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The QL general formulation is

(2.36a)

subject to

(2.36b)

(2.36c)

(2.36d)

Figure 2.5 depicts the objective function for QL estimator in the case of one single error. The violet and green curves correspond to a high and low weighting factor ω_i, respectively. Note that the behavior of function J(x) for the QL estimator is the same as WLS if the measurement error y_i(x) is within given bounds.

Figure 2.5 Objective function for the QL estimator as a function of the error (one measurement).

The objective function value of QL estimator for a given measurement has a quadratic shape (like WLS estimator) if the measurement error value is within a tolerance T; otherwise, this objective function component has an absolute value shape like LAV estimator (see Figure 2.5). Note that a tolerance T must be selected in advance.