Jianhua Z. Huang · Charles Kooperberg · Charles J. Stone · Young K. Truong·刊期 2000-08-01 · 卷 28（4）·The Annals of Statistics已编入正刊
比例风险回归的函数型 ANOVA 建模AI 解读
The logarithm of the relative risk function in a proportional hazards model involving one or more possibly time-dependent covariates is treated as a specified sum of a constant term, main effects, and selected interaction terms. Maximum partial likelihood estimation is used, where the maximization is taken over a suitably chosen finite-dimensional estimation space, whose dimension increases with the sample size and which is constructed from linear spaces of functions of one…
An automatic procedure that uses linear splines and their tensor products is proposed for fitting a regression model to data involving a polychotomous response variable and one or more predictors. The fitted model can be used for multiple classification. The automatic fitting procedure involves maximum likelihood estimation, stepwise addition, stepwise deletion, and model selection by the Akaike information criterion, cross-validation, or an independent test set. A modified…
1. RANDOM VARIABLES AND THEIR DISTRIBUTION Introduction / Sample Distributions / Distributions / Random Variables / Probability Functions and Density Functions / Distribution Functions and Quantiles / Univariate Transformations / Independence 2. EXPECTATION Introduction / Properties of Expectation / Variance / Weak Law of Large Numbers Simulation and the Monte Carlo Method 3. SPECIAL CONTINUOUS MODELS Gamma and Beta Distributions / The Normal Distribution / Normal…
Linear splines and their tensor products are used to estimate the conditional log-hazard function based on possibly censored, positive response data and one or more covariates. An automatic procedure involving the maximum likelihood method, stepwise addition, stepwise deletion, and the Bayes Information Criterion is used to select the final model. The possible models contain proportional hazards models as a subclass, which makes it possible to diagnose departures from…
Let $X_1, \ldots, X_M, Y_1,\ldots, Y_N$ be random variables, and set $\mathbf{X} = (X_1, \ldots, X_M)$ and $\mathbf{Y} = (Y_1, \ldots, Y_N)$. Let $\varphi$ be the regression or logistic or Poisson regression function of $\mathbf{Y}$ on $\mathbf{X}(N = 1)$ or the logarithm of the density function of $\mathbf{Y}$ or the conditional density function of $\mathbf{Y}$ on $\mathbf{X}$. Consider the approximation $\varphi^\ast$ to $\varphi$ having a suitably defined form involving a…
Consider a stationary time series $(\mathbf{X}_t, Y_t), t = 0, \pm 1,\ldots,$ with $\mathbf{X}_t$ being $\mathbb{R}^d$-valued and $Y_t$ real-valued. The conditional mean function is given by $\theta(\mathbf{X}_0) = E(Y_0\mid\mathbf{X}_0)$. Under appropriate regularity conditions, a local average estimator of this function based on a finite realization $(\mathbf{X}_1, Y_1),\ldots,(\mathbf{X}_n, Y_n)$ can be chosen to achieve the optimal rate of convergence $n^{-1/(2 + d)}$ bot…
Consider a $\mathscr{Y}$-valued response variable having a density function $f(\cdot\mid x)$ that depends on an $\mathscr{X}$-valued input variable $x.$ It is assumed that $\mathscr{X}$ and $\mathscr{Y}$ are compact intervals and that $f(\cdot\mid\cdot)$ is continuous and positive on $\mathscr{X} \times \mathscr{Y}.$ Let $F(\cdot\mid x)$ denote the distribution function of $f(\cdot\mid x)$ and let $Q(\cdot\mid x)$ denote its quantile function. A finite-parameter exponential…
Let $f$ be a continuous and positive unknown density on a known compact interval $\mathscr{Y}$. Let $F$ denote the distribution function of $f$ and let $Q = F^{-1}$ denote its quantile function. A finite-parameter exponential family model based on $B$-splines is constructed. Maximum-likelihood estimation of the parameters of the model based on a random sample of size $n$ from $f$ yields estimates $\hat{f, F}$ and $\hat{Q}$ of $f, F$ and $Q$, respectively. Under mild…
Let $(X, Y)$ be a pair of random variables such that $X = (X_1,\cdots, X_J)$ ranges over $C = \lbrack 0, 1\rbrack^J$. The conditional distribution of $Y$ given $X = x$ is assumed to belong to a suitable exponential family having parameter $\eta \in \mathbb{R}$. Let $\eta = f(x)$ denote the dependence of $\eta$ on $x$. Let $f^\ast$ denote the additive approximation to $f$ having the maximum possible expected log-likelihood under the model. Maximum likelihood is used to fit an…
Let $(X, Y)$ be a pair of random variables such that $X = (X_1, \cdots, X_J)$ and let $f$ by a function that depends on the joint distribution of $(X, Y).$ A variety of parametric and nonparametric models for $f$ are discussed in relation to flexibility, dimensionality, and interpretability. It is then supposed that each $X_j \in \lbrack 0, 1\rbrack,$ that $Y$ is real valued with mean $\mu$ and finite variance, and that $f$ is the regression function of $Y$ on $X.$ Let $f^\ast,$…
Kernel estimates of an unknown multivariate density are investigated, with mild restrictions being placed on the kernel. A window selection rule is considered, which can be interpreted in terms of cross-validation. Under the mild assumption that the unknown density and its one-dimensional marginals are bounded, the rule is shown to be asymptotically optimal. This strengthens recent results of Peter Hall.
