In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. The variance of a real-valued random variable is its second central moment, and it also happens to be its second cumulant. The variance of a random variable is the square of its standard deviation. Variance - Definition. If μ = E(X) is the expected value (mean) of the random variable X, then the variance is ...

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Variance - Definition Variance - Properties Variance - Population variance and sample variance
Variance - An unbiased estimator
Variance - Generalizations Variance - History Variance - Moment of inertia

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Variance, Variance - Definition, Variance - Properties, Variance - Population variance and sample variance, Variance - An unbiased estimator, Variance - Generalizations, Variance - History, Variance - Moment of inertia

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If X is a vector-valued random variable, with values in Rn, and thought of as a column vector, then the natural generalization of variance is E[(X − μ)(X − μ)T], where μ = E(X) and XT is the transpose of X, and so is a row vector. This variance is a nonnegative-definite square matrix, commonly referred to as the covariance matrix. If X is a complex-valued random variable, then its variance is E[(X − μ)(X − μ)*], where X* is the complex conjugate of X. ...

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Variance, Variance - Definition, Variance - Properties, Variance - Population variance and sample variance, Variance - An unbiased estimator, Variance - Generalizations, Variance - History, Variance - Moment of inertia

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In statistics, the Cramér-Rao inequality, named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, expresses a lower bound on the accuracy of a statistical estimator, based on Fisher information. It states that the reciprocal of the Fisher information, , of a parameter θ, is a lower bound on the variance of an unbiased estimator of the parameter (denoted ). In some cases ...

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Cramér-Rao inequality - Regularity conditions Cramér-Rao inequality - Multiple parameters Cramér-Rao inequality - Single-parameter proof Cramér-Rao inequality - Multivariate normal distribution

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For a point estimator of parameter θ, The error of is The bias of is defined as the expected value of the is an unbiased estimator of θ iff for all θ, or, equivalently, iff for all θ. The mean squared error of is defined as i.e. mean squared error = variance + square of bias. where var(X) is the variance of X and ...

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Estimator, Estimator - Point estimators, Estimator - Consistency, Estimator - Efficiency, Estimator - Other properties

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For a point estimator of parameter θ, The error of is The bias of is defined as is an unbiased estimator of θ iff for all θ, or, equivalently, iff for all θ. The mean squared error of is defined as i.e. mean squared error = variance + square of bias. where var(X) is the variance of X and ...

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Estimator, Estimator - Point estimators, Estimator - Consistency, Estimator - Efficiency, Estimator - Other properties

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The quality of an estimator is generally judged by its mean squared error. However, occasionally one chooses the unbiased estimator with the lowest variance. Efficient estimators are those that have the lowest possible variance among all unbiased estimators. In some cases, a biased estimator may have a uniformly smaller mean squared error than does any unbiased estimator, so one should not make too much of this concept. For that and other reasons, it is sometimes preferable not to limit oneself to unbiased estimators; see bias (statis ...

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Estimator, Estimator - Point estimators, Estimator - Consistency, Estimator - Efficiency, Estimator - Other properties

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A consistent estimator is an estimator that converges in probability to the quantity being estimated as the sample size grows. An estimator tn (where n is the sample size) is a consistent estimator for parameter θ if and only if, for all ε > 0, no matter how small, we have It is called strongly consistent, if it c ...

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Estimator, Estimator - Point estimators, Estimator - Consistency, Estimator - Efficiency, Estimator - Other properties

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There are numerous fields that require the use of estimation theory. Some of these fields include (but by no means limited to): Medicine Clinical trials Imaging: CAT EEG EKG/ECG MRI Medical ultrasonography Opinion polls Quality control Radar, sonar Localization of objects Telecommunications Channel parameters Noise variance DC gain (see example below) See also:

Estimation theory, Estimation theory - Fields that use estimation theory, Estimation theory - Estimation process, Estimation theory - Basics, Estimation theory - Estimators, Estimation theory - Example: DC gain in white Gaussian noise, Estimation theory - Maximum likelihood, Estimation theory - Cramér-Rao lower bounds, Estimation theory - Books

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Consider a received discrete signal, x[n], of N independent samples that consists of a DC gain A with Additive white Gaussian noise w[n] with known variance σ2 (i.e., ). Since the variance is known then the only unknown parameter is A. The model for the signal is then Two possible (of ...

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Estimation theory, Estimation theory - Fields that use estimation theory, Estimation theory - Estimation process, Estimation theory - Basics, Estimation theory - Estimators, Estimation theory - Example: DC gain in white Gaussian noise, Estimation theory - Maximum likelihood, Estimation theory - Cramér-Rao lower bounds, Estimation theory - Books

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The entire purpose of estimation theory is to arrive at an estimator, and preferably an implementable one that could actually be used. The estimator takes the measured data as input and produces an estimate of the parameters. It is also preferable to derive an estimator that exhibits optimality. An optimal estimator would indicate that all available information in the measured data has been extracted, for if there was unused informat ...

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Estimation theory, Estimation theory - Fields that use estimation theory, Estimation theory - Estimation process, Estimation theory - Basics, Estimation theory - Estimators, Estimation theory - Example: DC gain in white Gaussian noise, Estimation theory - Maximum likelihood, Estimation theory - Cramér-Rao lower bounds, Estimation theory - Books

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To build a model, several statistical "ingredients" need to be known. These are needed to ensure the estimator has some mathematical tractability instead of being based on "gut feel." The first is a set of statistical samples taken from a random vector (RV) of size N which can be put into a vector and their M parameters need to be established with their probability density function (pdf) or probabil ...

