statsmodels.robust.norms.MQuantileNorm

class statsmodels.robust.norms.MQuantileNorm(q, base_norm)

M-quantiles objective function based on a base norm

This norm has the same asymmetric structure as the objective function in QuantileRegression but replaces the L1 absolute value by a chosen base norm.

rho_q(u) = abs(q - I(u < 0)) * rho_base(u)

or, equivalently,

rho_q(u) = q * rho_base(u) if u >= 0 rho_q(u) = (1 - q) * rho_base(u) if u < 0

Parameters:

qfloat

M-quantile, must be between 0 and 1, excluding the end points.

base_normRobustNorm instance

Basic norm that is transformed into an asymmetric M-quantile norm

Methods

__call__(z)

Return the value of estimator rho applied to an input

Raises:

TypeError

If q is not a real number.

ValueError

If q is not in the open interval (0, 1), or is nan.

Notes

Changed in version 0.15.1: The M-quantile q is validated.

The constant q is validated when the norm is created. Plain attribute assignment after construction, norm.q = value, is not validated.

This is mainly for base norms that are not redescending, like HuberT or LeastSquares. (See Jones for the relationship of M-quantiles to quantiles in the case of non-redescending Norms.) See [BianchiEtAl2015], [BrecklingChambers1988], [Jones1994], and [NeweyPowell1987] for more information.

Expectiles are M-quantiles with the LeastSquares as base norm.

References

[BianchiEtAl2015]

Bianchi, Annamaria, and Nicola Salvati. 2015. “Asymptotic Properties and Variance Estimators of the M-Quantile Regression Coefficients Estimators.” Communications in Statistics - Theory and Methods 44 (11): 2416-29. doi:10.1080/03610926.2013.791375.

[BrecklingChambers1988]

Breckling, Jens, and Ray Chambers. 1988. “M-Quantiles.” Biometrika 75 (4): 761-71. doi:10.2307/2336317.

[Jones1994]

Jones, M. C. 1994. “Expectiles and M-Quantiles Are Quantiles.” Statistics & Probability Letters 20 (2): 149-53. doi:10.1016/0167-7152(94)90031-0.

[NeweyPowell1987]

Newey, Whitney K., and James L. Powell. 1987. “Asymmetric Least Squares Estimation and Testing.” Econometrica 55 (4): 819-47. doi:10.2307/1911031.

Methods

psi(z)	The psi function for MQuantileNorm estimator
psi_deriv(z)	The derivative of MQuantileNorm function
rho(z)	The robust criterion function for MQuantileNorm
weights(z)	MQuantileNorm weighting function for the IRLS algorithm

Properties

continuous