Simple conditions on the observed information ensure asymptotic normality of the conditional distributions of the randomly normed score statistic and maximum likelihood estimator given a suitable asymptotically ancillary statistic. In particular, asymptotic normality holds conditional on any asymptotically ancillary statistic asymptotically equivalent to observed information. The results apply to inference from a general stochastic process and are of particular relevance in…
In the intuitive approach, a distribution function $F$ is said to be not more heavily tailed than $G$ if $\lim \sup_{x \rightarrow \infty} \bar{F}/\bar{G} < \infty$. An alternative is to consider the behavior of the ratio $F^{-1}(u)/G^{-1}(u)$, in a neighborhood of one. The present paper examines the relationship between these two criteria and concludes that the intuitive approach gives a more thorough comparison of distribution functions than the ratio of the quantile…
A sequential procedure for estimating the mean of a normal distribution is proposed. The procedure is shown to have a negative regret at the origin and the same regret as the usual procedure at the other point asymptotically.
White noise models often renormalize exactly yielding optimal rates of convergence for pointwise nonparametric functional estimation problems. Similar rescaling ideas lead to a sequence of experiments appropriate for pointwise density estimation problems.
In this paper we consider the product-limit estimator of the survival distribution function in the context of independent but nonidentically distributed censoring times. An upper bound on the mean square increment of the stopped Kaplan-Meier process is obtained. Also, a representation is given for the ratio of the survival distribution function to the product-limit estimator as the product of a bounded process and a martingale. From this representation bounds on the mean…
Normal approximations, as provided by permutational central limit theorems, conditionally can be arbitrarily bad. Such approximations therefore are poorly suited to the construction of critical values for Pitman (permutation) tests. A classical remedy consists in substituting a beta approximation (over the appropriate conditional interval range) for the normal one. Whereas deriving permutational extreme values for usual, nonserial statistics is generally straightforward, the…
Consider the problem of estimating the total of some finite population. Suppose the labels attached to the units of the population are such that members of the population whose labels are close together are more alike than units whose labels are far apart. For such a population a sensible estimator of the population total is one that interpolates linearly between successive members of the sample. The admissibility of this estimator will be demonstrated. The related interval…
Suppose $\hat{\theta}_n$ is a strongly consistent estimator for $\theta_0$ in some i.i.d. situation. Let $N_\varepsilon$ and $Q_\varepsilon$ be, respectively, the last $n$ and the total number of $n$ for which $\hat{\theta}_n$ is at least $\varepsilon$ away from $\theta_0$. The limit distributions for $\varepsilon^2N_\varepsilon$ and $\varepsilon^2Q_\varepsilon$ as $\varepsilon$ goes to zero are obtained under natural and weak conditions. The theory covers both parametric…
We prove strong consistency of a class of maximum objective estimators for exponential parametric families of Markov random fields on $\mathbb{Z}^d$, including both maximum likelihood and pseudolikelihood estimators, using large deviation estimates. We also obtain the optimality property for the maximum likelihood estimator in the sense of Bahadur.
In this paper stochastic integrals with respect to the basic martingale are approximated by Gaussian processes. The probability inequalities governing this approximation are used to study goodness-of-fit tests based on sublinear functionals of weighted versions of these stochastic integrals. As special cases of these tests, generalized rank and supremum-type tests are considered.
An adaptive maximum likelihood estimator based on the estimation of the log-density by $B$-splines is introduced. A data-driven method of selecting the smoothing parameter involved in the consequent density estimation is demonstrated. A Monte Carlo study is conducted to evaluate the small sample performance of the estimator in a location and a regression problem. The adaptive estimator is seen to compare favorably to some standard estimates. We show that the estimator is…
We propose an affine equivariant estimator of multivariate location that combines a high breakdown point and a bounded influence function with high asymptotic efficiency. This proposal is basically a location $M$-estimator based on the observations obtained after scaling with an affine equivariant high breakdown covariance estimator. The resulting location estimator will inherit the breakdown point of the initial covariance estimator and within the location-covariance model…
This paper discusses robust estimation for structural errors-in-variables (EV) linear regression models. Such models have important applications in many areas. Under certain assumptions, including normality, the maximum likelihood estimates for the EV model are provided by orthogonal regression (OR) which minimizes the orthogonal distance from the regression line to the data points instead of the vertical distance used in ordinary regression. OR is very sensitive to…
It is known that the standard delete-1 jackknife and the classical bootstrap are in general equally efficient for estimating the mean-square-error of a statistic in the i.i.d. setting. However, this equivalence no longer holds in the linear regression model. It turns out that the bootstrap is more efficient when error variables are homogeneous and the jackknife is more robust when they are heterogeneous. In fact, we can divide all the commonly used resampling procedures for…
P. J. Bickel's approach to and results on estimating the parameter vector $\beta$ of a conditionally contaminated linear regression model by asymptotically linear (AL) estimators $\hat\beta^\ast$ which have minimum trace of the asymptotic covariance matrix among all AL estimators with a given bound $b$ on their asymptotic bias (MT-AL estimators with bias bound $b$) is here extended to conditionally contaminated general linear models and in particular for estimating arbitrary…
We show that regression quantiles, which could be computed as solutions of a linear programming problem, and the solutions of the corresponding dual problem, which we call the regression rank-scores, generalize the duality of order statistics and of ranks from the location to the linear model. Noting this fact, we study the regression quantile and regression rank-score processes in the heteroscedastic linear regression model, obtaining some new estimators and interesting…
D. L. Hawkins · Subhash Kochar · Clive Loader·刊期 1992-03-01 · 卷 20（1）·The Annals of Statistics已编入正刊
转折点未知时针对平均剩余寿命先增后减分布的指数性检验AI 解读
Guess, Hollander and Proschan proposed tests for exponentiality versus IDMRL (increasing initially and then decreasing mean residual life) distributions when the change point, or corresponding quantile, is known. In this paper we propose two tests which do not require such knowledge of the change point. The tests are based on estimates of functionals of the $\operatorname{cdf}$ which discriminate between the exponential and IDMRL families.
