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Register a new userAnalytic Bias Reduction for $k$-Sample Functionals
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Added by
Saralees Nadarajah
Publication date
2009
fields
Mathematical Statistics
and research's language is
English
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We give analytic methods for nonparametric bias reduction that remove the need for computationally intensive methods like the bootstrap and the jackknife. We call an estimate {it $p$th order} if its bias has magnitude $n_0^{-p}$ as $n_0 to infty$, where $n_0$ is the sample size (or the minimum sample size if the estimate is a function of more than one sample). Most estimates are only first order and require O(N) calculations, where $N$ is the total sample size. The usual bootstrap and jackknife estimates are second order but they are computationally intensive, requiring $O(N^2)$ calculations for one sample. By contrast Jaeckels infinitesimal jackknife is an analytic second order one sample estimate requiring only O(N) calculations. When $p$th order bootstrap and jackknife estimates are available, they require $O(N^p)$ calculations, and so become even more computationally intensive if one chooses $p>2$. For general $p$ we provide analytic $p$th order nonparametric estimates that require only O(N) calculations. Our estimates are given in terms of the von Mises derivatives of the functional being estimated, evaluated at the empirical distribution. For products of moments an unbiased estimate exists: our form for this polykay is much simpler than the usual form in terms of power sums.
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Let $X_1,dots, X_n$ be i.i.d. random variables sampled from a normal distribution $N(mu,Sigma)$ in ${mathbb R}^d$ with unknown parameter $theta=(mu,Sigma)in Theta:={mathbb R}^dtimes {mathcal C}_+^d,$ where ${mathcal C}_+^d$ is the cone of positively definite covariance operators in ${mathbb R}^d.$ Given a smooth functional $f:Theta mapsto {mathbb R}^1,$ the goal is to estimate $f(theta)$ based on $X_1,dots, X_n.$ Let $$ Theta(a;d):={mathbb R}^dtimes Bigl{Sigmain {mathcal C}_+^d: sigma(Sigma)subset [1/a, a]Bigr}, ageq 1, $$ where $sigma(Sigma)$ is the spectrum of covariance $Sigma.$ Let $hat theta:=(hat mu, hat Sigma),$ where $hat mu$ is the sample mean and $hat Sigma$ is the sample covariance, based on the observations $X_1,dots, X_n.$ For an arbitrary functional $fin C^s(Theta),$ $s=k+1+rho, kgeq 0, rhoin (0,1],$ we define a functional $f_k:Theta mapsto {mathbb R}$ such that begin{align*} & sup_{thetain Theta(a;d)}|f_k(hat theta)-f(theta)|_{L_2({mathbb P}_{theta})} lesssim_{s, beta} |f|_{C^{s}(Theta)} biggr[biggl(frac{a}{sqrt{n}} bigvee a^{beta s}biggl(sqrt{frac{d}{n}}biggr)^{s} biggr)wedge 1biggr], end{align*} where $beta =1$ for $k=0$ and $beta>s-1$ is arbitrary for $kgeq 1.$ This error rate is minimax optimal and similar bounds hold for more general loss functions. If $d=d_nleq n^{alpha}$ for some $alphain (0,1)$ and $sgeq frac{1}{1-alpha},$ the rate becomes $O(n^{-1/2}).$ Moreover, for $s>frac{1}{1-alpha},$ the estimators $f_k(hat theta)$ is shown to be asymptotically efficient. The crucial part of the construction of estimator $f_k(hat theta)$ is a bias reduction method studied in the paper for more general statistical models than normal.
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Many proposals have emerged as alternatives to the Heckman selection model, mainly to address the non-robustness of its normal assumption. The 2001 Medical Expenditure Panel Survey data is often used to illustrate this non-robustness of the Heckman model. In this paper, we propose a generalization of the Heckman sample selection model by allowing the sample selection bias and dispersion parameters to depend on covariates. We show that the non-robustness of the Heckman model may be due to the assumption of the constant sample selection bias parameter rather than the normality assumption. Our proposed methodology allows us to understand which covariates are important to explain the sample selection bias phenomenon rather than to only form conclusions about its presence. We explore the inferential aspects of the maximum likelihood estimators (MLEs) for our proposed generalized Heckman model. More specifically, we show that this model satisfies some regularity conditions such that it ensures consistency and asymptotic normality of the MLEs. Proper score residuals for sample selection models are provided, and model adequacy is addressed. Simulated results are presented to check the finite-sample behavior of the estimators and to verify the consequences of not considering varying sample selection bias and dispersion parameters. We show that the normal assumption for analyzing medical expenditure data is suitable and that the conclusions drawn using our approach are coherent with findings from prior literature. Moreover, we identify which covariates are relevant to explain the presence of sample selection bias in this important dataset.
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We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition in the u-channel Regge limit. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.
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