Bodhisattva Sen · Moulinath Banerjee · Michael Woodroofe·刊期 2010-08-01 · 卷 38（4）·The Annals of Statistics已编入正刊
Bootstrap 的不一致性：Grenander 估计量AI 解读
In this paper, we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate n1/3. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function f on [0, ∞), is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for f(t0), where t0 ∈ (0, ∞) is an…
A new nonparametric method is described to estimate the distribution of mass within spherical galaxies. The problem of estimating the mass, M(r), within radius r is converted into a problem of estimating a regression function nonparametrically, subject to shape restrictions. We represent the restrictions by the interception of quadratic cones and use the second-order cone programming to estimate the unknown parameters. We establish asymptotic results that are used to…
We extend the isotonic analysis for Wicksell’s problem to estimate a regression function, which is motivated by the problem of estimating dark matter distribution in astronomy. The main result is a version of the Kiefer–Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a convergence rate $n^{-1}\log n$ faster than $n^{-2/3}\log n$. The main result is useful in obtaining asymptotic distributions for estimators, such as isotonic and…
For the problem of estimating a regression function, $\mu$ say, subject to shape constraints, like monotonicity or convexity, it is argued that the divergence of the maximum likelihood estimator provides a useful measure of the effective dimension of the model. Inequalities are derived for the expected mean squared error of the maximum likelihood estimator and the expected residual sum of squares. These generalize equalities from the case of linear regression. As an…
The paper is concerned with testing uniformity versus a monotone den-sity. This problem arises in two important contexts, after transformations, testing whether a sample is a simple random sample or a biased sample, and testing whether the intensity function of a nonhomogeneous Poisson process is constant against monotone alternatives. A penalized likelihood ratio test ($P$-test) and a Dip likelihood test ($D$-test) are developed. The $D$-test is analogous to Hartigan and…
Let $S_1$, $S_2,\cdots$ denote a driftless random walk with values in an inner product space $\mathscr{W}$; let $Z_1$, $Z_2,\cdots$ denote a perturbed random walk of the form $Z_n=n+\langle c,S_n \rangle+\xi_n$, $n = 1, 2,\cdots$, where $\xi_1,\xi_2,\cdots$ are slowly changing, $\langle\centerdot,\centerdot\rangle$ denotes the inner product, and $c\in\mathscr{W}$; and let $t=t_a=inf{n\geq1:Z_n>a}$ for $0\leq a<\infty$. Conditions are developed under which the first four…
One observes $n$ data points, $(\mathbf{t}_i, Y_i),$ with the mean of $Y_i$, conditional on the regression function $f,$ equal to $f(\mathbf{t}_i).$ The prior distribution of the vector $\mathbf{f} = (f(\mathbf{t}_1), \ldots, f(\mathbf{t}_n))^t$ is unknown, but lies in a known class $\Omega.$ An estimator, $\hat{\mathbf{f}},$ of $\mathbf{f}$ is found which minimizes the maximum $E\|\hat{\mathbf{f}} - \mathbf{f}\|^2.$ The maximum is taken over all priors in $\Omega$ and the…
In sequentially designed experiments with linear models, each design variable may depend on previous responses. The use of such sequential designs does not affect the likelihood function or the functional form of the maximum likelihood estimator, but it may affect sampling distributions. In this paper, asymptotic expansions for sampling distributions are obtained. The expansions are very weak ones in which a confidence curve (a function of the unknown parameters) is replaced…
Let $X_1, X_2, \cdots$ be i.i.d. random variables with mean $\theta$ and finite, positive variance $\sigma^2,$ depending on unknown parameters $\omega\in\Omega.$ The problem addressed is that of finding a stopping time $t$ for which the risk $R_A(t, \omega) = E_\omega\{A \gamma^2_0(\omega)(\bar{X}_t - \theta)^2 + t\}$ is as small as possible (in a suitable sense), where $A > 0, \gamma_0$ is a positive function on $\Omega$, and $\bar{X}_t = (X_1 + \cdots + X_t)/t.$ For fixed…
Let $\mathscr{P}$ be a finite population with $N \geq 1$ elements; for each $e \in \mathscr{P}$, let $X_e$ and $Y_e$ be independent, positive random variables with unknown distribution functions $F$ and $G$; and suppose that the pairs $(X_e, Y_e)$ are i.i.d. We consider the problem of estimating $F, G$, and $N$ when the data consist of those pairs $(X_e, Y_e)$ for which $e \in \mathscr{P}$ and $Y_e \leq X_e$. The nonparametric maximum likelihood estimators (MLEs) of $F$ and $G$…
Generalizations of the arc sine laws are shown to provide insight into the operating characteristics of certain techniques for selecting models to fit a given data set, when the available models are nested. As a corollary, one sees that a popular technique may be expected to include about one superfluous parameter, even if the sample size is large.
