Randall J. Olsen·刊期 1980-07 · 卷 48（5）·Econometrica已编入正刊
用矩方法近似截断正态回归AI 解读
A NUMBER OF ARTICLES have been written in recent as well as more distant times on the subject of limited dependent variables. This broad classification includes two particular special cases. First, the dependent variable has finite probability mass concentrated at some limit point-the so called Tobit model. This case is referred to as the censored model in the statistical literature. The second case is when the dependent variable simply has a limited range and follows a…
The existence and some properties of maximum likelihood estimators (MLE's) are studied for a minimum-type distribution function corresponding to a minimum of two independent random variables having exponential and Weibull distributions. It is shown that if all three parameters are unknown, then there is a path in the parameter space along which the likelihood function (LF) tends to infinity. It is also proved that if the Weibull shape parameter is known, then the LF is…
Relative efficiency of estimators of the common mean of possibly different normal populations N(μ, σ i 2) is investigated empirically. A weighted least squares estimator , with weights based on a modification of minimum norm quadratic unbiased (MINQU) estimators of the σ i 2, is found to be substantially more efficient than the maximum likelihood (ML) estimator of μ when the heterogeneity in the σ i , is small to moderate and the number of sample observations from a…
Consider a life-testing experiment subject to Type I censoring, where m items are put on test at the outset and are not replaced on failure. Assume an exponential failure distribution F(t) = 1 – exp (– t/θ). A test of H 0: θ ≥ ξ versus H 1: θ < ξ based on the maximum likelihood estimator of θ is proposed. It is shown that early termination of the test resulting in either acceptance or rejection of H 0 is possible. Critical points are presented. The test is shown to compare…
This article is primarily concerned with modifications of local maximum likelihood estimators and modifications of moment estimators for parameters of the three-parameter lognormal distribution. First, second, and third order statistics are employed to facilitate estimation of the distribution lower limit. Computational procedures are developed to simplify the practical application of these estimators. Results of a simulation study provide insight into the sampling behavior…
Relative efficiency of estimators of the common mean of possibly different normal populations N(μ, σ i 2) is investigated empirically. A weighted least squares estimator , with weights based on a modification of minimum norm quadratic unbiased (MINQU) estimators of the σ i 2, is found to be substantially more efficient than the maximum likelihood (ML) estimator of μ when the heterogeneity in the σ i , is small to moderate and the number of sample observations from a…
The maximum likelihood estimators (m.l.e.) are obtained for the parameters of a bivariate normal distribution with equal variances when some of the observations are missing on one of the variables. The likelihood equation for estimating $\rho$, the correlation coefficient, may have multiple roots but a result proved here provides a unique root which is the m.l.e. of $\rho$. The problem of estimating the difference $\delta$ of the two means is also considered and it is shown…
Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random variables, each having density $f(x, \theta_0)$ where $\{f(x, \theta)\}$ is a family of densities with respect to a dominating measure $\mu$. Suppose $n^{\frac{1}{2}}(\hat{\theta} - \theta)$ and $n^{\frac{1}{2}}(T - \theta)$, where $\hat{\theta}$ is the mle and $T$ is any other efficient estimate, have Edgeworth expansions up to $o(n^{-1})$ uniformly in a compact neighbourhood of $\theta_0$. Then (under certain regularity…
The $n^{-2}$ order mean squared errors of the maximum likelihood and the minimum chi-square estimator of the logit regression model are derived and the latter is shown to be superior for many parameter values considered. The maximum likelihood is shown to be better if the bias of each estimator is corrected to the order of $n^{-1}$; however, the difference is shown to be negligibly small in many practical situations.
The sovereignty of MLE is questioned. Minimum $\chi^2_\lambda$ yields the same estimating equations as MLE. For many cases, as illustrated in presented examples, and further algorithmic exploration in progress may show that for all cases, minimum $\chi^2_\lambda$ estimates are available. In this sense minimum $\chi^2$ is the basic principle of estimation. The criterion of asymptotic sufficiency which has been called "second order efficiency" is rejected as a criterion of…
Gregory M. Duncan·刊期 1980-05 · 卷 48（4）·Econometrica已编入正刊
经典生产理论中连续与离散混合因变量模型的构建与统计分析AI 解读
Data sets which contain jointly endogenous discrete and continuous variables often occur in practice. This paper presents a model of the economic and stochastic processes generating such data as well as methods of estimation. A maximum likelihood estimator is examined and found to exhibit the usual optimality but it is computationally burdensome. A simpler estimator, the QREG, which is a simple weighted average of separate probit (or logit) and regression estimates is…
Ronald M. Schrader · Thomas P. Hettmansperger·刊期 1980-04 · 卷 67（1）·Biometrika已编入正刊
基于似然比准则的稳健方差分析AI 解读
Robust tests of general linear hypotheses in linear models are developed. These are likelihood ratio type tests in the same sense that M -estimates are maximum likelihood type estimates. Construction of the tests suggests a decomposition of the data into terms analogous to classical sums of squares, providing a robust analysis of variance. Asymptotic efficiency and robustness properties of the tests are the same as those of the M -estimates upon which they are based.
