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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 model assumptions. In statistics, classical estimation methods rely heavily on assumptions which are often ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German: '' Statistik'', "description of a 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 surveys and 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 samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sampling Distribution
In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were separately used in order to compute one value of a statistic (such as, for example, the sample mean or sample variance) for each sample, then the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample is observed, but the sampling distribution can be found theoretically. Sampling distributions are important in statistics because they provide a major simplification en route to statistical inference. More specifically, they allow analytical considerations to be based on the probability distribution of a statistic, rather than on the joint probability distribution of all the individual sample values. Introduction The sampling distribution of a statistic is the distribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in the Social Sciences, p.2 Colloquially, measures of central tendency are often called '' averages.'' The term ''central tendency'' dates from the late 1920s. The most common measures of central tendency are the arithmetic mean, the median, and the mode. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."Upton, G.; Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP (entry for "central tendency")Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP for International Statistical Institute. (entry for "centra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is great ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the data set. An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses. Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in the data-set, measurement error, or that the population has a heavy-tailed distribution. In the case of measurement error, one wishes to discard them or use statistics that are robust to outliers, while in the case of heavy-tailed distributions, they indicate that the distribution has high skewness and that one should be very cautious in using tools or intuitions that assume a normal distribution. A frequent cause of outliers is a mixture of two distributions, which may be two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotically Unbiased
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bias (statistics)
Statistical bias is a systematic tendency which causes differences between results and facts. The bias exists in numbers of the process of data analysis, including the source of the data, the estimator chosen, and the ways the data was analyzed. Bias may have a serious impact on results, for example, to investigate people's buying habits. If the sample size is not large enough, the results may not be representative of the buying habits of all the people. That is, there may be discrepancies between the survey results and the actual results. Therefore, understanding the source of statistical bias can help to assess whether the observed results are close to the real results. Bias can be differentiated from other mistakes such as accuracy (instrument failure/inadequacy), lack of data, or mistakes in transcription (typos). Bias implies that the data selection may have been skewed by the collection criteria. Bias does not preclude the existence of any other mistakes. One may have a po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Efficiency (statistics)
In statistics, efficiency is a measure of quality of an estimator, of an experimental design, or of a hypothesis testing procedure. Essentially, a more efficient estimator, needs fewer input data or observations than a less efficient one to achieve the Cramér–Rao bound. An ''efficient estimator'' is characterized by having the smallest possible variance, indicating that there is a small deviance between the estimated value and the "true" value in the L2 norm sense. The relative efficiency of two procedures is the ratio of their efficiencies, although often this concept is used where the comparison is made between a given procedure and a notional "best possible" procedure. The efficiencies and the relative efficiency of two procedures theoretically depend on the sample size available for the given procedure, but it is often possible to use the asymptotic relative efficiency (defined as the limit of the relative efficiencies as the sample size grows) as the principal compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypothesis. The average (or mean) of sample values is a statistic. The term statistic is used both for the function and for the value of the function on a given sample. When a statistic is being used for a specific purpose, it may be referred to by a name indicating its purpose. When a statistic is used for estimating a population parameter, the statistic is called an '' estimator''. A population parameter is any characteristic of a population under study, but when it is not feasible to directly measure the value of a population parameter, statistical methods are used to infer the likely value of the parameter on the basis of a statistic computed from a sample taken from the population. For example, the sample mean is an unbiased estimator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kalman Filter
For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory. This digital filter is sometimes termed the ''Stratonovich–Kalman–Bucy filter'' because it is a special case of a more general, nonlinear filter developed somewhat earlier by the Soviet mathematician Ruslan Stratonovich. In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before summer 1960, when Kalman met with Stratonovich during a conference in Moscow. Kalman filtering has numerous te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space (controls)
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve over time in a way that depends on the values they have at any given time and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables. The " state space" is the Euclidean space in which the variables on the axes are the state variables. The state of the system can be represented as a ''state vector'' within that space. To abstract from the number of inputs, outputs and states, these variables are expressed as vectors. If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. The state-space method is characterized by significant algebraization of general syst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixture Density
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution. In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other distribution functions and density fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