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Estimation theory, Estimation theory - Fields that use estimation theory, Estimation theory - Estimation process, Estimation theory - Basics, Estimation theory - Estimators, Estimation theory - Example: DC gain in white Gaussian noise, Estimation theory - Maximum likelihood, Estimation theory - Cramér-Rao lower bounds, Estimation theory - Books

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If an estimator of a parameter, , attains e(T) = 1 for all values of the parameter, then the estimator is called efficient. Equivalently, the estimator attains the equality on the Cramér-Rao inequality . Furthermore, an efficient estimator that is unbiased is also a minimum variance unbiased estimator. This is because an efficient estimator maintains equality on the Cramér-Rao inequality for all parameter values, which means it attains the minimum variance for all parameters (the def ...

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Efficiency statistics, Efficiency statistics - Efficient estimator, Efficiency statistics - Asymptotic efficiency, Efficiency statistics - Examples, Efficiency statistics - Relative efficiency

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Given a sub-set of samples from a population, the sample kurtosis above is a biased estimator of the population kurtosis. The usual estimator of the population kurtosis (used in SAS, SPSS, and Excel but not by MINITAB or BMDP) is G2, defined as follows: where k4 is the unique symmetric unbiased estimator of the fourth cumulant, k2 is the unbiased estimator of the population variance, m4 is the fourth sample moment about the mean, m2 is the sample ...

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Kurtosis, Kurtosis - Definition of kurtosis, Kurtosis - Terminology and examples, Kurtosis - Sample kurtosis, Kurtosis - Estimators of population kurtosis

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The small sample variance properties of G are not known, and large sample approximations to the variance of G are poor. In order for G to be an unbiased estimate of the true population value, it should be multiplied by n/(n-1). The Gini coefficient is calculated as a ratio of the areas on the Lorenz curve diagram. If the area between the line of perfect equality and Lorenz curve is A, and the area underneath the Lorenz curve is B, then the Gini coefficient is A/(A+B). This ratio is expressed as a percentage or as the numerical equivalent of that perce ...

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Gini coefficient, Gini coefficient - Calculation, Gini coefficient - Gini coefficients in the world, Gini coefficient - Development of Gini coefficients in the US over time, Gini coefficient - Advantages of the Gini coefficient as a measure of inequality, Gini coefficient - Disadvantages of the Gini coefficient as a measure of inequality

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Normal distribution - Maximum likelihood estimation of parameters. Suppose are independent and identically distributed, and are normally distributed with expectation μ and variance σ2. In the language of statisticians, the observed values of these random variables make up a "sample from a normally distributed population." It is desired to estimate the "population mean" μ and the "population standard deviation" σ, based on observed values of this sample. The joint probability de ...

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Normal distribution, Normal distribution - Overview, Normal distribution - History, Normal distribution - Specification of the normal distribution, Normal distribution - Probability density function, Normal distribution - Cumulative distribution function, Normal distribution - Generating functions, Normal distribution - Properties, Normal distribution - Standardizing normal random variables, Normal distribution - Moments, Normal distribution - Generating normal random variables, Normal distribution - The central limit theorem, Normal distribution - Infinite divisibility, Normal distribution - Stability, Normal distribution - Standard deviation, Normal distribution - Normality tests, Normal distribution - Related distributions, Normal distribution - Estimation of parameters, Normal distribution - Maximum likelihood estimation of parameters, Normal distribution - Unbiased estimation of parameters, Normal distribution - Occurrence, Normal distribution - Photon counting, Normal distribution - Measurement errors, Normal distribution - Physical characteristics of biological specimens, Normal distribution - Financial variables, Normal distribution - Lifetime, Normal distribution - Test scores

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Some of the properties of the normal distribution: If and a and b are real numbers, then (see expected value and variance). If and are independent normal random variables, then: Their sum is normally distributed with (proof). Their difference is normally distributed with . Both U and V are independent of each other. If and a ...

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Normal distribution, Normal distribution - Overview, Normal distribution - History, Normal distribution - Specification of the normal distribution, Normal distribution - Probability density function, Normal distribution - Cumulative distribution function, Normal distribution - Generating functions, Normal distribution - Properties, Normal distribution - Standardizing normal random variables, Normal distribution - Moments, Normal distribution - Generating normal random variables, Normal distribution - The central limit theorem, Normal distribution - Infinite divisibility, Normal distribution - Stability, Normal distribution - Standard deviation, Normal distribution - Normality tests, Normal distribution - Related distributions, Normal distribution - Estimation of parameters, Normal distribution - Maximum likelihood estimation of parameters, Normal distribution - Unbiased estimation of parameters, Normal distribution - Occurrence, Normal distribution - Photon counting, Normal distribution - Measurement errors, Normal distribution - Physical characteristics of biological specimens, Normal distribution - Financial variables, Normal distribution - Lifetime, Normal distribution - Test scores

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In general, the population variance of a finite population is given by where is the population mean. This is merely a special case of the general definition of variance introduced above, but restricted to finite populations. In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with large finite populations, it is almost never possible to find the exact value of the population variance, due to time, cost, and other resource constraints. W ...

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Variance, Variance - Definition, Variance - Properties, Variance - Population variance and sample variance, Variance - An unbiased estimator, Variance - Generalizations, Variance - History, Variance - Moment of inertia

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If μ = E(X) is the expected value (mean) of the random variable X, then the variance is That is, it is the expected value of the square of the deviation of X from its own mean. In plain language, it can be expressed as "The average of the square of the distance of each data point from the mean". It is thus the mean squared deviation. The variance of random variable X is typically designated as , , or simply σ2. Note that the above definition can be used for both di ...

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Variance, Variance - Definition, Variance - Properties, Variance - Population variance and sample variance, Variance - An unbiased estimator, Variance - Generalizations, Variance - History, Variance - Moment of inertia

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