I. V. Basawa · P. J. Brockwell·刊期 1980-04 · 卷 67（1）·Biometrika已编入正刊
伽马过程与稳定过程参数估计的一则札记AI 解读
Maximum likelihood estimators of the parameters of nondecreasing pure-jump gamma and stable processes based on the jumps of size greater than or equal to ε were studied in a previous paper (Basawa & Brockwell, 1978). In particular, for a single realization {X( u ), 0 ≤ u ≤ t }, the estimators for the stable process are consistent as ε ↓ 0, but this is not so for the gamma process. In the present note we derive asymptotic distributions as ε ↓ 0 for these estimators, centred…
A. Ronald Gallant · Alberto Holly·刊期 1980-04 · 卷 48（3）·Econometrica已编入正刊
隐式非线性联立方程模型中基于最大似然估计的统计推断AI 解读
A. Ronald Gallant, Alberto Holly, Statistical Inference in an Implicit, Nonlinear, Simultaneous Equation Mode in the Context of Maximum Likelihood Estimation, Econometrica, Vol. 48, No. 3 (Apr., 1980), pp. 697-720
Generalised residuals, as defined by Cox and Snell, may be thought of as residuals from generalised regression. Under regularity conditions on the regression and on the estimator of the unknown parameters, the asymptotic behaviour of the empirical distribution of these residuals is determined. The addition of a random adjustment to the maximum likelihood estimator leads to the familiar Brownian bridge as the limit.
Ann Cohen Brandwein · William E. Strawderman·刊期 1980-03-01 · 卷 8（2）·The Annals of Statistics已编入正刊
凹损失下球对称分布位置参数的极小极大估计AI 解读
For $p \geqslant 4$ and one observation $X$ on a $p$-dimensional spherically symmetric distribution, minimax estimators of $\theta$ whose risks are smaller than the risk of $X$ (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss. For $n$ observations $X_1, X_2, \cdots, X_n$, we have classes of minimax estimators which are better than the usual procedures, such as the best invariant estimator, $\bar{X}$, or a maximum…
Inference about the correlation coefficient $\rho$ in a bivariate normal distribution is considered when observations from the distribution are available only in the form of a broken random sample. In other words, a random sample of $n$ pairs is drawn from the distribution but the observed data are only the first components of the $n$ pairs and, separately, some unknown permutation of the second components of the $n$ pairs. Under these conditions, the estimation of $\rho$ is,…
We propose an improvement of the maximum likelihood (ML) estimate in linear functional relationships. The improved estimate is a linear combination of the ML and the least squares estimate so as to remove the bias of the former. Approximations to the distribution of the estimate are derived for two alternative parameter sequences: a sequence in which the noncentrality parameter (the spread of the true values) increases while the number of observations stays fixed, and that…
In this paper we consider four alternative forms of two-parameter normal and nonnormal error distributions and report on a Monte Carlo study of the small-sample properties of least squares, two-stage least squares, three-stage least squares and full information maximum likelihood estimators. On the basis of 1,000 replications of sample size 20 in two experiments on an overidentified model, we found that the small-sample rankings of econometric estimators of both structural…
Let X be the sum of independent variables Y and Z, where Y is binomially distributed and Z is a nonnegative integer-valued variable whose distribution does not depend on the binomial parameter p. Convoluted binomial distributions describing variables like X are characterized by regularity conditions and equations of the form ∂/∂p P(X = x)=c[P(X = x − 1) − P(X = x)]. The characterization, together with a monotonicity property for probability ratios, is shown to facilitate…
Abstract When repetitive estimations are to be made under field conditions using data that follow a nonlinear regression law, a simple polynomial function of the observations has considerable appeal as an estimator. The polynomial estimator of finite degree with smallest average mean squared error is found. Conditions are given such that as degree increases it converges in probability to the Bayes estimator and its average mean squared error converges to the lower bound of…
The maximum likelihood estimators (m.l.e.) are obtained for the parameters of a bivariate normal distribution with equal variances when some of the observations are missing on one of the variables. The likelihood equation for estimating ρ, the correlation coefficient, may have multiple roots but a result proved here provides a unique root which is the m.l.e. of ρ. The problem of estimating the difference δ of the two means is also considered and it is shown…
Let X1,X2,⋯ be a sequence of i.i.d. random variables, each having density f(x,θ0) where {f(x,θ)} is a family of densities with respect to a dominating measure μ. Suppose n21(θ^−θ) and n21(T−θ), where θ^ is the mle and T is any other efficient estimate, have Edgeworth expansions up to o(n−1) uniformly in a compact neighbourhood of θ0. Then (under certain regularity…
The n−2-Order Mean Squared Errors of the Maximum Likelihood and the Minimum Logit Chi-Square Estimator
The n−2 order mean squared errors of the maximum likelihood and the minimum chi-square estimator of the logit regression model are derived and the latter is shown to be superior for many parameter values considered. The maximum likelihood is shown to be better if the bias of each estimator is corrected to the order of n−1; however, the difference is shown to be negligibly small in many practical situations.
The sovereignty of MLE is questioned. Minimum χλ2 yields the same estimating equations as MLE. For many cases, as illustrated in presented examples, and further algorithmic exploration in progress may show that for all cases, minimum χλ2 estimates are available. In this sense minimum χ2 is the basic principle of estimation. The criterion of asymptotic sufficiency which has been called "second order efficiency" is rejected as a criterion of…
For p⩾4 and one observation X on a p-dimensional spherically symmetric distribution, minimax estimators of θ whose risks are smaller than the risk of X (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss. For n observations X1,X2,⋯,Xn, we have classes of minimax estimators which are better than the usual procedures, such as the best invariant estimator, Xˉ, or a maximum…
Inference about the correlation coefficient ρ in a bivariate normal distribution is considered when observations from the distribution are available only in the form of a broken random sample. In other words, a random sample of n pairs is drawn from the distribution but the observed data are only the first components of the n pairs and, separately, some unknown permutation of the second components of the n pairs. Under these conditions, the estimation of ρ is,…