Estimation accuracy of non-standard maximum likelihood estimators

Abstract

In many deterministic estimation problems, the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on additional random variables. Unfortunately, this marginalization is often mathematically intractable, which prevents from using standard maximum likelihood estimators (MLEs) or any standard lower bound on their mean squared error (MSE). To circumvent this problem, the use of joint MLEs of deterministic and random parameters are proposed as being a substitute. It is shown that, regarding the deterministic parameters: 1) the joint MLEs provide generally suboptimal estimates in any asymptotic regions of operation yielding unbiased efficient estimates, 2) any representative of the two general classes of lower bounds, respectively the Small-Error bounds and the Large-Error bounds, has a “non-standard” version lower bounding the MSE of the deterministic parameters estimate.