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

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
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

Notes

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