Pollard's rho algorithm

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, expresses a lower bound on the variance of estimators of a deterministic parameter. The bound is also known as the Cramér–Rao inequality or the information inequality.

In its simplest form, the bound states that the variance of any 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) 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 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 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 which is to be estimated from measurements $x$, distributed according to some probability density function $f(x;\theta )$. The variance of any unbiased estimator ${\hat {\theta }}$ of $\theta$ is then bounded by the reciprocal of the Fisher information $I(\theta )$:

$\mathrm {var} ({\hat {\theta }})\geq {\frac {1}{I(\theta )}}$

where the Fisher information $I(\theta )$ is defined by

$I(\theta )=\mathrm {E} \left[\left({\frac {\partial \ell (x;\theta )}{\partial \theta }}\right)^{2}\right]=-\mathrm {E} \left[{\frac {\partial ^{2}\ell (x;\theta )}{\partial \theta ^{2}}}\right]$

and $\ell (x;\theta )=\log f(x;\theta )$ is the natural logarithm of the likelihood function and $\mathrm {E}$ denotes the expected value (over $x$).

The efficiency of an unbiased estimator ${\hat {\theta }}$ measures how close this estimator's variance comes to this lower bound; estimator efficiency is defined as

$e({\hat {\theta }})={\frac {I(\theta )^{-1}}{{\rm {var}}({\hat {\theta }})}}$

or the minimum possible variance for an unbiased estimator divided by its actual variance. The Cramér–Rao lower bound thus gives

$e({\hat {\theta }})\leq 1.\$

General scalar case

A more general form of the bound can be obtained by considering an unbiased estimator $T(X)$ of a function $\psi (\theta )$ of the parameter $\theta$. Here, unbiasedness is understood as stating that $E\{T(X)\}=\psi (\theta )$. In this case, the bound is given by

$\mathrm {var} (T)\geq {\frac {[\psi '(\theta )]^{2}}{I(\theta )}}$

where $\psi '(\theta )$ is the derivative of $\psi (\theta )$ (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 ${\hat {\theta }}$ with bias $b(\theta )=E\{{\hat {\theta }}\}-\theta$, and let $\psi (\theta )=b(\theta )+\theta$. By the result above, any unbiased estimator whose expectation is $\psi (\theta )$ has variance greater than or equal to $(\psi '(\theta ))^{2}/I(\theta )$. Thus, any estimator ${\hat {\theta }}$ whose bias is given by a function $b(\theta )$ satisfies

$\mathrm {var} \left({\hat {\theta }}\right)\geq {\frac {[1+b'(\theta )]^{2}}{I(\theta )}}.$

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

$\mathrm {E} \left(({\hat {\theta }}-\theta )^{2}\right)\geq {\frac {[1+b'(\theta )]^{2}}{I(\theta )}}+b(\theta )^{2},$

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

${\boldsymbol {\theta }}=\left[\theta _{1},\theta _{2},\dots ,\theta _{d}\right]^{T}\in \mathbb {R} ^{d}$

with probability density function $f(x;{\boldsymbol {\theta }})$ which satisfies the two regularity conditions below.

The Fisher information matrix is a $d\times d$ matrix with element $I_{m,k}$ defined as

$I_{m,k}=\mathrm {E} \left[{\frac {\partial }{\partial \theta _{m}}}\log f\left(x;{\boldsymbol {\theta }}\right){\frac {\partial }{\partial \theta _{k}}}\log f\left(x;{\boldsymbol {\theta }}\right)\right]=-\mathrm {E} \left[{\frac {\partial ^{2}}{\partial \theta _{m}\partial \theta _{k}}}\log f\left(x;{\boldsymbol {\theta }}\right)\right].$

Let ${\boldsymbol {T}}(X)$ be an estimator of any vector function of parameters, ${\boldsymbol {T}}(X)=(T_{1}(X),\ldots ,T_{n}(X))^{T}$, and denote its expectation vector $\mathrm {E} [{\boldsymbol {T}}(X)]$ by ${\boldsymbol {\psi }}({\boldsymbol {\theta }})$. The Cramér–Rao bound then states that the covariance matrix of ${\boldsymbol {T}}(X)$ satisfies

$\mathrm {cov} _{\boldsymbol {\theta }}\left({\boldsymbol {T}}(X)\right)\geq {\frac {\partial {\boldsymbol {\psi }}\left({\boldsymbol {\theta }}\right)}{\partial {\boldsymbol {\theta }}}}[I\left({\boldsymbol {\theta }}\right)]^{-1}\left({\frac {\partial {\boldsymbol {\psi }}\left({\boldsymbol {\theta }}\right)}{\partial {\boldsymbol {\theta }}}}\right)^{T}$

where

• The matrix inequality $A\geq B$ is understood to mean that the matrix $A-B$ is positive semidefinite, and
• $\partial {\boldsymbol {\psi }}({\boldsymbol {\theta }})/\partial {\boldsymbol {\theta }}$ is the Jacobian matrix whose $ij$th element is given by $\partial \psi _{i}({\boldsymbol {\theta }})/\partial \theta _{j}$.

