In estimation theory and statistics, the Cramér–Rao bound (CRB) relates to estimation of a deterministic (fixed, though unknown) parameter. The result is named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, but has also been derived independently by Maurice Fréchet, Georges Darmois, and by Alexander Aitken and Harold Silverstone. It is also known as Fréchet-Cramér–Rao or Fréchet-Darmois-Cramér-Rao lower bound. It states that the precision of any unbiased estimator is at most the Fisher information; or (equivalently) the reciprocal of the Fisher information is a lower bound on its variance. An unbiased estimator that achieves this bound is said to be (fully) efficient. Such a solution achieves the lowest possible mean squared error among all unbiased methods, and is, therefore, the minimum variance unbiased (MVU) estimator. However, in some cases, no unbiased technique exists which achieves the bound. This may occur either if for any unbiased estimator, there exists another with a strictly smaller variance, or if an MVU estimator exists, but its variance is strictly greater than the inverse of the Fisher information. The Cramér–Rao bound can also be used to bound the variance of estimators of given bias. In some cases, a biased approach can result in both a variance and a mean squared error that are the unbiased Cramér–Rao lower bound; see estimator bias. Significant progress over the Cramér–Rao lower bound was proposed by Anil Kumar Bhattacharyya through a series of works, called Bhattacharyya bound.

Statement

The Cramér–Rao bound is stated in this section for several increasingly general cases, beginning with the case in which the parameter is a scalar and its estimator is unbiased. All versions of the bound require certain regularity conditions, which hold for most well-behaved distributions. These conditions are listed later in this section.

Scalar unbiased case

Suppose \theta is an unknown deterministic parameter that is to be estimated from n independent observations (measurements) of x, each from a distribution according to some probability density function f(x;\theta). The variance of any unbiased estimator of \theta is then bounded by the reciprocal of the Fisher information I(\theta): where the Fisher information I(\theta) is defined by and is the natural logarithm of the likelihood function for a single sample x and denotes the expected value with respect to the density f(x;\theta) of X. If not indicated, in what follows, the expectation is taken with respect to X. If is twice differentiable and certain regularity conditions hold, then the Fisher information can also be defined as follows: The efficiency of an unbiased estimator measures how close this estimator's variance comes to this lower bound; estimator efficiency is defined as or the minimum possible variance for an unbiased estimator divided by its actual variance. The Cramér–Rao lower bound thus gives

General scalar case

A more general form of the bound can be obtained by considering a biased estimator T(X), whose expectation is not \theta but a function of this parameter, say,. Hence is not generally equal to 0. In this case, the bound is given by where is the derivative of (by \theta), and I(\theta) is the Fisher information defined above.

Bound on the variance of biased estimators

Apart from being a bound on estimators of functions of the parameter, this approach can be used to derive a bound on the variance of biased estimators with a given bias, as follows. Consider an estimator with bias, and let. By the result above, any unbiased estimator whose expectation is has variance greater than or equal to. Thus, any estimator whose bias is given by a function b(\theta) satisfies The unbiased version of the bound is a special case of this result, with b(\theta)=0. It's trivial to have a small variance − an "estimator" that is constant has a variance of zero. But from the above equation, we find that the mean squared error of a biased estimator is bounded by using the standard decomposition of the MSE. Note, however, that if this bound might be less than the unbiased Cramér–Rao bound 1/I(\theta). For instance, in the example of estimating variance below,.

Multivariate case

Extending the Cramér–Rao bound to multiple parameters, define a parameter column vector with probability density function which satisfies the two regularity conditions below. The Fisher information matrix is a d \times d matrix with element I_{m, k} defined as Let be an estimator of any vector function of parameters,, and denote its expectation vector by. The Cramér–Rao bound then states that the covariance matrix of satisfies where If is an unbiased estimator of (i.e., ), then the Cramér–Rao bound reduces to If it is inconvenient to compute the inverse of the Fisher information matrix, then one can simply take the reciprocal of the corresponding diagonal element to find a (possibly loose) lower bound.

Regularity conditions

The bound relies on two weak regularity conditions on the probability density function,, and the estimator T(X):

Proof

Proof for the general case based on the Chapman–Robbins bound

Proof based on.

A standalone proof for the general scalar case

For the general scalar case: Assume that T=t(X) is an estimator with expectation (based on the observations X), i.e. that. The goal is to prove that, for all \theta, Let X be a random variable with probability density function. Here T = t(X) is a statistic, which is used as an estimator for. Define V as the score: where the chain rule is used in the final equality above. Then the expectation of V, written, is zero. This is because: where the integral and partial derivative have been interchanged (justified by the second regularity condition). If we consider the covariance of V and T, we have, because. Expanding this expression we have again because the integration and differentiation operations commute (second condition). The Cauchy–Schwarz inequality shows that therefore which proves the proposition.

Examples

Multivariate normal distribution

For the case of a d-variate normal distribution the Fisher information matrix has elements where "tr" is the trace. For example, let w[j] be a sample of n independent observations with unknown mean \theta and known variance \sigma^2. Then the Fisher information is a scalar given by and so the Cramér–Rao bound is

Normal variance with known mean

Suppose X is a normally distributed random variable with known mean \mu and unknown variance \sigma^2. Consider the following statistic: Then T is unbiased for \sigma^2, as. What is the variance of T? (the second equality follows directly from the definition of variance). The first term is the fourth moment about the mean and has value ; the second is the square of the variance, or. Thus Now, what is the Fisher information in the sample? Recall that the score V is defined as where L is the likelihood function. Thus in this case, where the second equality is from elementary calculus. Thus, the information in a single observation is just minus the expectation of the derivative of V, or Thus the information in a sample of n independent observations is just n times this, or The Cramér–Rao bound states that In this case, the inequality is saturated (equality is achieved), showing that the estimator is efficient. However, we can achieve a lower mean squared error using a biased estimator. The estimator obviously has a smaller variance, which is in fact Its bias is so its mean squared error is which is less than what unbiased estimators can achieve according to the Cramér–Rao bound. When the mean is not known, the minimum mean squared error estimate of the variance of a sample from Gaussian distribution is achieved by dividing by n+1, rather than n-1 or n+2.

References and notes