Asymptotic expansions for the error probabilities of repeated significance tests about a normal mean are developed. Use of the expansions appears to result in substantially improved numerical accuracy, when compared to the use of the leading term, at least in some important special cases. The expansions are sufficiently refined to show the effect of some simple modifications of the basic procedure, such as requiring an initial sample size.
We consider a Bayesian sequential allocation problem that incorporates a concomitant variable in the model. Although the model is highly simplified, our results do have some mildly surprising aspects: (a) the approximate solution to the allocation problem is simpler in the presence of concomitant information than in its absence, (b) the exact value of the discount factor is less important than in previous work, and (c) the myopic procedure is asymptotically optimal.
Michael Woodroofe·刊期 1979-12 · 卷 66（3）·Biometrika已编入正刊
重复似然比检验AI 解读
We study the actual significance level which results from performing repeated likelihood ratio tests over time. We present a general method for estimating the actual significance level when sampling from an exponential family, and apply it to problems of testing hypotheses about means, variance, correlations and failure rates. Our results confirm and extend those of earlier authors that the actual significance level may exceed the nominal one by several times.
MICHAEL WOODROOFE·刊期 1979 · 卷 66（3）·Biometrika已编入正刊
重复似然比检验AI 解读
We study the actual significance level which results from performing repeated likelihood ratio tests over time. We present a general method for estimating the actual significance level when sampling from an exponential family, and apply it to problems of testing hypotheses about means, variance, correlations and failure rates. Our results confirm and extend those of earlier authors that the actual significance level may exceed the nominal one by several times.
We study the tail of the null distribution of the $\log$ likelihood ratio statistic for testing sharp hypotheses about the parameters of an exponential family. We show that the classical chisquare approximation is of exactly the right order of magnitude, although it may be off by a constant factor. We then apply our results and techniques to find the error probabilities of a sequential version of the likelihood ratio test. The sequential version rejects if the likelihood…
Several stopping times which arise from problems of sequential estimation may be written in the form $t_c = \inf\{n \geqq m: S_n < cn^\alpha L(n)\}$ where $S_n, n \geqq 1,$ are the partial sums of i.i.d. positive random variables, $\alpha > 1, L(n)$ is a convergent sequence, and $c$ is a positive parameter which is often allowed to approach zero. In this paper we find the asymptotic distribution of the excess $R_c = ct_c^\alpha - S_{t_c}$ as $c \rightarrow 0$ and use it to…
Michael Woodroofe·刊期 1976-04 · 卷 63（1）·Biometrika已编入正刊
贝叶斯序贯检验的频率学派性质AI 解读
Let X1 , X2 , … denote independent random variables which are normally distributed with unknown mean θ and unit variance. We consider sequential tests of the hypothesis H0 : θ < –δ versus H1 : θ > δ. The tests which we consider were shown by Schwarz (1962) to approximate the optimal Bayesian tests with respect to a general loss structure and any prior density which is everywhere positive. Their continuation regions are bounded subsets of the ( n , S n ) plane, where S n is…
MICHAEL WOODROOFE·刊期 1976 · 卷 63（1）·Biometrika已编入正刊
贝叶斯序贯检验的频率学性质AI 解读
Let X1 , X2 , … denote independent random variables which are normally distributed with unknown mean θ and unit variance. We consider sequential tests of the hypothesis H0 : θ < –δ versus H1 : θ > δ. The tests which we consider were shown by Schwarz (1962) to approximate the optimal Bayesian tests with respect to a general loss structure and any prior density which is everywhere positive. Their continuation regions are bounded subsets of the ( n , S n ) plane, where S n is…
Let $f$ be a density which vanishes for negative values of its argument and varies regularly with exponent $\alpha - 1$ at zero, where $1 < \alpha < 2$. Further, let $f_\theta$ denote $f$ translated by $\theta$. We find and study the asymptotic distribution of the MLE $\hat{\theta}_n$ based on a sample size $n$ as $n \rightarrow \infty$.
