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Completeness (statistics)
In statistics, completeness is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. It is opposed to the concept of an ancillary statistic. While an ancillary statistic contains no information about the model parameters, a complete statistic contains only information about the parameters, and no ancillary information. It is closely related to the concept of a sufficient statistic which contains all of the information that the dataset provides about the parameters. Definition Consider a random variable ''X'' whose probability distribution belongs to a parametric model ''P''''θ'' parametrized by ''θ''. Say ''T'' is a statistic; that is, the composition of a measurable function with a random sample ''X''1,...,''X''n. The statistic ''T'' is said to be complete for the distribution of ''X'' if, for every measurable function ''g,'' :\text\operatorname_\theta(g(T))=0\text\theta\text\mathbf_\theta(g(T)=0)=1\text\theta. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "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. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Independence
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other. When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Sufficient
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It is closely related to the concepts of an ancillary statistic which contains no information about the model parameters, and of a complete statistic which only contains information about the parameters and no ancillary information. A related concept is that of linear sufficiency, which is weaker than ''sufficiency'' but can be applied in some cases where there is no sufficient statistic, although it is restricted to linear estimators. The Kolmogorov structure function deals with individual finite data; the related notion there is the algorithmic sufficient statistic. The concept is due to Sir Ronald Fisher in 1920. Stephen Stigler noted in 1973 that the concept of sufficiency had fallen out of favor in descriptive statistics because of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basu's Theorem
In statistics, Basu's theorem states that any boundedly complete and sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata Basu. It is often used in statistics as a tool to prove independence of two statistics, by first demonstrating one is complete sufficient and the other is ancillary, then appealing to the theorem. An example of this is to show that the sample mean and sample variance of a normal distribution are independent statistics, which is done in the Example section below. This property (independence of sample mean and sample variance) characterizes normal distributions. Statement Let (P_\theta; \theta \in \Theta) be a family of distributions on a measurable space (X, \mathcal) and a statistic T maps from (X, \mathcal) to some measurable space (Y, \mathcal). If T is a boundedly complete sufficient statistic for \theta, and A is ancillary to \theta, then conditional on \theta, T is independent of A. That is, T\perp\!\!\!\p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum-variance Unbiased Estimator
In 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. While combining the constraint of unbiasedness with the desirability metric of least variance leads to good results in most practical settings—making MVUE a natural starting point for a broad range of analyses—a targeted specification may perform better for a given problem; thus, MVUE is not always the best stopping point. Definition Consider estimation of g(\theta) based on data X_1, X_2, \ldots, X_n i.i.d. from some member of a family of densities p_\theta, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by Integral, integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (mathematics), ''epigraph'' (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup \cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap \cap. A twice-differentiable function, differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain of a function, domain. Well-known examples of convex functions of a single variable include a linear function f(x) = cx (where c is a real number), a quadratic function cx^2 (c as a nonnegative real number) and an exponential function ce^x (c as a nonnegative real number). Convex functions pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sufficiency (statistics)
In statistics, sufficiency is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. A sufficient statistic contains all of the information that the dataset provides about the model parameters. It is closely related to the concepts of an ancillary statistic which contains no information about the model parameters, and of a complete statistic which only contains information about the parameters and no ancillary information. A related concept is that of linear sufficiency, which is weaker than ''sufficiency'' but can be applied in some cases where there is no sufficient statistic, although it is restricted to linear estimators. The Kolmogorov structure function deals with individual finite data; the related notion there is the algorithmic sufficient statistic. The concept is due to Sir Ronald Fisher in 1920. Stephen Stigler noted in 1973 that the concept of sufficiency had fallen out of favor in descriptive statistics because of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Statistic
In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with Ranking (statistics), rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and non-parametric inference, inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other quantile, sample quantiles. When using probability theory to analyze order statistics of random samples from a continuous probability distribution, continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution (continuous), uniform distribution. Notation and examples For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. If the sample values are :6, 9, 3, 7, the order statistics would be denoted : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lehmann–Scheffé Theorem
In statistics, the Lehmann–Scheffé theorem ties together completeness, sufficiency, uniqueness, and best unbiased estimation. The theorem states that any estimator that is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers. If T is a complete sufficient statistic for \theta and \operatorname (T)\tau(\theta) then g(T) is the uniformly minimum-variance unbiased estimator (UMVUE) of \tau(\theta). Statement Let \vec= X_1, X_2, \dots, X_n be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) f(x:\theta) where \theta \in \Omega is a parameter in the parameter space. Suppose Y = u(\vec) is a sufficient statistic for ''θ'', and let \ be a complete family. If \varphi:\operatornamevarphi(Y)= \theta then \varphi(Y) is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
