zbMATH — the first resource for mathematics

Asymptotic efficiency in estimation with conditional moment restrictions. (English) Zbl 0618.62040
Suppose data are i.i.d. according to some unknown $$F\in {\mathcal F}$$ and suppose $$\theta$$ is a parameter, real or vector valued, defined on $${\mathcal F}$$. Let (z,$$\theta)$$ be an estimating function, a function of parameter as well as of observations. As in regression, the restriction is that the conditional expectation of (z,$$\theta)$$ is zero at a particular point $$\theta_ 0$$ which is to be estimated.
The author first proves the result that if F is a multinomial distribution with known finite support then the lower bound $$\Lambda_ 0$$, for the variance-covariance matrix of the estimator is obtained from the information matrix. This lower bound is w.r.t. the partial order defined by non-negative definiteness of the difference between the variance-covariance matrices of the competing estimators.
It is shown that this bound depends only on certain conditional moments and not on the support of the multinomial F. Since a general F can be approximated by a multinomial distribution the lower bound $$\Lambda_ 0$$ provides the lower bound in general.
Reviewer: B.K.Kale

MSC:
 62F12 Asymptotic properties of parametric estimators 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation
Full Text:
References:
 [1] Amemiya, T., Generalized least squares with an estimated autocovariance matrix, Econometrica, 41, 723-732, (1973) · Zbl 0305.62046 [2] Amemiya, T., The nonlinear two-stage least-squares estimator, Journal of econometrics, 2, 105-110, (1974) · Zbl 0282.62089 [3] Amemiya, T., The maximum likelihood and the nonlinear three-stage least squares estimator in the general nonlinear simultaneous equation model, Econometrica, 45, 955-968, (1977) · Zbl 0359.62026 [4] Amemiya, T., Partially generalized least squares and two-stage least squares estimators, Journal of econometrics, 23, 275-283, (1983) · Zbl 0517.62060 [5] Berge, C., Topological spaces, (1963), Macmillan New York · Zbl 0114.38602 [6] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201 [7] Burguete, J.F.; Gallant, A.R.; Souza, G., On unification of the asymptotic theory of nonlinear econometric models, Econometric reviews, 1, 151-190, (1982) · Zbl 0496.62093 [8] Chamberlain, G., Multivariate regression models for panel data, Journal of econometrics, 18, 5-46, (1982) · Zbl 0512.62115 [9] Cragg, J.G., More efficient estimation in the presence of heteroskedasticity of unknown form, Econometrica, 51, 751-763, (1983) · Zbl 0513.62069 [10] Ferguson, T.S., A method of generating best asymptotically normal estimates with an application to the estimation of bacterial densities, Annals of mathematical statistics, 29, 1046-1062, (1958) · Zbl 0089.15402 [11] Hájek, J., Local asymptotic minimax and admissibility in estimation, () · Zbl 0281.62010 [12] Hansen, L.P., Large sample properties of generalized method of moments estimators, Econometrica, 50, 1029-1054, (1982) · Zbl 0502.62098 [13] Hayashi, F.; Sims, C.A., Nearly efficient estimation of time series models with predetermined, but not exogenous, instruments, Econometrica, 51, 783-798, (1983) · Zbl 0536.62097 [14] Huber, P.J., The behavior of maximum likelihood estimates under nonstandard conditions, () · Zbl 0212.21504 [15] Ibragimov, I.A.; Khas’minskii, R.Z., Local asymptotic normality for non-identically distributed observations, Theory of probability and its applications, 20, 246-260, (1975) · Zbl 0332.62012 [16] Ibragimov, I.A.; Khas’minskii, R.Z., Statistical estimation - asymptotic theory, (1981), Springer-Verlag New York · Zbl 0516.62061 [17] Jorgenson, D.W.; Laffont, J., Efficient estimation of nonlinear simultaneous equations with additive disturbances, Annals of economic and social measurement, 3, 615-640, (1974) [18] Le Cam, L., (), 277-330 [19] Levit, B.Y., On optimality of some statistical estimates, (), 215-238 [20] Levit, B.Y., On the efficiency of a class of nonparametric estimates, Theory of probability and its applications, 20, 723-740, (1975) · Zbl 0367.62041 [21] Loomis, L.H.; Sternberg, S., Advanced calculus, (1968), Addison-Wesley Reading, MA · Zbl 0162.35301 [22] MaCurdy, T.E., Using information on the moments of disturbances to increase the efficiency of estimation, (1982), Stanford University Stanford, CA, Unpublished manuscript [23] Malinvaud, E., Statistical methods of econometrics, (1970), North-Holland Amsterdam · Zbl 0276.62095 [24] Manski, C.F., Closest empirical distribution estimation, Econometrica, 51, 305-319, (1983) · Zbl 0541.62016 [25] Manski, C.F., Adaptive estimation of nonlinear regression models, Econometric reviews, 3, 145-194, (1984) · Zbl 0607.62034 [26] Nevel’son, M.B., One informational lower bound, Problemy peredachi informatsii, 13, 26-31, (1977) · Zbl 0368.94019 [27] Neyman, J., Contribution to the theory of the χ^2 test, () · Zbl 0039.14302 [28] Roussas, G.G., Contiguity of probability measures, (1972), Cambridge University Press Cambridge · Zbl 0233.62008 [29] Rudin, W., Real and complex analysis, (1974), McGraw-Hill New York [30] Sims, C.A., Distributed lags, () · Zbl 0338.90023 [31] White, H., Using least squares to approximate unknown regression functions, International economic review, 21, 149-170, (1980) · Zbl 0444.62119 [32] White, H., A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica, 48, 817-838, (1980) · Zbl 0459.62051 [33] White, H., Instrumental variables regression with independent observations, Econometrica, 50, 483-499, (1982) · Zbl 0518.62094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.