1. Introduction and Summary.- 1.1. History and main contributions.- 1.2. Why robust estimations?.- 1.3. Summary.- A The Theoretical Background.- 2. Sample spaces, distributions, estimators.- 2.1. Introduction.- 2.2. Example.- 2.3. Metrics for probability distributions.- 2.4. Estimators seen as functionals of distributions.- 3. Robustness, breakdown point and influence function.- 3.1. Definition of robustness.- 3.2. Definition of breakdown point.- 3.3. The influence…
A family of estimators based upon M specified quantiles is defined. These procedures take as the estimate the vector that minimizes a quadratic distance measure between M sample quantiles and a parametric family of quantile functions. Under regularity conditions these estimators are consistent, asymptotically normal, and robust. For a specific quadratic form the estimator is optimal among a class of asymptotically normal estimators, and it approaches full efficiency as M…
TOPICS IN LARGE SAMPLE THEORY AND APPROXIMATION METHODS. A Review of Asymptotic Distribution Theory and Maximum Likelihood Estimation. Methods of Approximating Distribution Functions, Density Functions, and Moments. TOPICS IN TIME SERIES ANALYSIS. An Introduction to Time Series Analysis. Identification, Estimation, and Prediction in Time Series Models. The Relationship Between Time Series Models and Econometric Models. TOPICS IN ESTIMATION UNDER A QUADRATIC LOSS CRITERION.…
In this paper we consider asymptotic approximations to the distribution function $F(x)$ of a linear combination of an estimate in a multivariate linear model. A method is given for obtaining an asymptotic expansion $F_{s - 1}(x)$ of $F(x)$ up to $O(n^{-s + 1})$ and a bound $c_s$ such that $|F(x) - F_{s - 1}(x)| \leq c_s$ uniformly in $x$ and $c_s = O(n^{-s})$.
It is shown that Hill's estimator (1975) for the exponent of regular variation is asymptotically normal if the number $k_n$ of extreme order statistics used to construct it tends to infinity appropriately with the sample size $n.$ As our main result, we derive a general condition which can be used to determine the optimal $k_n$ explicitly, provided that some prior knowledge is available on the underlying distribution function with regularly varying upper tail. This condition…
The problem of testing the underlying distribution in a general regression model is considered when the data are right censored. It is proposed that tests be based on efficient scores from a certain class of parametric alternatives. The connection between tests based on the scores and more traditional approaches using empirical distribution functions is explored. Neyman's smooth test is extended to allow for censored data.
Linear regression is a common method for analyzing continuous, cardinal data, but it is inappropriate when the dependent variable is an ordinal ranking. The model proposed for analyzing these data sets is the general linear model u = Xβ + ε, where the rank of the dependent variable u is observed instead of its value. A description is given for a numerical algorithm to evaluate the likelihood function that is efficient enough to permit maximum likelihood estimation of…
Linear regression is a common method for analyzing continuous, cardinal data, but it is inappropriate when the dependent variable is an ordinal ranking. The model proposed for analyzing these data sets is the general linear model u = Xβ + ε, where the rank of the dependent variable u is observed instead of its value. A description is given for a numerical algorithm to evaluate the likelihood function that is efficient enough to permit maximum likelihood estimation of…
The article by Kakwani (1984) introduced a measure of inequality that is weighted to take account of a sense of deprivation an individual may feel on seeing others who are better off. In the second part of the article, Kakwani presented estimates of deprivation using Australian data. I would like, first, to comment on the welfare implications of the measure of relative deprivation proposed in the article and, second, to point out some crucial assumptions underlying the use…
Multidimensional asymptotic normality of the kernel quantile estimator is established under fairly general conditions on the underlying distribution function and on the kernel. Sharpening these assumptions, one can utilize the proof to achieve also a bound for the rate of convergence which entails the comparison of the kernel estimator with the empirical quantile on the basis of their covering probabilities.