We extend the isotonic analysis for Wicksell’s problem to estimate a regression function, which is motivated by the problem of estimating dark matter distribution in astronomy. The main result is a version of the Kiefer–Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a convergence rate n−1logn faster than n−2/3logn. The main result is useful in obtaining asymptotic distributions for estimators, such as isotonic and…
For the problem of estimating a regression function, μ say, subject to shape constraints, like monotonicity or convexity, it is argued that the divergence of the maximum likelihood estimator provides a useful measure of the effective dimension of the model. Inequalities are derived for the expected mean squared error of the maximum likelihood estimator and the expected residual sum of squares. These generalize equalities from the case of linear regression. As an…
The paper is concerned with testing uniformity versus a monotone den-sity. This problem arises in two important contexts, after transformations, testing whether a sample is a simple random sample or a biased sample, and testing whether the intensity function of a nonhomogeneous Poisson process is constant against monotone alternatives. A penalized likelihood ratio test (P-test) and a Dip likelihood test (D-test) are developed. The D-test is analogous to Hartigan and…
Let S1, S2,⋯ denote a driftless random walk with values in an inner product space W; let Z1, Z2,⋯ denote a perturbed random walk of the form Zn=n+⟨c,Sn⟩+ξn, n=1,2,⋯, where ξ1,ξ2,⋯ are slowly changing, ⟨⋅,⋅⟩ denotes the inner product, and c∈W; and let t=ta=infn≥1:Zn>a for 0≤a<∞. Conditions are developed under which the first four…
One observes n data points, (ti,Yi), with the mean of Yi, conditional on the regression function f, equal to f(ti). The prior distribution of the vector f=(f(t1),…,f(tn))t is unknown, but lies in a known class Ω. An estimator, f^, of f is found which minimizes the maximum E∥f^−f∥2. The maximum is taken over all priors in Ω and the…
Let X1,X2,⋯ be i.i.d. random variables with mean θ and finite, positive variance σ2, depending on unknown parameters ω∈Ω. The problem addressed is that of finding a stopping time t for which the risk RA(t,ω)=Eω{Aγ02(ω)(Xˉt−θ)2+t} is as small as possible (in a suitable sense), where A>0,γ0 is a positive function on Ω, and Xˉt=(X1+⋯+Xt)/t. For fixed…
Let P be a finite population with N≥1 elements; for each e∈P, let Xe and Ye be independent, positive random variables with unknown distribution functions F and G; and suppose that the pairs (Xe,Ye) are i.i.d. We consider the problem of estimating F,G, and N when the data consist of those pairs (Xe,Ye) for which e∈P and Ye≤Xe. The nonparametric maximum likelihood estimators (MLEs) of F and G…
We study the tail of the null distribution of the log likelihood ratio statistic for testing sharp hypotheses about the parameters of an exponential family. We show that the classical chisquare approximation is of exactly the right order of magnitude, although it may be off by a constant factor. We then apply our results and techniques to find the error probabilities of a sequential version of the likelihood ratio test. The sequential version rejects if the likelihood…
Several stopping times which arise from problems of sequential estimation may be written in the form tc=inf{n≧m:Sn<cnαL(n)} where Sn,n≧1, are the partial sums of i.i.d. positive random variables, α>1,L(n) is a convergent sequence, and c is a positive parameter which is often allowed to approach zero. In this paper we find the asymptotic distribution of the excess Rc=ctcα−Stc as c→0 and use it to…
Let f be a density which vanishes for negative values of its argument and varies regularly with exponent α−1 at zero, where 1<α<2. Further, let fθ denote f translated by θ. We find and study the asymptotic distribution of the MLE θ^n based on a sample size n as n→∞.