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| In [[estimation theory]] and [[statistics]], the '''Cramér–Rao bound (CRB)''' or '''Cramér–Rao lower bound (CRLB)''', named in honor of [[Harald Cramér]] and [[Calyampudi Radhakrishna Rao]] who were among the first to derive it,<ref name="Cramèr">{{cite book | last = Cramér | first = Harald | title = Mathematical Methods of Statistics | place = Princeton, NJ | publisher = Princeton Univ. Press | year = 1946 | isbn = 0-691-08004-6 | oclc = 185436716 }}</ref><ref name="Rao">{{cite journal | last = Rao | first = Calyampudi Radakrishna | title = Information and the accuracy attainable in the estimation of statistical parameters | journal = Bulletin of the [[Calcutta Mathematical Society]] |mr=0015748 | volume = 37 | pages = 81–89 | year = 1945 }}</ref><ref name="Rao papers">{{cite book | last = Rao | first = Calyampudi Radakrishna | title = Selected Papers of C. R. Rao | editor = S. Das Gupta | place = New York | publisher = Wiley | year = 1994 | isbn = 978-0-470-22091-7 | oclc = 174244259 }}</ref> expresses a lower bound on the [[variance]] of [[estimator]]s of a deterministic parameter. The bound is also known as the '''Cramér–Rao inequality''' or the '''information inequality'''.
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| | |
| In its simplest form, the bound states that the variance of any [[bias of an estimator|unbiased]] estimator is at least as high as the inverse of the [[Fisher information]]. An unbiased estimator which achieves this lower bound is said to be (fully) [[Efficiency (statistics)|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 even when an MVU estimator exists.
| |
| | |
| The Cramér–Rao bound can also be used to bound the variance of [[estimator bias|''biased'' estimators]] of given bias. In some cases, a biased approach can result in both a variance and a [[mean squared error]] that are ''below'' the unbiased Cramér–Rao lower bound; see [[estimator bias]]. | |
| | |
| == 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 (mathematics)|scalar]] and its estimator is [[estimator bias|unbiased]]. All versions of the bound require certain regularity conditions, which hold for most well-behaved distributions. These conditions are listed [[#Regularity conditions|later in this section]].
| |
| | |
| === Scalar unbiased case ===
| |
| Suppose <math>\theta</math> is an unknown deterministic parameter which is to be estimated from measurements <math>x</math>, distributed according to some [[probability density function]] <math>f(x;\theta)</math>. The [[variance]] of any ''unbiased'' estimator <math>\hat{\theta}</math> of <math>\theta</math> is then bounded by the [[multiplicative inverse|reciprocal]] of the [[Fisher information]] <math>I(\theta)</math>:
| |
| | |
| :<math>\mathrm{var}(\hat{\theta})
| |
| \geq
| |
| \frac{1}{I(\theta)}
| |
| </math>
| |
| where the Fisher information <math>I(\theta)</math> is defined by
| |
| :<math>
| |
| I(\theta) = \mathrm{E}
| |
| \left[
| |
| \left(
| |
| \frac{\partial \ell(x;\theta)}{\partial\theta}
| |
| \right)^2
| |
| \right] = -\mathrm{E}\left[ \frac{\partial^2 \ell(x;\theta)}{\partial\theta^2} \right]
| |
| </math>
| |
| and <math>\ell(x;\theta)=\log f(x;\theta)</math> is the [[natural logarithm]] of the [[likelihood function]] and <math>\mathrm{E}</math> denotes the [[expected value]] (over <math>x</math>).
| |
| | |
| The [[efficiency (statistics)|efficiency]] of an unbiased estimator <math>\hat{\theta}</math> measures how close this estimator's variance comes to this lower bound; estimator efficiency is defined as
| |
| | |
| :<math>e(\hat{\theta}) = \frac{I(\theta)^{-1}}{{\rm var}(\hat{\theta})}</math>
| |
| | |
| or the minimum possible variance for an unbiased estimator divided by its actual variance.
