Draft:Epanechnikov distribution
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Parameters | scale (real) | ||
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In probability theory and statistics, the Epanechnikov distribution, also known as the Epanechnikov kernel, is a continuous probability distribution that is defined on a finite interval. It is named after V. A. Epanechnikov, who introduced it in 1969 in the context of kernel density estimation [1].
Definition
[edit]A random variable has an Epanechnikov distribution if its probability density function is given by:
where is a scale parameter. Setting gives the unit variance probability distribution originally considered by Epanechnikov.
Properties
[edit]Cumulative distribution function
[edit]The cumulative distribution function (CDF) of the Epanechnikov distribution is:
- for
Moments and other properties
[edit]- Mean:
- Median:
- Mode:
- Variance:
Applications
[edit]The Epanechnikov distribution has applications in various fields, including:
- Kernel density estimation: It is widely used as a kernel function in non-parametric statistics, particularly in kernel density estimation. In this context, it is often referred to as the Epanechnikov kernel. For more information, see Kernel functions in common use.
Related distributions
[edit]- The Epanechnikov distribution can be viewed as a special case of a Beta distribution that has been shifted and scaled along the x-axis.
References
[edit]- ^ Epanechnikov, V. A. (January 1969). "Non-Parametric Estimation of a Multivariate Probability Density". Theory of Probability & Its Applications. 14 (1): 153–158. doi:10.1137/1114019.
- Continuous distributions
- [[:Category:Probability distributions with support [-1,1|Probability distributions with support [-1,1]]
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