Gauss–Markov theoremIn statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero. The errors do not need to be normal, nor do they need to be independent and identically distributed (only uncorrelated with mean zero and homoscedastic with finite variance).
Logistic regressionIn statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression analysis, logistic regression (or logit regression) is estimating the parameters of a logistic model (the coefficients in the linear combination).
Minimum-variance unbiased estimatorIn statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter. For practical statistics problems, it is important to determine the MVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal. This has led to substantial development of statistical theory related to the problem of optimal estimation.
Local regressionLocal regression or local polynomial regression, also known as moving regression, is a generalization of the moving average and polynomial regression. Its most common methods, initially developed for scatterplot smoothing, are LOESS (locally estimated scatterplot smoothing) and LOWESS (locally weighted scatterplot smoothing), both pronounced ˈloʊɛs. They are two strongly related non-parametric regression methods that combine multiple regression models in a k-nearest-neighbor-based meta-model.
Extremum estimatorIn statistics and econometrics, extremum estimators are a wide class of estimators for parametric models that are calculated through maximization (or minimization) of a certain objective function, which depends on the data. The general theory of extremum estimators was developed by . An estimator is called an extremum estimator, if there is an objective function such that where Θ is the parameter space. Sometimes a slightly weaker definition is given: where op(1) is the variable converging in probability to zero.
General linear modelThe general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as where Y is a matrix with series of multivariate measurements (each column being a set of measurements on one of the dependent variables), X is a matrix of observations on independent variables that might be a design matrix (each column being a set of observations on one of the independent variables), B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors (noise).
Elastic net regularizationIn statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods. The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on Use of this penalty function has several limitations. For example, in the "large p, small n" case (high-dimensional data with few examples), the LASSO selects at most n variables before it saturates.
Weight functionA weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average. Weight functions occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings.
Truncated meanA truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, and typically discarding an equal amount of both. This number of points to be discarded is usually given as a percentage of the total number of points, but may also be given as a fixed number of points. For most statistical applications, 5 to 25 percent of the ends are discarded.
Influential observationIn statistics, an influential observation is an observation for a statistical calculation whose deletion from the dataset would noticeably change the result of the calculation. In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter estimates. Various methods have been proposed for measuring influence. Assume an estimated regression , where is an n×1 column vector for the response variable, is the n×k design matrix of explanatory variables (including a constant), is the n×1 residual vector, and is a k×1 vector of estimates of some population parameter .
Quasi-likelihoodIn statistics, quasi-likelihood methods are used to estimate parameters in a statistical model when exact likelihood methods, for example maximum likelihood estimation, are computationally infeasible. Due to the wrong likelihood being used, quasi-likelihood estimators lose asymptotic efficiency compared to, e.g., maximum likelihood estimators. Under broadly applicable conditions, quasi-likelihood estimators are consistent and asymptotically normal. The asymptotic covariance matrix can be obtained using the so-called sandwich estimator.
Lasso (statistics)In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso or LASSO) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. It was originally introduced in geophysics, and later by Robert Tibshirani, who coined the term. Lasso was originally formulated for linear regression models. This simple case reveals a substantial amount about the estimator.
