Questions tagged [statistics]
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.
11,528
questions with no upvoted or accepted answers
5
votes
0
answers
5k
views
A question about the Laplace transform of the probability distribution function
I am an undergraduate, who has just begun getting into probability theory. Recently we learned of the moment generating function of a random variable can be used to find moments, and the moment ...
- 119
5
votes
0
answers
2k
views
Any known relationship between the eigenvalues of correlation matrix and covariance matrix?
For principal component analysis we usually use correlation matrix when the data are measured in different units. In case where data units are consistent we can use covariance matrix instead.
I want ...
- 5,596
5
votes
1
answer
132
views
On average, where is the lift?
This started as a computing problem with several variables, and I'd like to know if there's a closed form formula for the average position of the lift.
Context: there's a building with $N$ floors and ...
- 5,480
5
votes
1
answer
1k
views
Derivative of expected log likelihood in a logistic regression model
Consider the univariate logistic regression model:
$$
P(Y = 1\mid X = x) = \psi(x\beta_0)\equiv \frac 1 {1+\exp\{-x \beta_0\}},\quad\text{for all $x$, and some unknown $\beta_0\in\mathbb{R}$.}
$$
...
- 168
5
votes
0
answers
151
views
Likelihood function & MLE without known values of observed data
Question: Let $X_1,\dots,X_n$ be iid exponential rate $\lambda$. Suppose we don't know the observed values of our experiments, but we know that $k$ values were $\le M$ and the remaining $n-k$ were $&...
- 1,986
5
votes
1
answer
635
views
Geometric mean, harmonic mean and loss functions
Consider a sequence $(x_i)_{i \in I}$ of real numbers indexed on a set $I$. The mode of the series is the minimizing argument for the $L_0$ loss
$$ \text{mode}[ \; (x_i)_{i \in I} \; ] = \arg\min_{u \...
- 151
5
votes
0
answers
481
views
What's the most accurate way to estimate a percentile from multiple partial percentiles?
There exists 3 sets of numbers. I have the 99th percentile (p99) of each set and the cardinality of the set, but not the values in the set themselves.
p99: 540, cardinality: 215
p99: 288, cardinality:...
- 153
5
votes
0
answers
536
views
If a class of functions $\mathcal{F}$ is a Glivenko-Cantelli class then it is also a Donsker class?
Definitions: Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ with probability distribution $P$. All functions ...
- 266
5
votes
0
answers
1k
views
Expected value of the sample covariance
Let $X = (X_1,\dots,X_p)$ is a random (column) vector with values in $\mathbb R^p$. The covariance matrix $\mathrm{Cov}(X,X)$ is defined by
$$\mathrm{Cov}(X) := E[(X-E[X])(X-E[X])^T]$$
By definition ...
- 2,070
5
votes
2
answers
140
views
Determining number of randomly picked people
Firstly I want to put big disclaimer here. This particular problem is a smaller part of my homework. Since even after discussion with my fellow classmates we are not sure how to handle it we decided ...
- 255
5
votes
0
answers
103
views
Random sphere-valued fields
I would like to generate random functions from an $m$-sphere $S^m$ to an $n$-sphere $S^n$ that are not too wild, some kind of generalization of random Gaussian fields.
More precisely, I want $f(x)$, ...
- 11.7k
5
votes
2
answers
1k
views
Difference of Ordered Uniform Random Variables
Let $X_1, X_2,..., X_n$ be $n$ random variables distributed uniform(0,1) and $X_{(1)},X_{(2)},..., X_{(n)}$ be the ordered statistics of $X_1,...,X_n$ such that:
$X_{(1)} < X_{(2)} < ... < ...
- 59
5
votes
1
answer
5k
views
Finding the mode of the negative binomial distribution
The negative binomial distribution is as follows: $\displaystyle f_X(k)=\binom{k-1}{n-1}p^n(1-p)^{k-n}.$
To find its mode, we want to find the $k$ with the highest probability.
