Edgeworth Expansions for Fisher­Consistent Estimators and Second Order Efficiency

1979 ◽  
Vol 28 (1-4) ◽  
pp. 1-18
Author(s):  
J. K. Ghosh ◽  
Bimal Kumar Sinha ◽  
K. Subramanyam
Keyword(s):  
Recent Result ◽  
Lebesgue Measure ◽  
Loss Function ◽  
Exponential Family ◽  
Second Order ◽  
Probability Inequality ◽  
Edgeworth Expansions ◽  
Second Order Efficiency ◽  
Curved Exponential Family

Using a recent result of Bhattacharya, R. N. and Ghosh, J. K. ( Annals of Statistics, 1978, 434­451), Edgeworth expansions of distributions of Fisherconsistent estimators for curved exponential family of parent distributions (dominated by the Lebesgue measure) can be obtained. We compare directly the first four cumulants of an arbitrary Fisher­consistent estimator with those of the MLE after correcting them for their bias. This leads to a key probability inequality which immediately implies the second order efficiency of the MLE wrt any bounded bowl­ shaped loss function. If the assumption of a dominating Lebesgue measure is dropped the formal Edgeworth expansions are no longer valid. However it turns out that if the loss function satisfies certain additional conditions the second order efficiency of the MLE holds. This modification takes care of the curved multinomial.

scholarly journals Second Order Efficiency of the MLE with Respect to any Bounded Bowl-Shape Loss Function

1980 ◽  
Vol 8 (3) ◽  
pp. 506-521 ◽  
Author(s):  
J. K. Ghosh ◽  
B. K. Sinha ◽  
H. S. Wieand
Keyword(s):  
Loss Function ◽  
Second Order ◽  
Second Order Efficiency ◽  
Shape Loss

scholarly journals RESIDUAL-BASED GARCH BOOTSTRAP AND SECOND ORDER ASYMPTOTIC REFINEMENT

2016 ◽  
Vol 33 (3) ◽  
pp. 779-790 ◽  
Author(s):  
Minsoo Jeong
Keyword(s):  
Second Order ◽  
Moment Condition ◽  
Garch Models ◽  
Exact Order ◽  
Block Bootstrap ◽  
Edgeworth Expansions ◽  
Coverage Probabilities ◽  
The Moment ◽  
Autoregressive Conditional Heteroscedasticity ◽  
Reliable Methods

The residual-based bootstrap is considered one of the most reliable methods for bootstrapping generalized autoregressive conditional heteroscedasticity (GARCH) models. However, in terms of theoretical aspects, only the consistency of the bootstrap has been established, while the higher order asymptotic refinement remains unproven. For example, Corradi and Iglesias (2008) demonstrate the asymptotic refinement of the block bootstrap for GARCH models but leave the results of the residual-based bootstrap as a conjecture. To derive the second order asymptotic refinement of the residual-based GARCH bootstrap, we utilize the analysis in Andrews (2001, 2002) and establish the Edgeworth expansions of the t-statistics, as well as the convergence of their moments. As expected, we show that the bootstrap error in the coverage probabilities of the equal-tailed t-statistic and the corresponding test-inversion confidence intervals are at most of the order of O(n−1), where the exact order depends on the moment condition of the process. This convergence rate is faster than that of the block bootstrap, as well as that of the first order asymptotic test.

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