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UMVUE
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] |
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] |
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] |
Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (i.e. ,b/math>) or open (i.e. (a,b)). Therefore, the distribution is often abbreviated U(a,b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is f(x) = \begin \dfrac & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample Mean
The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger population of numbers, where "population" indicates not number of people but the entirety of relevant data, whether collected or not. A sample of 40 companies' sales from the Fortune 500 might be used for convenience instead of looking at the population, all 500 companies' sales. The sample mean is used as an estimator for the population mean, the average value in the entire population, where the estimate is more likely to be close to the population mean if the sample is large and representative. The reliability of the sample mean is estimated using the standard error, which in turn is calculated using the variance of the sample. If the sample is random, the standard error falls with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
U-statistic
In statistical theory, a U-statistic is a class of statistics defined as the average over the application of a given function applied to all tuples of a fixed size. The letter "U" stands for unbiased. In elementary statistics, U-statistics arise naturally in producing minimum-variance unbiased estimators. The theory of U-statistics allows a minimum-variance unbiased estimator to be derived from each unbiased estimator of an ''estimable parameter'' (alternatively, ''statistical functional'') for large classes of probability distributions. An estimable parameter is a measurable function of the population's cumulative probability distribution: For example, for every probability distribution, the population median is an estimable parameter. The theory of U-statistics applies to general classes of probability distributions. History Many statistics originally derived for particular parametric families have been recognized as U-statistics for general distributions. In non-parame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bias–variance Tradeoff
In statistics and machine learning, the bias–variance tradeoff describes the relationship between a model's complexity, the accuracy of its predictions, and how well it can make predictions on previously unseen data that were not used to train the model. In general, as the number of tunable parameters in a model increase, it becomes more flexible, and can better fit a training data set. That is, the model has lower error or lower Bias of an estimator, bias. However, for more flexible models, there will tend to be greater variance to the model fit each time we take a set of sample (statistics), samples to create a new training data set. It is said that there is greater variance in the model's estimation theory, estimated statistical parameter, parameters. The bias–variance dilemma or bias–variance problem is the conflict in trying to simultaneously minimize these two sources of Errors and residuals in statistics, error that prevent supervised learning algorithms from general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Best Linear Unbiased Estimator
Best or The Best may refer to: People * Best (surname), people with the surname Best * Best (footballer, born 1968), retired Portuguese footballer Companies and organizations * Best & Co., an 1879–1971 clothing chain * Best Lock Corporation, a lock manufacturer * Best Manufacturing Company, a farm machinery company * Best Products, a chain of catalog showroom retail stores * Brihanmumbai Electric Supply and Transport, a public transport and utility provider * Best High School (other) Acronyms * Berkeley Earth Surface Temperature, a project to assess global temperature records * BEST Robotics, a student competition * BioEthanol for Sustainable Transport * Bootstrap error-adjusted single-sample technique, a statistical method * Bringing Examination and Search Together, a European Patent Office initiative * Bronx Environmental Stewardship Training, a program of the Sustainable South Bronx organization * Smart BEST, a Japanese experimental train * Brihanmumba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cramér–Rao Bound
In estimation theory and statistics, the Cramér–Rao bound (CRB) relates to estimation of a deterministic (fixed, though unknown) parameter. The result is named in honor of Harald Cramér and Calyampudi Radhakrishna Rao, but has also been derived independently by Maurice Fréchet, Georges Darmois, and by Alexander Aitken and Harold Silverstone. It is also known as Fréchet-Cramér–Rao or Fréchet-Darmois-Cramér-Rao lower bound. It states that the precision of any unbiased estimator is at most the Fisher information; or (equivalently) the reciprocal of the Fisher information is a lower bound on its variance. An unbiased estimator that achieves this bound is said to be (fully) '' efficient''. Such a solution achieves the lowest possible mean squared error among all unbiased methods, and is, therefore, the minimum variance unbiased (MVU) estimator. However, in some cases, no unbiased technique exists which achieves the bound. This may occur either if for any unbiased ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German Tank Problem
German(s) may refer to: * Germany, the country of the Germans and German things **Germania (Roman era) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizenship in Germany, see also German nationality law **Germanic peoples (Roman era) * German diaspora * German language * German cuisine, traditional foods of Germany People * German (given name) * German (surname) * Germán, a Spanish name Places * German (parish), Isle of Man * German, Albania, or Gërmej * German, Bulgaria * German, Iran * German, North Macedonia * German, New York, U.S. * Agios Germanos, Greece Other uses * German (mythology), a South Slavic mythological being * Germans (band), a Canadian rock band * "German" (song), a 2019 song by No Money Enterprise * ''The German'', a 2008 short film * "The Germans", an episode of ''Fawlty Towers'' * ''The German'', a nickname for Congolese rebel André Kisase Ngandu See also * Germanic (di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample Maximum
In statistics, the sample maximum and sample minimum, also called the largest observation and smallest observation, are the values of the greatest and least elements of a sample. They are basic summary statistics, used in descriptive statistics such as the five-number summary and Bowley's seven-figure summary and the associated box plot. The minimum and the maximum value are the first and last order statistics (often denoted ''X''(1) and ''X''(''n'') respectively, for a sample size of ''n''). If the sample has outliers, they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum need not be outliers, if they are not unusually far from other observations. Robustness The sample maximum and minimum are the ''least'' robust statistics: they are maximally sensitive to outliers. This can either be an advantage or a drawback: if extreme values are real (not measurement errors), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Uniform Distribution
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein each of some finite whole number ''n'' of outcome values are equally likely to be observed. Thus every one of the ''n'' outcome values has equal probability 1/''n''. Intuitively, a discrete uniform distribution is "a known, finite number of outcomes all equally likely to happen." A simple example of the discrete uniform distribution comes from throwing a fair six-sided die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of each given value is 1/6. If two dice were thrown and their values added, the possible sums would not have equal probability and so the distribution of sums of two dice rolls is not uniform. Although it is common to consider discrete uniform distributions over a contiguous range of integers, such as in this six-sided die example, one can define discrete uniform distributions over any finite set. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mid-range
In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set: :M=\frac. The mid-range is closely related to the range, a measure of statistical dispersion defined as the difference between maximum and minimum values. The two measures are complementary in sense that if one knows the mid-range and the range, one can find the sample maximum and minimum values. The mid-range is rarely used in practical statistical analysis, as it lacks efficiency as an estimator for most distributions of interest, because it ignores all intermediate points, and lacks robustness, as outliers change it significantly. Indeed, for many distributions it is one of the least efficient and least robust statistics. However, it finds some use in special cases: it is the maximally efficient estimator for the center of a uniform distribution, trimmed mid-ranges address robustness, and as an L-estima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
