An autoregressive process $y_n = \beta_1y_{n-1} + \cdots + \beta_py_{n-p} + \varepsilon_n$ is said to be unstable if the characteristic polynomial $\phi(z) = 1 - \beta_1z - \cdots - \beta_pz^p$ has all roots on or outside the unit circle. The limiting distribution of the least squares estimate of $(\beta_1, \cdots, \beta_p)$ is derived and characterized as a functional of stochastic integrals under a $2 + \delta$ moment assumption on $\varepsilon_n$. Up to the present,…
Consider a sample of size $n$ from a regular exponential family in $p_n$ dimensions. Let $\hat\theta_n$ denote the maximum likelihood estimator, and consider the case where $p_n$ tends to infinity with $n$ and where $\{\theta_n\}$ is a sequence of parameter values in $R^{p_n}$. Moment conditions are provided under which $\|\hat\theta_n - \theta_n\| = O_p(\sqrt{p_n/n})$ and $\|\hat\theta_n - \theta_n - \overline{X}_n\| = O_p (p_n/n)$, where $\overline{X}_n$ is the sample…
It is proved in this paper that covariance hypotheses which are linear in both the covariance and the inverse covariance are products of models each of which consists of either (i) independent identically distributed random vectors which have a covariance with a real, complex or quaternion structure or (ii) independent identically distributed random vectors with a parametrization of the covariance which is given by means of the Clifford algebra. The models (i) are well…
This paper characterizes admissible linear estimators of multiple Poisson parameters under entropy loss. Estimators dominating some of the standard estimators are given. Further, hierarchical Bayes estimators are generated and conditions under which they dominate the standard estimators are also given. Monte Carlo simulations are undertaken to indicate the extent of the risk dominance.
Simultaneous confidence regions and simultaneous prediction regions are considered using the expected coverage measure criterion of Naiman. Invariance methods are used to construct minimax procedures in multivariate linear models.
The problem of detecting a change in drift of Brownian motion is considered in the Bayesian framework with the time of change having a (prior) exponential distribution. To the well known problem of finding an optimal stopping rule for "declaring a change," we add the option of continuously controlling the sampling rates--resulting in controlling the variance coefficient of the process. The combined problem of finding an optimal rate function (dynamic sampling) together with…
The LBI (locally best invariant) test is suggested under normality for the constancy of regression coefficients against the alternative hypothesis that one component of the coefficients follows a random walk process. We discuss the limiting null behavior of the test statistic without assuming normality under two situations, where the initial value of the random walk process is known or unknown. The limiting distribution is that of a quadratic functional of Brownian motion…
A sequence of independent random variables $X_1, X_2, \cdots, X_N$ is said to have a change point if $X_1, X_2, \cdots, X_n$ have a common distribution $F$ and $X_{n+1}, \cdots, X_N$ have a common distribution $G, G \neq F$. Consider the problem of testing the null hypothesis of no change against the alternative of a change $G < F$ at an unknown change point $n$. Two classes of statistics based upon two-sample linear rank statistics (max- and sum-type) are compared in terms…
Consider a sequence of independent random variables $\{X_i: 1 \leq i \leq n\}$ having cdf $F$ for $i \leq \theta n$ and cdf $G$ otherwise. A class of strongly consistent estimators for the change-point $\theta \in (0, 1)$ is proposed. The estimators require no knowledge of the functional forms or parametric families of $F$ and $G$. Furthermore, $F$ and $G$ need not differ in their means (or other measure of location). The only requirement is that $F$ and $G$ differ on a set…
The multivariate calibration problem is considered, in which a sample of $n$ observations on vectors $\xi_{(i)}$ (of "true values") and $Y_{(i)}$ (of less accurate but more easily obtained values) are to be used to estimate the unknown $\xi$ corresponding to a future $Y$. It is assumed that $Y = BX + \varepsilon$, where $\varepsilon$ is multivariate normal and $X = h(\xi)$ for known $h$. Current methods for obtaining a confidence region $C$ for $\xi$, which consist of…