Consider a $p$-times differentiable unknown regression function $\theta$ of a $d$-dimensional measurement variable. Let $T(\theta)$ denote a derivative of $\theta$ of order $m$ and set $r = (p - m)/(2p + d)$. Let $\hat{T}_n$ denote an estimator of $T(\theta)$ based on a training sample of size $n$, and let $\| \hat{T}_n - T(\theta)\|_q$ be the usual $L^q$ norm of the restriction of $\hat{T}_n - T(\theta)$ to a fixed compact set. Under appropriate regularity conditions, it is…
Let $M_0$ be a normal linear regression model and let $M_1,\cdots, M_K$ be distinct proper linear submodels of $M_0$. Let $\hat k \in \{0,\cdots, K\}$ be a model selection rule based on observed data from the true model. Given $\hat k$, let the unknown parameters of the selected model $M_{\hat k}$ be fitted by the maximum likelihood method. A loss function is introduced which depends additively on two parts: (i) a measure of the difference between the fitted model $M_{\hat k}$…
Let $d$ denote a positive integer, $\|x\| = (x^2_1 + \cdots + x^2_d)^{1/2}$ the Euclidean norm of $x = (x_1, \cdots, x_d) \in \mathbb{R}^d, k$ a nonnegative integer, $\mathscr{C}_k$ the collection of $k$ times continuously differentiable functions on $\mathbb{R}^d$, and $g_k$ the Taylor polynomial of degree $k$ about the origin corresponding to $g \in \mathscr{C}_k$. Let $M$ and $p > k$ denote positive constants and let $U$ be an open neighborhood of the origin of $\mathbb{R}^d$…
Let $(X, Y)$ be a pair of random variables such that $X$ is $\mathbb{R}^d$-valued and $Y$ is $\mathbb{R}^{d'}$-valued. Given a random sample $(X_1, Y_1), \cdots, (X_n, Y_n)$ from the distribution of $(X, Y)$, the conditional distribution $P^Y(\bullet \mid X)$ of $Y$ given $X$ can be estimated nonparametrically by $\hat{P}_n^Y(A \mid X) = \sum^n_1 W_{ni}(X)I_A(Y_i)$, where the weight function $W_n$ is of the form $W_{ni}(X) = W_{ni}(X, X_1, \cdots, X_n), 1 \leqq i \leqq n$.…
Consider estimators $\hat{\theta}_n$ of the location parameter $\theta$ based on a sample of size $n$ from $\theta + X$, where the random variable $X$ has an unknown distribution $F$ which is symmetric about the origin but otherwise arbitrary. Let $\mathscr{F}$ denote the Fisher information on $\theta$ contained in $\theta + X$. We show that there is a nonrandomized translation and scale invariant adaptive maximum likelihood estimator $\hat{\theta}_n$ of $\theta$ which doe…
Consider the problem of estimating the location parameter $\theta \in R^d$ based on a sample of size $n$ from $(\theta + X, Y)$, where $X$ is a $d$-dimensional random vector, $Y$ is a random element of some measure space, and $(X, Y)$ has a known distribution. Let $\mathscr{J}^-$ denote the corresponding inverse Fisher information matrix. We show that there is always an invariant estimator $\hat{\theta}_n$ such that $\mathscr{L}(n^{\frac{1}{2}}(\hat{\theta}_n - \theta)) \rightarrow N(0, \mathscr{J}^-)$…
In this paper we consider estimation of the location parameter $\theta \in R^d$ based on a random sample from $(\theta + X, Y),$ where $X$ is a $d$-dimensional random vector, $Y$ is a random element of some measure space $\mathscr{Y},$ and $(X, Y)$ has a known distribution. We first define the Fisher information $\mathscr{J}(\theta + X, Y)$ and the inverse information $\mathscr{J}^-(\theta + X, Y)$ under no regularity conditions. The properties of these quantities are…
Let X1,…,XM,Y1,…,YN be random variables, and set X=(X1,…,XM) and Y=(Y1,…,YN). Let φ be the regression or logistic or Poisson regression function of Y on X(N=1) or the logarithm of the density function of Y or the conditional density function of Y on X. Consider the approximation φ∗ to φ having a suitably defined form involving a…
Consider a stationary time series (Xt,Yt),t=0,±1,…, with Xt being Rd-valued and Yt real-valued. The conditional mean function is given by θ(X0)=E(Y0∣X0). Under appropriate regularity conditions, a local average estimator of this function based on a finite realization (X1,Y1),…,(Xn,Yn) can be chosen to achieve the optimal rate of convergence n−1/(2+d) bot…
