Unanswered Questions

21
votes
0answers
2k views

Inverting the Fourier Transform for a Fisher distribution

The characteristic function of Fisher $\mathcal{F}(1,\alpha)$ distribution is: $$C(t)=\frac{\Gamma \left(\frac{\alpha +1}{2}\right) U\left(\frac{1}{2},1-\frac{\alpha }{2},-i t \alpha \right)}{\Gamma \...
21
votes
2answers
1k views

Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s

I'm working with two independent normal distributions $X$ and $Y$, with means $\mu_x$ and $\mu_y$ and variances $\sigma^2_x$ and $\sigma^2_y$. I'm interested in the distribution of their ratio $Z=X/...
20
votes
0answers
1k views

MLEs from glmer {lme4} in R

Numerically deriving the MLEs of GLMM is difficult and, in practice, I know, we should not use the brute force optimization (e.g., using optim in a simple way). But ...
20
votes
2answers
330 views

Has the journal Science endorsed the Garden of Forking Pathes Analyses?

The idea of adaptive data analysis is that you alter your plan for analyzing the data as you learn more about it. In the case of exploratory data analysis (EDA), this is generally a good idea (you are ...
19
votes
1answer
164 views

Is there such a thing as an adjusted $R^2$ for a quantile regression model?

Having included an quantile regression model in a paper, the reviewers want me to include adjusted $R^2$ in the paper. I have calculated the pseudo-$R^2$s (from Koenker and Machado's 1999 JASA paper) ...
17
votes
0answers
675 views

Degrees of freedom of $\chi^2$ in Hosmer-Lemeshow test

The test statistic for the Hosmer-Lemeshow test (HLT) for goodness of fit of a logistic regression model is defined as follows: the sample is then split into $d=10$ deciles, $D_1, D_2, \dots , D_{d}$...
16
votes
0answers
384 views

Fisher information in a hierarchical model

Given the following hierarchical model, $$ X \sim {\mathcal N}(\mu,1), $$ and, $$ \mu \sim {\rm Laplace}(0, c) $$ where $\mathcal{N}(\cdot,\cdot)$ is a normal distribution. Is there a way to get an ...
16
votes
2answers
1k views

How to test equality of variances with circular data

I am interested in comparing the amount of variability within 8 different samples (each from a different population). I am aware that this can be done by several methods with ratio data: F-test ...
15
votes
0answers
607 views

How can we bound the probability that a random variable is maximal?

Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_n$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$. I am looking for ...
14
votes
1answer
223 views

Second moment method, Brownian motion?

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...
13
votes
0answers
145 views

Is there a result that provides the bootstrap is valid if and only if the statistic is smooth?

Throughout we assume our statistic $\theta(\cdot)$ is a function of some data $X_1, \ldots X_n$ which is drawn from the distribution function $F$; the empirical distribution function of our sample is $...
13
votes
0answers
202 views

If the LASSO is equivalent to linear regression with a Laplace prior how can there be mass on sets with components at zero?

We are all familiar with the notion, well documented in the literature, that LASSO optimization (for sake of simplicity confine attention here to the case of linear regression) $$ {\rm loss} = || y - ...
12
votes
0answers
176 views

Is frequentist conditional inference still being used in practice?

I've recently reviewed some old papers by Nancy Reid, Barndorff-Nielsen, Richard Cox and, yes, a little Ronald Fisher on the concept of "conditional inference" in the frequentist paradigm, which ...
12
votes
0answers
106 views

What does the Akaike Information Criterion (AIC) score of a model mean?

I have seen some questions here about what it means in layman terms, but these are too layman for for my purpose here. I am trying to mathematically understand what does the AIC score mean. But at ...
12
votes
0answers
269 views

Help in how the paper derives the CRLB for Gaussian ARMA model

An univariate autoregressive process AR(p) process is expressed as $$y(n) = \sum_{j=1}^p a_jy(n-j) + u(n) $$ is excited by Gaussian sequence, $u$. Paper : On the Computation of the Cramer-Rao Bound ...

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