Estimation of the parameters of the nonlinear functional model with known error covariance matrix is discussed. Asymptotic properties of the maximum likelihood estimator for the implicit functional model are presented. The approximate bias in the maximum likelihood estimator due to the nonlinearity of the relationship is given and a bias-adjusted estimator is suggested. Numerical and theoretical results support the superiority of the bias-adjusted estimator relative to the…
Consider the regression model $Y_i = X'_i\beta + g(t_i) + e_i$ for $i = 1, \cdots, n$. Here $g$ is an unknown Holder continuous function of known order $p$ in $R, \beta$ is a $k \times 1$ parameter vector to be estimated and $e_i$ is an unobserved disturbance. Such a model is often encountered in situations in which there is little real knowledge about the nature of $g$. A piecewise polynomial $g_n$ is proposed to approximate $g$. The least-squares estimator $\hat\beta$ is…
Let $\{X_i, Y_i\}^n_{i=1} \subset \mathbb{R}^d \times \mathbb{R}$ be independent identically distributed random variables. If the conditional distribution $F(y \mid x)$ can be parametrized by $F(y \mid x) = F_0((y - m(x))/\sigma(x))$ with a fixed and known distribution $F_0$, the regression curve $m(x)$ and scale curve $\sigma(x)$ could be estimated by some parametric method. More generally, we assume that $F$ is unknown and consider nonparametric simultaneous $M$-type…
Dennis Cox · Eunmee Koh · Grace Wahba · Brian S. Yandell·刊期 1988-03-01 · 卷 16（1）·The Annals of Statistics已编入正刊
半参数部分样条与广义样条模型中的参数零模型检验AI 解读
Cox and Koh (1986) considered the model $y_i = f(x(i)) + \varepsilon_i, \varepsilon_i$ i.i.d. $N(0, \sigma^2)$, with the (parametric) null hypothesis $f(x), x \in \lbrack 0, 1 \rbrack$, a polynomial of degree $m - 1$ or less, versus the alternative $f$ is "smooth," based on the Bayesian model for $f$ which leads to polynomial smoothing spline estimates for $f$. They showed that there was no uniformly most powerful test and found the locally most powerful (LMP) test. We…
This paper is concerned with parametric regression models of the form $Y_{ij} = f(t_{ij}, \theta_i) + \text{error}, i = 1, \ldots, n, j = 1, \ldots, T_i$, where the continuous function $f$ may depend nonlinearly on the known regressors $t_{ij}$ and the unknown parameter vectors $\theta_i$. The assumption of an a priori known $f$ is dropped and replaced by the requirement that qualitative information about the structure of the model is available or can be generated by a…
Best linear unbiased predictors of a random field can be obtained if the covariance function of the random field is specified correctly. Consider a random field defined on a bounded region $R$. We wish to predict the random field $z(\cdot)$ at a point $x$ in $R$ based on observations $z(x_1), z(x_2), \ldots, z(x_N)$ in $R$, where $\{x_i\}^\infty_{i = 1}$ has $x$ as a limit point but does not contain $x$. Suppose the covariance function is misspecified, but has an equivalent…
Examples are presented of statistical techniques for the analysis of random process data and of their uses in the substantive fields of seismology and neurophysiology. The problems addressed include frequency estimation for decaying cosinusoids, signal estimation, association measurement, causal connection assessment, estimation of speed and direction and structural modeling. The techniques employed include complex demodulation, nonlinear regression, probit analysis,…
The selection of the order, i.e., number of vanishing moments, of the kernel in a kernel density estimator is considered from two points of view. First, theoretical properties are investigated by a mean integrated squared error analysis of the problem. Second, and perhaps more importantly, cross validation is proposed as a practical method of choice, and theoretical backing for this is provided through an asymptotic optimality result.
A case-cohort design was recently proposed [Prentice (1986)] as a means of reducing cost in large epidemiologic cohort studies. A "pseudolikelihood" procedure was described for relative risk regression parameter estimation. This procedure involves covariate data only on subjects who develop disease and on a random subset of the entire cohort. In contrast, the usual partial likelihood estimation procedure requires covariate histories on the entire cohort. Accordingly, a…
The phenomenon of partial $n^{-1/2}$-term correction by the bootstrap in the estimation of sampling distributions of nonstandardized statistics is explained and studied in this note.
It is shown that the convergence in law of estimators entails convergence uniformly over all Borel sets whenever the estimators are asymptotically equivariant in a suitable sense and the likelihood ratios of the sample are appropriately smooth. This result generalizes a recent result of Boos in many directions.
