|
M-estimation
In 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 likelihood estimation are special cases of M-estimators. The definition of M-estimators was motivated by robust statistics, which contributed new types of M-estimators. The statistical procedure of evaluating an M-estimator on a data set is called M-estimation. 48 samples of robust M-estimators can be found in a recent review study. More generally, an M-estimator may be defined to be a zero of an estimating function. This estimating function is often the derivative of another statistical function. For example, a maximum-likelihood estimate is the point where the derivative of the likelihood function with respect to the parameter is zero; thus, a maximum-likelihood estimator is a critical point of the score function. In many applications, such M-estimators can be thought of as estimating characteristics of the popula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly. Introduction Robust statistics seek to provide methods that emulate popular statistical methods, but which are not unduly affected by outliers or other small departures from Statistical assumption, model assumptions. In statistics, classical estimation methods rely heavily on assumpti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly. Introduction Robust statistics seek to provide methods that emulate popular statistical methods, but which are not unduly affected by outliers or other small departures from Statistical assumption, model assumptions. In statistics, classical estimation methods rely heavily on assumpti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sara Van De Geer
Sara Anna van de Geer (born 7 May 1958, Leiden) is a Dutch statistician who is a professor in the department of mathematics at ETH Zurich.. She is the daughter of psychologist John P. van de Geer. Education She earned a master's degree in 1982 and a doctorate in mathematics in 1987 from Leiden University. Her dissertation, entitled ''Regression Analysis and Empirical Processes'', was supervised by Willem Rutger van Zwet and Richard D. Gill. Career She taught at the University of Bristol from 1987 to 1988, at Utrecht University from 1989 to 1990, at Leiden University from 1990 to 1997 and 1999 to 2005, and at Paul Sabatier University in Toulouse, France, from 1997 to 1999, before moving to ETH Zurich in September 2005 as Full Professor. There, she has been the first female professor at the Department of Mathematics. Recognition She was an invited speaker at the International Congress of Mathematicians in 2010 and received the Van Wijngaarden Award in 2016. She is a mem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Large Numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed. The LLN is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a ''large number'' of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seemingly Unrelated Regressions
In econometrics, the seemingly unrelated regressions (SUR) or seemingly unrelated regression equations (SURE) model, proposed by Arnold Zellner in (1962), is a generalization of a linear regression model that consists of several regression equations, each having its own dependent variable and potentially different sets of exogenous explanatory variables. Each equation is a valid linear regression on its own and can be estimated separately, which is why the system is called ''seemingly unrelated'', although some authors suggest that the term ''seemingly related'' would be more appropriate, since the error terms are assumed to be correlated across the equations. The model can be estimated equation-by-equation using standard ordinary least squares (OLS). Such estimates are consistent, however generally not as efficient as the SUR method, which amounts to feasible generalized least squares with a specific form of the variance-covariance matrix. Two important cases when SUR is in fact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seemingly Unrelated Regressions
In econometrics, the seemingly unrelated regressions (SUR) or seemingly unrelated regression equations (SURE) model, proposed by Arnold Zellner in (1962), is a generalization of a linear regression model that consists of several regression equations, each having its own dependent variable and potentially different sets of exogenous explanatory variables. Each equation is a valid linear regression on its own and can be estimated separately, which is why the system is called ''seemingly unrelated'', although some authors suggest that the term ''seemingly related'' would be more appropriate, since the error terms are assumed to be correlated across the equations. The model can be estimated equation-by-equation using standard ordinary least squares (OLS). Such estimates are consistent, however generally not as efficient as the SUR method, which amounts to feasible generalized least squares with a specific form of the variance-covariance matrix. Two important cases when SUR is in fact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Median Absolute Deviation
In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample. For a univariate data set ''X''1, ''X''2, ..., ''Xn'', the MAD is defined as the median of the absolute deviations from the data's median \tilde=\operatorname(X) : : \operatorname = \operatorname( , X_i - \tilde, ) that is, starting with the residuals (deviations) from the data's median, the MAD is the median of their absolute values. Example Consider the data (1, 1, 2, 2, 4, 6, 9). It has a median value of 2. The absolute deviations about 2 are (1, 1, 0, 0, 2, 4, 7) which in turn have a median value of 1 (because the sorted absolute deviations are (0, 0, 1, 1, 2, 4, 7)). So the median absolute deviation for this data is 1. Uses The median absolute deviation is a measure of statistical dispersion. Moreover, the MAD is a robus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result. Finite data set of numbers The median of a finite list of numbers is the "middle" number, when those numbers are list ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Redescending M-estimator
In statistics, redescending M-estimators are ''Ψ''-type M-estimators which have ''ψ'' functions that are non-decreasing near the origin, but decreasing toward 0 far from the origin. Their ''ψ'' functions can be chosen to redescend smoothly to zero, so that they usually satisfy ''ψ''(''x'') = 0 for all x with , ''x'', > ''r'', where ''r'' is referred to as the minimum rejection point. Due to these properties of the ''ψ'' function, these kinds of estimators are very efficient, have a high breakdown point and, unlike other outlier rejection techniques, they do not suffer from a masking effect. They are efficient because they completely reject gross outliers, and do not completely ignore moderately large outliers (like median). Advantages Redescending M-estimators have high breakdown points (close to 0.5), and their ''Ψ'' function can be chosen to redescend smoothly to 0. This means that moderately large outliers are not ignored completely, and greatly improves the efficiency ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iteratively Re-weighted Least Squares
The method of iteratively reweighted least squares (IRLS) is used to solve certain optimization problems with objective functions of the form of a ''p''-norm: :\underset \sum_^n \big, y_i - f_i (\boldsymbol\beta) \big, ^p, by an iterative method in which each step involves solving a weighted least squares problem of the form:C. Sidney Burrus, Iterative Reweighted Least Squares' :\boldsymbol\beta^ = \underset \sum_^n w_i (\boldsymbol\beta^) \big, y_i - f_i (\boldsymbol\beta) \big, ^2. IRLS is used to find the maximum likelihood estimates of a generalized linear model, and in robust regression to find an M-estimator, as a way of mitigating the influence of outliers in an otherwise normally-distributed data set. For example, by minimizing the least absolute errors rather than the least square errors. One of the advantages of IRLS over linear programming and convex programming is that it can be used with Gauss–Newton and Levenberg–Marquardt numerical algorithms. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
