Approximating an AL profile

3. Replace the integral in by a summation.

Normal

\[\begin{equation} \widehat{g}_{AL}^{j}(z) = \sum_{k=1}^{k_j(z)} \frac{1}{n_j(k)} \sum_{i: x_i^j \in N_j(k)} \left\{ f\left(\underline{x}_i^{j| = z_k^j}\right) - f\left(\underline{x}_i^{j| = z_{k-1}^j}\right) \right\} - \hat{c} \end{equation}\]

• \(n_j(k)\) denote the number of observations \(x_i^j\) falling into \(N_j(k)\)
• The value \(z_0^j\) must be just below \(\min(x_1^j,\ldots,x_N^j)\) and \(z_K^j=\max(x_1^j,\ldots,x_N^j)\)
• \(k_j(z)\) is the index of interval \(N_j(k)\) in which \(z\) falls as \(z \in N_j\{k_j(z)\}\)
• \(\hat{c}\) is selected so that \(\sum_{i=1}^n \widehat{g}_{AL}^{f,j}(x_i^j)=0\)