| |
| The Cramér–Rao lower bound thus gives
| |
| :<math>e(\hat{\theta}) \le 1.\ </math>
| |
| | |
| === General scalar case ===
| |
| A more general form of the bound can be obtained by considering an unbiased estimator <math>T(X)</math> of a function <math>\psi(\theta)</math> of the parameter <math>\theta</math>. Here, unbiasedness is understood as stating that <math>E\{T(X)\} = \psi(\theta)</math>. In this case, the bound is given by
| |
| :<math>
| |
| \mathrm{var}(T)
| |
| \geq
| |
| \frac{[\psi'(\theta)]^2}{I(\theta)}
| |
| </math>
| |
| where <math>\psi'(\theta)</math> is the derivative of <math>\psi(\theta)</math> (by <math>\theta</math>), and <math>I(\theta)</math> 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 <math>\hat{\theta}</math> with bias <math>b(\theta) = E\{\hat{\theta}\} - \theta</math>, and let <math>\psi(\theta) = b(\theta) + \theta</math>. By the result above, any unbiased estimator whose expectation is <math>\psi(\theta)</math> has variance greater than or equal to <math>(\psi'(\theta))^2/I(\theta)</math>. Thus, any estimator <math>\hat{\theta}</math> whose bias is given by a function <math>b(\theta)</math> satisfies
| |
| :<math>
| |
| \mathrm{var} \left(\hat{\theta}\right)
| |
| \geq
| |
| \frac{[1+b'(\theta)]^2}{I(\theta)}.
| |
| </math>
| |
| The unbiased version of the bound is a special case of this result, with <math>b(\theta)=0</math>.
| |
| | |
| 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 | |
| | |
| :<math>\mathrm{E}\left((\hat{\theta}-\theta)^2\right)\geq\frac{[1+b'(\theta)]^2}{I(\theta)}+b(\theta)^2,</math>
| |
| | |
| using the standard decomposition of the MSE. Note, however, that this bound can be less than the unbiased Cramér–Rao bound 1/''I''(θ). See the example of estimating variance below.
| |
| | |
| === Multivariate case ===
| |
| Extending the Cramér–Rao bound to multiple parameters, define a parameter column [[vector space|vector]]
| |
| :<math>\boldsymbol{\theta} = \left[ \theta_1, \theta_2, \dots, \theta_d \right]^T \in \mathbb{R}^d</math>
| |
| with probability density function <math>f(x; \boldsymbol{\theta})</math> which satisfies the two [[#Regularity conditions|regularity conditions]] below.
| |
| | |
| The [[Fisher information matrix]] is a <math>d \times d</math> matrix with element <math>I_{m, k}</math> defined as | |
| : <math> | |
| I_{m, k}
| |
| = \mathrm{E} \left[
| |
| \frac{\partial }{\partial \theta_m} \log f\left(x; \boldsymbol{\theta}\right)
| |
| \frac{\partial }{\partial \theta_k} \log f\left(x; \boldsymbol{\theta}\right)
| |
| \right] = -\mathrm{E} \left[
| |
| \frac{\partial ^2}{\partial \theta_m \partial \theta_k} \log f\left(x; \boldsymbol{\theta}\right)
| |
| \right].
| |
| </math>
| |
| | |
| Let <math>\boldsymbol{T}(X)</math> be an estimator of any vector function of parameters, <math>\boldsymbol{T}(X) = (T_1(X), \ldots, T_n(X))^T</math>, and denote its expectation vector <math>\mathrm{E}[\boldsymbol{T}(X)]</math> by <math>\boldsymbol{\psi}(\boldsymbol{\theta})</math>. The Cramér–Rao bound then states that the [[covariance matrix]] of <math>\boldsymbol{T}(X)</math> satisfies
| |
| : <math>
| |
| \mathrm{cov}_{\boldsymbol{\theta}}\left(\boldsymbol{T}(X)\right)
| |
| \geq
| |
| \frac
| |
| {\partial \boldsymbol{\psi} \left(\boldsymbol{\theta}\right)}
| |
| {\partial \boldsymbol{\theta}}
| |
| [I\left(\boldsymbol{\theta}\right)]^{-1}
| |
| \left(
| |
| \frac
| |
| {\partial \boldsymbol{\psi}\left(\boldsymbol{\theta}\right)}
| |
| {\partial \boldsymbol{\theta}}
| |
| \right)^T
| |
| </math>
| |
| where
| |
| * The matrix inequality <math>A \ge B</math> is understood to mean that the matrix <math>A-B</math> is [[positive semidefinite matrix|positive semidefinite]], and
| |
| * <math>\partial \boldsymbol{\psi}(\boldsymbol{\theta})/\partial \boldsymbol{\theta}</math> is the [[Jacobian matrix]] whose <math>ij</math>th element is given by <math>\partial \psi_i(\boldsymbol{\theta})/\partial \theta_j</math>.