R. A. Boyles · A. W. Marshall · F. Proschan·刊期 1985-03-01 · 卷 13（1）·The Annals of Statistics已编入正刊
平均失效率递增分布的最大似然估计不一致性AI 解读
Marshall and Proschan (1965) showed that the MLE for a life distribution with increasing failure rate is strongly consistent. In this note we show that the MLE for a life distribution with increasing failure rate average is not consistent; in fact, maximum likelihood estimation in the IFRA case yields estimators of the average failure rate and of the distribution function which, in general, converge a.s. to values other than the true values. In the decreasing failure rate…
Consider the problem of estimating an unknown distribution function $F$ from the class of all distribution functions given a random sample of size $n$ from $F$. It is proved that the empirical distribution function is admissible for the loss functions $L(F, \hat{F}) = \int (\hat{F}(t) - F(t))^2(F(t))^\alpha(1 - F(t))^b dW(t)$ for any $a < 1$ and $b < 1$ and finite measure $W$. Related results for simultaneous estimation of distribution functions and for finite population…
Let $\mathscr{P}$ be a finite population with $N \geq 1$ elements; for each $e \in \mathscr{P}$, let $X_e$ and $Y_e$ be independent, positive random variables with unknown distribution functions $F$ and $G$; and suppose that the pairs $(X_e, Y_e)$ are i.i.d. We consider the problem of estimating $F, G$, and $N$ when the data consist of those pairs $(X_e, Y_e)$ for which $e \in \mathscr{P}$ and $Y_e \leq X_e$. The nonparametric maximum likelihood estimators (MLEs) of $F$ and $G$…
The dip test measures multimodality in a sample by the maximum difference, over all sample points, between the empirical distribution function, and the unimodal distribution function that minimizes that maximum difference. The uniform distribution is the asymptotically least favorable unimodal distribution, and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform.
SUMMARY We consider the estimation of the three parameters of the lower tail of a distribution function based on the k smallest out of n observations. The likelihood function has a singularity but it is argued that a local maximum, when it exists, should be taken as the m.l.e. Asymptotic results as k → ∞, n → ∞, k/n → 0 show that such a local maximum does exist, and provides consistent estimators whenever the shape parameter is greater than one; otherwise there is no local…
This article proposes new distribution-free methods for estimating the economic distance (advantage) between one group and another, especially for interval-to-interval comparisons, which can be more informative than a single aggregative measure. Novel techniques for computing projected quantiles, smoothing of empirical cumulative distribution functions, maximum entropy methods for finding the upper bound of the richest income interval, and statistical confidence intervals…
Paul Janssen · Robert Serfling · Noel Veraverbeke·刊期 1984-12-01 · 卷 12（4）·The Annals of Statistics已编入正刊
一类统计泛函的渐近正态性及其离散程度度量应用AI 解读
A general class of statistical functionals is introduced and the asymptotic normality of the corresponding estimators is established. Included are certain measures of spread, proposed by Bickel and Lehmann (1979), for which the asymptotic distribution theory has been an open problem. The proofs involve some new results, of independent interest, for empirical and quantile processes.
An estimate of the upper tail of a distribution function which is based on the upper $m$ order statistics from a sample of size $n(m \rightarrow \infty, m/n \rightarrow 0$ as $n \rightarrow \infty)$ is shown to be consistent for a wide class of distribution functions. The empirical mean residual life of the $\log$ transformed data and the sample $1 - m/n$ quantile play a key role in the estimate. The joint asymptotic behavior of the empirical mean residual life and sample $1 - m/n$…
This article introduces the relative deprivation curve to represent the size distribution of income and wealth. The curve has many useful applications in the measurement of poverty and inequality, which are explored. The methodology developed is then applied to the data obtained from the Australian Household Expenditure Survey, 1975–1976.
