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Spectral Estimation and Modeling
Published in Richard C. Dorf, Circuits, Signals, and Speech and Image Processing, 2018
S. Unnikrishna Pillai, Theodore I. Shim, Stella N. Batalama, Dimitri Kazakos, Ping Xiong, David D. Sworder, John E. Boyd
Since the estimate θ^(xn) is a random variable, we have to specify in what sense Equation (17.21) holds. Thus, if the above limit holds w.p. 1, we say that the estimator is strongly consistent or consistent w.p. 1. In a similar way we can define a weakly consistent estimator. As far as the asymptotic distribution of θ(xn) as n→∞ to is concerned, it turns out that the central limit theorem can often be applied to θ^(xn) to infer that n[θ^(xn)]−θ is asymptotically normal with zero mean as n → ∞.
Broad Scope of Sufficient Dimension Reduction
Published in Bing Li, Sufficient Dimension Reduction, 2018
where Σ $ \Sigma $ is the covariance matrix var(X) $ \mathrm{var}(X) $ , and Y represents the labels of a pair of slices; that is, Y=1 $ Y=1 $ if X belongs to one slice, and Y=-1 $ Y=-1 $ if X belongs to the other slice. Li et al. (2011) showed that this method is unbiased under the linear conditional mean condition. They also developed the asymptotic distribution of this estimator.
Estimation and inference for variance functions
Published in Raymond J. Carroll, David Ruppert, Transfor mation and Weighting in Regression, 2017
Raymond J. Carroll, David Ruppert
For symmetrically distributed errors or as an approximation for small σ, the weighted squared residual estimate θ^WLS is known to be asymptotically more efficient than the unweighted estimator minimizing (3.17) (see Davidian and Carroll, 1987). Under very general conditions, they show that any weighted estimator based on (3.18) has the same asymptotic distribution as the pseudo-likelihood estimator θ^PL.
Bayesian and Classical Estimation of the Inverse Pareto Distribution and Its Application to Strength-Stress Models
Published in American Journal of Mathematical and Management Sciences, 2018
This asymptotic distribution of can be used to construct asymptotic confidence intervals for R. Using the MLE estimate of R in place of R in , the asymptotic 100(1 − γ)% confidence interval for R would be (L2, U2), where, where z1 − γ/2 is the (1 − γ/2)th percentile of the standard normal distribution and is given by Equation (7).
Sensor fault detection and isolation: a game theoretic approach
Published in International Journal of Systems Science, 2018
Hamed Habibi, Ian Howard, Reza Habibi
Here, without loss of generality, assume that the magnitude of change is negative, i.e. According to the results of Section 2, the null hypothesis is rejected (against AMOC alternative hypothesis ) for large value of test statistic , where where for , and see Section 2. Values of are attainable if the 's are known. This approach is referred to as the player test statistic in Section 2. For the unknown states cases, following (Chung and Speyer, 1998) and using a conservative approach, the null hypothesis is rejected if, for some thresholds . These are computed using the null distribution of , such that for some desired significant level Following Section 2, when the above-mentioned method is repeated for two players 1 and 2, a two-way FDI is made. To make inference, the asymptotic distribution of is needed. However, an alternative approach is the use of the LMI technique. is rejected if for all 's