Minimum Variance Estimation without Regularity Assumptions

Poisson Point Processes

“…The notion of a lower bound for parameters that are inherently discrete is available, but not discussed here, even though such bounds are potentially useful in applications. The Hammersley-Chapman-Robbins (HCR) bound holds for discrete parameters [14,45]. It is a special case of the earlier Barankin bound [5,134].…”

On lower bounds for deterministic parameter estimation

“…Block inversion of G in (11) le8ds to: Remark: In order to exploit the Barankin's works at a low computational cost, Chapman and Robbins ( [5]) proposed to work with K = 1. On finds, thanks to (23), the same lower bound noted MSEohapRo6 as Chapman and Robbins:…”

Breakthroughs in Statistics

“…The inequality has been strengthened and generalized from numerous angles. Some of the major contributions on this inequality have been made by Bhattacharya (1946), Blyth (1974), Blyth and Roberts (1972), Chapman and Robbins (1951), Fabian and Hannan (1977), Hammersley (1950), Simons (1980), Simons and Woodroofe (1983), , and others. In addition to its applications in establishing that in certain standard applications the inequality is attained, its most useful applications in statistics have been in the area of decision theory and probability.…”