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statistics, minimum chi-square estimation is a method of estimation of unobserved quantities based on observed data. In certain chi-square tests, one rejects
Minimum_chi-square_estimation
Statistical hypothesis test
Chi-squared test nomogram Cramér's V GEH statistic G-test Minimum chi-square estimation Nonparametric statistics Wald test Wilson score interval "Chi-Square
Chi-squared_test
Evaluates how likely it is that any difference between data sets arose by chance
minimizing the chi-squared statistic. More generally however, when maximum likelihood estimation does not coincide with minimum chi-squared estimation, the distribution
Pearson's_chi-squared_test
Approximation method in statistics
October 2023. van de Geer, Sara (June 1987). "A New Approach to Least-Squares Estimation, with Applications". Annals of Statistics. 15 (2): 587–602. doi:10
Least_squares
Method for fitting a statistical model to data
ordinary least squares can be thought of as special cases of minimum-distance estimation. While consistent and asymptotically normal, minimum-distance estimators
Minimum-distance_estimation
Interval bounded by an upper and a lower limit statistics
In statistics, interval estimation is the use of sample data to estimate an interval of possible values of a (sample) parameter of interest. This is in
Interval_estimation
Branch of statistics
is the uniform distribution, then MAP estimation is equivalent to maximum likelihood estimation. Uniformly minimum-variance unbiased estimators (UMVUE)
Parametric_statistics
Measure of the error of an estimator
error Mean square quantization error Reduced chi-squared statistic Mean squared displacement Mean squared prediction error Minimum mean square error Overfitting
Mean_squared_error
(clinical trials) Minimum chi-square estimation Minimum distance estimation Minimum mean square error Minimum-variance unbiased estimator Minimum viable population
List_of_statistics_articles
Parameter estimation via sample statistics
In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate, since it identifies a point rather
Point_estimation
Statistical test used on paired nominal data
2 {\displaystyle \chi ^{2}} has a chi-squared distribution with 1 degree of freedom. If the χ 2 {\displaystyle \chi ^{2}} result is significant, this
McNemar's_test
Statistical property
Rao–Blackwell procedure for mean-unbiased estimation but for a larger class of loss-functions. Any minimum-variance mean-unbiased estimator minimizes
Bias_of_an_estimator
Procedure to estimate standard deviation from a sample
In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated
Unbiased estimation of standard deviation
Unbiased_estimation_of_standard_deviation
Concept in statistics
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method
Kernel_density_estimation
Statistical modeling method
parameters of linear regression models with standard estimation techniques such as ordinary least squares, it is necessary to make a number of assumptions
Linear_regression
Unbiased statistical estimator minimizing variance
minimum mean square error (MMSE). An efficient estimator need not exist, but if it does and if it is unbiased, it is the MVUE. Since the mean squared
Minimum-variance unbiased estimator
Minimum-variance_unbiased_estimator
Danish statistician (1878–1939)
produced an influential paper in the journal Biometrika on minimum chi-square estimation of the correlation coefficient. Disagreements about aspects
Kirstine_Smith
Least squares approximation of linear functions to data
This special case of GLS is called "weighted least squares". The GLS solution to an estimation problem is β ^ = ( X T Ω − 1 X ) − 1 X T Ω − 1 y , {\displaystyle
Linear_least_squares
Form of causal modeling that fit networks of constructs to data
but it is also used in econometrics (where it's known as structural estimation), epidemiology, business, and other fields. By a standard definition,
Structural_equation_modeling
Data analysis approach in frequentist statistics
Estimation statistics, or simply estimation, is a data analysis framework that uses a combination of effect sizes, confidence intervals, precision planning
Estimation_statistics
Statistical measure of association
Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the minimum dimension minus 1: V = φ 2 min (
Cramér's_V
Measure of variation in statistics
of uncertainty Percentile Raw data Reduced chi-squared statistic Robust standard deviation Root mean square Sample size Samuelson's inequality Six Sigma
Standard_deviation
Method of estimating the parameters of a statistical model, given observations
using a chi-squared distribution Generalized method of moments: methods related to the likelihood equation in maximum likelihood estimation M-estimator:
Maximum_likelihood_estimation
Method for estimating the unknown parameters in a linear regression model
uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional
Ordinary_least_squares
Statistical test
involves the observation that Pearson's chi-squared test is an approximate test. Suppose Pearson's chi-squared test is used to ascertain whether a six-sided
Exact_test
Metric for fit of statistical models
Pearson's chi-square test). In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares. In
Goodness_of_fit
Statistical regression where the dependent variable can take only two values