A class of procedures is proposed for testing the composite hypothesis that a stationary stochastic process is Gaussian. Requiring very limited prior knowledge about the structure of the process, the tests rely on quadratic forms in deviations of certain sample statistics from their population counterparts, minimized with respect to the unknown parameters. A specific test is developed, which employs differences between components of the sample and Gaussian characteristic…
Herein we consider the asymptotic performance of the least squares predictors $\hat{y}_n$ of the stochastic regression model $y_n = \beta_1 x_{n1} + \cdots + \beta_p x_{np} + \varepsilon_n$. In particular, the accumulated cost function $\sum^n_{k=1} (y_k - \hat{y}_k - \varepsilon_k)^2$ is studied. The results are then applied to nonstationary autoregressive time series. A statistic is also constructed to show how many times one should difference a nonstationary time series…
We consider the problem of estimating parameter matrices which occur in the noncentral Wishart, noncentral multivariate $F$ and canonical correlations distributions. A decision-theoretic approach is taken with squared error as the loss function. In these three settings the eigenvalues of the parameter matrices are of primary interest. Sensible estimates of these are obtained by restricting attention to orthogonally invariant estimates of the parameter matrices, whose…
An autoregressive process yn=β1yn−1+⋯+βpyn−p+εn is said to be unstable if the characteristic polynomial ϕ(z)=1−β1z−⋯−βpzp has all roots on or outside the unit circle. The limiting distribution of the least squares estimate of (β1,⋯,βp) is derived and characterized as a functional of stochastic integrals under a 2+δ moment assumption on εn. Up to the present,…
Consider a sample of size n from a regular exponential family in pn dimensions. Let θ^n denote the maximum likelihood estimator, and consider the case where pn tends to infinity with n and where {θn} is a sequence of parameter values in Rpn. Moment conditions are provided under which ∥θ^n−θn∥=Op(pn/n) and ∥θ^n−θn−Xn∥=Op(pn/n), where Xn is the sample…
A sequence of independent random variables X1,X2,⋯,XN is said to have a change point if X1,X2,⋯,Xn have a common distribution F and Xn+1,⋯,XN have a common distribution G,G=F. Consider the problem of testing the null hypothesis of no change against the alternative of a change G<F at an unknown change point n. Two classes of statistics based upon two-sample linear rank statistics (max- and sum-type) are compared in terms…
Consider a sequence of independent random variables {Xi:1≤i≤n} having cdf F for i≤θn and cdf G otherwise. A class of strongly consistent estimators for the change-point θ∈(0,1) is proposed. The estimators require no knowledge of the functional forms or parametric families of F and G. Furthermore, F and G need not differ in their means (or other measure of location). The only requirement is that F and G differ on a set…
The multivariate calibration problem is considered, in which a sample of n observations on vectors ξ(i) (of "true values") and Y(i) (of less accurate but more easily obtained values) are to be used to estimate the unknown ξ corresponding to a future Y. It is assumed that Y=BX+ε, where ε is multivariate normal and X=h(ξ) for known h. Current methods for obtaining a confidence region C for ξ, which consist of…
Consider the regression model Yi=Xi′β+g(ti)+ei for i=1,⋯,n. Here g is an unknown Holder continuous function of known order p in R,β is a k×1 parameter vector to be estimated and ei is an unobserved disturbance. Such a model is often encountered in situations in which there is little real knowledge about the nature of g. A piecewise polynomial gn is proposed to approximate g. The least-squares estimator β^ is…
Let {Xi,Yi}i=1n⊂Rd×R be independent identically distributed random variables. If the conditional distribution F(y∣x) can be parametrized by F(y∣x)=F0((y−m(x))/σ(x)) with a fixed and known distribution F0, the regression curve m(x) and scale curve σ(x) could be estimated by some parametric method. More generally, we assume that F is unknown and consider nonparametric simultaneous M-type…
Cox and Koh (1986) considered the model yi=f(x(i))+εi,εi i.i.d. N(0,σ2), with the (parametric) null hypothesis f(x),x∈[0,1], a polynomial of degree m−1 or less, versus the alternative f is "smooth," based on the Bayesian model for f which leads to polynomial smoothing spline estimates for f. They showed that there was no uniformly most powerful test and found the locally most powerful (LMP) test. We…
This paper is concerned with parametric regression models of the form Yij=f(tij,θi)+error,i=1,…,n,j=1,…,Ti, where the continuous function f may depend nonlinearly on the known regressors tij and the unknown parameter vectors θi. The assumption of an a priori known f is dropped and replaced by the requirement that qualitative information about the structure of the model is available or can be generated by a…
Best linear unbiased predictors of a random field can be obtained if the covariance function of the random field is specified correctly. Consider a random field defined on a bounded region R. We wish to predict the random field z(⋅) at a point x in R based on observations z(x1),z(x2),…,z(xN) in R, where {xi}i=1∞ has x as a limit point but does not contain x. Suppose the covariance function is misspecified, but has an equivalent…
The phenomenon of partial n−1/2-term correction by the bootstrap in the estimation of sampling distributions of nonstandardized statistics is explained and studied in this note.
Herein we consider the asymptotic performance of the least squares predictors y^n of the stochastic regression model yn=β1xn1+⋯+βpxnp+εn. In particular, the accumulated cost function ∑k=1n(yk−y^k−εk)2 is studied. The results are then applied to nonstationary autoregressive time series. A statistic is also constructed to show how many times one should difference a nonstationary time series…
We consider the problem of estimating parameter matrices which occur in the noncentral Wishart, noncentral multivariate F and canonical correlations distributions. A decision-theoretic approach is taken with squared error as the loss function. In these three settings the eigenvalues of the parameter matrices are of primary interest. Sensible estimates of these are obtained by restricting attention to orthogonally invariant estimates of the parameter matrices, whose…