So we want to find $P(...
- 6,534
5
votes
0
answers
562
views
Skellam CDF Increasing vs Decreasing in a parameter
I'm working with the following Poisson difference distribution:
$$\text{Prob}\{X_1-X_2 \geq 0\} $$
where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$.
I need to ...
- 51
5
votes
0
answers
222
views
Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound
this question relates to the requirement for achieving CRLB.
I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also $\operatorname{...
- 329
5
votes
1
answer
3k
views
Kruskal Wallis - Effect size
I analyse 4 algorithms and 3 sets of metrics for each algorithm in which I apply the non-parametric Kruskal-Wallis test for each metric to detect any differences in performance between these ...
- 153
5
votes
0
answers
604
views
How to compute uniformly distributed points on an ellipse
The ellipse can be parametrized in polar coordinates by
$$r(\theta)=\frac{1}{a+\cos\theta}$$
up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the $\...
- 1,467
5
votes
1
answer
1k
views
Consistency of kernel density estimator with constant bandwidth
Let ($x_1, ..., x_n$) be i.i.d. samples drawn from some distribution $P$ with an unknown probability density function $f$. Its kernel density estimator is
\begin{align}
\hat{f}_h(x) = \frac{1}{n}\...
- 840
5
votes
0
answers
367
views
Rate of convergence of a martingale
I have a question related to convergence rates of martingales:
Assume that there is a sequence of maximized likelihood ratios:
$ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) }{...
- 71
5
votes
0
answers
708
views
What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?
The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$
What is ${\rm cov}(e_i, \hat y_i)$?
What is ${\rm cov}(\epsilon_i, \hat \beta_1)$?
What is ${\rm cov}(e_i, \epsilon_i)$?
For 1, I am writing $...
- 141
5
votes
0
answers
696
views
Intuition behind (statistical) completeness
I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, ...
- 10.5k
5
votes
0
answers
6k
views
95% confidence interval around sum of random variables
Suppose I have two random variables, $X$ and $Y$. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. Suppose that $Y$ is not ...
- 311
5
votes
0
answers
511
views
Multilinear or Tensor Regression?
Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
- 1,129
5
votes
1
answer
281
views
A equivalent definition of the Feller Process.
I saw this on Liggett's Book (P.95).
Let $S=%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,$ and suppose $\left( X_{t}\right) _{t\geq 0}$ is a continuous-time Markov
process with ...
- 151
5
votes
0
answers
153
views
Clarification in a paper
This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari.
In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} D_{\alpha}\{q(x,t),p(x,\...
- 706
5
votes
0
answers
1k
views
Solved problems book in Mathematical statistics
Are there some solved problems book in Mathematical statistics?
Or some set of solutions for qualifying exams preparation?
Say to the level of
Casella, Berger, Statistical Inference, or,
Hogg et al.,...
user14108
5
votes
1
answer
335
views
Quantitative Analysis of Structure of Gaussian Mixture Model
I am fitting a Gaussian Mixture Model to high-dimensional data (40 dimensions).
I have trained the model using EM, learned the parameters and now I want to know quantitatively:
What is most ...
- 51
5
votes
1
answer
647
views
Concentration inequalities for product of gaussians
Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $n$ independent gaussian random variables with zero means ...
- 1,079
5
votes
1
answer
362
views
Coin Flips and Hypothesis Tests
Here's a problem I thought of that I don't know how to approach:
You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with ...
- 311
4
votes
0
answers
55
views
Consistency of bootstrap estimator that is continuously $\rho_\infty$-Frechet differentiable
Theorem. Suppose $T$ is continuously $\rho_r$-Frechet differentiable at $F$ with the influence function satisfying $0<E[\phi_F(X_1)]^2<\infty$ and that $\int F(x)[1-F(x)]^{r/2}dx<\infty$. ...
- 2,224