Constrained Maximum Lilkihood Estimation Given the joint PDF $h_{X,Y}(x,y)$ of the correlated variables $X$ and $Y$, and given that the joint PDF is a function of a parameter $\lambda$ such that $|\lambda|\le 1$, I need to find the maximum likelihood estimator for the parameter $\lambda$. The Joint PDF is given by
$$h_{X,Y}(x,y) = f_X(x)f_Y(y) \left[1+\lambda\left(2F_X(x)-1\right)\left(2F_Y(y)-1\right)\right],\quad |\lambda|\le 1 $$
The marginal PDFs $f_X(x)$ and $f_Y(y)$ are known as well as the marginal CDFs $F_X(x)$ and $F_Y(y)$. So how can I apply constrained maximum likelihood estimation to obtain the estimated value of $\lambda$?
 A: For a sample of size $n$, $\{x_i,y_i|i=1,...,n\}$ we write for compactness 
$$\left(2F_X(x_i)-1\right)\left(2F_Y(y_i)-1\right) \equiv g(x_i,y_i)$$
and we note that $g(x_i,y_i) =0$ if and only if both $x_i$ and $y_i$ equal the medians of the respective marginal distributions. Moreover $0<g(x_i,y_i) <1$ if both are above or below the medians, while $-1<g(x_i,y_i) <0$ if the one is above and the other below the respective median.
The the log-likelihood is
$$\ln L(\lambda \mid \mathbf x, \mathbf y) = \sum_{i=1}^n\ln\left[f_X(x_i)f_Y(y_i)\right] + \sum_{i=1}^n\ln[1+\lambda g(x_i,y_i)]$$
To proceed with the maximization we need to take into account the constraint on the value of $\lambda$. We have
$$|\lambda|\leq 1 \Rightarrow -1\leq \lambda , \lambda \leq 1$$
In so called "normal form" these constraints are written
$$\lambda +1 \geq 0, \;\;1-\lambda \geq 0$$
So the lagrangean of the maximization problem is
$$\Lambda = \ln L(\lambda \mid \mathbf x, \mathbf y) + \xi_1(\lambda +1)+\xi_2(1-\lambda)$$
and we have a standard Karush-Kuhn-Tucker problem of maximization under (linear) inequality constraints), where $\xi_1, \xi_2$ are non-negative multipliers. Note that 
$$\frac {\partial \ln L(\lambda \mid \mathbf x, \mathbf y)}{\partial \lambda} = \sum_{i=1}^n\frac{g(x_i,y_i)}{1+\lambda g(x_i,y_i)}$$
and
$$\frac {\partial^2\ln L(\lambda \mid \mathbf x, \mathbf y)}{\partial \lambda^2}=\sum_{i=1}^n\frac{-[g(x_i,y_i)]^2}{(1+\lambda g(x_i,y_i))^2} <0$$
So the log-likelihood is a strictly concave function of $\lambda$. Given also that the constraints are linear, this means that the first-order necessary conditions for a maximum will also be sufficient for a global maximum. The basic condition is
$$\frac {\partial\Lambda}{\partial \lambda} \leq 0\Rightarrow \sum_{i=1}^n\frac{g(x_i,y_i)}{1+\lambda g(x_i,y_i)} +\xi_1-\xi_2 \leq 0$$
The multipliers will be both zero if $\lambda$ is strictly inside $(-1,1)$, and they cannot be both non-zero. Also, given the constraints on the value of $\lambda$ and the range of $g()$, one can find that the sign of the ratio $\frac{g(x_i,y_i)}{1+\lambda g(x_i,y_i)}$ is governed by the sign of $g(x_i,y_i)$, irrespective of the sign of $\lambda$ at the optimum.  
This permits as to single out a special case that also has some intuition: For, assume that it so happens that our sample is such that all $g(x_i,y_i)$ are positive, meaning that all pairs of realizations are either either below or above the medians of the marginal distributions. In such a case $\sum_{i=1}^n\frac{g(x_i,y_i)}{1+\lambda g(x_i,y_i)} >0$. Then, given also the non-negativity of the multipliers, the only way that the first-order condition can be satisfied, is for $\xi_2>0$, meaning that the optimal $\lambda$ will be equal to $1$. So a solution here is
$$g(x_i,y_i) >0\; \forall i \Rightarrow \{\lambda^*=1, \xi^*_1=0, \xi^*_2>0\}$$
For the specific joint density function, the sign and value of $\lambda$ is related to the direction and strength respectively, of the dependence between two random variables: if all realizations show that both variables are realized together either above, or below, their medians, this indicates a strong and positive relationship, and this is what the solution tells us.  
In general, as is usually the case with problems under inequality constraints, rarely can we derive a final solution analytically, since we have to algorithmically compute the values of the objective function at the various candidate solutions, given the actual sample.