| |
| | |
| <!-- please leave this extra space as it improves legibility. -->
| |
| | |
| If <math>\boldsymbol{T}(X)</math> is an [[estimator bias|unbiased]] estimator of <math>\boldsymbol{\theta}</math> (i.e., <math>\boldsymbol{\psi}\left(\boldsymbol{\theta}\right) = \boldsymbol{\theta}</math>), then the Cramér–Rao bound reduces to
| |
| : <math>
| |
| \mathrm{cov}_{\boldsymbol{\theta}}\left(\boldsymbol{T}(X)\right)
| |
| \geq
| |
| I\left(\boldsymbol{\theta}\right)^{-1}.
| |
| </math>
| |
| | |
| 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 | |
| (For the Bayesian case, see eqn. (11) of Bobrovsky, Mayer-Wolf, Zakai,
| |
| "Some classes of global Cramer-Rao bounds", Ann. Stats., 15(4):1421-38, 1987). | |
| | |
| : <math>
| |
| \mathrm{var}_{\boldsymbol{\theta}}\left(T_m(X)\right)
| |
| =
| |
| \left[\mathrm{cov}_{\boldsymbol{\theta}}\left(\boldsymbol{T}(X)\right)\right]_{mm}
| |
| \geq
| |
| \left[I\left(\boldsymbol{\theta}\right)^{-1}\right]_{mm}
| |
| \geq
| |
| \left(\left[I\left(\boldsymbol{\theta}\right)\right]_{mm}\right)^{-1}.
| |
| </math>
| |
| | |
| === Regularity conditions ===
| |
| The bound relies on two weak regularity conditions on the [[probability density function]], <math>f(x; \theta)</math>, and the estimator <math>T(X)</math>:
| |
| * The Fisher information is always defined; equivalently, for all <math>x</math> such that <math>f(x; \theta) > 0</math>,
| |
| ::<math> \frac{\partial}{\partial\theta} \log f(x;\theta)</math>
| |
| :exists, and is finite.
| |
| * The operations of integration with respect to <math>x</math> and differentiation with respect to <math>\theta</math> can be interchanged in the expectation of <math>T</math>; that is,
| |
| ::<math>
| |
| \frac{\partial}{\partial\theta}
| |
| \left[
| |
| \int T(x) f(x;\theta) \,dx
| |
| \right]
| |
| =
| |
| \int T(x)
| |
| \left[
| |
| \frac{\partial}{\partial\theta} f(x;\theta)
| |
| \right]
| |
| \,dx
| |
| </math>
| |
| :whenever the right-hand side is finite.
| |
| :This condition can often be confirmed by using the fact that integration and differentiation can be swapped when either of the following cases hold:
| |
| :# The function <math>f(x;\theta)</math> has bounded support in <math>x</math>, and the bounds do not depend on <math>\theta</math>;
| |
| :# The function <math>f(x;\theta)</math> has infinite support, is [[continuously differentiable]], and the integral converges uniformly for all <math>\theta</math>.
| |
| | |
| === Simplified form of the Fisher information ===
| |
| Suppose, in addition, that the operations of integration and differentiation can be swapped for the second derivative of <math>f(x;\theta)</math> as well, i.e.,
| |
| :<math> \frac{\partial^2}{\partial\theta^2}
| |
| \left[
| |
| \int T(x) f(x;\theta) \,dx
| |
| \right]
| |
| =
| |
| \int T(x)
| |
| \left[
| |
| \frac{\partial^2}{\partial\theta^2} f(x;\theta)
| |
| \right]
| |
| \,dx.