If ${\boldsymbol {T}}(X)$ is an unbiased estimator of ${\boldsymbol {\theta }}$ (i.e., ${\boldsymbol {\psi }}\left({\boldsymbol {\theta }}\right)={\boldsymbol {\theta }}$), then the Cramér–Rao bound reduces to

$\mathrm {cov} _{\boldsymbol {\theta }}\left({\boldsymbol {T}}(X)\right)\geq I\left({\boldsymbol {\theta }}\right)^{-1}.$

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).

$\mathrm {var} _{\boldsymbol {\theta }}\left(T_{m}(X)\right)=\left[\mathrm {cov} _{\boldsymbol {\theta }}\left({\boldsymbol {T}}(X)\right)\right]_{mm}\geq \left[I\left({\boldsymbol {\theta }}\right)^{-1}\right]_{mm}\geq \left(\left[I\left({\boldsymbol {\theta }}\right)\right]_{mm}\right)^{-1}.$

Regularity conditions

The bound relies on two weak regularity conditions on the probability density function, $f(x;\theta )$, and the estimator $T(X)$:

• The Fisher information is always defined; equivalently, for all $x$ such that $f(x;\theta )>0$,

${\frac {\partial }{\partial \theta }}\log f(x;\theta )$

exists, and is finite.

• The operations of integration with respect to $x$ and differentiation with respect to $\theta$ can be interchanged in the expectation of $T$; that is,

${\frac {\partial }{\partial \theta }}\left[\int T(x)f(x;\theta )\,dx\right]=\int T(x)\left[{\frac {\partial }{\partial \theta }}f(x;\theta )\right]\,dx$

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:

1. The function $f(x;\theta )$ has bounded support in $x$, and the bounds do not depend on $\theta$;
2. The function $f(x;\theta )$ has infinite support, is continuously differentiable, and the integral converges uniformly for all $\theta$.

Simplified form of the Fisher information

Suppose, in addition, that the operations of integration and differentiation can be swapped for the second derivative of $f(x;\theta )$ as well, i.e.,

${\frac {\partial ^{2}}{\partial \theta ^{2}}}\left[\int T(x)f(x;\theta )\,dx\right]=\int T(x)\left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}f(x;\theta )\right]\,dx.$

In this case, it can be shown that the Fisher information equals

$I(\theta )=-\mathrm {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\log f(X;\theta )\right].$

The Cramèr–Rao bound can then be written as

$\mathrm {var} \left({\widehat {\theta }}\right)\geq {\frac {1}{I(\theta )}}={\frac {1}{-\mathrm {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\log f(X;\theta )\right]}}.$

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 above; namely, that if the expectation of $T$ is denoted by $\psi (\theta )$, then, for all $\theta$,

${\rm {var}}(t(X))\geq {\frac {[\psi ^{\prime }(\theta )]^{2}}{I(\theta )}}.$

Let $X$ be a random variable with probability density function $f(x;\theta )$. Here $T=t(X)$ is a statistic, which is used as an estimator for $\psi (\theta )$. If $V$ is the score, i.e.

$V={\frac {\partial }{\partial \theta }}\ln f(X;\theta )$

then the expectation of $V$, written ${\rm {E}}(V)$, is zero. If we consider the covariance ${\rm {cov}}(V,T)$ of $V$ and $T$, we have ${\rm {cov}}(V,T)={\rm {E}}(VT)$, because ${\rm {E}}(V)=0$. Expanding this expression we have

${\rm {cov}}(V,T)={\rm {E}}\left(T\cdot {\frac {\partial }{\partial \theta }}\ln f(X;\theta )\right)$

This may be expanded using the chain rule

${\frac {\partial }{\partial \theta }}\ln Q={\frac {1}{Q}}{\frac {\partial Q}{\partial \theta }}$

and the definition of expectation gives, after cancelling $f(x;\theta )$,

${\rm {E}}\left(T\cdot {\frac {\partial }{\partial \theta }}\ln f(X;\theta )\right)=\int t(x)\left[{\frac {\partial }{\partial \theta }}f(x;\theta )\right]\,dx={\frac {\partial }{\partial \theta }}\left[\int t(x)f(x;\theta )\,dx\right]=\psi ^{\prime }(\theta )$

because the integration and differentiation operations commute (second condition).

The Cauchy–Schwarz inequality shows that

${\sqrt {{\rm {var}}(T){\rm {var}}(V)}}\geq \left|{\rm {cov}}(V,T)\right|=\left|\psi ^{\prime }(\theta )\right|$

therefore

${\rm {var\ }}T\geq {\frac {[\psi ^{\prime }(\theta )]^{2}}{{\rm {var}}(V)}}={\frac {[\psi ^{\prime }(\theta )]^{2}}{I(\theta )}}=\left[{\frac {\partial }{\partial \theta }}{\rm {E}}(T)\right]^{2}{\frac {1}{I(\theta )}}$

which proves the proposition.

