I want to estimate the variables $a$ and $b$ ($\theta = \left[ {\begin{array}{*{20}{c}} a\\ b \end{array}} \right]$) in the nonlinear model:
$$y\left( t \right) = au\left( t \right) + b\exp (u(t)) + e(t)$$
where $e\left( t \right) \sim N\left( {0,1} \right)$ and $u(t)$ are random independent samples drawn from a uniform distribution in the interval $\left[ {0,1} \right]$. I have $N$ available measurements $1 \cdots N$ and I want to get an estimate through least squares.
attempt: if I accumulate the measurements in the matrices
$$Y = \left[ {\begin{array}{*{20}{c}} {y\left( 1 \right)}\\ \vdots \\ {y\left( 2 \right)} \end{array}} \right],X = \left[ {\begin{array}{*{20}{c}} {u\left( 1 \right)}&{\exp (u\left( 1 \right))}\\ \vdots & \vdots \\ {u\left( N \right)}&{\exp (u\left( 1 \right))} \end{array}} \right],\bar E = Y = \left[ {\begin{array}{*{20}{c}} {e\left( 1 \right)}\\ \vdots \\ {e\left( N \right)} \end{array}} \right]$$ Then the least squares is ${{\hat \theta }_N} = {\left( {{X^T}X} \right)^{ - 1}}{X^T}Y$
taking the expected value of estimator gives $E\left( {{{\hat \theta }_N}} \right) = E\left( {{{\left( {{X^T}X} \right)}^{ - 1}}{X^T}Y} \right) = E\left( {{{\left( {{X^T}X} \right)}^{ - 1}}{X^T}X\theta + \bar E} \right) = \theta $
hence it is unbiased. As for covariance of error I get
$$E\left( {\left( {{{\hat \theta }_N} - \theta } \right){{\left( {{{\hat \theta }_N} - \theta } \right)}^T}} \right) = {\left( {{X^T}X} \right)^{ - 1}}$$
Could I simplify the result for covariance further and write it in terms of $N$ and hence to see what I get as $N \to \infty$ to see if it is consistent? if not, how could I conclude it is consistent or not?
Thanks in advance:)