This paper concerns nonparametric inference for hazard rates with censored serial data. The focus is upon "delta sequence" estimators of the form $h_n(x) = \int K_b(x, y) dH_n(y)$ with $K_b$ integrating to 1 and concentrating mass near $x$ as $b \rightarrow 0. H_n$ is the Nelson-Aalen empirical cumulative hazard. Strong approximation and simultaneous confidence bands are derived for Rosenblatt-Parzen estimators, with $K_b(x, y) = w((x - y)/b)/b, b = o(n^{-1}),$ and $w(\cdot)$…
Suppose that a test of fit to a parametric family of distributions is employed, with critical points determined from the limiting null distribution of the test statistic for IID observations. It is shown that if the observations are in fact a stationary process satisfying a positive dependence condition, the test will reject a true null hypothesis too often. This result is established for a broad class of chi squared and empiric df tests, including the Pearson,…
Necessary and sufficient conditions for an invariant test to be admissible among invariant tests in the general multivariate analysis of variance problem are presented. It is shown that in many cases the popular tests based on the likelihood ratio matrix are inadmissible. Other tests are shown admissible. Numerical work suggests that the inadmissibility of the likelihood ratio test is not serious. The results are given for the multivariate analysis of variance problem as a…
Stable laws are often fit to outlier-prone data and, if the index $\alpha$ is estimated to be much less than two, then the normal law is rejected in favor of an infinite-variance stable law. This paper derives the theoretical properties of such a procedure. When the true distribution is stable, the distribution of the m.l.e. of $\alpha$ is non-regular if $\alpha = 2$. When the true distribution is not stable, the estimate of $\alpha$ is not a robust measure of the rate of…
The problem of optimal experimental design for estimating parameters in linear regression models is placed in a general convex analysis setting. Duality results are obtained using two approaches, one based on subgradients and the other on Lagrangian theory. The subgradient concept is also used to derive a potentially useful equivalence theorm for establishing the optimality of a singular design and, finally, general versions of the original equivalence theorems of Kiefer and…
We introduce and describe orthogonal series methods for estimating the density of qualitative, quantitative or mixed data. The techniques are completely nonparametric in character, and so may be used in situations where parametric models are difficult to construct. Just this situation arises in the context of mixed--both qualitative and quantitative--data, where there are few parametric models.
In a recent paper (Tanner and Wong, 1983b), a family of data-based nonparametric hazard estimators was introduced. Several of these estimators were studied in an extensive simulation experiment. The estimator which allows for variable bandwidth was found to have a superior performance. In this note, sufficient conditions for the variable kernel estimator to be strongly consistent are presented.
By convolution smoothing of the empirical hazards, a kernel estimate of the hazard function from censored data is obtained. Small and large sample expressions for the mean and the variance of the estimator are given. Conditions for asymptotic normality are investigated using the Hajek projection method.
In some problems of directional data, finite mixtures of simple distributions have been proposed as statistical models. In this paper we show that for a wide class of directional distributions, all such mixtures are identifiable.
The affine type of distributions on the real line are represented as sequences of distributions of maximal invariants on spheres. It is shown that such a representation characterizes the affine type. A consistency condition is introduced, and it is shown that any sequence of maximal invariant distributions satisfying the condition is generated by some affine type on $\mathbf{R}$.
