On Limits of Uniquely Best Linear Estimators

In a general linear model, it is shown that all admissible linear estimators are limits of linear estimators that are uniquely best at some point in an extended parameter set. The principal result shows that a linear estimator that is uniquely best at a pointW ₂ among multiple linear estimators that are best at a pointW ₁ is the limit of uniquely best estimators at points approachingW ₁ along the line joiningW ₁ andW ₂. Research supported in part by U.S.A.F. Aerospace Research Laboratories under contract F33615-71-C-1463, summer 1973, and in part by Grant DMS-9104811 from the National Science Foundation.     

Quadratic semiparametric Von Mises calculus

We discuss a new method of estimation of parameters in semiparametric and nonparametric models. The method is based on U-statistics constructed from quadratic influence functions. The latter extend ordinary linear influence functions of the parameter of interest as defined in semiparametric theory, and represent second order derivatives of this parameter. For parameters for which the matching cannot be perfect the method leads to a bias-variance trade-off, and results in estimators that converge at a slower than n ^?1/2-rate. In a number of examples the resulting rate can be shown to be optimal. We are particularly interested in estimating parameters in models with a nuisance parameter of high dimension or low regularity, where the parameter of interest cannot be estimated at n ^?1/2-rate.     

Generalized M-fluctuation tests for parameter instability

A general class of fluctuation tests for parameter instability in an M-estimation framework is suggested. Tests from this framework can be constructed by first choosing an appropriate estimation technique, deriving a partial sum process of the estimation scores that captures instabilities over time, and aggregating this process to a test statistic by using a suitable scalar functional. Inference for these tests is based on functional central limit theorems, which are derived under the null hypothesis of parameter stability and local alternatives. For (generalized) linear regression models, concrete tests are derived, which cover several known tests for (approximately) normal data but also allow for testing for parameter instability in regressions with binary or count data. The usefulness of the test procedures—complemented by powerful visualizations derived from these—is illustrated using Dow Jones industrial average stock returns, youth homicides in Boston, USA, and illegitimate births in Grossarl, Austria.     

Standardization of Variables and Collinearity Diagnostic in Ridge Regression

Ridge estimation (RE) is an alternative method to ordinary least squares when there exists a collinearity problem in a linear regression model. The variance inflator factor (VIF) is applied to test if the problem exists in the original model and is also necessary after applying the ridge estimate to check if the chosen value for parameter k has mitigated the collinearity problem. This paper shows that the application of the original data when working with the ridge estimate leads to non‐monotone VIF values. García et al. (2014) showed some problems with the traditional VIF used in RE. We propose an augmented VIF, VIF_R(j,k), associated with RE, which is obtained by standardizing the data before augmenting the model. The VIF_R(j,k) will coincide with the VIF associated with the ordinary least squares estimator when k = 0. The augmented VIF has the very desirable properties of being continuous, monotone in the ridge parameter and higher than one.     

Beta spatial linear mixed model with variable dispersion using Monte Carlo maximum likelihood

We propose a beta spatial linear mixed model with variable dispersion using Monte Carlo maximum likelihood. The proposed method is useful for those situations where the response variable is a rate or a proportion. An approach to the spatial generalized linear mixed models using the Box–Cox transformation in the precision model is presented. Thus, the parameter optimization process is developed for both the spatial mean model and the spatial variable dispersion model. All the parameters are estimated using Markov chain Monte Carlo maximum likelihood. Statistical inference over the parameters is performed using approximations obtained from the asymptotic normality of the maximum likelihood estimator. Diagnosis and prediction of a new observation are also developed. The method is illustrated with the analysis of one simulated case and two studies: clay and magnesium contents. In the clay study, 147 soil profile observations were taken from the research area of the Tropenbos Cameroon Programme, with explanatory variables: elevation in metres above sea level, agro‐ecological zone, reference soil group and land cover type. In the magnesium content, the soil samples were taken from 0‐ to 20‐cm‐depth layer at each of the 178 locations, and the response variable is related to the spatial locations, altitude and sub‐region.     

Estimation of the parameters of log-gamma distribution using order statistics

In this work we propose a technique of estimating the location parameter μ and scale parameter σ of log-gamma distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.     

