Mean integrated squared error

In statistics, the mean integrated squared error (MISE) is used in density estimation. The MISE of an estimate of an unknown probability density is given by E ⁡ ‖ f n − f ‖ 2 2 = E ⁡ ∫ ( f n ( x ) − f ( x ) ) 2 d x {\displaystyle \operatorname {E} \|f_{n}-f\|_{2}^{2}=\operatorname {E} \int (f_{n}(x)-f(x))^{2}\,dx} where ƒ is the unknown density, ƒn is its estimate based on a sample of n independent and identically distributed random variables. Here, E denotes the expected value with respect to that sample. The MISE is also known as L2 risk function.

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"MISE" redirects here. For other uses, see Mise.

In statistics, the mean integrated squared error (MISE) is used in density estimation. The MISE of an estimate of an unknown probability density is given by^[1]

E ⁡ ‖ f n − f ‖ 2 2 = E ⁡ ∫ ( f n ( x ) − f ( x ) ) 2 d x {\displaystyle \operatorname {E} \|f_{n}-f\|_{2}^{2}=\operatorname {E} \int (f_{n}(x)-f(x))^{2}\,dx}

where ƒ is the unknown density, ƒ_n is its estimate based on a sample of n independent and identically distributed random variables. Here, E denotes the expected value with respect to that sample.

The MISE is also known as L² risk function.

See also

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• Minimum distance estimation
• Mean squared error

References

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1. ^ Wand, M. P.; Jones, M. C. (1994). Kernel smoothing. CRC press. p. 15.