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Estimation method that minimizes the mean square error
signal processing, a minimum mean square error estimator (MMSE estimator) is an estimation method which minimizes the mean square error (MSE), which is a
Minimum mean square error estimator
Minimum_mean_square_error_estimator
Measure of the error of an estimator
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures
Mean_squared_error
Unbiased statistical estimator minimizing variance
} A Bayesian analog is a Bayes estimator, particularly with minimum mean square error (MMSE). An efficient estimator need not exist, but if it does and
Minimum-variance unbiased estimator
Minimum-variance_unbiased_estimator
Statistical measure
The root mean square deviation (RMSD) or root mean square error (RMSE) is a frequently used measure of the distances between actual observed values and
Root_mean_square_deviation
Statistical property
because a biased estimator gives a lower value of some loss function (particularly mean squared error) compared with unbiased estimators (notably in shrinkage
Bias_of_an_estimator
Approximation method in statistics
where the errors have a mean of zero, are uncorrelated, normally distributed, and have equal variances, the best linear unbiased estimator of the coefficients
Least_squares
Condition for optimality of Bayesian estimator
Bayesian estimator. Loosely stated, the orthogonality principle says that the error vector of the optimal estimator (in a mean square error sense) is
Orthogonality_principle
Rule for calculating an estimate of a given quantity based on observed data
the "estimators". The attractiveness of different estimators can be judged by looking at their properties, such as unbiasedness, mean square error, consistency
Estimator
Statistics concept
observable prediction errors: The mean squared error (MSE) refers to the amount by which the values predicted by an estimator differ from the quantities
Errors_and_residuals
Statistical property
standard error (SE) of a statistic is the standard deviation of its sampling distribution. It is the square root of the variance of an estimator of a parameter
Standard_error
Method for estimating the unknown parameters in a linear regression model
unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased
Ordinary_least_squares
Middle quantile of a data set or probability distribution
medians of the subsamples. Any mean-unbiased estimator minimizes the risk (expected loss) with respect to the squared-error loss function, as observed by
Median
Statistical amount
al. (1988) when treating the weighted mean as a combination of a weighted total estimator divided by an estimator of the population size, based on the
Weighted_arithmetic_mean
Theorem related to ordinary least squares
restricting to unbiased estimators, minimum mean squared error implies minimum variance. The goal is therefore to show that such an estimator has a variance no
Gauss–Markov_theorem
Measure of variation in statistics
but it is still consistent. Its mean squared error, on the other hand, may be lower than that of the unbiased estimator. If the population of interest
Standard_deviation
Mathematical decision rule
\theta \,p(\theta |x)\,d\theta .} This is known as the minimum mean square error (MMSE) estimator. If there is no inherent reason to prefer one prior probability
Bayes_estimator
Summary statistic of variability
accuracy is very closely related to the mean squared error (MSE) method which is just the average squared error of the forecasts. Although these methods
Average_absolute_deviation
Statistical algorithm
finding the filter coefficients that relate to producing the least mean square of the error signal (difference between the desired and the actual signal).
Least_mean_squares_filter
Statistical theorem
transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar
Rao–Blackwell_theorem
Quality measure of a statistical method
calculated by finding the mean squared error. More formally, let T be an estimator for the parameter θ. The mean squared error of T is the value MSE ( T ) =
Efficiency_(statistics)
Probability distribution
biased estimator σ ^ 2 {\displaystyle \textstyle {\hat {\sigma }}^{2}} is better than the s 2 {\textstyle s^{2}} in terms of the mean squared error (MSE)
Normal_distribution
Statistical method for resampling
other than the mean. This simple example for the case of mean estimation is just to illustrate the construction of a jackknife estimator, while the real
Jackknife_resampling
Method for fitting a statistical model to data
empirical distribution. Often-used estimators such as ordinary least squares can be thought of as special cases of minimum-distance estimation. While consistent
Minimum-distance_estimation
Type of statistics
based on the mean, are typically bounded above by the nominal size of the test. The same is not true of M-estimators and the type I error rate can be substantially
Robust_statistics
Statistical property
which assume that the modelling errors all have the same variance. While the ordinary least squares (OLS) estimator is still unbiased in the presence
Homoscedasticity and heteroscedasticity
Homoscedasticity_and_heteroscedasticity
Statistical measure of the magnitude of a phenomenon
education research. A similar effect size estimator for multiple comparisons (e.g., ANOVA) is the Ψ root-mean-square standardized effect: Ψ = 1 k − 1 ⋅ ∑ j