| |
| </math>
| |
| In this case, it can be shown that the Fisher information equals
| |
| :<math> | |
| I(\theta)
| |
| =
| |
| -\mathrm{E}
| |
| \left[
| |
| \frac{\partial^2}{\partial\theta^2} \log f(X;\theta)
| |
| \right].
| |
| </math>
| |
| The Cramèr–Rao bound can then be written as
| |
| :<math>
| |
| \mathrm{var} \left(\widehat{\theta}\right)
| |
| \geq
| |
| \frac{1}{I(\theta)}
| |
| =
| |
| \frac{1}
| |
| {
| |
| -\mathrm{E}
| |
| \left[
| |
| \frac{\partial^2}{\partial\theta^2} \log f(X;\theta)
| |
| \right]
| |
| }.
| |
| </math>
| |
| In some cases, this formula gives a more convenient technique for evaluating the bound.
| |
| | |
| == Single-parameter proof ==
| |
| The following is a proof of the general scalar case of the Cramér–Rao bound, which was described [[#General scalar case|above]]; namely, that if the expectation of <math>T</math> is denoted by <math>\psi (\theta)</math>, then, for all <math>\theta</math>,
| |
| :<math>{\rm var}(t(X)) \geq \frac{[\psi^\prime(\theta)]^2}{I(\theta)}.</math>
| |
| | |
| Let <math>X</math> be a [[random variable]] with probability density function <math>f(x; \theta)</math>.
| |
| Here <math>T = t(X)</math> is a [[statistic]], which is used as an [[estimator]] for <math>\psi (\theta)</math>. If <math>V</math> is the [[score (statistics)|score]], i.e.
| |
| | |
| :<math>V = \frac{\partial}{\partial\theta} \ln f(X;\theta)</math>
| |
| | |
| then the [[expected value|expectation]] of <math>V</math>, written <math>{\rm E}(V)</math>, is zero. | |
| If we consider the [[covariance]] <math>{\rm cov}(V, T)</math> of <math>V</math> and <math>T</math>, we have <math>{\rm cov}(V, T) = {\rm E}(V T)</math>, because <math>{\rm E}(V) = 0</math>. Expanding this expression we have
| |
| | |
| :<math> | |
| {\rm cov}(V,T)
| |
| =
| |
| {\rm E}
| |
| \left( | |
| T \cdot \frac{\partial}{\partial\theta} \ln f(X;\theta)
| |
| \right)
| |
| </math>
| |
| | |
| This may be expanded using the [[chain rule]]
| |
| | |
| :<math>\frac{\partial}{\partial\theta} \ln Q = \frac{1}{Q}\frac{\partial Q}{\partial\theta}</math>
| |
| | |
| and the definition of expectation gives, after cancelling <math>f(x; \theta)</math>,
| |
| | |
| :<math>
| |
| {\rm E} \left(
| |
| T \cdot \frac{\partial}{\partial\theta} \ln f(X;\theta)
| |
| \right)
| |
| =
| |
| \int
| |
| t(x)
| |
| \left[
| |
| \frac{\partial}{\partial\theta} f(x;\theta)
| |
| \right]
| |
| \, dx
| |
| =
| |
| \frac{\partial}{\partial\theta}
| |
| \left[
| |
| \int t(x)f(x;\theta)\,dx
| |
| \right]
| |
| =
| |
| \psi^\prime(\theta)
| |
| </math>
| |
| | |
| because the integration and differentiation operations commute (second condition).
| |
| | |
| The [[Cauchy–Schwarz inequality]] shows that
| |
| | |
| :<math>
| |
| \sqrt{ {\rm var} (T) {\rm var} (V)} \geq \left| {\rm cov}(V,T) \right| = \left | \psi^\prime (\theta)
| |
| \right |</math>
| |
| | |
| therefore
| |
| | |
| :<math>
| |
| {\rm var\ } T \geq \frac{[\psi^\prime(\theta)]^2}{{\rm var} (V)}
| |
| =
| |
| \frac{[\psi^\prime(\theta)]^2}{I(\theta)}
| |
| =
| |
| \left[
| |
| \frac{\partial}{\partial\theta}
| |
| {\rm E} (T)
| |
| \right]^2
| |
| \frac{1}{I(\theta)}
| |
| </math>
| |
| which proves the proposition.
