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…
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…
For each $t$ in an index set $T$, let $P_t$ be a probability measure with mode $M(t)$. In this paper we consider a maximum likelihood nonparametric estimator $\hat{M}_n(t)$ of $M(t)$ subject to the constraint that $\hat{M}_n(\cdot)$ be isotonic with respect to an order on $T$. The estimator is a solution to a minimization problem with zero-one loss. The estimator is not a max-min or min-max representation of "naive" estimators. Naive modal estimators are used but they are…
An easily computable predictive density is considered. Although involving a plain maximum likelihood technique, it coincides with the posterior predictive density with a noninformative prior. For standard situations achievements are comparable with classical inference.
Many times, aspects of populations exist that must satisfy a stochastic ordering requirement. Nevertheless, estimates may not bear out this stochastic ordering because of the inherent variability of the observations. This article will consider the problem of finding maximum likelihood estimates of stochastically ordered survival functions for the cases (a) one survival function being fixed in advance and (b) estimating two survival functions when the data include censored…
P. C. B. Phillips·刊期 1982-09 · 卷 50（5）·Econometrica已编入正刊
非线性完全信息极大似然估计的一致性AI 解读
Examples are given which show that:(i) normality is not Necessary for the consistency of the quasi maximum likelihood estimator in the nonlinear simultaneous equations model (nonlinear FIML) even when there are major departures from linearity; and (ii) the lemma which is used extensively by Amemiya [2] in the theoretical development of the properties of nonlinear FIML under the assumption of normality is, as presently stated, incorrect.
E. J. Godolphin · J. G. De Gooijer·刊期 1982-08 · 卷 69（2）·Biometrika已编入正刊
高斯移动平均过程参数的最大似然估计AI 解读
Several authors have described methods for computing the exact maximum likelihood estimates for the parameters of a Gaussian moving average process. In this paper, an iterative procedure for solving the likelihood equations is derived which includes the covariance determinant. The procedure expresses the estimator of each parameter as a linear combination of a suitably large set of sample serial correlations which has many of the computational advantages of the direct…
C. L. Mallows · Y. Vardi·刊期 1982-08 · 卷 69（2）·Biometrika已编入正刊
基于重叠数据的估计效率界AI 解读
If we have several estimates of the same quantity, based on partially overlapping sets of data, we can contemplate combining them, linearly, to give a single estimate. We give bounds, which depend only on the degrees of overlap but not on the individual variances, for the efficiency of the best linear unbiased combination of these estimates. with and without covariance information, relative to the overall best linear unbiased estimate. In particular, if the sets have no more…
T. W. Anderson · Naoto Kunitomo · Takamitsu Sawa·刊期 1982-07 · 卷 50（4）·Econometrica已编入正刊
有限信息极大似然估计量分布函数的计算AI 解读
T. W. Anderson, Naoto Kunitomo, Takamitsu Sawa, Evaluation of the Distribution Function of the Limited Information Maximum Likelihood Estimator, Econometrica, Vol. 50, No. 4 (Jul., 1982), pp. 1009-1027
Robert F. Engle·刊期 1982-07 · 卷 50（4）·Econometrica已编入正刊
自回归条件异方差与英国通胀方差估计AI 解读
Traditional econometric models assume a constant one-period forecast variance. To generalize this implausible assumption, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes are introduced in this paper. These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period…
Let $\{X_i\}$ be a sample from $F(x - \theta)$ where $F$ is in a class $\mathscr{F}$ of symmetric distributions on the line and $\theta$ is the location parameter to be estimated. Huber has shown that maximum likelihood estimation has a minimax property over a convex $\mathscr{F}$. Here a simple convex $\mathscr{F}$ is given for which neither $L$- nor $R$-estimation has the minimax property. In particular, this example shows that a recent assertion concerning $L$-estimation…
We derive the nonparametric maximum likelihood estimate, $\hat{F}$ say, of a lifetime distribution $F$ on the basis of two independent samples, one a sample of size $m$ from $F$ and the other a sample of size $n$ from the length-biased distribution of $F$, i.e. from $G_F(x) = \int^x_0 u dF(u)/\mu, \mu = \int^\infty_0 x dF(x)$. We further show that $(m + n)^{1/2}(\hat{F} - F)$ converges weakly to a pinned Gaussian process with a simple covariance function, when $m + n \rightarrow \infty$…
The product limit estimator due to Kaplan and Meier (1958) is well-known to be the nonparametric maximum likelihood estimator of a distribution function based on censored data. It is shown here that the product limit estimator is an asymptotically optimal estimator in two senses: in the sense of a Hajek-Beran type representation theorem for regular estimators; and in an asymptotic minimax sense similar to the classical result for the uncensored case due to Dvoretzky, Kiefer,…
We propose a method of estimating the endpoint, $\theta$, of a distribution when only limited information is available about the behaviour of the distribution in the neighbourhood of $\theta$. By using increasing numbers of extreme order statistics we obtain an estimator which improves on earlier estimators based on only a bounded number of extremes. In a certain particular model our estimator is equal to a maximum likelihood estimator, but it is robust against departures…
