I. Asymptotic Consistency

The maximum likelihood estimates are asymptotically consistent, that is, converges in probability to the true values ?. In the following, we investigate this property.

We know from the properties of the probability functions that

Partial differentiation of Eq. (D.1) with respect to ?, and interchanging the order of integration and differentiation assuming sufficient regularity conditions,1 yields

Equation (D.2) can be rewritten as

or equivalently,

Differentiating Eq. (D.3) and rewriting yields

or

The Fisher information matrix J, defined in Eq. (D.6) is generally positive definite. However, if the observations are independent of ?, that is, p( z ?) is not a function of ? as assumed, then J in this case will reduce to zero. In practice, this implies that it would not be possible to estimate ? from sample observations which do not contain information about ?.

The Taylor series expansion of the term [ ? ln p( zQ)/ ? ?] in the above equation about the true values ? evaluated at , leads to

where ?* = ? ? + (1 ? ?) ; 0 ? ? ? 1.

Since is the solution of the likelihood equation, equating Eq. (D.7) to zero yields

We recall our assumption that the measurements at different time points are assumed to be statistically independent. This provides

Similarly,

Now, from Eqs. (D.9) and (D.10), the strong law of...