 Isotonic least squares estimator

For observations $y_1,\dots ,y_n$ in a known covariate order, the isotonic least squares estimator solvesIt is the orthogonal projection of the observation vector onto a closed convex cone of nondecreasing sequences. Strict convexity of the squared distance gives a unique estimate.

For data $(3,1,2)$, the first pair violates monotonicity. Pooling it gives mean two, producing $(2,2,2)$ with squared error two. The pool adjacent violators algorithm repeatedly merges neighboring blocks whose means are out of order; the fitted value is constant at the block mean on each final block.

Positive observation weights replace squared error by ${\sum }_i{w}_i(y_i-{\theta }_i{)}^2$ and block means by weighted means. The order restriction regularizes the estimate without specifying a fixed polynomial form. The method requires the ordering to be meaningful; it does not discover an unknown covariate order from the response values.