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Improving Point and Interval Estimates of Monotone Functions by Rearrangement
Suppose that a target function is monotonic, namely, weakly increasing, and
an available original estimate of this target function is not weakly
increasing. Rearrangements, univariate and multivariate, transform the original
estimate to a monotonic estimate that always lies closer in common metrics to
the target function. Furthermore, suppose an original simultaneous confidence
interval, which covers the target function with probability at least
, is defined by an upper and lower end-point functions that are not
weakly increasing. Then the rearranged confidence interval, defined by the
rearranged upper and lower end-point functions, is shorter in length in common
norms than the original interval and also covers the target function with
probability at least . We demonstrate the utility of the improved
point and interval estimates with an age-height growth chart example.Comment: 24 pages, 4 figures, 3 table
Improving point and interval estimates of monotone functions by rearrangement
Suppose that a target function is monotonic, namely weakly increasing, and an original estimate of this target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates. We show that these estimates can always be improved with no harm by using rearrangement techniques: The rearrangement methods, univariate and multivariate, transform the original estimate to a monotonic estimate, and the resulting estimate is closer to the true curve in common metrics than the original estimate. The improvement property of the rearrangement also extends to the construction of confidence bands for monotone functions. Let l and u be the lower and upper endpoint functions of a simultaneous confidence interval [l,u] that covers the true function with probability (1-a), then the rearranged confidence interval, defined by the rearranged lower and upper end-point functions, is shorter in length in common norms than the original interval and covers the true function with probability greater or equal to (1-a). We illustrate the results with a computational example and an empirical example dealing with age-height growth charts. Please note: This paper is a revised version of cemmap working Paper CWP09/07.
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