The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set

We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator's asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function

Estimation of a kk-monotone density: limit distribution theory and the spline connection

We study the asymptotic behavior of the Maximum Likelihood and Least Squares Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$. We find that the $j$th derivative of the estimators at $x_0$ converges at the rate $n^{-(k-j)/(2k+1)}$ for $j=0,...,k-1$. The limiting distribution depends on an almost surely uniquely defined stochastic process $H_k$ that stays above (below) the $k$-fold integral of Brownian motion plus a deterministic drift when $k$ is even (odd). Both the MLE and LSE are known to be splines of degree $k-1$ with simple knots. Establishing the order of the random gap $\tau_n^+-\tau_n^-$, where $\tau_n^{\pm}$ denote two successive knots, is a key ingredient of the proof of the main results. We show that this ``gap problem'' can be solved if a conjecture about the upper bound on the error in a particular Hermite interpolation via odd-degree splines holds.Comment: Published in at http://dx.doi.org/10.1214/009053607000000262 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

MONOTONICITY IMPLIES STRATEGY-PROOFNESS FOR CORRESPONDENCES

We show that Maskin monotone social choice correspondences on sufficiently rich domains satisfy a generalized strategy-proofness property, thus generalizing Muller and Satterthwaite''s (1977) theorem to correspondences. From the point of view of Nash implementation theory, the result yields a partial characterization of the restrictions entailed by Nash implementability. Alternatively, the result can be viewed as a possibility theorem on the dominant-strategy-implementability of monotone SCCs via set-valued mechanisms for agents who are completely ignorant about the finally selected outcome. It is shown by examples that stronger strategy-proofness properties fail easily.

Stochastic efficiency measurement: The curse of theoretical consistency

The availability of efficiency estimation software – freely distributed via the internet and relatively easy to use – recently inflated the number of corresponding applications. The resulting efficiency estimates are used without a critical assessment with respect to the literature on theoretical consistency, flexibility and the choice of the appropriate functional form. The robustness of policy suggestions based on inferences from efficiency measures nevertheless crucially depends on theoretically well-founded estimates. This paper adresses stochastic efficiency measurement by critically reviewing the theoretical consistency of recently published technical efficiency estimates. The results confirm the need for a posteriori checking the regularity of the estimated frontier by the researcher and, if necessary, the a priori imposition of the theoretical requirements.functional form, stochastic efficiency analysis, theoretical consistency