| |
| | |
| ==Examples==
| |
| | |
| ===Multivariate normal distribution===
| |
| For the case of a [[multivariate normal distribution|''d''-variate normal distribution]]
| |
| : <math>
| |
| \boldsymbol{x}
| |
| \sim
| |
| N_d
| |
| \left(
| |
| \boldsymbol{\mu} \left( \boldsymbol{\theta} \right)
| |
| ,
| |
| {\boldsymbol C} \left( \boldsymbol{\theta} \right)
| |
| \right)
| |
| </math>
| |
| the [[Fisher information matrix]] has elements<ref>{{cite book
| |
| | last = Kay
| |
| | first = S. M.
| |
| | title = Fundamentals of Statistical Signal Processing: Estimation Theory
| |
| | year = 1993
| |
| | publisher = Prentice Hall
| |
| | page = 47
| |
| | isbn = 0-13-042268-1 }}
| |
| </ref>
| |
| :<math>
| |
| I_{m, k}
| |
| =
| |
| \frac{\partial \boldsymbol{\mu}^T}{\partial \theta_m}
| |
| {\boldsymbol C}^{-1}
| |
| \frac{\partial \boldsymbol{\mu}}{\partial \theta_k}
| |
| +
| |
| \frac{1}{2}
| |
| \mathrm{tr}
| |
| \left(
| |
| {\boldsymbol C}^{-1}
| |
| \frac{\partial {\boldsymbol C}}{\partial \theta_m}
| |
| {\boldsymbol C}^{-1}
| |
| \frac{\partial {\boldsymbol C}}{\partial \theta_k}
| |
| \right)
| |
| </math>
| |
| where "tr" is the [[trace (matrix)|trace]].
| |
| | |
| For example, let <math>w[n]</math> be a sample of <math>N</math> independent observations) with unknown mean <math>\theta</math> and known variance <math>\sigma^2</math>
| |
| :<math>w[n] \sim \mathbb{N}_N \left(\theta {\boldsymbol 1}, \sigma^2 {\boldsymbol I} \right).</math>
| |
| Then the Fisher information is a scalar given by
| |
| :<math>
| |
| I(\theta)
| |
| =
| |
| \left(\frac{\partial\boldsymbol{\mu}(\theta)}{\partial\theta}\right)^T{\boldsymbol C}^{-1}\left(\frac{\partial\boldsymbol{\mu}(\theta)}{\partial\theta}\right)
| |
| = \sum^N_{i=1}\frac{1}{\sigma^2} = \frac{N}{\sigma^2},
| |
| </math>
| |
| and so the Cramér–Rao bound is | |
| :<math>
| |
| \mathrm{var}\left(\hat \theta\right)
| |
| \geq
| |
| \frac{\sigma^2}{N}.
| |
| </math>
| |
| | |
| ===Normal variance with known mean===
| |
| Suppose ''X'' is a [[normal distribution|normally distributed]] random variable with known mean <math>\mu</math> and unknown variance <math>\sigma^2</math>. Consider the following statistic:
| |
| | |
| :<math>
| |
| T=\frac{\sum_{i=1}^n\left(X_i-\mu\right)^2}{n}.
| |
| </math> | |
| | |
| Then ''T'' is unbiased for <math>\sigma^2</math>, as <math>E(T)=\sigma^2</math>. What is the variance of ''T''?
| |
| | |
| :<math>
| |
| \mathrm{var}(T) = \frac{\mathrm{var}(X-\mu)^2}{n}=\frac{1}{n}
| |
| \left[
| |
| E\left\{(X-\mu)^4\right\}-\left(E\left\{(X-\mu)^2\right\}\right)^2
| |
| \right]
| |
| </math>
| |
| | |
| (the second equality follows directly from the definition of variance). The first term is the fourth [[moment about the mean]] and has value <math>3(\sigma^2)^2</math>; the second is the square of the variance, or <math>(\sigma^2)^2</math>.
