19
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 ...
17
votes
0answers
194 views

I'm more likely to read “interesting” studies, so do all p-values I read suffer a “readership bias”?

Do I bias p-values simply by reading them — or, more precisely, by having chosen to read the study that contains them? I generally take most p-values with a sceptical pinch of salt. Sometimes ...
16
votes
0answers
991 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 ...
15
votes
0answers
196 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, ...
14
votes
0answers
554 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 ...
13
votes
0answers
345 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 ...
12
votes
0answers
111 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 ...
12
votes
0answers
558 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 split into $d=10$ deciles, $D_1, D_2, \dots , D_{d}$, ...
11
votes
0answers
261 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 ...
11
votes
0answers
458 views

Model selection with Firth logistic regression

In a small data set ($n\sim100$ ) that I am working with, several variables give me perfect prediction/separation. I thus use Firth logistic regression to deal with the issue. If I select the best ...
10
votes
0answers
217 views

Switch from Modelling a Process using a Poisson Distribution to use a Negative Binomial Distribution?

We have a random process that may-or-may-not occur multiple times in a set period of time $T$. We have a data feed from a pre-existing model of this process, that provides the probability of a number ...
10
votes
0answers
169 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 - ...
10
votes
0answers
446 views

Link Anomaly Detection in Temporal Network

I came across this paper that uses link anomaly detection to predict trending topics, and I found it incredibly intriguing: The paper is "Discovering Emerging Topics in Social Streams via Link Anomaly ...
9
votes
0answers
145 views

Special probability distribution

If $p(x)$ is a probability distribution with non-zero values on $[0,+\infty)$, for what type(s) of $p(x)$ there exist a constant $c>0$ such that $\int_0^{\infty}p(x)\log{\frac{ ...
9
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 ...
9
votes
0answers
127 views

Derivation of normalizing transform for GLMs

How is the $A(\cdot) = \int\frac{du}{V^{1/3}(\mu)}$ normalizing transform for the exponential family derived? More specifically: I tried to follow the Taylor expansion sketch on page 3, slide 1 of ...
8
votes
0answers
115 views

Restricted maximum likelihood with less than full column rank of $X$

This question deals with restricted maximum likelihood (REML) estimation in a particular version of the linear model, namely: $$ Y = X(\alpha)\beta + \epsilon, \epsilon\sim N_n(0, \Sigma(\alpha)), $$ ...
8
votes
0answers
41 views

Algorithm: Binary search when values are uncertain

I need an algorithm to do a binary search when the test at each step may give the wrong result. Background: I need to place students on the most appropriate of 12 difficulty levels. The current ...
8
votes
0answers
99 views

Jaynes' $A_p$ distribution

In Jaynes' book "Probability Theory: The Logic of Science", he has a chapter (Ch 18) entitled "The $A_p$ distribution and rule of succession" in which he introduces the idea of $A_p$ distribution; ...
8
votes
0answers
137 views

Speed, computational expenses of PCA, LASSO, elastic net

I am trying to compare computational complexity / estimation speed of three groups of methods for linear regression as distinguished in Hastie et al. "Elements of Statistical Learning" (2nd ed.), ...
8
votes
0answers
114 views

10 % false positives from nonlinear mixed effect models : Why?

I've run a simulation study in order to estimate type I error rate of the test of group effect in a nonlinear mixed effects model, using nlmer from lme4 package. The results show there is 8-10 % ...
8
votes
0answers
114 views

Tail of the inverse cdf

I am almost sure I have already seen the following result in statistics but I can't remember where. If $X$ is a positive random variable and $E(X)<\infty$ then $\epsilon F^{-1}(1-\epsilon) \to 0$ ...
8
votes
0answers
189 views

A question related to Borel-Cantelli Lemma

Note: Borel-Cantelli Lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ $$\sum_{n=1}^\infty P(A_n) =\infty \textrm{ and } ...
8
votes
0answers
405 views

Goodness of fit test: question about Anderson–Darling test and Cramér–von Mises criterion

I'm reading web pages for goodness of fit tests, when I came to the Anderson–Darling test and the Cramér–von Mises criterion. So far I got the point; it seems the Anderson–Darling test ...
8
votes
0answers
255 views

Bound for Arithmetic Harmonic mean inequality for matrices?

NOTE: This question has originally been posted in MSE, but it did not generate any interest. It was first posted there, because the question itself is a pure matrix-algebra question. Nevertheless, ...
8
votes
0answers
519 views

Algorithm for real-time normalization of time-series data?

I'm working on an algorithm that takes in a vector of the most recent data point from a number of sensor streams and compares the euclidean distance to previous vectors. The problem is that the ...
8
votes
0answers
2k views

pdf of the product of two independent random variables, normal and chi-square

what is the pdf of the product of two independent random variables X and Y, if X and Y are independent? X is normal distributed and Y is chi-square distributed. Z = XY if $X$ has normal distribution ...
8
votes
0answers
191 views

Penalized spline confidence intervals based on cluster-sandwich VCV

This is my first post here, but I've benefited a lot from this forum's results popping up in google search results. I've been teaching myself semi-parametric regression using penalized splines. ...
8
votes
0answers
867 views

Simulating time-series given power and cross spectral densities

I am having trouble generating a set of stationary colored time-series, given the covariance matrix (their PSDs and CSDs). I know that, given two time-series $y_{I}(t)$ and $y_{J}(t)$, I can ...
8
votes
0answers
113 views

Estimation and functional space

In the first chapter of the book Algebraic Geometry and Statistical Learning Theory which talks about the convergence of estimations in different functional space, it mentions that the Bayesian ...
8
votes
0answers
424 views

Physical/pictoral interpretation of higher-order moments

I'm preparing a presentation about parallel statistics. I plan to illustrate the formulas for distributed computation of the mean and variance with examples involving center of gravity and moment of ...
7
votes
0answers
85 views

Specifying prior for effect size in meta-analysis

My question concerns priors on effect sizes, in my project the measure is Cohen's D. Through reading the literature, it seems vague priors are often used, such as in the well-know eight schools ...
7
votes
0answers
57 views

How can we simulate from a geometric mixture?