This paper concerns robust estimation of the parameter $\theta$ which indexes a parametric model for independent non-identically distributed data. For reasonable choices of contamination neighborhood and of what is to be estimated when the parametric model does not hold, we characterize asymptotically minimax robust estimates of $\theta$. When applied to the normal regression model, the theory yields recipes for the influence curves of optimal robust regression and scale…
Maximum likelihood estimation often fails when the parameter takes values in an infinite dimensional space. For example, the maximum likelihood method cannot be applied to the completely nonparametric estimation of a density function from an $\operatorname{iid}$ sample; the maximum of the likelihood is not attained by any density. In this example, as in many other examples, the parameter space (positive functions with area one) is too big. But the likelihood method can often…
The differential-geometrical framework is given for analyzing statistical problems related to multi-parameter families of distributions. The dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint. The duality connected by the Legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures. The second-order information loss is calculated for Fisher-efficient…
This paper discusses five questions concerning maximum likelihood estimation: What kind of theory is maximum likelihood? How is maximum likelihood used in practice? To what extent can this theory and practice be justified from a decision-theoretic viewpoint? What are maximum likelihood's principal virtues and defects? What improvements have been suggested by decision theory?
Strong consistency and asymptotic normality of the maximum likelihood estimators are established in the context of jointly type-II censored samples from two exponential populations. Large-sample confidence intervals are derived for the individual scale parameters as well as their ratio, and some applications to series and parallel systems are discussed. For the ratio of the scale parameters, an exact confidence procedure is developed on the basis of the failure counts in the…
It is well known that unconditional and conditional least squares estimators are asymptotically maximum likelihood for invertible moving average (MA) processes. By writing the criteria functions for these three estimators in comparable form, we establish simple inequalities between them and also between the corresponding disturbance variance estimates. Further analysis is carried out by examining the expected values of the criteria functions for given (true) coefficients in…
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…
For each t in an index set T, let Pt be a probability measure with mode M(t). In this paper we consider a maximum likelihood nonparametric estimator M^n(t) of M(t) subject to the constraint that M^n(⋅) be isotonic with respect to an order on T. The estimator is a solution to a minimization problem with zero-one loss. The estimator is not a max-min or min-max representation of "naive" estimators. Naive modal estimators are used but they are…
L- and R-Estimation and the Minimax Property
Let {Xi} be a sample from F(x−θ) where F is in a class F of symmetric distributions on the line and θ is the location parameter to be estimated. Huber has shown that maximum likelihood estimation has a minimax property over a convex F. Here a simple convex F is given for which neither L- nor R-estimation has the minimax property. In particular, this example shows that a recent assertion concerning L-estimation…
We derive the nonparametric maximum likelihood estimate, F^ say, of a lifetime distribution F on the basis of two independent samples, one a sample of size m from F and the other a sample of size n from the length-biased distribution of F, i.e. from GF(x)=∫0xudF(u)/μ,μ=∫0∞xdF(x). We further show that (m+n)1/2(F^−F) converges weakly to a pinned Gaussian process with a simple covariance function, when m+n→∞…
We propose a method of estimating the endpoint, θ, of a distribution when only limited information is available about the behaviour of the distribution in the neighbourhood of θ. By using increasing numbers of extreme order statistics we obtain an estimator which improves on earlier estimators based on only a bounded number of extremes. In a certain particular model our estimator is equal to a maximum likelihood estimator, but it is robust against departures…
This paper concerns robust estimation of the parameter θ which indexes a parametric model for independent non-identically distributed data. For reasonable choices of contamination neighborhood and of what is to be estimated when the parametric model does not hold, we characterize asymptotically minimax robust estimates of θ. When applied to the normal regression model, the theory yields recipes for the influence curves of optimal robust regression and scale…
Maximum likelihood estimation often fails when the parameter takes values in an infinite dimensional space. For example, the maximum likelihood method cannot be applied to the completely nonparametric estimation of a density function from an iid sample; the maximum of the likelihood is not attained by any density. In this example, as in many other examples, the parameter space (positive functions with area one) is too big. But the likelihood method can often…