5 Section 8.1 Consistency We first want to show that if we have a sample of i.i.d.data from a common distribution which belongs to a probability model,then under some regularity conditions on the form of the density,the sequence of estimators,{(Xn)},will converge in probability to 00. So far,we have not discussed the issue of whether a maximum likelihood estimator exists or,if one does,whether it is unique.We will get to this,but first we start with a heuristic proof of consistency.5 Section 8.1 Consistency We first want to show that if we have a sample of i.i.d. data from a common distribution which belongs to a probability model, then under some regularity conditions on the form of the density, the sequence of estimators, { ˆ θ(Xn)}, will converge in probability to θ0. So far, we have not discussed the issue of whether a maximum likelihood estimator exists or, if one does, whether it is unique. We will get to this, but first we start with a heuristic proof of consistency