The main purpose of this paper is the development of an asymptotic theory of systematic sampling from a stochastic population. The superpopulation model assumed is that the population arises from a second-order stationary process. A comparison among the multiple random start systematic sampling schemes is made in terms of the limiting expected variance of the sample mean. Asymptotic normality of the systematic sampling mean is obtained, both unconditionally and conditionally…
In this paper the multivariate normal linear patterned mean and covariance matrix testing and estimation problem is studied in the presence of missing data for general one-population hypotheses. The Newton-Raphson, Method of Scoring and EM algorithms are given for finding the maximum likelihood estimates. The asymptotic joint distribution of the maximum likelihood estimates under null and alternative hypotheses are derived along with the form of the likelihood ratio…
It is well known that the Kolmogorov-Smirnov (K-S) test exhibits poor sensitivity to deviations from the hypothesized distribution that occur in the tails. A modified version of the K-S test is introduced that is more sensitive than the K-S test to deviations in the tails. The finite and infinite sample distribution along with the consistency properties of the proposed test are studied. Tables of critical values are provided for two versions of the test (one sensitive to…
This paper discusses a class of minimum distance Cramer-Von Mises type estimators of the slope parameter in a linear regression model. These estimators are obtained by minimizing an integral of squared difference between weighted empiricals of the residuals and their expectations with respect to a large class of integrating measures. The estimator corresponding to the weights proportional to the design variable is shown to be asymptotically efficient within the class at a…
We derive the joint asymptotic distribution of the $k$ midranges formed by averaging the $i$th smallest normalized order statistic with the $i$th largest normalized order statistic, $i = 1, \cdots, k$. We then derive the distribution of the maximum midrange among these $k$ extreme midranges and the limiting distribution of this maximum as $k \rightarrow \infty$. These results imply that, even in infinite samples, different distributions in the class of symmetric, unimodal…
The coverage of a multinomial random sample is the sum of the probabilities of the observed classes. A normal limit law is rigorously proved for Good's (1953) coverage estimator. The result is valid under very general conditions and all terms except the coverage itself are observable. Nevertheless the implied confidence intervals are not much wider than those developed under restrictive assumptions such as in the classical occupancy problem. The asymptotic variance is…
Let $f$ be a density on $R^d$, and let $f_n$ be the kernel estimate of $f$, $f_n(x) = (nh^d)^{-1} \sum^n_{i=1} K((x - X_i)/h)$ where $h = h_n$ is a sequence of positive numbers, and $K$ is an absolutely integrable function with $\int K(x) dx = 1$. Let $J_n = \int |f_n(x) - f(x)| dx$. We show that when $\lim_nh = 0$ and $\lim_nnh^d = \infty$, then for every $\varepsilon > 0$ there exist constants $r, n_0 > 0$ such that $P(J_n \geq \varepsilon) \leq \exp(-rn), n \geq n_0$.…
A unified treatment of the consistency properties of the ordinary least squares estimates in an autoregressive fitting of time series from nonstationary or stationary autoregressive moving average models is given. For a given model, the orders of autoregressions which produce consistent estimates are obtained and the limiting values, hence the biases, of the estimates of other autoregressions are investigated.
Nicholas P. Jewell · Peter Bloomfield · Flavio C. Bartmann·刊期 1983-09-01 · 卷 11（3）·The Annals of Statistics已编入正刊
时间序列过去与未来的典型相关：界与计算AI 解读
This paper continues an investigation into the canonical correlations and canonical components of the past and future of a stationary Gaussian time series which were introduced in Jewell and Bloomfield (1983). Bounds for the maximum canonical correlation are provided under specified conditions on the spectrum of the series. A computational scheme is described for estimating the canonical correlations and components and the procedure is illustrated on the well-known sunspot…
The concepts of canonical correlations and canonical components are familiar ideas in multivariate statistics. In this paper we extend these notions to stationary time series with a view to determining the most predictable aspect of the future of a time series. We relate properties of the canonical description of a time series to well known structural properties of the series such as (i) rational spectra (i.e., ARMA series), (ii) strong mixing, (iii) absolute regularity, etc.
