The Cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life-time distribution of an individual. In this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process. This permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring…
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…
In a Borel setting, every classically sufficient statistic is Bayes sufficient, but not vice versa. The example is a hypothesis testing problem in which Bayesians, but not classicists, can achieve zero error probabilities.
Let $X$ be distributed according to a non-central Chi squared distribution with $p$ degrees of freedom. The non-central Chi squared distribution arises in various statistical analyses and the estimation of the non-centrality parameter is of importance in some problems. This paper deals with the admissibility of certain estimates of the non-centrality parameter. It is shown that $(X - p)^+$, the positive part of $X - p$ dominates the maximum likelihood estimator with squared…
Simar (1976) suggested an iteration procedure for finding the maximum likelihood estimate of a compound Poisson process, but he could not show convergence. Here the more general case of maximizing a concave functional on the set of all probability measures is treated. As a generalization of Simar's procedure, an algorithm is given for solving this problem, including assumptions to ensure convergence to an optimum. Finally, it is shown that Simar's functional fulfills these…
A family of probability measures $\mathscr{P}$ on some measurable space $(X, \mathscr{A})$ and a class of estimator sequences $\hat{P}_n: X^n \rightarrow \mathscr{P}, n \in \mathbb{N}$, containing maximum likelihood estimators are considered. For $P \in \mathscr{P}$ it is proved that there are numbers $c > 0, h_0 > 0$ fulfilling $P^n\{n^{1/2} d(\hat{P}_n, P) > h\} \leq \exp(-ch^2)$ for $n \in \mathbb{N}, h \geq h_0$, where $d$ denotes the Hellinger distance of probability…
In this paper we present minimal complete class theorems for testing problems with simple null hypotheses and multivariate one-sided alternative hypotheses. The results extend previous ones by not requiring that the tests be based on exponentially distributed random variables or that the null and alternative parameter spaces be topologically separated.
The general outlier problem for a multivariate normal random sample with mean slippage is defined and is shown to be invariant under a natural group of transformations. A family of maximal invariants is obtained, and the common distribution of its members is derived. The critical region for the locally best invariant test of the null hypothesis, that there are no outliers, versus the alternative hypothesis, that some outliers are present, is found. Under very general…
The idea of smoothing-spline interpolation is generalized to propose an estimator for the mean function of a stochastic process. A minimax property of the proposed estimator is then demonstrated under the usual squared loss function.
Some stationary and non-stationary time series arise from mixed distributions, the probabilities attached to the occurrence of certain values being positive, while a continuum of possible values is also involved. Such series are modeled in terms of a stationary Gaussian process $X_t$, which is censored when it crosses certain thresholds. Procedures are proposed for estimating the autocorrelation function of $X_t$. Their strong consistency and asymptotic normality are…
Let $X_1, X_2, \cdots$ be independent observations from a population with mean $\mu$ and variance $\sigma^2$, and suppose that given a sample of size $n$ one wishes to estimate $\mu$ by $\bar{X}_n$, subject to the loss function $L_n = A\sigma^{2\beta-2}(\bar{X}_n - \mu)^2 + n, A > 0, \beta > 0$. If $\sigma$ is known, then the optimal fixed sample size $n_0$ for minimizing the risk $R_n = EL_n$ can be computed, but if $\sigma$ is unknown there is no fixed sample size…
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 sequential tests with parabolic stopping boundary for the hypothesis that the drift of a Wiener process is positive. Formulas are derived for the power and expected sample size. The performance of minimax parabolic tests is compared with that of other tests considered in the literature.
For the problem of estimating a $p$-variate normal mean, the existence of confidence procedures which dominate the usual one, a sphere centered at the observations, has long been known. However, no explicit procedure has yet been shown to dominate. For $p \geq 4$, we prove that if the usual confidence sphere is recentered at the positive-part James Stein estimator, then the resulting confidence set has uniformly higher coverage probability, and hence is a minimax confidence…
Assume that $X_1, \cdots, X_p$ are independent random observations having discrete exponential densities $\rho_i(\theta_i)t_i(x_i)\theta^{xi}_i, i = 1, \cdots, p$ respectively. A general technique of improving upon the uniform minimum variance unbiased estimator (UMVUE) of $(\theta_1, \cdots, \theta_p)$ is developed under possibly weighted squared error loss functions. It is shown that improved estimators can be constructed by solving a difference inequality. Typical…
James O. Berger · L. Mark Berliner · Asad Zaman·刊期 1982-09-01 · 卷 10（3）·The Annals of Statistics已编入正刊
控制问题中估计量可容许性与不可容许性的一般结果AI 解读
Let $\mathbf{X} = (X_1, \cdots, X_p)^t$ be an observation from a $p$-variate normal distribution with unknown mean $\mathbf{\theta} = (\theta_1, \cdots, \theta_p)^t$ and identity covariance matrix. We consider a control problem which, in canonical form, is the problem of estimating $\mathbf{\theta}$ under the loss $L(\mathbf{\theta, \delta}) = (\mathbf{\theta}^t \mathbf{\delta} - 1)^2$, where $\mathbf{\delta(x)} = (\delta_1(\mathbf{x}), \cdots, \delta_p(\mathbf{x}))^t$ is…
We study the consistency properties of a nonparametric estimator $f_n$ of a density function $f$ on the real line, which is known as the "first MPLE of Good and Gaskins," and which is obtained by maximizing the likelihood functional multiplied by the roughness penality $\exp\{- \alpha \int (f'/f)^2 f\}$ with $\alpha > 0$. Under modest assumptions on the density function $f$, and letting $\alpha = \alpha_n \rightarrow \infty$ and $\alpha_n/n \rightarrow 0$ a.s. as $n \rightarrow \infty$…
A class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density. The limiting case of the estimates as the amount of smoothing increases has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty. The estimates are shown to be the solution of an unconstrained convex optimization problem,…
A new methodology is proposed for discrimination among stationary time-series. The time series are transformed into binary arrays by clipping (retaining only the signs of) the $j$th difference series, $j = 0, 1, 2, \cdots$. The degeneracy of clipped $j$th differences is studied as $j$ becomes large. A new goodness of fit statistic is defined as a quadratic form in the counts of axis-crossings by each of the first $k$ differences of the series. Simulations and the degeneracy…
Data collected from many independent identically distributed renewal processes, each of which is observed for an arbitrary period of time, is usually affected by censoring coupled with length biased sampling. In this paper we derive an algorithm that produces the nonparametric maximum likelihood estimator (i.e., the analog of the single-sample empirical distribution function) of the common lifetime distribution, based on such data.