Malay Ghosh · William C. Parr · Kesar Singh · G. Jogesh Babu·刊期 1984-09-01 · 卷 12（3）·The Annals of Statistics已编入正刊
关于样本中位数自助法的一则说明AI 解读
Efron (1979, 1982), in his treatment of the bootstrap, discusses its use for estimation of the asymptotic variance of the sample median, in the sampling situation of independent and identically distributed random variables with common distribution function $F$ having a positive derivative continuous in a neighborhood of the true median $\mu$. The natural conjecture that the bootstrap variance estimator converges almost surely to the asymptotic variance is shown by an example…
Introduction Acceptance Quality Control Acceptance Control and Process Control Process Quality Control Background of Acceptance Quality Control NEW! Top 10 Reasons for Acceptance Sampling Probability and the Operating Characteristic Curve Probability Random Samples and Random Numbers Counting Possibilities Probability Calculus Operating Characteristic Curve Probability Functions Probability Distributions Measures of Distribution Functions Hypergeometric Distribution Binomial…
SUMMARY Cornish and Fisher (1937) gave the first few terms of the formal expansions for the distribution and percentiles of an asymptotically normal random variable (r.v.) in terms of its cumulants. They also showed that these expansions can sometimes produce extremely accurate approximations. In practice the cumulants are first expanded as a power series in a known parameter n (such as the sample size or degrees of freedom), and these expansions are substituted into the…
The question is discussed when of two ordered sets of continuous distribution functions one shows less positive trend than the other. The resulting ordering has applications to power function monotonicity of isotonic rank tests, and is related to certain orderings in parametric models.
In this paper the asymptotic relative deficiency of the sample $q$-quantile with respect to kernel type estimators of the $q$-quantile is evaluated. The comparison is based on the mean square errors of the estimators. The result suggests a purely analytic measure of performance within the class of kernels. It is notable that a similar situation occurs when kernel estimators of a distribution function are studied.
In this paper we consider asymptotic approximations to the distribution function F(x) of a linear combination of an estimate in a multivariate linear model. A method is given for obtaining an asymptotic expansion Fs−1(x) of F(x) up to O(n−s+1) and a bound cs such that ∣F(x)−Fs−1(x)∣≤cs uniformly in x and cs=O(n−s).
It is shown that Hill's estimator (1975) for the exponent of regular variation is asymptotically normal if the number kn of extreme order statistics used to construct it tends to infinity appropriately with the sample size n. As our main result, we derive a general condition which can be used to determine the optimal kn explicitly, provided that some prior knowledge is available on the underlying distribution function with regularly varying upper tail. This condition…
Consider the problem of estimating an unknown distribution function F from the class of all distribution functions given a random sample of size n from F. It is proved that the empirical distribution function is admissible for the loss functions L(F,F^)=∫(F^(t)−F(t))2(F(t))α(1−F(t))bdW(t) for any a<1 and b<1 and finite measure W. Related results for simultaneous estimation of distribution functions and for finite population…
Let P be a finite population with N≥1 elements; for each e∈P, let Xe and Ye be independent, positive random variables with unknown distribution functions F and G; and suppose that the pairs (Xe,Ye) are i.i.d. We consider the problem of estimating F,G, and N when the data consist of those pairs (Xe,Ye) for which e∈P and Ye≤Xe. The nonparametric maximum likelihood estimators (MLEs) of F and G…
An estimate of the upper tail of a distribution function which is based on the upper m order statistics from a sample of size n(m→∞,m/n→0 as n→∞) is shown to be consistent for a wide class of distribution functions. The empirical mean residual life of the log transformed data and the sample 1−m/n quantile play a key role in the estimate. The joint asymptotic behavior of the empirical mean residual life and sample 1−m/n…
Efron (1979, 1982), in his treatment of the bootstrap, discusses its use for estimation of the asymptotic variance of the sample median, in the sampling situation of independent and identically distributed random variables with common distribution function F having a positive derivative continuous in a neighborhood of the true median μ. The natural conjecture that the bootstrap variance estimator converges almost surely to the asymptotic variance is shown by an example…
In this paper the asymptotic relative deficiency of the sample q-quantile with respect to kernel type estimators of the q-quantile is evaluated. The comparison is based on the mean square errors of the estimators. The result suggests a purely analytic measure of performance within the class of kernels. It is notable that a similar situation occurs when kernel estimators of a distribution function are studied.