Effect_size
Class of statistical estimators
statistics, M-estimators are a broad class of extremum estimators for which the objective function is a sample average. Both non-linear least squares and maximum
M-estimator
Method of estimating the parameters of a statistical model, given observations
instance, the ordinary least squares estimator for a linear regression model maximizes the likelihood when the random errors are assumed to have normal
Maximum_likelihood_estimation
Indicator for how well data points fit a line or curve
the errors and the dependent variable instead of estimating them. Ingram Olkin and John W. Pratt derived the minimum-variance unbiased estimator for the
Coefficient_of_determination
Number taken as representative of a list of numbers
mean Logarithmic mean Moving average Neuman–Sándor mean Quasi-arithmetic mean Root mean square (quadratic mean) Rényi's entropy (a generalized f-mean)
Average
Single measure of some attribute of a sample
sample mean is an unbiased estimator of the population mean. This means that the expected value of the sample mean equals the true population mean. A descriptive
Statistic
Branch of statistics
all unbiased estimators. Due to the bias-variance decomposition, they are optimal in the sense that they minimise the mean squared error among all unbiased
Parametric_statistics
Statistical measure of how far values spread from their average
the population (see Mean squared error § Variance) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number
Variance
Concept in statistics
characteristic-function estimator therefore converges pointwise at the parametric n − 1 {\displaystyle n^{-1}} mean squared error rate, although the underlying
Kernel_density_estimation
Measure of linear correlation
\quad } therefore r is a biased estimator of ρ . {\displaystyle \rho .} The unique minimum variance unbiased estimator radj is given by where: r , n {\displaystyle
Pearson correlation coefficient
Pearson_correlation_coefficient
Linear regression model with a single explanatory variable
that of the normality of the error terms, the estimator of the slope coefficient will itself be normally distributed with mean β and variance σ 2 / ∑ i (
Simple_linear_regression
Asymptotic variances under heteroskedasticity
the OLS point estimator remains unbiased, it is not "best" in the sense of having minimum mean square error, and the OLS variance estimator V ^ [ β ^ O
Heteroskedasticity-consistent standard errors
Heteroskedasticity-consistent_standard_errors
Loss function used in robust regression
the mean-unbiased, minimum-variance estimator of the mean (using the quadratic loss function) and the robustness of the median-unbiased estimator (using
Huber_loss
Probability distribution
\end{aligned}}} Other estimators also exist, such as Finney's UMVUE estimator, the "Approximately Minimum Mean Squared Error Estimator", the "Approximately
Log-normal_distribution
Parameter estimation via sample statistics
unbiased estimators. According to the bias-variance decomposition, the variance of an unbiased estimator is equal to its mean squared error (MSE), which
Point_estimation
Branch of statistics to estimate models based on measured data
Maximum likelihood estimators Bayes estimators Method of moments estimators Cramér–Rao bound Least squares Minimum mean squared error (MMSE), also known
Estimation_theory
Graphical representation of the distribution of numerical data
gives the minimum number of bins required for an asymptotically optimal histogram, where optimality is measured by the integrated mean squared error. The bound
Histogram
Complete set of items that share at least one property in common
likely it is that the sample mean will be close to the population mean. Data collection system Horvitz–Thompson estimator Sample (statistics) Stratum (statistics)
Statistical_population
Lower bound on variance of an estimator
unbiased estimator that achieves this bound is said to be (fully) efficient. Such a solution achieves the lowest possible mean squared error among all
Cramér–Rao_bound
Statistical model to calculate the value of multiple quantities as they change over time
maximum likelihood estimator (MLE) of the covariance matrix differs from the ordinary least squares (OLS) estimator. MLE estimator:[citation needed] Σ
Vector_autoregression
non-robust L-estimators include the minimum, maximum, mean, and mid-range. The trimmed equivalents are robust, however. Robust L-estimators used to measure
L-estimator
Algorithm that estimates unknowns from a series of measurements over time
the best possible linear estimator in the minimum mean-square-error sense, although there may be better nonlinear estimators. It is a common misconception
Kalman_filter
Statistical considerations on how many observations to make
the observations are independent, this estimator has a (scaled) binomial distribution (and is also the sample mean of data from a Bernoulli distribution)
Sample_size_determination
Statistical modeling method
Bayes method Errors and residuals Lack-of-fit sum of squares Line fitting Linear classifier Linear equation Logistic regression M-estimator Multivariate
Linear_regression
formalising the idea that an estimator should have certain intuitively appealing qualities. Strictly speaking, "invariant" would mean that the estimates themselves
Invariant_estimator
Statistical measure of association
