What question did this study set out to answer?

To formulate and evaluate non-parametric kernel estimators for multivariate probability density functions without assuming a predetermined distribution shape.

January 1, 1969Theory of Probability and Its Applications

Non-Parametric Estimation of a Multivariate Probability Density

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Authors

VEV. A. Epanechnikov

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Implication

Theoretical analysis demonstrates optimal kernel functions for multivariate probability density estimation, highlighting minimal mean integrated square error across multidimensional data.

Key Points

To formulate and evaluate non-parametric kernel estimators for multivariate probability density functions without assuming a predetermined distribution shape.
Extended univariate kernel density approaches pioneered by Rosenblatt and Parzen to multidimensional random vectors.
Assessed estimator variance and bias using asymptotic mean integrated squared error criteria under smoothness constraints.
Derived an optimal parabolic kernel function that minimizes asymptotic mean integrated squared error across multidimensional space.
Demonstrated that efficiency losses from selecting other standard kernel shapes are relatively small compared to the parabolic optimum.

Cite This Study

V. A. Epanechnikov (1969) studied this question.

synapsesocial.com/papers/69d7d81105ee2ba81dbee531 https://doi.org/10.1137/1114019

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Implication

Theoretical analysis demonstrates optimal kernel functions for multivariate probability density estimation, highlighting minimal mean integrated square error across multidimensional data.

Non-Parametric Estimation of a Multivariate Probability Density

Discussion

Member takes

Implication

Key Points

Cite This Study

Non-Parametric Estimation of a Multivariate Probability Density

Discussion

Member takes

Implication

Key Points

Cite This Study

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