Product partition models assume that observations in different components of a random partition of the data are independent. If the probability distribution of random partitions is in a certain product form prior to making the observations, it is also in product form given the observations. The product model thus provides a convenient machinery for allowing the data to weight the partitions likely to hold; and inference about particular future observations may then be made…
Bayes $A$-optimal designs for the one-way and the two-way elimination of heterogeneity models and optimal $\Gamma$-minimax designs for the one-way elimination of heterogeneity model for experiments to compare test treatments with a control are given. Properties such as robustness, of these designs are studied. Optimality results are derived in the exact theory setup.
In this paper a method of identifying stationary and invertible vector autoregressive moving-average time series is proposed. The models are presumed to be represented in (reversed) echelon canonical form. Consideration is given to both parameter estimation and the determination of structural indices, the evaluations being based on the use of closed form least squares calculations. Consistency of the technique is shown and the operational characteristics of the procedure…
The problem of estimating the integral of a stochastic process from observations at a finite number of sampling points is considered. Sacks and Ylvisaker found a sequence of asymptotically optimal sampling designs for general processes with exactly 0 and 1 quadratic mean (q.m.) derivatives using optimal-coefficient estimators, which depend on the process covariance. These results were extended to a restricted class of processes with exactly $K$ q.m. derivatives, for all $K = 0,1,2,\ldots$…
Ordinary smoothing splines have an integrated mean squared error which is dominated by bias contributions at the boundaries. When the estimated function has additional derivatives, the boundary contribution to the bias affects the asymptotic rate of convergence unless the derivatives of the estimated function meet the natural boundary conditions. This paper introduces relaxed boundary smoothing splines and shows that they obtain the optimal asymptotic rate of convergence…
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…
In the intuitive approach, a distribution function F is said to be not more heavily tailed than G if limsupx→∞Fˉ/Gˉ<∞. An alternative is to consider the behavior of the ratio F−1(u)/G−1(u), in a neighborhood of one. The present paper examines the relationship between these two criteria and concludes that the intuitive approach gives a more thorough comparison of distribution functions than the ratio of the quantile…
On the Last Time and the Number of Times an Estimator is More than ε From its Target Value
Suppose θ^n is a strongly consistent estimator for θ0 in some i.i.d. situation. Let Nε and Qε be, respectively, the last n and the total number of n for which θ^n is at least ε away from θ0. The limit distributions for ε2Nε and ε2Qε as ε goes to zero are obtained under natural and weak conditions. The theory covers both parametric…
We prove strong consistency of a class of maximum objective estimators for exponential parametric families of Markov random fields on Zd, including both maximum likelihood and pseudolikelihood estimators, using large deviation estimates. We also obtain the optimality property for the maximum likelihood estimator in the sense of Bahadur.
An adaptive maximum likelihood estimator based on the estimation of the log-density by B-splines is introduced. A data-driven method of selecting the smoothing parameter involved in the consequent density estimation is demonstrated. A Monte Carlo study is conducted to evaluate the small sample performance of the estimator in a location and a regression problem. The adaptive estimator is seen to compare favorably to some standard estimates. We show that the estimator is…
We propose an affine equivariant estimator of multivariate location that combines a high breakdown point and a bounded influence function with high asymptotic efficiency. This proposal is basically a location M-estimator based on the observations obtained after scaling with an affine equivariant high breakdown covariance estimator. The resulting location estimator will inherit the breakdown point of the initial covariance estimator and within the location-covariance model…
Generalized M-Estimators for Errors-in-Variables Regression
P. J. Bickel's approach to and results on estimating the parameter vector β of a conditionally contaminated linear regression model by asymptotically linear (AL) estimators β^∗ which have minimum trace of the asymptotic covariance matrix among all AL estimators with a given bound b on their asymptotic bias (MT-AL estimators with bias bound b) is here extended to conditionally contaminated general linear models and in particular for estimating arbitrary…
Guess, Hollander and Proschan proposed tests for exponentiality versus IDMRL (increasing initially and then decreasing mean residual life) distributions when the change point, or corresponding quantile, is known. In this paper we propose two tests which do not require such knowledge of the change point. The tests are based on estimates of functionals of the cdf which discriminate between the exponential and IDMRL families.
Bayes A-optimal designs for the one-way and the two-way elimination of heterogeneity models and optimal Γ-minimax designs for the one-way elimination of heterogeneity model for experiments to compare test treatments with a control are given. Properties such as robustness, of these designs are studied. Optimality results are derived in the exact theory setup.
The problem of estimating the integral of a stochastic process from observations at a finite number of sampling points is considered. Sacks and Ylvisaker found a sequence of asymptotically optimal sampling designs for general processes with exactly 0 and 1 quadratic mean (q.m.) derivatives using optimal-coefficient estimators, which depend on the process covariance. These results were extended to a restricted class of processes with exactly K q.m. derivatives, for all K=0,1,2,……
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…