population quantity as Cramér's V but with typically much smaller mean squared error. The rationale for the correction is that under independence, E [
Cramér's_V
Regularization technique for ill-posed problems
ridge regression estimator (RR). This provides a more precise ridge parameters estimate, as its variance and mean square estimator are often smaller
Ridge_regression
Procedure to estimate standard deviation from a sample
the standard error of s is σ 1 − c 4 2 {\displaystyle \sigma {\sqrt {1-c_{4}^{2}}}} , while the standard error of the unbiased estimator is σ c 4 − 2
Unbiased estimation of standard deviation
Unbiased_estimation_of_standard_deviation
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMSD), and relative standard deviation (RSD), is a standardized
Coefficient_of_variation
Mathematical relation assigning a probability event to a cost
{\theta }})^{2}\right].} An estimator found by minimizing the mean squared error estimates the posterior distribution's mean. In density estimation, the
Loss_function
Method for model fitting in statistics
{\boldsymbol {\beta }}}=X^{\textsf {T}}Wy} .} If the errors are correlated, the resulting estimator is the BLUE if the weight matrix is equal to the inverse
Weighted_least_squares
Family of statistical methods based on sampling of available data
of an estimator by sampling with replacement from the original sample, most often with the purpose of deriving robust estimates of standard errors and confidence
Resampling_(statistics)
Probability distribution
it equal to zero yields the maximum likelihood estimator of the θ parameter, which equals the sample mean x ¯ {\displaystyle {\bar {x}}} divided by the
Gamma_distribution
Statistical estimator for ratio of means
The ratio estimator is a statistical estimator for the ratio of means of two random variables. Ratio estimates are biased and corrections must be made
Ratio_estimator
Statistical hypothesis test
uncorrelated). Let α ^ , β ^ = least-squares estimators , S E α ^ , S E β ^ = the standard errors of least-squares estimators . {\displaystyle {\begin{aligned}{\hat
Student's_t-test
Correction for sample variance bias
unbiased estimator of standard deviation. The corrected estimator often has a higher mean squared error (MSE) than the uncorrected estimator. Furthermore
Bessel's_correction
Method of estimating the parameters of a statistical model
posterior mean or median instead, together with credible intervals. This is both because these estimators are optimal under squared-error and linear-error loss
Maximum a posteriori estimation
Maximum_a_posteriori_estimation
Organized raw data that has not been otherwise processed or transformed
Discretization of continuous features Logistic regression § Minimum chi-squared estimator for grouped data Newbold, P.; Carlson, W.; Thorne, B. (2009)
Grouped_data
Statistical phenomenon
In statistics, regression toward the mean (also called regression to the mean, reversion to the mean, and reversion to mediocrity) is the phenomenon where
Regression_toward_the_mean
Uniform distribution on an interval
sample mid-range, i.e. the arithmetic mean of the sample maximum and the sample minimum, which is the UMVU estimator of the midpoint (and also the maximum
Continuous uniform distribution
Continuous_uniform_distribution
Concepts from statistical hypothesis testing
Type I error, or a false positive, is the incorrect rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false
Type_I_and_type_II_errors
Phenomenon in statistics
population, as discussed at mean squared error: variance, but one can always do better (in terms of MSE) than the unbiased estimator; for the normal distribution
Shrinkage_(statistics)
Study of collection and analysis of data
estimators, a widely used class of estimators. Root mean square error is simply the square root of mean squared error. Many statistical methods seek to
Statistics
Least squares approximation of linear functions to data
{\boldsymbol {\beta }}}} is known, then a Bayes estimator can be used to minimize the mean squared error, E { ‖ β − β ^ ‖ 2 } {\displaystyle E\left\{\|{\boldsymbol
Linear_least_squares
Minimax estimator Minimisation (clinical trials) Minimum chi-square estimation Minimum distance estimation Minimum mean square error Minimum-variance
List_of_statistics_articles
Statistical method
estimators. Popular families of point-estimators include mean-unbiased minimum-variance estimators, median-unbiased estimators, Bayesian estimators (for
Bootstrapping_(statistics)
Statistical model validation technique
continuously distributed, the mean squared error, root mean squared error or median absolute deviation could be used to summarize the errors. When users apply cross-validation
Cross-validation_(statistics)
Statistical model
data analysis the term fixed effects estimator (also known as the within estimator) is used to refer to an estimator for the coefficients in the regression
Fixed_effects_model
Estimate of an interval in which future observations will fall
unbiased estimate, while dividing by n yields the maximum likelihood estimator, and either might be used. One then uses the quantile function with these
Prediction_interval
Statistical hypothesis test
by the continuous chi-squared distribution. This assumption is not quite correct and introduces some error. To reduce the error in approximation, Frank
Chi-squared_test
Method of data analysis
from the mean Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating
