The l-th absolute moment of a zero-mean normal

This is the fact that makes the sign quantizer work: the estimator turns an inner product into E|x| for a Gaussian with variance ||k||^2, and E|x| = sigma sqrt(2/pi) – which is exactly why sqrt(pi/2) sits in front of the estimator, cancelling it.

Usage

Tqmom(sigma, l)

Arguments

sigma

Standard deviation.

l

Moment order; need not be an integer.

Value

List with moment, estimate, sigma, l.

Details

Formula: E|X|^l = sigma^l 2^(l/2) Gamma((l+1)/2) / sqrt(pi).

References

Zandieh, A., Daliri, M. & Han, I. (2024). arXiv:2406.03482, fact 3.4.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Tqmom(V, V)
#> $moment
#> [1] 7.978846e-01 4.000000e+00 4.308577e+01 7.680000e+02 1.994711e+04
#> [6] 6.998400e+05 3.154043e+07 1.761608e+09
#> 
#> $estimate
#> [1] 7.978846e-01 4.000000e+00 4.308577e+01 7.680000e+02 1.994711e+04
#> [6] 6.998400e+05 3.154043e+07 1.761608e+09
#> 
#> $sigma
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $l
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $method
#> [1] "Absolute moment of a centred normal"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"