^{-1}({\hat {p}}_{t}){\big )}}}} Then Berkson's minimum chi-square estimator is a generalized least squares estimator in a regression of Φ − 1 ( p ^ t )
Probit_model
Lower bound on variance of an estimator
In estimation theory and statistics, the Cramér–Rao bound (CRB) relates to estimation of a deterministic (fixed, though unknown) parameter. The result
Cramér–Rao_bound
Statistical model for a binary dependent variable
Using the 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
Logistic_regression
Graphical representation of the distribution of numerical data
density of the underlying distribution of the data, and often for density estimation: estimating the probability density function of the underlying variable
Histogram
Estimator for quality of a statistical model
model with minimum AIC is equivalent to selecting the model with minimum RSS—which is the usual objective of model selection based on least squares. Leave-one-out
Akaike_information_criterion
Relative measure of dispersion expressed as the ratio of standard deviation to the mean
coefficient of variation in normally distributed data is often based on McKay's chi-square approximation for the coefficient of variation. Liu (2012) reviews methods
Coefficient_of_variation
Statistical test
RVR')} Recalling that a quadratic form of normal distribution has a Chi-squared distribution: n ( R θ ^ n − r ) ′ [ R V R ′ ] − 1 n ( R θ ^ n − r ) →
Wald_test
Statistical property
term "standard error" can also be used to refer to the square root of the reduced chi-squared statistic in addition to the more common use in describing
Standard_error
Statistical measure of the magnitude of a phenomenon
square root of the chi-squared statistic divided by the sample size. Similarly, Cramér's V is computed by taking the square root of the chi-squared statistic
Effect_size
Statistical property
non-linear transformations of the X variables). Apply a weighted least squares estimation method, in which OLS is applied to transformed or weighted values
Homoscedasticity and heteroscedasticity
Homoscedasticity_and_heteroscedasticity
Statistical method
deliver the local treatment effect. The two most common approaches to estimation using an RDD are non-parametric and parametric (normally polynomial regression)
Regression discontinuity design
Regression_discontinuity_design
Table that displays the frequency of variables
{\displaystyle \phi =\pm {\sqrt {\frac {\chi ^{2}}{N}}},} where χ2 is computed as in Pearson's chi-squared test, and N is the grand total of observations
Contingency_table
Number of values in the final calculation of a statistic that are free to vary
associated with the squared lengths (or "sum of squares" of the coordinates) of such vectors, and the parameters of chi-squared and other distributions
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Statistical considerations on how many observations to make
Sample size determination or estimation is the act of choosing the number of observations or replicates to include in a statistical sample. The sample
Sample_size_determination
Statistical method for resampling
a form of resampling. It is especially useful for bias and variance estimation. The jackknife pre-dates other common resampling methods such as the bootstrap
Jackknife_resampling
Indian statistician (1915–1996)
the sum of chi-squares", Sankhya, 7 (1945), 27 - 28. In this paper, an expression of the distribution function of sum two dependent Chi-square random variables
Anil_Kumar_Bhattacharyya
Kind of numerical parameter of a parametric family of probability distributions
linear estimators also exist, such as the L-moments. Maximum likelihood estimation can also be used. The following continuous probability distributions have
Shape_parameter
Indicator for how well data points fit a line or curve
Gwowen (2008-04-01). "Improved shrinkage estimation of squared multiple correlation coefficient and squared cross-validity coefficient". Organizational
Coefficient_of_determination
Statistical test
significance tests that are increasingly being used in situations where chi-squared tests were previously recommended. The general formula for test statistics
G-test
Specialized form of regression analysis, in statistics
process as the rest of the data, least squares estimation is inefficient and can be biased. Because the least squares predictions are dragged towards the
Robust_regression
Study of high-dimensional data
deterioration in estimation performance in high dimensions observed in the previous paragraph is not limited to the ordinary least squares estimator. In
High-dimensional_statistics
Branch of statistics focusing on spatial data sets
random variable) theory to model the uncertainty associated with spatial estimation and simulation. A number of simpler interpolation methods/algorithms,
Geostatistics
Method of estimating the parameters of a statistical model
the quantity one wants to estimate. MAP estimation is therefore a regularization of maximum likelihood estimation. Assume that we want to estimate an unobserved
Maximum a posteriori estimation
Maximum_a_posteriori_estimation
Regression analysis
find the global minimum of a sum of squares. For details concerning nonlinear data modeling see least squares and non-linear least squares. The assumption
Nonlinear_regression
Statistics concept
a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate
Estimation of covariance matrices
Estimation_of_covariance_matrices
