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

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.