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 $\sum_{i=1}^n \widehat{g}_{AL}^{f,j}(x_i^j)=0$