delicatessen.estimating_equations.basic.ee_mean_robust

ee_mean_robust(theta, y, k, loss='huber', lower=None, upper=None)

Estimating equation for the (unscaled) robust mean. The estimating equation for the robust mean is

\[\sum_{i=1}^n f_k(Y_i - \theta) = 0\]

where \(f_k(x)\) is the corresponding robust loss function. Options for the loss function include: Huber, Tukey’s biweight, Andrew’s Sine, and Hampel. See robust_loss_function for further details on the loss functions for the robust mean.

Parameters

  • theta (ndarray, list, vector) – Theta in the case of the robust mean consists of a single value. Therefore, an initial value like the form of [0, ] is should be provided.

  • y (ndarray, vector, list) – 1-dimensional vector of n observed values.

  • k (int, float) – Tuning or hyperparameter for the chosen loss function. Notice that the choice of hyperparameter depends on the loss function.

  • loss (str, optional) – Robust loss function to use. Default is 'huber'. Options include 'andrew', 'hampel', 'tukey'.

  • lower (int, float, None, optional) – Lower parameter for the Hampel loss function. This parameter does not impact the other loss functions. Default is None.

  • upper (int, float, None, optional) – Upper parameter for the Hampel loss function. This parameter does not impact the other loss functions. Default is None.

Returns

Returns a 1-by-n NumPy array evaluated for the input theta and y.

Return type

array

Examples

Construction of a estimating equation(s) with ee_mean_robust should be done similar to the following

>>> from delicatessen import MEstimator
>>> from delicatessen.estimating_equations import ee_mean_robust

Some generic data to estimate the mean for

>>> y_dat = [-10, 1, 2, 4, 1, 2, 3, 1, 5, 2, 33]

Defining psi, or the stacked estimating equations for Huber’s robust mean

>>> def psi(theta):
>>>     return ee_mean_robust(theta=theta, y=y_dat, k=9, loss='huber')

Calling the M-estimation procedure

>>> estr = MEstimator(stacked_equations=psi, init=[0, ])
>>> estr.estimate()

Inspecting the parameter estimates, the variance, and the asymptotic variance

>>> estr.theta
>>> estr.variance
>>> estr.asymptotic_variance

References

Andrews DF. (1974). A robust method for multiple linear regression. Technometrics, 16(4), 523-531.

Beaton AE & Tukey JW (1974). The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics, 16(2), 147-185.

Boos DD, & Stefanski LA. (2013). M-estimation (estimating equations). In Essential Statistical Inference (pp. 297-337). Springer, New York, NY.

Hampel FR. (1971). A general qualitative definition of robustness. The Annals of Mathematical Statistics, 42(6), 1887-1896.

Huber PJ. (1964). Robust Estimation of a Location Parameter. The Annals of Mathematical Statistics, 35(1), 73–101.

Huber PJ, Ronchetti EM. (2009) Robust Statistics 2nd Edition. Wiley. pgs 98-100