Principal_component_analysis
Type of average of a collection of numbers
{x}})^{2}} . The sample mean is also the best single predictor because it has the lowest root mean squared error. If the arithmetic mean of a population of
Arithmetic_mean
Non-parametric statistic used to estimate the survival function
The Kaplan–Meier estimator, also known as the product limit estimator, is a non-parametric statistic used to estimate the survival function from lifetime
Kaplan–Meier_estimator
Probability distribution
the skewness and kurtosis estimators used in BMDP and in MINITAB (at that time) had smaller variance and mean-squared error in normal samples, but the
Beta_distribution
Measure of the asymmetry of random variables
symmetric unbiased estimator of the third cumulant and k 2 = s 2 {\displaystyle k_{2}=s^{2}} is the symmetric unbiased estimator of the second cumulant
Skewness
Statistical measure of variability
is a more robust estimator of scale than the sample variance or standard deviation, it works better with distributions without a mean or variance, such
Median_absolute_deviation
Property of a model
y_{n})\}} . We make "as well as possible" precise by measuring the mean squared error between y {\displaystyle y} and f ^ ( x ; D ) {\displaystyle {\hat
Bias–variance_tradeoff
Range to estimate an unknown parameter
the true value of an unknown statistical parameter, such as a population mean. Rather than reporting a single point estimate (e.g. "the average screen
Confidence_interval
Statistical technique
which the corresponding estimator β ^ L {\displaystyle {\widehat {\boldsymbol {\beta }}}_{L}} achieves the minimum prediction error is given by: L ( p −
Principal component regression
Principal_component_regression
Inverse of the average of the inverses of a set of numbers
In numerical experiments H3 is generally a superior estimator of the harmonic mean than H1. H2 produces estimates that are largely similar to H1
Harmonic_mean
Statistical model for a binary dependent variable
chi-squared test, we may then estimate how many of these permuted sets of yk will yield a minimum error less than or equal to the minimum error using
Logistic_regression
Fundamental theorem in probability theory and statistics
least squares, specifies that a dependent variable depends according to some function upon one or more independent variables, with an additive error term
Central_limit_theorem
Robust and nonparametric estimator of a population's location parameter
In statistics, the Hodges–Lehmann estimator is a robust and nonparametric estimator of a population's location parameter. For populations that are symmetric
Hodges–Lehmann_estimator
Regression analysis
are Jacobian matrix elements. It follows from this that the least squares estimators are given by β ^ ≈ ( J T J ) − 1 J T y , {\displaystyle {\hat {\boldsymbol
Nonlinear_regression
representative of the larger population, the sample mean is often used as an estimator of the population mean, which in this example would be the average test
Glossary of probability and statistics
Glossary_of_probability_and_statistics
N-th root of the product of n numbers
numbers is the square root of their product, for example with numbers 2 {\displaystyle 2} and 8 {\displaystyle 8} the geometric mean is 2 ⋅ 8 = {\displaystyle
Geometric_mean
Statistical method
average is no longer an optimal estimator, since the error in X ¯ {\displaystyle {\overline {X}}} might actually exceed the error in the least noisy measurement
Inverse-variance_weighting
Diagnostic plot of binary classifier ability
Type I Error of the decision rule (when the performance is calculated from just a sample of the population, it can be thought of as estimators of these
Receiver operating characteristic
Receiver_operating_characteristic
Data visualization
In addition, the box plot allows one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean
Box_plot
Number of values in the final calculation of a statistic that are free to vary
William Sealy Gosset in his 1908 Biometrika article "The Probable Error of a Mean", published under the pen name "Student". While Gosset did not actually
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Statistical method
the data. In factor analysis, the best fit is defined as the minimum of the mean square error in the off-diagonal residuals of the correlation matrix: ε
Factor_analysis
Fourth standardized moment in statistics
{\displaystyle g_{2}} above is a biased estimator of the population excess kurtosis. An alternative estimator of the population excess kurtosis, which
Kurtosis
Specialized form of regression analysis, in statistics
trimmed squares (LTS) is a viable alternative and is currently (2007) the preferred choice of Rousseeuw and Ryan (1997, 2008). The Theil–Sen estimator has
Robust_regression
Process of using data analysis for predicting population data from sample data
median-unbiased estimators are optimal under absolute value loss functions, in that they minimize expected loss, and least squares estimators are optimal
Statistical_inference
Value that appears most often in a set of data
distributions, the mode is within √3 standard deviations of the mean, and the root mean square deviation about the mode is between the standard deviation and
Mode_(statistics)
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MINIMUM MEAN-SQUARE-ERROR-ESTIMATOR
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