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Multivariate Pareto Distribution
In statistics, a multivariate Pareto distribution is a multivariate extension of a univariate Pareto distribution. There are several different types of univariate Pareto distributions including Pareto distribution#Pareto types I–IV, Pareto Types I−IV and Pareto distribution#Feller–Pareto distribution, Feller−Pareto. Chapter 3. Multivariate Pareto distributions have been defined for many of these types. Bivariate Pareto distributions Bivariate Pareto distribution of the first kind Mardia (1962) defined a bivariate distribution with cumulative distribution function (CDF) given by : F(x_1, x_2) = 1 -\sum_^2\left(\frac\right)^+ \left(\sum_^2 \frac - 1\right)^, \qquad x_i > \theta_i > 0, i=1,2; a>0, and joint density function : f(x_1, x_2) = (a+1)a(\theta_1 \theta_2)^(\theta_2x_1 + \theta_1x_2 - \theta_1 \theta_2)^, \qquad x_i \geq \theta_i>0, i=1,2; a>0. The marginal distributions are Pareto distribution, Pareto Type 1 with density functions : f(x_i)=a\theta_i^a x_i^, \q ... [...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 ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareto Distribution
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population. The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value () of log45 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena and human activities. Definitions If ''X'' is a random variable with a Pareto (Type I) distribution, then the probability that ''X'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
