Algorithms And Applications Of Isotonic Regression

The classic isotonic regression is a constrained optimization problem which is based on the quadratic loss. The corresponding algorithms are PAVA, the Min-max formulas and MLS algorithm and so on. In the exponential family distributions, the relationship between the classic isotonic regression and the maximum likelihood estimate subjected to the constraint, the properties of isotonic regression and its large sample properties are discussed. The main application of isotonic regression is to solve the estimation problem on various parameters and non-parameters in a class of statistical inference, because of the relationship between the classic isotonic regression and the maximum likelihood estimate. A typical example is the estimation of survival function under the caseⅠi nterval-censored data.Instead of the classical isotonic regression, L₁ isotonic regression is discussed in this paper. L₁ isotonic regression is a constrained optimization problem based on absolute loss. We can also use the idea of PAVA to solve this problem. It is proved that the iterative algorithm is convergent and it converges to the corresponding optimal solution of the problem; The maximum likelihood estimate of the parameter with simple order restrictions in the bilateral exponential distribution can be transformed into a problem of L₁ isotonic regression. The consistency of parameter estimates and asymptotic normality of the estimator are proved in this paper.In the final, using the idea of PAVA and the relationships between the two isotonic regressions and the maximum likelihood estimate, we give a more direct iterative method for the solution of a class of constrained maximum likelihood estimation.