Mathematical problem
orthogonal Latin squares is Graeco-Latin square, introduced by Euler. A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Statistical theorem
arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria. The Rao–Blackwell
Rao–Blackwell_theorem
Statistical model validation technique
Cross-validation, sometimes called rotation estimation or out-of-sample testing, is any of various similar model validation techniques for assessing how
Cross-validation_(statistics)
Numerical measure of a statistical relationship between variables
x^{2}} and ∑ y 2 {\displaystyle \sum y^{2}} are the sums of squared x-scores and squared y-scores. Intraclass correlation (ICC) is a descriptive statistic
Correlation_coefficient
Data visualization
on the five-number summary: the minimum, the maximum, the sample median, and the first and third quartiles. Minimum (Q0 or 0th percentile): the lowest
Box_plot
Statistical test that compares goodness of fit
approximations thereof: e.g. the Z-test, the F-test, the G-test, and Pearson's chi-squared test; for an illustration with the one-sample t-test, see below. If the
Likelihood-ratio_test
Statistical analysis where the sample size is not fixed in advance
stage than would be possible with more classical hypothesis testing or estimation, at consequently lower financial and/or human cost. The method of sequential
Sequential_analysis
Estimate of an unobservable underlying probability density function
In statistics, probability density estimation or simply density estimation is the construction of an estimate, based on observed data, of an unobservable
Density_estimation
Method of statistical inference
be equal given "conventional wisdom". 1900: Karl Pearson develops the chi squared test to determine "whether a given form of frequency curve will effectively
Statistical_hypothesis_test
Nonparametric measure of rank correlation
)^{2}}}\leq \chi _{1,\alpha }^{2}\right\},} where χ 1 , α 2 {\displaystyle \chi _{1,\alpha }^{2}} is the α {\displaystyle \alpha } quantile of a chi-square distribution
Spearman's rank correlation coefficient
Spearman's_rank_correlation_coefficient
Statistical methods for comparing samples
sample-size for a minimum-detectable-effect calculations. The test is related to other well known tests such as Pearson's chi-squared test, Fisher's exact
Two-proportion_Z-test
Concept in inferential statistics
table, or in some other way. Mathematics portal A/B testing, ABX test Estimation statistics Fisher's method for combining independent tests of significance
Statistical_significance
Single measure of some attribute of a sample
statistics, such as t-statistic, chi-squared statistic, f statistic Order statistics, including sample maximum and minimum Sample moments and functions thereof
Statistic
Statistics method
this provides an intuitive justification for this type of approach to estimation. Generalized estimating equations Method of moments (statistics) Generalized
Estimating_equations
Signal processing technique
statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the
Spectral_density_estimation
Kth smallest value in a statistical sample
Garg, Vikram V.; Tenorio, Luis; Willcox, Karen (2017). "Minimum local distance density estimation". Communications in Statistics - Theory and Methods. 46
Order_statistic
Concept in statistics
the largest and smallest values (also known as the sample maximum and minimum). It is expressed in the same units as the data. The range provides an
Range_(statistics)
Statistical technique to aid interpretation of data
Linear trend estimation is a statistical technique used to analyze data patterns. Data patterns, or trends, occur when the information gathered tends to
Linear_trend_estimation
Method of estimating a statistical model's parameters
In statistics, maximum spacing estimation (MSE or MSP), or maximum product of spacing estimation (MPS), is a method for estimating the parameters of a
Maximum_spacing_estimation
Sequence of data points over time
cannot adequately represent. Estimation of TVAR models typically involves methods such as kernel smoothing, recursive least squares, or Kalman filtering. Non-linear
Time_series
Family of statistical methods based on sampling of available data
to the sample variance tends to be distributed as one half the square of a chi square distribution with two degrees of freedom. Instead of using the jackknife
Resampling_(statistics)
Function related to statistics and probability theory
equal to cPr[x | θ] for some positive value c. In maximum likelihood estimation, the model parameter(s) or argument that maximizes the likelihood function
Likelihood_function
Statistical test of variance
refers to an overall or a global test. Other names include F-test or Chi-squared test. It is a statistical test implemented on an overall hypothesis that
Omnibus_test
Time series model
Chi-square table value, we reject the null hypothesis and conclude there is an ARCH effect in the ARMA model. If T'R² is smaller than the Chi-square table
Autoregressive conditional heteroskedasticity
Autoregressive_conditional_heteroskedasticity
Probability distribution
distribution Log-normal distribution O'Hagan, A.; Leonard, Tom (1976). "Bayes estimation subject to uncertainty about parameter constraints". Biometrika. 63 (1):
Skew_normal_distribution
Process of using data analysis for predicting population data from sample data