In this paper the exponential rates of decrease and bounds on tail probabilities for consistent estimators are studied using large deviation methods. The asymptotic expansions of Bahadur bounds and exponential rates in the case of the maximum likelihood estimator are obtained. Based on these results we have obtained a result parallel to the Fisher-Rao-Efron result concerning second-order efficiency (see Efron, 1975). Our results also substantiate the geometric observation…
Let $\Theta$ be a convex subset of $R^d$, and for each $\theta \in \Theta$ let $F(\theta; dt)$ be a probability on the line. For any vector $a_n = (a_{n1}, a_{n2}, \cdots, a_{nn})$ where $a_{ni} \in \Theta$, let $X_{n1}, \cdots, X_{nn}$ be independent observations, the distribution of $X_{ni}$ being $F(a_{ni}; dt)$. The main results give a method of estimating the unknown regression function $a_n$ based on a minimum distance recipe. Under regularity assumptions, the proposed…
Algorithms for solving the isotonic regression problem in more than one dimension are difficult to implement because of the large number of lower sets present or because they involve search techniques which require a significant amount of checking and readjustment. Here a new algorithm for solving this problem based on a simple iterative technique is proposed and shown to converge to the correct solution.
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 X be distributed according to a non-central Chi squared distribution with p degrees of freedom. The non-central Chi squared distribution arises in various statistical analyses and the estimation of the non-centrality parameter is of importance in some problems. This paper deals with the admissibility of certain estimates of the non-centrality parameter. It is shown that (X−p)+, the positive part of X−p dominates the maximum likelihood estimator with squared…
A family of probability measures P on some measurable space (X,A) and a class of estimator sequences P^n:Xn→P,n∈N, containing maximum likelihood estimators are considered. For P∈P it is proved that there are numbers c>0,h0>0 fulfilling Pn{n1/2d(P^n,P)>h}≤exp(−ch2) for n∈N,h≥h0, where d denotes the Hellinger distance of probability…
Some stationary and non-stationary time series arise from mixed distributions, the probabilities attached to the occurrence of certain values being positive, while a continuum of possible values is also involved. Such series are modeled in terms of a stationary Gaussian process Xt, which is censored when it crosses certain thresholds. Procedures are proposed for estimating the autocorrelation function of Xt. Their strong consistency and asymptotic normality are…
Let X1,X2,⋯ be independent observations from a population with mean μ and variance σ2, and suppose that given a sample of size n one wishes to estimate μ by Xˉn, subject to the loss function Ln=Aσ2β−2(Xˉn−μ)2+n,A>0,β>0. If σ is known, then the optimal fixed sample size n0 for minimizing the risk Rn=ELn can be computed, but if σ is unknown there is no fixed sample size…
For the problem of estimating a p-variate normal mean, the existence of confidence procedures which dominate the usual one, a sphere centered at the observations, has long been known. However, no explicit procedure has yet been shown to dominate. For p≥4, we prove that if the usual confidence sphere is recentered at the positive-part James Stein estimator, then the resulting confidence set has uniformly higher coverage probability, and hence is a minimax confidence…
Assume that X1,⋯,Xp are independent random observations having discrete exponential densities ρi(θi)ti(xi)θixi,i=1,⋯,p respectively. A general technique of improving upon the uniform minimum variance unbiased estimator (UMVUE) of (θ1,⋯,θp) is developed under possibly weighted squared error loss functions. It is shown that improved estimators can be constructed by solving a difference inequality. Typical…
Let X=(X1,⋯,Xp)t be an observation from a p-variate normal distribution with unknown mean θ=(θ1,⋯,θp)t and identity covariance matrix. We consider a control problem which, in canonical form, is the problem of estimating θ under the loss L(θ,δ)=(θtδ−1)2, where δ(x)=(δ1(x),⋯,δp(x))t is…
We study the consistency properties of a nonparametric estimator fn of a density function f on the real line, which is known as the "first MPLE of Good and Gaskins," and which is obtained by maximizing the likelihood functional multiplied by the roughness penality exp{−α∫(f′/f)2f} with α>0. Under modest assumptions on the density function f, and letting α=αn→∞ and αn/n→0 a.s. as n→∞…
A new methodology is proposed for discrimination among stationary time-series. The time series are transformed into binary arrays by clipping (retaining only the signs of) the jth difference series, j=0,1,2,⋯. The degeneracy of clipped jth differences is studied as j becomes large. A new goodness of fit statistic is defined as a quadratic form in the counts of axis-crossings by each of the first k differences of the series. Simulations and the degeneracy…
Let Θ be a convex subset of Rd, and for each θ∈Θ let F(θ;dt) be a probability on the line. For any vector an=(an1,an2,⋯,ann) where ani∈Θ, let Xn1,⋯,Xnn be independent observations, the distribution of Xni being F(ani;dt). The main results give a method of estimating the unknown regression function an based on a minimum distance recipe. Under regularity assumptions, the proposed…