Partially varying coefficient instrumental variables models

In this article, we study a new class of semiparametric instrumental variables models, in which the structural function has a partially varying coefficient functional form. Under this specification, the model is linear in the endogenous/exogenous components with unknown constant or functional coefficients. As a result, the ill‐posed inverse problem in a general non‐parametric model with continuous endogenous variables can be avoided. We propose a three‐step estimation procedure for estimating both constant and functional coefficients and establish their asymptotic properties such as consistency and asymptotic normality. We develop consistent estimators for their error variances. We demonstrate that the constant coefficient estimators achieve the optimal ‐convergence rate, and the functional coefficient estimators are oracle. In addition, efficiency issue of the parameter estimation is discussed and a simple efficient estimator is proposed. The proposed procedure is illustrated via a Monte Carlo simulation and an application to returns to education.     

Estimation of a linear regression under microaggregation with the response variable as a sorting variable

Microaggregation is a popular statistical disclosure control technique for continuous data. The basic principle of microaggregation is to group the observations in a data set and to replace them by their corresponding group means. However, while reducing the disclosure risk of data files, the technique also affects the results of statistical analyses. The paper deals with the impact of microaggregation on a multiple linear regression in continuous variables. We show that parameter estimates are biased if the dependent variable is used to form the groups. Using this result, we develop a consistent estimator that removes the aggregation bias, and derive its asymptotic covariance matrix.     

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.     

A regularization approach to the many instruments problem

This paper focuses on the estimation of a finite dimensional parameter in a linear model where the number of instruments is very large or infinite. In order to improve the small sample properties of standard instrumental variable (IV) estimators, we propose three modified IV estimators based on three different ways of inverting the covariance matrix of the instruments. These inverses involve a regularization or smoothing parameter. It should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation. We show that the three estimators are asymptotically normal and attain the semiparametric efficiency bound. Higher-order analysis of the MSE reveals that the bias of the modified estimators does not depend on the number of instruments. Finally, we suggest a data-driven method for selecting the regularization parameter. Interestingly, our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument.     

Testing a linear dynamic panel data model against nonlinear alternatives

The most popular econometric models in the panel data literature are the class of linear panel data models with unobserved individual- and/or time-specific effects. The consistency of parameter estimators and the validity of their economic interpretations as marginal effects depend crucially on the correct functional form specification of the linear panel data model. In this paper, a new class of residual-based tests is proposed for checking the validity of dynamic panel data models with both large cross-sectional units and time series dimensions. The individual and time effects can be fixed or random, and panel data can be balanced or unbalanced. The tests can detect a wide range of model misspecifications in the conditional mean of a dynamic panel data model, including functional form and lag misspecification. They check a large number of lags so that they can capture misspecification at any lag order asymptotically. No common alternative is assumed, thus allowing for heterogeneity in the degrees and directions of functional form misspecification across individuals. Thanks to the use of panel data with large $N$ and $T$, the proposed nonparametric tests have an asymptotic normal distribution under the null hypothesis without requiring the smoothing parameters to grow with the sample sizes. This suggests better nonparametric asymptotic approximation for the panel data than for time series or cross sectional data. This is confirmed in a simulation study. We apply the new tests to test linear specification of cross-country growth equations and found significant nonlinearities in mean for OECD countries’ growth equation for annual and quintannual panel data.     

Three Corrected Score Tests for Generalized Linear Models with Dispersion Covariates

We develop three corrected score tests for generalized linear models with dispersion covariates, thus generalizing the results of Cordeiro , Ferrari and Paula (1993) and Cribari-Neto and Ferrari (1995) . We present, in matrix notation, general formulae for the coefficients which define the corrected statistics. The formulae only require simple operations on matrices and can be used to obtain analytically closed-form corrections for score test statistics in a variety of special generalized linear models with dispersion covariates. They also have advantages for numerical purposes since our formulae are readily computable using a language supporting numerical linear algebra. Two examples, namely, iid sampling without covariates on the mean or dispersion parameter oand one-way classification models, are given. We also present some simulations where the three corrected tests perform better than the usual score test, the likelihood ratio test and its Bartlett corrected version. Finally, we present a numerical example for a data set discussed by Simonoff and Tsai (1994) .     

Application of one‐step method to parameter estimation in ODE models

In this paper, we study application of Le Cam's one‐step method to parameter estimation in ordinary differential equation models. This computationally simple technique can serve as an alternative to numerical evaluation of the popular non‐linear least squares estimator, which typically requires the use of a multistep iterative algorithm and repetitive numerical integration of the ordinary differential equation system. The one‐step method starts from a preliminary ‐consistent estimator of the parameter of interest and next turns it into an asymptotic (as the sample size n →∞ ) equivalent of the least squares estimator through a numerically straightforward procedure. We demonstrate performance of the one‐step estimator via extensive simulations and real data examples. The method enables the researcher to obtain both point and interval estimates. The preliminary ‐consistent estimator that we use depends on non‐parametric smoothing, and we provide a data‐driven methodology for choosing its tuning parameter and support it by theory. An easy implementation scheme of the one‐step method for practical use is pointed out.     

