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"In the usage of present-day statistics 'statistical inference' is a profoundly ambiguous expression. In some literature a statistical inference is a "decision made under risk', in other literature it is 'a conclusion drawn from given data', and most of the literature displays no awareness that the two meanings might be different. This book concerns the problem of drawing conclusions from given data, in which respect we have to ask: Does there exist a need for the term 'statistical inference'? If so, does there also exist a corresponding need for every other science? If so, how does, for example, agronomy then manage to reason in terms of botanical inference, soil scientific inference, meteorological inference, biochemical inference, molecular biological inference, entomological inference, plant pathological inference, etc. without incoherence or self-contradiction? Consider the possibility that agronomy does not reason in terms of such a motley of special kinds of inference. Consider the possibility that, apart from subject matter, botany, soil science, entomology, etc. all employ the same kind of reasoning. If so, must we then believe that statistics, alone among all the sciences, is the only one that requires its own special kind of inference?"--Page i.
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This paper considers the estimation of the variance of coefficients in time varying parameter models with stationary regressors. The maximum likelihood estimator has large point mass at zero. We therefore develop asymptotically median unbiased estimators and confidence intervals by inverting median functions of regression-based parameter stability test statistics, computed under the constant-parameter null. These estimators have good asymptotic relative efficiencies for small to moderate amounts of parameter variability. We apply these results to an unobserved components model of trend growth in postwar U.S. GDP: the MLE implies that there has been no change in the trend rate, while the upper range of the median-unbiased point estimates imply that the annual trend growth rate has fallen by 0.7 percentage points over the postwar period.
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We examine the implications of arbitrage in a market with many assets. The absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous; hence there exist unique portfolios that represent these functionals. The mean variance efficient set is a cone generated by these portfolios. Ross [16, 18J showed that if there is a factor structure, then the distance between the vector or mean returns and the space spanned by the factor loadings is bounded as the number of assets increases. We show that if the covariance matrix of asset returns has only K unbounded eigenvalues, then the corresponding K eigenvectors converge and play the role of factor loadings in Ross' result. Hence only a principal components analysis is needed to test the arbitrage pricing theory. Our eigenvalue conditional can hold even though conventional measures of the approximation error in a K factor model are unbounded. We also resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available.
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Mathematical statistics --- Sampling (Statistics) --- Asymptotic theory. --- Asymptotic theory
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Asymptotic expansions --- 517.91 --- Numerical solutions --- Asymptotic theory
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Partial differential equations (PDEs) describe technological phenomena and processes used for the analysis, design, and modeling of technical products. Solutions of spatial and transient PDEs are realized by using the PDE Toolbox included in the MATLAB® software. MATLAB® is introduced here as an essential foundation for PDE, and the Modeler of the PDE Toolbox, with appropriate explanatory solutions, is applied to engineering problems in mechanics, heat/mass transfer, tribology, materials science, physics, and biotechnology. The appendixes contain collections of commands and functions used to solve actual engineering problems.
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