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Mar
30
revised Real approximation to the maximum using Laplace's method integral
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Mar
30
revised Laplace's method with nontrivial parameter dependency
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Mar
30
revised Laplace's method with nontrivial parameter dependency
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Mar
30
revised Laplace's method with nontrivial parameter dependency
added precise form for the error order
Mar
28
awarded  Revival
Mar
28
comment Laplace's method with nontrivial parameter dependency
@AntonioVargas, It should say "...still underestimates". My only evidence is numerical, starting from the suspicion that the order must be lower than in the common Laplace Method. Empirically I find that the order of the error is $O(2^{-\lambda}\lambda^{-2}\log^2\lambda)$ and that the prefactor is $11/10$. In summary: $\int = \sqrt{2\pi}\lambda^{-3/2}2^{-\lambda} + \frac{11}{10} 2^{-\lambda}\lambda^{-2}\log^2\lambda + \cdots$. Perhaps the extra order in the error is because you have a infinite series of errors that add up to a higher error (the convergence inside the series may not be uniform)
Mar
28
awarded  Explainer
Mar
27
comment Laplace's method with nontrivial parameter dependency
@AntonioVargas, so if it is $O(2^{-\lambda}\dots)$ then it is much better than $O(\lambda^-1 \log\lambda)$, no?. Also, from my numerical experiment I get that $O(2^{-\lambda}\lambda^{-5/2}\log\lambda)$ still overestimates the error. It looks like the error is in practice $O(2^{-\lambda}\lambda^{-2}\log\lambda$.
Mar
27
comment Laplace's method with nontrivial parameter dependency
@AntonioVargas, Thanks, I think your answer is more formal. I also get to the same approximation and also find that some weaker variant of the Laplace method needs to be used. What I couldn't find is the "order" of the approximation. Did you?
Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
revised Laplace's method with nontrivial parameter dependency
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Mar
27
answered Laplace's method with nontrivial parameter dependency