Method-of-moment view of linear simultaneous equation systems

In this paper, we review the modern method-of-moment-based approaches to identification and estimation of linear simultaneous equation systems. First, we present the rank condition for the structural form (SF) parameter identification. The rank condition comes naturally and is much easier to understand than that in the conventional reduced-form-based indirect approach. Then, we show how to estimate all SF parameters jointly (in a single step) with method-of-moment estimators. As it turns out, using only unconditional moments, but not any conditional moments, greatly simplifies the identification and estimation issues, and makes light work of conveying the essential ideas involved.     

An outlier test for linear processes — II. Large contamination

In Flak/Schmid (1993) an outlier test for linear processes was introduced. The test statistic bases on a comparison of each observation with a one-step predictor. It was assumed that an upper bound for the total number of outlierss _n is known, wheren denotes the sample size. The asymptotic distribution of the test statistic was derived under the assumption thats _n/n → 0 ands _n → ∞ asn → ∞. This note deals with the asymptotic behaviour of this quantity, ifs _n/n →p ₀ ∈ (0, 1).     

A Note on the Prediction Intervals for a Future Ordered Observation from a Pareto Distribution

In the researching of products' reliability, the result of life testing is used as the basis for the evaluation and improvement of reliability. During life testing, however, the future observation in an ordered sample is often expected to be predicted so as to show how long a sample of units might run until all fail in life testing. Therefore, we propose five new pivotal quantities to obtain prediction intervals of future order statistics based on right type II censored samples from the Pareto distribution with known shape parameter, then compares the lengths of the prediction intervals when using the pivotal quantity of Ouyang and Wu (1994) based on best linear unbiased estimator (BLUE) of scale parameter, and these five pivotal quantities. An advantage of these five pivotal quantities is that these are easier to calculate than the pivotal quantity of Ouyang and Wu (1994) based on BLUE of scale parameter, since they need to compute the tables of coefficients of BLUE of scale parameter.     

An outlier test for linear processes

In this paper we introduce an outlier test for linear processes. It is assumed that an upper bound for the number of outliers is known which is not too big in relation to the sample size. The test statistic bases on the comparison of the observations with certain predictors. We discuss the asymptotical behaviour of the test statistic under the null hypothesis ‘no outlier’ and derive the asymptotic distribution for the case that the distribution of the squared white noise process belongs to a certain subset of the domain of attraction of the Gumbel distribution. Especially the most important case in applications, the Gaussian white noise is included.     

Quantiles as optimal point forecasts

Loss functions play a central role in the theory and practice of forecasting. If the loss function is quadratic, the mean of the predictive distribution is the unique optimal point predictor. If the loss is symmetric piecewise linear, any median is an optimal point forecast. Quantiles arise as optimal point forecasts under a general class of economically relevant loss functions, which nests the asymmetric piecewise linear loss, and which we refer to as generalized piecewise linear (GPL). The level of the quantile depends on a generic asymmetry parameter which reflects the possibly distinct costs of underprediction and overprediction. Conversely, a loss function for which quantiles are optimal point forecasts is necessarily GPL. We review characterizations of this type in the work of Thomson, Saerens and Komunjer, and relate to proper scoring rules, incentive-compatible compensation schemes and quantile regression. In the empirical part of the paper, the relevance of decision theoretic guidance in the transition from a predictive distribution to a point forecast is illustrated using the Bank of England’s density forecasts of United Kingdom inflation rates, and probabilistic predictions of wind energy resources in the Pacific Northwest.     

线性合同招标中投标人数的博弈模型研究

在线性合同的假设前提下，论文通过建立模型，求解博弈的贝叶斯均衡得到了风险中性的理性招标人的最优合同形式以及投标人的最优报价策略。最后，论文通过引入招标成本函数，进一步提出了对于招标人控制投标人数从而达到自身目标最优的策略。     

Identifiability criteria for Muth-rational expectations models

The authors deal with complete static linear models that contain current Muth-rational expectations. Rank, order and variety conditions for the identifiability of the structural parameter are derived under general restrictions. We also correct statements that appeared in the literature. Our main finding is that, in general, the standard rank and order conditions are sufficient also for the identifiability of the Muth-rational expectations model parameter, whenever there are enough not fully anticipated exogenous variables. If the number of imperfectly forecasted exogenous variables falls short of the number of endogenous variables by g, then g extra restrictions are needed on every equation and the restrictions must meet easily verifiable variety conditions as well as an augmented rank criterion.