**Figure 1: **Flow chart of computational algorithm to fit the integrated model
based on adaptive Gaussian quadrature method and dual quasi-Newton
algorithm. Step 0: estimate the initial value of Ø
,0,σ_{1}, denoted as š„(0), by using
the naĆÆve model; Step 1: generate the approximate likelihood function by using
adaptive Gaussian Quadrature method; Step 2: compute the quasi-Newton
direction Δx, determine the step size š” to satisfy the Goldstein conditions;
Step 3: update parameters š„ value; Step 4: update Hessian matrix; Step 5:
check if the iteration stops. If not, go to Step 2, if yes, the iteration stops. A
grid searching with center from the estimates of naĆÆve model was applied in
our algorithm. |