Examples

Multivariate normal distribution

For the case of a d-variate normal distribution

${\boldsymbol {x}}\sim N_{d}\left({\boldsymbol {\mu }}\left({\boldsymbol {\theta }}\right),{\boldsymbol {C}}\left({\boldsymbol {\theta }}\right)\right)$

the Fisher information matrix has elements

$I_{m,k}={\frac {\partial {\boldsymbol {\mu }}^{T}}{\partial \theta _{m}}}{\boldsymbol {C}}^{-1}{\frac {\partial {\boldsymbol {\mu }}}{\partial \theta _{k}}}+{\frac {1}{2}}\mathrm {tr} \left({\boldsymbol {C}}^{-1}{\frac {\partial {\boldsymbol {C}}}{\partial \theta _{m}}}{\boldsymbol {C}}^{-1}{\frac {\partial {\boldsymbol {C}}}{\partial \theta _{k}}}\right)$

where "tr" is the trace.

For example, let $w[n]$ be a sample of $N$ independent observations) with unknown mean $\theta$ and known variance $\sigma ^{2}$

$w[n]\sim \mathbb {N} _{N}\left(\theta {\boldsymbol {1}},\sigma ^{2}{\boldsymbol {I}}\right).$

Then the Fisher information is a scalar given by

$I(\theta )=\left({\frac {\partial {\boldsymbol {\mu }}(\theta )}{\partial \theta }}\right)^{T}{\boldsymbol {C}}^{-1}\left({\frac {\partial {\boldsymbol {\mu }}(\theta )}{\partial \theta }}\right)=\sum _{i=1}^{N}{\frac {1}{\sigma ^{2}}}={\frac {N}{\sigma ^{2}}},$

and so the Cramér–Rao bound is

$\mathrm {var} \left({\hat {\theta }}\right)\geq {\frac {\sigma ^{2}}{N}}.$

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:

$T={\frac {\sum _{i=1}^{n}\left(X_{i}-\mu \right)^{2}}{n}}.$

Then T is unbiased for $\sigma ^{2}$, as $E(T)=\sigma ^{2}$. What is the variance of T?

$\mathrm {var} (T)={\frac {\mathrm {var} (X-\mu )^{2}}{n}}={\frac {1}{n}}\left[E\left\{(X-\mu )^{4}\right\}-\left(E\left\{(X-\mu )^{2}\right\}\right)^{2}\right]$

(the second equality follows directly from the definition of variance). The first term is the fourth moment about the mean and has value $3(\sigma ^{2})^{2}$; the second is the square of the variance, or $(\sigma ^{2})^{2}$. Thus

$\mathrm {var} (T)={\frac {2(\sigma ^{2})^{2}}{n}}.$

Now, what is the Fisher information in the sample? Recall that the score V is defined as

$V={\frac {\partial }{\partial \sigma ^{2}}}\log L(\sigma ^{2},X)$

where $L$ is the likelihood function. Thus in this case,

$V={\frac {\partial }{\partial \sigma ^{2}}}\log \left[{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(X-\mu )^{2}/{2\sigma ^{2}}}\right]={\frac {(X-\mu )^{2}}{2(\sigma ^{2})^{2}}}-{\frac {1}{2\sigma ^{2}}}$

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

$I=-E\left({\frac {\partial V}{\partial \sigma ^{2}}}\right)=-E\left(-{\frac {(X-\mu )^{2}}{(\sigma ^{2})^{3}}}+{\frac {1}{2(\sigma ^{2})^{2}}}\right)={\frac {\sigma ^{2}}{(\sigma ^{2})^{3}}}-{\frac {1}{2(\sigma ^{2})^{2}}}={\frac {1}{2(\sigma ^{2})^{2}}}.$

Thus the information in a sample of $n$ independent observations is just $n$ times this, or ${\frac {n}{2(\sigma ^{2})^{2}}}.$

The Cramer Rao bound states that

$\mathrm {var} (T)\geq {\frac {1}{I}}.$

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

$T={\frac {\sum _{i=1}^{n}\left(X_{i}-\mu \right)^{2}}{n+2}}.$

obviously has a smaller variance, which is in fact

$\mathrm {var} (T)={\frac {2n(\sigma ^{2})^{2}}{(n+2)^{2}}}.$

Its bias is

$\left(1-{\frac {n}{n+2}}\right)\sigma ^{2}={\frac {2\sigma ^{2}}{n+2}}$

so its mean squared error is

$\mathrm {MSE} (T)=\left({\frac {2n}{(n+2)^{2}}}+{\frac {4}{(n+2)^{2}}}\right)(\sigma ^{2})^{2}={\frac {2(\sigma ^{2})^{2}}{n+2}}$

which is clearly less than the Cramér–Rao bound found above.

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.

See also

• Chapman–Robbins bound
• Kullback's inequality

References and notes

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Further reading

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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang. Chapter 3.
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang. Section 3.1.3.

External links

• FandPLimitTool a GUI-based software to calculate the Fisher information and Cramer-Rao Lower Bound with application to single-molecule microscopy.