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 ...
19
votes
0answers
256 views

Anscombe-like datasets with the same box and whiskers plot (mean/std/median/MAD/min/max)

EDIT: As this question has been inflated, a summary: finding different meaningful and interpretable datasets with the same mixed statistics (mean, median, midrange and their associated dispersions, ...
16
votes
0answers
628 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
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
586 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 ...
15
votes
0answers
362 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 ...
14
votes
1answer
367 views

Expected numbers of distinct colors when drawing without replacement

Consider an urn containing $N$ balls of $P$ different colors, with $p_i$ being the proportion of balls of color $i$ among the $N$ balls ($\sum_i p_i = 1$). I draw $n \leq N$ balls from the urn without ...
14
votes
2answers
1k views

Contrast matrix definition

What exactly is contrast matrix and how exactly is contrast matrix specified? I.e. what are columns, what are rows, what are the constraints on that matrix and what does number in column ...
13
votes
0answers
128 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
1answer
206 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}} ...
12
votes
1answer
188 views

Assessments of “Approximately Normal” for t-tests

I am testing equality of means using Welch's t-test. The underlying distribution is far from normal (more skewed than the example in a related discussion here). I can obtain more data but would like ...
12
votes
0answers
266 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 ...
12
votes
0answers
1k views

How does Krizhevsky's '12 CNN get 253,440 neurons in the first layer?

In Alex Krizhevsky, et al. Imagenet classification with deep convolutional neural networks they enumerate the number of neurons in each layer (see diagram below). The network’s input is 150,528-...

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