| |
| Thus
| |
| | |
| :<math>\mathrm{var}(T)=\frac{2(\sigma^2)^2}{n}.</math>
| |
| | |
| Now, what is the [[Fisher information]] in the sample? Recall that the [[score (statistics)|score]] ''V'' is defined as
| |
| | |
| :<math>
| |
| V=\frac{\partial}{\partial\sigma^2}\log L(\sigma^2,X)
| |
| </math>
| |
| | |
| where <math>L</math> is the [[likelihood function]]. Thus in this case,
| |
| | |
| :<math>
| |
| V=\frac{\partial}{\partial\sigma^2}\log\left[\frac{1}{\sqrt{2\pi\sigma^2}}e^{-(X-\mu)^2/{2\sigma^2}}\right]
| |
| =\frac{(X-\mu)^2}{2(\sigma^2)^2}-\frac{1}{2\sigma^2}
| |
| </math>
| |
| | |
| 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
| |
| | |
| :<math>
| |
| I
| |
| =-E\left(\frac{\partial V}{\partial\sigma^2}\right)
| |
| =-E\left(-\frac{(X-\mu)^2}{(\sigma^2)^3}+\frac{1}{2(\sigma^2)^2}\right)
| |
| =\frac{\sigma^2}{(\sigma^2)^3}-\frac{1}{2(\sigma^2)^2}
| |
| =\frac{1}{2(\sigma^2)^2}.</math>
| |
| | |
| Thus the information in a sample of <math>n</math> independent observations is just <math>n</math> times this, or <math>\frac{n}{2(\sigma^2)^2}.</math>
| |
| | |
| The Cramer Rao bound states that
| |
| | |
| :<math>
| |
| \mathrm{var}(T)\geq\frac{1}{I}.</math>
| |
| | |
| In this case, the inequality is saturated (equality is achieved), showing that the [[estimator]] is [[efficiency (statistics)|efficient]].
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| | |
| However, we can achieve a lower [[mean squared error]] using a biased estimator. The estimator
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| :<math>
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| T=\frac{\sum_{i=1}^n\left(X_i-\mu\right)^2}{n+2}.
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| </math>
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| obviously has a smaller variance, which is in fact
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| :<math>\mathrm{var}(T)=\frac{2n(\sigma^2)^2}{(n+2)^2}.</math>
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| Its bias is
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| <math>\left(1-\frac{n}{n+2}\right)\sigma^2=\frac{2\sigma^2}{n+2}</math>
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| so its mean squared error is
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| :<math>\mathrm{MSE}(T)=\left(\frac{2n}{(n+2)^2}+\frac{4}{(n+2)^2}\right)(\sigma^2)^2
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| =\frac{2(\sigma^2)^2}{n+2}</math>
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| which is clearly less than the Cramér–Rao bound found above.
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| 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.
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| == See also ==
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| * [[Chapman–Robbins bound]]
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| * [[Kullback's inequality]]
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| == References and notes ==
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| {{reflist}}
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| == Further reading ==
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| * {{Cite journal
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| | last = Kay
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| | first = Steven M.
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| | title = Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory
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| | publisher = Prentice Hall
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| | year = 1993
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| | isbn = 0-13-345711-7 }}. Chapter 3.
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| * {{Cite journal
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| | last = Shao
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| | first = Jun
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| | title = Mathematical Statistics
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| | place = New York
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| | publisher = Springer
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| | year = 1998
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| | isbn = 0-387-98674-X }}. Section 3.1.3.
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| == External links ==
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| *[http://www4.utsouthwestern.edu/wardlab/fandplimittool.asp FandPLimitTool] a GUI-based software to calculate the Fisher information and Cramer-Rao Lower Bound with application to single-molecule microscopy.
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| {{DEFAULTSORT:Cramer-Rao bound}}
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| [[Category:Articles containing proofs]]
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| [[Category:Statistical inequalities]]
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| [[Category:Estimation theory]]
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