Mixture of K Normal Distributions by Dyar's Law

The problem considered in this paper can be defiened as follows. Consider observations $x_1, x_2, \cdot, x_n$ which are assumed to come from a mixed population of the density function, $$f(x) = \sum^m_{k=1} pkf_k(x)$$ where m is the number of subpoulations and $p_k$ is the proportion of subpopulation k such that $\sum^m_{k=1} pk=1, 0