descriptive complexity), MDL estimation is similar to maximum likelihood estimation and maximum a posteriori estimation (using maximum-entropy Bayesian
Statistical_inference
Branch of statistics
chi-squared test) Student's t distribution, the distribution of the ratio of a standard normal variable and the square root of a scaled chi squared variable;
Mathematical_statistics
Probabilistic problem-solving algorithm
Moral, G. Rigal, and G. Salut. "Estimation and nonlinear optimal control: Particle resolution in filtering and estimation: Experimental results". Convention
Monte_Carlo_method
Mathematical decision rule
In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value
Bayes_estimator
Model selection principle
extended to other forms of inductive inference and learning, for example to estimation and sequential prediction, without explicitly identifying a single model
Minimum_description_length
Statistical measure of how far values spread from their average
}}},\end{aligned}}} where ymin is the minimum of the sample. The F-test of equality of variances and the chi square tests are adequate when the sample is
Variance
Model for generating observable data in probability and statistics
as synthetic data generation. Generative models are used for density estimation, simulation, and learning with missing or partially labeled data. In classification
Generative_model
Statistical estimator for ratio of means
imaging. Cytometry 39:300–305 Ogliore RC, Huss GR, Nagashima K (2011) Ratio estimation in SIMS analysis. Nuclear Instruments and Methods in Physics Research
Ratio_estimator
Class of statistical estimators
objective function is a sample average. Both non-linear least squares and maximum likelihood estimation are special cases of M-estimators. The definition of M-estimators
M-estimator
Measure of linear correlation
to robust estimation and hypothesis testing. Academic Press. Devlin, Susan J.; Gnanadesikan, R.; Kettenring J.R. (1975). "Robust estimation and outlier
Pearson correlation coefficient
Pearson_correlation_coefficient
central limit theorem central moment characteristic function chi-squared distribution chi-squared test cluster analysis cluster sampling complementary event
Glossary of probability and statistics
Glossary_of_probability_and_statistics
Complete set of items that share at least one property in common
Point estimation Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance
Statistical_population
Statistics concept
_{i=1}^{n}r_{i}^{2}\sim \chi _{n-1}^{2}.} This difference between n and n − 1 degrees of freedom results in Bessel's correction for the estimation of sample variance
Errors_and_residuals
Non-parametric statistical test
approximated by that of a chi-squared distribution. In this case the p-value is given by P ( χ k − 1 2 ≥ Q ) {\displaystyle \mathbf {P} (\chi _{k-1}^{2}\geq Q)}
Friedman_test
Parameter estimation technique in statistics, particularly econometrics
conditions, and can therefore be thought of as a special case of minimum-distance estimation. The GMM estimators are known to be consistent, asymptotically
Generalized_method_of_moments
Statistical theorem
{\displaystyle -2\log(\Lambda )} asymptotically approaches the chi-squared ( χ 2 {\displaystyle \chi ^{2}} ) distribution under the null hypothesis H 0 {\displaystyle
Wilks'_theorem
Sampling from a population which can be partitioned into subpopulations
across these towns and hence is biased, causing a significant error in estimation (when the outcome of interest has a different distribution, in terms of
Stratified_sampling
Statistical test
{\displaystyle Q>\chi _{1-\alpha ,h}^{2}} where χ 1 − α , h 2 {\displaystyle \chi _{1-\alpha ,h}^{2}} is the (1 − α)-quantile of the chi-squared distribution
Ljung–Box_test
Simultaneous observation and analysis of more than one outcome variable
variables that summarise the original set. The underlying model assumes chi-squared dissimilarities among records (cases). Canonical (or "constrained") correspondence
Multivariate_statistics
Test used in the analysis of stratified or matched categorical data
doi:10.1093/jnci/22.4.719. PMID 13655060. Nathan Mantel (September 1963). "Chi-Square Tests with One Degree of Freedom, Extensions of the Mantel–Haenszel Procedure"
Cochran–Mantel–Haenszel statistics
Cochran–Mantel–Haenszel_statistics
Term in statistical hypothesis testing
combined through a meta-analysis. Many statistical analyses involve the estimation of several unknown quantities. In simple cases, all but one of these quantities
Power_(statistics)
Non-parametric method for testing whether samples originate from the same distribution
from this chi-squared distribution. If a table of the chi-squared probability distribution is available, the critical value of chi-squared, χ α : g −
Kruskal–Wallis_test
Method of plotting numeric data
Point estimation Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance
Violin_plot
Overview of and topical guide to statistics
function Mean squared error Mean absolute error Estimation theory Estimator Bayes estimator Maximum likelihood Trimmed estimator M-estimator Minimum-variance
Outline_of_statistics
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MINIMUM CHI-SQUARE-ESTIMATION
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MINIMUM CHI-SQUARE-ESTIMATION
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MINIMUM CHI-SQUARE-ESTIMATION
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