Consider a Y-valued response variable having a density function f(⋅∣x) that depends on an X-valued input variable x. It is assumed that X and Y are compact intervals and that f(⋅∣⋅) is continuous and positive on X×Y. Let F(⋅∣x) denote the distribution function of f(⋅∣x) and let Q(⋅∣x) denote its quantile function. A finite-parameter exponential…
Let f be a continuous and positive unknown density on a known compact interval Y. Let F denote the distribution function of f and let Q=F−1 denote its quantile function. A finite-parameter exponential family model based on B-splines is constructed. Maximum-likelihood estimation of the parameters of the model based on a random sample of size n from f yields estimates f,F^ and Q^ of f,F and Q, respectively. Under mild…
Let (X,Y) be a pair of random variables such that X=(X1,⋯,XJ) ranges over C=[0,1]J. The conditional distribution of Y given X=x is assumed to belong to a suitable exponential family having parameter η∈R. Let η=f(x) denote the dependence of η on x. Let f∗ denote the additive approximation to f having the maximum possible expected log-likelihood under the model. Maximum likelihood is used to fit an…
Let (X,Y) be a pair of random variables such that X=(X1,⋯,XJ) and let f by a function that depends on the joint distribution of (X,Y). A variety of parametric and nonparametric models for f are discussed in relation to flexibility, dimensionality, and interpretability. It is then supposed that each Xj∈[0,1], that Y is real valued with mean μ and finite variance, and that f is the regression function of Y on X. Let f∗,…
Consider a p-times differentiable unknown regression function θ of a d-dimensional measurement variable. Let T(θ) denote a derivative of θ of order m and set r=(p−m)/(2p+d). Let T^n denote an estimator of T(θ) based on a training sample of size n, and let ∥T^n−T(θ)∥q be the usual Lq norm of the restriction of T^n−T(θ) to a fixed compact set. Under appropriate regularity conditions, it is…
Let M0 be a normal linear regression model and let M1,⋯,MK be distinct proper linear submodels of M0. Let k^∈{0,⋯,K} be a model selection rule based on observed data from the true model. Given k^, let the unknown parameters of the selected model Mk^ be fitted by the maximum likelihood method. A loss function is introduced which depends additively on two parts: (i) a measure of the difference between the fitted model Mk^…
Let d denote a positive integer, ∥x∥=(x12+⋯+xd2)1/2 the Euclidean norm of x=(x1,⋯,xd)∈Rd,k a nonnegative integer, Ck the collection of k times continuously differentiable functions on Rd, and gk the Taylor polynomial of degree k about the origin corresponding to g∈Ck. Let M and p>k denote positive constants and let U be an open neighborhood of the origin of Rd…
Let (X,Y) be a pair of random variables such that X is Rd-valued and Y is Rd′-valued. Given a random sample (X1,Y1),⋯,(Xn,Yn) from the distribution of (X,Y), the conditional distribution PY(∙∣X) of Y given X can be estimated nonparametrically by P^nY(A∣X)=∑1nWni(X)IA(Yi), where the weight function Wn is of the form Wni(X)=Wni(X,X1,⋯,Xn),1≦i≦n.…
Consider estimators θ^n of the location parameter θ based on a sample of size n from θ+X, where the random variable X has an unknown distribution F which is symmetric about the origin but otherwise arbitrary. Let F denote the Fisher information on θ contained in θ+X. We show that there is a nonrandomized translation and scale invariant adaptive maximum likelihood estimator θ^n of θ which doe…
Consider the problem of estimating the location parameter θ∈Rd based on a sample of size n from (θ+X,Y), where X is a d-dimensional random vector, Y is a random element of some measure space, and (X,Y) has a known distribution. Let J− denote the corresponding inverse Fisher information matrix. We show that there is always an invariant estimator θ^n such that L(n21(θ^n−θ))→N(0,J−)…
In this paper we consider estimation of the location parameter θ∈Rd based on a random sample from (θ+X,Y), where X is a d-dimensional random vector, Y is a random element of some measure space Y, and (X,Y) has a known distribution. We first define the Fisher information J(θ+X,Y) and the inverse information J−(θ+X,Y) under no regularity conditions. The properties of these quantities are…