Maximum-Likelihood Non-Decreasing Response Estimates

Let $x_{i,j}$, $1\le i\le m$, $1\le j\le n_i$, be observations from a doubly-indexed sequence $\{X_{i,j}\}$ of independent random variables (all of them discrete, or all of them absolutely continuous). Suppose that each $X_{i,j}$ has the PDF $f(x\mid {\theta }_i)$ from a one-parameter family of PDFs $f(x\mid \theta )$. Mild assumptions are described under which there is a unique, non-decreasing compound response estimate of $\mathbf \theta=<\theta_1, \hdots \theta_m>$ that maximizes the compound likelihood function among all non-decreasing response estimates. An efficient algorithm is described to compute this unique estimate. The same theory and algorithm also give the unique non-increasing compound response estimate that maximizes likelihood among all non-increasing response estimates. One simply reverses the order represented by the index $i$.