If $f_1,\ldots,f_k$ are known densities from which I can simulate, i.e., for which an algorithm is available. and if the product $$\prod_{i=1}^k f_i(x)^{\alpha_i}\qquad \alpha_1,\ldots,\alpha_k>0$$ ...
7
votes
0answers
119 views

How to test if a cross-covariance matrix is non-zero

The background of my study: In a Gibbs sampling where we sample $X$ (the variable of interests) and $Y$ from $P(X|Y)$ and $P(Y|X)$ respectively, where $X$ and $Y$ are $k$-dimensional random vectors. ...
7
votes
0answers
112 views

Implementation of CoVaR (a systemic risk measure) in R

I'm trying to estimate CoVaR using bivariate DCC GARCH in R. The concept of CoVaR is the dependence adjusted of VaR, which was first introduced by Adrian and Brunnermeier (2011). However, this ...
7
votes
0answers
60 views

Are contours $h^{-1}(y)$ interesting features of a function $h:X\to \mathbb R^n$ obtained by regression?

I assume a general setup of regression, that is, a continuous function $h_\theta:X\to \mathbb R^n$ is chosen from a family $\{h_\theta\}_\theta$ to fit given data $(x_i,y_i)\in X\times \mathbb R^n, ...
7
votes
0answers
149 views

State of art streaming learning

I have been working with large data sets lately and found a lot of papers of streaming methods. To name a few: Follow-the-Regularized-Leader and Mirror Descent: Equivalence Theorems and L1 ...
7
votes
0answers
318 views

ANOVA: testing assumption of normality for many groups with few samples per group

Assume the following situation: we have a large number (e.g. 20) with small group sized (e.g. n = 3). I noticed that if I generate values from the uniform distribution, the residuals will look ...
7
votes
0answers
248 views

Blind source separation of convex mixture?

Suppose I have $n$ independent sources, $X_1, X_2, ..., X_n$ and I observe $m$ convex mixtures: \begin{align} Y_1 &= a_{11}X_1 + a_{12}X_2 + \cdots + a_{1n}X_n\\ ...&\\ Y_m &= a_{m1}X_1 + ...
7
votes
0answers
1k views

Has the reported state-of-the-art performance of using paragraph vectors for sentiment analysis been replicated?

I was impressed by the results in the ICML 2014 paper "Distributed Representations of Sentences and Documents" by Le and Mikolov. The technique they describe, called "paragraph vectors", learns ...
7
votes
0answers
311 views

Convolutional neural networks: Aren't the central neurons over-represented in the output?

[This question was also posed at stack overflow] The question in short I'm studying convolutional neural networks, and I believe that these networks do not treat every input neuron ...
7
votes
0answers
453 views

Understanding Singular Value Decomposition in the context of LSI

My question is generally on Singular Value Decomposition (SVD), and particularly on Latent Semantic Indexing (LSI). Say, I have $ A_{word \times document} $ that contains frequencies of 5 words for ...
7
votes
0answers
148 views

What if probabilities are not equal in the “.632 Rule?”

This question is derived from this one about the ".632 Rule." I am writing with particular reference to user603's answer/notation to the extent it simplifies matters. That answer begins with a ...
7
votes
0answers
4k views

R - how to let glmnet choose lambda range when using caret?

To fit a lasso model using glmnet, you can simply do the following and glmnet will automatically calculate a reasonable range of ...
7
votes
0answers
2k views

Trouble finding good model fit for count data with mixed effects - ZINB or something else?

I have a very small data set on solitary bee abundance that I am having trouble analysing. It’s count data, and almost all the counts are in one treatment with most of the zeroes in the other ...
7
votes
0answers
248 views

Negative binomial Jeffreys prior

The negative binomial distribution NB($m,r$) is defined as $$\Pr(X = k) = \left(\frac{r}{r+m}\right)^r \frac{\Gamma(r+k)}{k! \, \Gamma(r)} \left(\frac{m}{r+m}\right)^k \quad\text{for }k = 0, 1, 2, ...
7
votes
0answers
2k views

Confusion with Vowpal Wabbit's multiple-pass behavior when performing ridge-regression

I have encountered many peculiarities/misunderstandings of Vowpal Wabbit when trying to do online multiple-pass learning. Specifically, I need to solve a Ridge Linear regression problem, with ...
7
votes
0answers
69 views

Example Of Strict von Neumann Inequality

Let $r(\pi, \delta)$ denote the Bayes risk of an estimator $\delta$ with respect to a prior $\pi$, let $\Pi$ denote the set of all priors on the parameter space $\Theta$, and let $\Delta$ denote the ...
7
votes
0answers
336 views

Luce choice axiom, question about conditional probability

I'm reading Luce (1959). Then I found this statement: When a person chooses among alternatives, very often their responses appear to be governed by probabilities that are conditioned on the ...
7
votes
0answers
10k views

Confusion with Augmented Dickey Fuller test

I am working on the data set electricity available in R package tsa. My aim is to find out if an ...

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