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Studentized Residual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's ''t''-statistic, with the estimate of error varying between points. This is an important technique in the detection of outliers. It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym ''Student''. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing. Motivation The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the ''residuals'' at different input variable values may differ, even if the variances of the ''errors'' at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions. Consider the simple linear regression model : Y = \al ... [...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] |
Univariate Distribution
In statistics, a univariate distribution is a probability distribution of only one random variable. This is in contrast to a multivariate distribution, the probability distribution of a random vector (consisting of multiple random variables). Examples One of the simplest examples of a discrete univariate distribution is the discrete uniform distribution, where all elements of a finite set are equally likely. It is the probability model for the outcomes of tossing a fair coin, rolling a fair die, etc. The univariate continuous uniform distribution on an interval 'a'', ''b''has the property that all sub-intervals of the same length are equally likely. Other examples of discrete univariate distributions include the binomial, geometric, negative binomial, and Poisson distributions.Johnson, N.L., Kemp, A.W., and Kotz, S. (2005) Discrete Univariate Distributions, 3rd Edition, Wiley, . At least 750 univariate discrete distributions have been reported in the literature.Wimmer G, A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The 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 (e.g. , b or open (e.g. (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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as exponents of the random variable and control the shape of the distribution. The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions. In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas ''beta distribution of the second kind'' is an alternative name for the beta prime distribution. The generalization to mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
