Malliavin calculus in statistical inference: Cramer-Rao lower bound for fuzzy random variables

Jafari, Hossein; Ebadi, Mohammad Javad

doi:10.22105/dmor.2020.234911.1152

Document Type : Original Article

Authors

Department of Mathematics, Chabahar Maritime University, Chabahar, Iran.

https://doi.org/10.22105/dmor.2020.234911.1152

Abstract

The Cramer-Rao lower bound is obtained by using integration by parts and the Cauchy-Schwarz inequality. The integration by parts formulas of Malliavin calculus plays a role in this study. The point estimation problem is very crucial and has a wide range of applications. When we deal with some concepts such as random variables, the parameters of interest and estimates may be observed as imprecise. Therefore, the theory of fuzzy sets is important in formulating such situations. Using the fuzzy set theory, we define a fuzzy-valued random variable and fuzzy stochastic process. We use the Malliavin derivative and Skorohod integral to study the asymptotic properties of the statistical model for fuzzy random variables. We show how to use the conditional expectations of certain expressions to derive Cramer-Rao lower bounds for Fuzzy valued Random Variables that they do not require the explicit expression of the likelihood function. As an example, we consider a fuzzy random sample of size n induced by independent standard normally distributed random variables with fuzzy parameter.

Keywords

Main Subjects

stochastic/Probabilistic/fuzzy/dynamic modeling

References

Akbari, M. G., & Rezaei, A. (2007). An uniformly minimum variance unbiased point estimator using fuzzy observations. Austrian journal of statistics, 36(4), 307-317.

Akbari, M.G., & Khanjari Sadegh, M. (2012). Estimators based on fuzzy random variables and their mathematical properties. Iranian journal of fuzzy systems, 9(1), 79-95.

Buckley, J. J. (2005). Fuzzy probabilities: new approach and applications (Vol. 115). Springer Science & Business Media.

Buckley, J. J. (2006). Fuzzy probability and statistics (pp. 223-234). Heidelberg: Springer.

Corcuera, J. M., & Kohatsu-Higa, A. (2011). Statistical inference and Malliavin calculus. In Seminar on stochastic analysis, random fields and applications, (pp. 59-82). Springer.

Ebadi, M. J., Suleiman, M., Ismail, F. B., Ahmadian, A., Shahryari, M. R., & Salahshour, S. (2013). A new distance measure for trapezoidal fuzzy numbers. Mathematical problems in engineering.

Farahani, H., Ebadi, M.J., & Jafari, H. (2019). Finding inverse of a fuzzy matrix using eigenvalue method, International Journal of innovative technology and exploring engineering, 9(2), 3030-3037.

Gobet, E. (2001). Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli, 7(6), 899-912.

Itô, K. (1951). Multiple wiener integral. Journal of the mathematical society of Japan, 3(1), 157-169.

Jafari, H., Paripour, M., & Farahani, H. (2018). Fuzzy Malliavin derivative and linear Skorohod fuzzy stochastic differential equation, Journal of intelligent & fuzzy systems, 35(2), 2447-2458.

Kumar Das, S., & Mandal, T. (2017). A new model for solving fuzzy linear fractional programming problem with ranking function. Journal of applied research on industrial engineering, 4(2), 89–96.

Montazeri, F. Z. (2019). The survey of data envelopment analysis models in fuzzy stochastic environments. International journal of research in industrial engineering, 8(4), 366–383.

Nualart, D. (2006). The Malliavin calculus and related topics (Vol. 1995). Berlin: Springer.

Sanz-Solé, M. (2005). Malliavin calculus with applications to stochastic partial differential equations. EPFL press.

Taghi-Nezhad, N., & Taleshian, F. (2018). A solution approach for solving fully fuzzy quadratic programming problems. Journal of applied research on industrial engineering, 5(1), 50–61.

Torabi, H. (2006). Cramér-Rao lower bound for fuzzy-valued random variables. Austrian journal of statistics, 35(4), 471-482.

Viertl, R. (2006). Univariate statistical analysis with fuzzy data. Computational statistics & data analysis, 51(1), 133-147.

Journal of Decisions and Operations Research

Malliavin calculus in statistical inference: Cramer-Rao lower bound for fuzzy random variables

References

References

Volume 5, Issue 2
September 2020
Pages 124-132

Malliavin calculus in statistical inference: Cramer-Rao lower bound for fuzzy random variables

References

References

Volume 5, Issue 2September 2020Pages 124-132

Volume 5, Issue 2
September 2020
Pages 124-132