Properties of the Conditional Mean Estimator in Poisson Noise This paper considers estimation of a random variable in Poisson noise. Specifically, the paper focuses on properties of the conditional mean estimator as a function of the scaling coefficient, the dark current parameter, the distribution of the input random variable and channel realizations.With respect to the scaling coefficient and the dark current, several identities in terms of derivatives are established. For example, it is shown that the derivative of the conditional mean estimator with respect to the dark current parameter is proportional to the conditional variance. Moreover, a version of score function is proposed and a Tweedylike formula for the conditional expectation is recovered.With respect to the distribution, several regularity conditions are shown. For instance, it is shown that the conditional mean estimator uniquely determines the input distribution. Moreover, it is shown that if the conditional expectation is close to a linear function in the mean squared error, then the input distribution is approximately gamma in the Levy metric.

Partial Derivative Estimation for Underlying FunctionalValued Process in a Unified Framework We consider functional data analysis when the observations at each location are functional rather than scalar. When the dynamic of underlying functionalvalued process at each location is of interest, it is desirable to recover partial derivatives of a sample function, especially from sparse and noisecontaminated measures. We propose a novel approach based on estimating derivatives of eigenfunctions of marginal kernels to obtain a representation for functionalvalued process and its partial derivatives in a unified framework in which the number of locations and number of observations at each location for each individual can be any rate relative to the sample size. We derive almost sure rates of convergence for the procedures and further establish consistency results for recovered partial derivatives.