav A.: Non-Parametric State Spase Models. Kendrick Press, USA (2012)
Lehmann, E.L.: Testing Statistical Hypotheses. Wiley, N.Y. (1959)
Dobrovidov, A.V.: Nonparametric methods of nonlinear filtering of stationary random sequences. Automat. Remote Control 44(6), 757–768 (1983)
Pensky, M.: A general approach to nonparametric empirical Bayes estimation. Statistics 29, 61–80 (1997)
Pensky, M., Singh, R.S.: Empirical Bayes estimation of reliability characteristics for an exponential family. Can. J. Stat. 27, 127–136 (1999)
Markovich, L.A.: The equation of optimal filtering, Kalman’s filter and theorem on normal correlation. In: Proceedings of the 11th International Vilnius Conference on Probability and Mathematical Statistics. Vilnius, Lithuania, 182, 30 June–4 July 2014. ISBN: 978-609-433-220-3
Dobrovidov, A.V.: Stable nonparametric signal filtration in nonlinear models. In: Topics in Nonparametric Statistics: Proceedings of the First Conference of the International Society for Nonparametric Statistics, vol. XVI, pp. 61–74. Springer, New York (2014)
Taoufik, B., Rambouts, J.: Nonparametric density estimation for multivariate bounded data. J. Stat. Plann. Infer. 140(1), 139–152 (2007)
Chen, S.X.: Probability density function estimation using gamma kernels. Ann. Inst. Statist. Math. 52(3), 471–480 (2000)
Markovich, L.A.: Gamma kernel estimation of multivariate density and its derivative on the nonnegative semi-axis by dependent data (2015). arXiv:1410.2507v2
Dobrovidov, A.V., Markovich, L.A.: Nonparametric gamma kernel estimators of density derivatives on positive semi-axis. In: Proceedings of IFAC MIM 2013, pp. 1–6. Petersburg, Russia, 19–21 June 2013
Hall, P., Marron, J.S., Park, B.U.: Smoothed cross-validation. Probab. Theory Relat. Fields 92, 1–20 (1992)
Devroye, L., Gyorfi, L.: Nonparametric Density Estimation. The L1 View. John Wiley, New York, N.Y. (1985)
Davydov, Y.A.: On Convergence of distributions induced by stationary random processes. Probab. Theory Appl. V. XIII(4), 730–737 (1968)
Masry, E.: Probability density estimation from sampled data. IEEE Trans. Inf. Theory. V. IT-29(5), 696–709 (1983)
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Appendix
Appendix
Proof of Lemma 2.1.3. We give here only outline of the proof. Provided the condition (1), from formula (13) it follows that for a sequence of statistically dependent random variables the variance of their sum is expressed through the covariance
This covariance is estimated from above by using the Davydov’s inequality [14]
where \(p^{-1}+q^{-1}+r^{-1}=1\), \(\alpha (i)\) is a strong mixing coefficient of the process \((X_n)\), and \(\parallel \cdot \parallel _q\) is a norm in the space \(L_q\). This norm can be estimated by the following expression
where a kernel \(\prod _{j=1}^dK_{{\rho _1(x_j),b_j}}\left( \xi _{j}\right) \) is used as density function and random variables \(\xi _{j}\) is distributed like \(Gamma(\rho _1(x_j),b_j)\)-distribution with expectation \(\mu _j=x_j\) and variance \(\sigma _{\xi }^2 = x_jb_j\). Expectation in parentheses of (42) is calculated by expanding the function \(f(\xi _1^d)\) in a Taylor series at the point \( \mathbf {\mu } = \mu _1^d \). After some algebra we get
Provided the condition (2), we use the technique of the proof from [15]. To do this, we divide (41) into two terms \(|C|=(2/N)\sum _{i=1}^{N-1}(\cdot )=(2/N)\big (\sum _{i=1}^{c(N)}(\cdot )+\sum _{i=c(N)+1}^{N-1}(\cdot )\big )=I_1+I_2\) and estimate each at \(N\rightarrow \infty .\) As a result we get
It remains to find c(N) satisfying conditions \(c(N)b^{d/2}\rightarrow 0\) and \(c(N)b^{d\upsilon /2}\rightarrow \infty .\) Let \(c(N)=b^{-\varepsilon }\). Then, these conditions are met simultaneously if \( \frac{d}{2}\!>\!\varepsilon \!> \!\frac{d}{2}\upsilon >0, \quad 0< \upsilon < 1,\) and \(|C|=o\left( Nb^{d/2}\right) ^{-1}\). \(\square \)
Proof of Lemma 2.2.3. Proof of this Lemma is carried out, in principle, in the same way as Lemma 2.1.3, but it occupies a lot of space because of a complex estimator expression containing a special functions (see [10]). \(\square \)
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Dobrovidov, A.V. (2016). Regularization of Positive Signal Nonparametric Filtering in Multiplicative Observation Model. In: Cao, R., González Manteiga, W., Romo, J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics & Statistics, vol 175. Springer, Cham. https://doi.org/10.1007/978-3-319-41582-6_7
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