delicatessen.estimating_equations.basic.ee_positive_mean_deviation

ee_positive_mean_deviation(theta, y)

Estimating equations for the positive mean deviation. The estimating equations are

\[\begin{split}\sum_{i=1}^n \begin{bmatrix} 2(Y_i - \theta_2)I(Y_i > \theta_2) - \theta_1 \\ 0.5 - I(Y_i \le \theta_2) \end{bmatrix} = 0\end{split}\]

where the first estimating equation is for the positive mean difference, and the second estimating equation is for the median. Notice that this estimating equation is non-smooth. Therefore, root-finding is difficult.

Note

As the derivative of the estimating equation for the median is not defined at \(\hat{\theta}\), the bread (and sandwich) cannot be used to estimate the variance. This estimating equation is offered for completeness, but is not generally recommended for applications.

Parameters

  • theta (ndarray, list, vector) – Theta in this case consists of two values. Therefore, initial values like the form of [0, 0] are recommended.

  • y (ndarray, list, vector) – 1-dimensional vector of n observed values. No missing data should be included (missing data may cause unexpected behavior when attempting to calculate the positive mean deviation).

Returns

Returns a 2-by-n NumPy array evaluated for the input theta and y.

Return type

array

Examples

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

>>> from delicatessen import MEstimator
>>> from delicatessen.estimating_equations import ee_positive_mean_deviation

Some generic data to estimate the mean for

>>> y_dat = [1, 2, 4, 1, 2, 3, 1, 5, 2]

Defining psi, or the stacked estimating equations

>>> def psi(theta):
>>>     return ee_positive_mean_deviation(theta=theta, y=y_dat)

Calling the M-estimation procedure (note that init has 2 values now).

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

Inspecting the parameter estimates

>>> estr.theta

References

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