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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
editA 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
editCumulative distribution function
editThe cumulative distribution function (CDF) of the Epanechnikov distribution is:
- for
Moments and other properties
edit- Mean:
- Median:
- Mode:
- Variance:
Applications
editThe 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]]
]