So I'm trying to develop a logistic model. It's a little different from traditional item response models. It models one person's response on $N$ items on an exam.
$P(X_{i} =x_{i}│θ,a_i,b_i,c,d )=c+\frac{d-c}{1+exp(-a_i (θ-b_i ) )} $
where $X_{i} =1$ indicates a correct response on item $i$, and $X_{i} =0$ indicates an incorrect response. The values $a_i$ (item discrimination) and $b_i$ (item difficulty) are known. Also, $c$ and $d$ have some range restrictions:
$0 \leq c \leq 1$
$0 \leq d \leq 1$
$d-c \geq 0$
How would I go about finding the joint maximum likelihood estimates for $c$, $d$, and $θ$? If it helps, here's some example data with 5 items.
a = {0.9, 2, 1.2, 1.2, 1}
b = {-2.5, -1.3, -0.6, 0, 0.5}
x = {0, 1, 1, 0, 0}
If you need more information, please let me know. Thank you!
EDIT: It's assumed that the conditional likelihood is independent. So the log-likelihood is
$\Sigma{lnP_i^{x_i}+ln(1-P_i)^{1-x_i}}$
where $P_i$ is the probability of correct response on item $i$