Given $X_1, X_2, \cdots$, i.i.d. with mean $\mu$ and variance $\sigma^2$, suppose that at stage $n$ one wishes to estimate $\mu$ by the sample mean $\bar{X}_n$, subject to the loss function $L_n = A(\bar{X}_n - \mu)^2 + n, A > 0$. If $\sigma$ is known, the optimal fixed sample size $n_0 \approx A^{1/2}\sigma$ can be used, with corresponding risk $R_{n_0}$, but if $\sigma$ is unknown there is no fixed sample size procedure that will achieve the risk $R_{n_0}$. For the…
Let $X$ have a $p$-variate normal distribution with unknown mean $\theta$ and identity covariance matrix. The following transformed version of a control problem (Zaman, 1981) is considered: estimate $\theta$ by $d$ subject to incurring a loss $L(d, \theta) = (\theta^t d - 1)^2$. The comparison of decision rules in terms of expected loss is reduced to the study of differential inequalities. Results establishing the minimaxity of a large class of estimators are obtained.…
The theory and application of the Cox (1972) failure time regression model has, almost without exception, assumed an exponential form for the dependence of the hazard function on regression variables. Other regression forms may be more natural or descriptive in some applications. For example, a linear relative risk regression model provides a convenient framework for studying epidemiologic risk factor interactions. This note uses the counting process formulation of Andersen…
This paper presents a sufficient condition for second order efficiency of an estimator. The condition is easily checked in the case of minimum contrast estimators. The $\alpha^\ast$-minimum contrast estimator is defined and proved to be second order efficient for every $\alpha, 0 < \alpha < 1$. The Fisher scoring method is also considered in the light of second order efficiency. It is shown that a contrast function is associated with the second order tensor and the affine…
This paper concerns nonparametric inference for hazard rates with censored serial data. The focus is upon "delta sequence" estimators of the form hn(x)=∫Kb(x,y)dHn(y) with Kb integrating to 1 and concentrating mass near x as b→0.Hn is the Nelson-Aalen empirical cumulative hazard. Strong approximation and simultaneous confidence bands are derived for Rosenblatt-Parzen estimators, with Kb(x,y)=w((x−y)/b)/b,b=o(n−1), and w(⋅)…
Estimating the Stable Index α in Order to Measure Tail Thickness: A Critique
Stable laws are often fit to outlier-prone data and, if the index α is estimated to be much less than two, then the normal law is rejected in favor of an infinite-variance stable law. This paper derives the theoretical properties of such a procedure. When the true distribution is stable, the distribution of the m.l.e. of α is non-regular if α=2. When the true distribution is not stable, the estimate of α is not a robust measure of the rate of…
The affine type of distributions on the real line are represented as sequences of distributions of maximal invariants on spheres. It is shown that such a representation characterizes the affine type. A consistency condition is introduced, and it is shown that any sequence of maximal invariant distributions satisfying the condition is generated by some affine type on R.
We derive the joint asymptotic distribution of the k midranges formed by averaging the ith smallest normalized order statistic with the ith largest normalized order statistic, i=1,⋯,k. We then derive the distribution of the maximum midrange among these k extreme midranges and the limiting distribution of this maximum as k→∞. These results imply that, even in infinite samples, different distributions in the class of symmetric, unimodal…
The Equivalence of Weak, Strong and Complete Convergence in L1 for Kernel Density Estimates
Let f be a density on Rd, and let fn be the kernel estimate of f, fn(x)=(nhd)−1∑i=1nK((x−Xi)/h) where h=hn is a sequence of positive numbers, and K is an absolutely integrable function with ∫K(x)dx=1. Let Jn=∫∣fn(x)−f(x)∣dx. We show that when limnh=0 and limnnhd=∞, then for every ε>0 there exist constants r,n0>0 such that P(Jn≥ε)≤exp(−rn),n≥n0.…
Given X1,X2,⋯, i.i.d. with mean μ and variance σ2, suppose that at stage n one wishes to estimate μ by the sample mean Xˉn, subject to the loss function Ln=A(Xˉn−μ)2+n,A>0. If σ is known, the optimal fixed sample size n0≈A1/2σ can be used, with corresponding risk Rn0, but if σ is unknown there is no fixed sample size procedure that will achieve the risk Rn0. For the…
Let X have a p-variate normal distribution with unknown mean θ and identity covariance matrix. The following transformed version of a control problem (Zaman, 1981) is considered: estimate θ by d subject to incurring a loss L(d,θ)=(θtd−1)2. The comparison of decision rules in terms of expected loss is reduced to the study of differential inequalities. Results establishing the minimaxity of a large class of estimators are obtained.…
This paper presents a sufficient condition for second order efficiency of an estimator. The condition is easily checked in the case of minimum contrast estimators. The α∗-minimum contrast estimator is defined and proved to be second order efficient for every α,0<α<1. The Fisher scoring method is also considered in the light of second order efficiency. It is shown that a contrast function is associated with the second order tensor and the affine…