parameters and the actual value of the parameters as the basis for optimality. This error term is then squared and the expected value of this squared value is minimized for the MMSE estimator. Estimators Commonly used estimators (estimation methods) and topics related to them include: Maximum likelihood estimators Bayes estimators Method of moments estimators Cramér–Rao bound Least squares Minimum mean squared error (MMSE), also known as Bayes least squared error (BLSE) Maximum a posteriori (MAP) Minimum variance unbiased estimator (MVUE) Nonlinear system identification Best linear unbiased estimator (BLUE) Unbiased estimators — see estimator bias. Particle filter Markov chain Monte Carlo (MCMC) Kalman filter, and its various derivatives Wiener filter Examples Unknown constant in additive white Gaussian noise Consider a received discrete signal, , of independent samples that consists of an unknown constant with additive white Gaussian noise (AWGN) with zero mean and known variance (i.e., ). Since the variance is known then the only unknown parameter is . The model for the signal is then Two possible (of many) estimators for the parameter are: which is the sample mean Both of these estimators have a mean of , which can be shown through taking the expected value of each estimator and At this point, these two estimators would appear to perform the same. However, the difference between them becomes apparent when comparing the variances. and It would seem that the sample mean is a better estimator since its variance is lower for every N > 1. Maximum likelihood Continuing the example using the maximum likelihood estimator, the probability density function (pdf) of the noise for one sample is and the probability of becomes ( can be thought of a ) By independence, the probability of becomes Taking the natural logarithm of the pdf and the maximum likelihood estimator is Taking the first derivative of the log-likelihood function and setting it to zero This results in the maximum likelihood estimator which is simply the sample mean. From this example, it was found that the sample mean is the maximum likelihood estimator for samples of a fixed, unknown parameter corrupted by AWGN. Cramér–Rao lower bound To find the Cramér–Rao lower bound (CRLB) of the sample mean estimator, it is first necessary to find the Fisher information number and copying from above Taking the second derivative and finding the negative expected value is trivial since it is now a deterministic constant Finally, putting