delicatessen.estimating_equations.basic.ee_mean_geometric

ee_mean_geometric(theta, y, weights=None, log_theta=True)

Estimating equations for the geometric mean. The geometric mean is defined as

\[\bar{\mu} = \left( \prod_{i=1}^n Y_i \right)^{1/n}\]

where \(Y_i\) is within the positive reals. This expression can be rewritten as the following estimating equation

\[\sum_{i=1}^n \left[ \log(Y_i) - \log(\hat{\mu}) \right] = 0\]

For the weighted geometric 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 geomtric mean consists of a single value. Therefore, an initial value like the form of [1, ] 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.

  • log_theta (bool, optional) – Whether to log-transform the input theta parameter internally. Default is True, which takes np.log(theta). The choice for this argument should not affect the point estimate, but it can change the confidence intervals.

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_geometric should be done similar to the following

>>> from delicatessen import MEstimator
>>> from delicatessen.estimating_equations import ee_mean_geometric

Some generic data to estimate the geometric mean for

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

Defining psi, or the stacked estimating equations for the geometric mean

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

Calling the M-estimation procedure

>>> estr = MEstimator(stacked_equations=psi, init=[2, ])
>>> estr.estimate()

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

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

References

Lesage É. (2011). The use of estimating equations to perform a calibration on complex parameters. Survey Methodology, 37(1), 103-108.