delicatessen.estimating_equations.basic.ee_mean

ee_mean(theta, y, weights=None)

Estimating equation for the mean. The estimating equation for the mean is

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

For the weighted mean, the difference in the previous estimating equation is multiplied by the corresponding weight.

Parameters

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

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

  • weights (ndarray, list, vector, None, optional) – 1-dimensional vector of n weights. Default is None, which assigns a weight of 1 to all observations.

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 should be done similar to the following

>>> from delicatessen import MEstimator
>>> from delicatessen.estimating_equations import ee_mean

Some generic data to estimate the mean for

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

Defining psi, or the estimating equation

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

Calling the M-estimator

>>> 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

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