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Mar
26
comment An example of a “pathological” power-spectral density function?
I'm confused, are you asking us to find a definite integral that does not evaluate to a constant? There doesn't appear to be additional variables of which this integral could be a function.
Mar
26
comment On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$
As an aside, I had a fun time assuming that $X$ was exponentially distributed with $\lambda=1$ and then computing the mean of $\log(1+X)$. It turns out to be Gompertz constant; I wonder if this coincidence has been noted before.
Mar
26
comment On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$
Yes, I agree with that last statement but only for $x\in[0,\infty]$ Given that $X$ has finite mean over $[0,\infty]$ then the transformation of variables $X\to Y=\log(1+aX)$ should also have a finite mean on the same region. This won't be true over $(-1,\infty]$ because $|\log(1+x)|$ on that region can be an arbitrarily large.
Mar
26
comment On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$
The space of functions that is integrable with respect to a given probability measure $\nu=f(x)dx$ is $L^{1}(\mathbb{R},\nu)$. The function $\log(1+x)$ doesn't decay towards $0$ as $x\to\infty$. This is a problem because then it's much harder to determine if $\log(1+x)\in L^{1}(\mathbb{R},\nu)$. If $f(x)$ does not die off to zero sufficiently quickly, then this function won't be integrable. You should look up some standard results in integrability theory to see why your stated conditions are not sufficient to determine the integrability of $\log(1+ax)$.
Mar
24
comment On hitting time of Brownian motion and Ito's lemma
This is a good approach. More generally, you could look at the standard proofs for hitting times of the interval $(a,b)$ and then just let $a\to-\infty$ in the arguments.
Mar
24
comment On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$
Even if $b'>-1$, that doesn't mean $\log(1+ab')$ is well-behaved. I think you have a backward inequality as well. I'm not sure that $\alpha$-stability buys you anything, because you're only ever working with one copy of the distribution. You're not using any central limit theorem type properties, and so all of the suggestions above about how there can be arbitrary probability density on $[b',b'']$ still applies. There's just not much you can possibly say without stronger assumptions.
Mar
24
answered On the integral $\int_{b'}^{b''}f(x)\ln(1+a x)\mathrm{d}x$
Mar
24
answered Likelihood Ratio Test for Linear Regression
Mar
23
comment Error of Pearson Correlation Coefficient
Just as an aside, there are many ways to evaluate fits. No single scoring method is ever "best", and in order to provide informative, truthful results to the readers of your work, you need to usually apply a wide array of model checks and fit scores. One place to start reading is Chapter 6 of the excellent resource book "Bayesian Data Analysis" by Gelman, Carlin, Stern, and Rubin.
Mar
23
awarded  Enlightened
Mar
22
awarded  Nice Answer
Mar
22
revised Gradient of a Mahalanobis distance
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Mar
22
revised Gradient of a Mahalanobis distance
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Mar
22
revised Gradient of a Mahalanobis distance
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Mar
22
answered Gradient of a Mahalanobis distance
Mar
22
revised Real analysis question
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Mar
22
answered Real analysis question
Mar
22
comment Finding displacement from origin of SVM hyperplane
Scikits.learn, which is a Python layer over LIBSVM, does return the bias parameter within SVM objects. It's easy to use; I would consider it.
Mar
22
comment How do you filter through published papers and find the ones you should read?
Yeah, like Cross-Validated, Sci-Comp, and Theoretical Computer Science for example.
Mar
22
comment Do non-mathematical fields use the appropriate level of analytic/probabilistic rigor?
Regarding your comments on probability theory, I think Jaynes wrote the definitive rebuke of dogmatic dependence on measure theory, in Appendix B of his "Probability Theory: The Logic of Science." Reading Jaynes in grad school revived my love of math. Measure theory is important, insofar as it is an expedient tool, and if other folks like it for aesthetic reasons, that's fine. But in general, I think graduate-level education in probability has forgotten that applied math is supposed to be in the service of something...