An estimate is said to be unbiased if the expected value of the estimate equals the true value of the parameter: E θ ^| θ =θ θ θ θ . Otherwise, the estimate is said to be biased: E θ ^| θ ≠θ θ θ θ . The bias b⁢θ b θ is usually considered to be additive, so that b⁢θ=E θ ^| θ −θ b θ θ θ θ . When we have a biased estimate, the bias usually depends on the number of observations L L. An estimate is said to be asymptotically unbiased if the bias tends to zero for large L L: limit   L → ∞ b=0 L b 0 . An estimate's variance equals the mean-squared estimation error only if the estimate is unbiased.

An unbiased estimate has a probability distribution where the mean equals the actual value of the parameter. Should the lack of bias be considered a desirable property? If many unbiased estimates are computed from statistically independent sets of observations having the same parameter value, the average of these estimates will be close to this value. This property does not mean that the estimate has less error than a biased one; there exist biased estimates whose mean-squared errors are smaller than unbiased ones. In such cases, the biased estimate is usually asymptotically unbiased. Lack of bias is good, but that is just one aspect of how we evaluate estimators.

We term an estimate consistent if the mean-squared estimation error tends to zero as the number of observations becomes large: limit   L → ∞ EεT⁢ε=0 L ε ε 0 . Thus, a consistent estimate must be at least asymptotically unbiased. Unbiased estimates do exist whose errors never diminish as more data are collected: Their variances remain nonzero no matter how much data are available. Inconsistent estimates may provide reasonable estimates when the amount of data is limited, but have the counterintuitive property that the quality of the estimate does not improve as the number of observations increases. Although appropriate in the proper circumstances (smaller mean-squared error than a consistent estimate over a pertinent range of values of L L, consistent estimates are usually favored in practice.

As estimators can be derived in a variety of ways, their error characteristics must always be analyzed and compared. In practice, many problems and the estimators derived for them are sufficiently complicated to render analytic studies of the errors difficult, if not impossible. Instead, numerical simulation and comparison with lower bounds on the estimation error are frequently used instead to assess the estimator performance. An efficient estimate has a mean-squared error that equals a particular lower bound: the Cramér-Rao bound. If an efficient estimate exists (the Cramér-Rao bound is the greatest lower bound), it is optimum in the mean-squared sense: No other estimate has a smaller mean-squared error (see Maximum Likelihood Estimators for details).

For many problems no efficient estimate exists. In such cases, the Cramér-Rao bound remains a lower bound, but its value is smaller than that achievable by any estimator. How much smaller is usually not known. However, practitioners frequently use the Cramér-Rao bound in comparisons with numerical error calculations. Another issue is the choice of mean-squared error as the estimation criterion; it may not suffice to pointedly assess estimator performance in a particular problem. Nevertheless, every problem is usually subjected to a Cramér-Rao bound computation and the existence of an efficient estimate considered.