# Questions tagged [st.statistics]

Applied and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments.

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### Sample complexity lower bound to learn the mode (the value with the highest probability) of a distribution with finite support

Say we have a black-box access to a distribution $\mathcal{D}$ with finite support $\{1,2,...,n\}$ with probability mass function $i \mapsto p_i$. How many samples of $\mathcal{D}$ are needed to learn ...
76 views

### Random probability following a log concave distribution of order p

In the article "Concentration of the information in data with Log-concave distributions" of Bobkov and Madiman, it is written that if $X$ is a positive random variable following a log ...
16 views

### How to determine how much variation was due to the differences between a group?

I am working on the data set that consists of Patients (after stroke), Time (then can walk after a going through the program), and Program they follow. ...
67 views

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### A question about the spread of uncertianty when an average value is calculated if the singular values have an uncertianty attached to them [migrated]

Given a set of N values, the error associated with their average will be: (standard deviation)/(N)^(1/2) But if the values themselves have an uncertianty attached to them eg 100 ± 1, 110 ± 1... is the ...
55 views

### What is the best method to calculate an adjustment value for a sample mean when I think it is too high? [closed]

I have the population mean (55) and SD (5) I have the sample mean (70), SD (10) and sample size (10) Based on the sample size being very small, I need to adjust the sample mean to be more ...
93 views

### $O(n^{2-\epsilon})$ bound on choosing $n$ points on the hypersphere to maximize $\pm 1$ weighted sum of their $\binom{n}{2}$ inner products

Given $n,d\in\mathbb{N}, n\gg d$, I'm looking for a bound on the maximum (or minimum) expected value of the following game: Draw a vector $\epsilon\in\{\pm 1\}^{\binom{n}{2}}$, uniformly at random. ...
124 views

### On finding an upper bound on the error of a sparse approximation

I posted this question on math.stackexchange earlier, but didn't see any response. So, I am posting it here, in case someone else has an answer. Original question: https://math.stackexchange.com/...
1 vote
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### Transforming a Poisson distribution into a power law

Consider the probability mass function of the Poisson distribution given a mean $\lambda$: \begin{equation} \mathbb{P}\left(Y=k|\lambda\right)=\frac{e^{-\lambda} \lambda^{k}}{k !} \end{equation} By ...
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### hypothesis testing and properties(biasedness/unbiasedness, consistency) of the OLS estimator [closed]

Consider the following equation * \begin{equation} \label{eq:1} C_{t} = \beta_{1} + \lambda   Y_{t} + \epsilon_{t} \end{equation} where, \begin{equation} \label{eq:2} E(\epsilon_{t}\mid Y_{t}) = 0 \...
69 views

### For a random sequence $X_0, X_1, X_2, \ldots$ and $F_n$ the empirical CDF, does $F_n(X_0)$ converge to a uniform random variable?

Let $X_0, X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with continuous CDF $F(x)$ and define a sequence of ...
1 vote
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### Find Kullback-Leibler distance between two densities [closed]

can someone help me with this exercise? (look at the image). How can I find Kullback-Leibler distance between this two densitie? I have no idea how to arrive at the solution. Every suggestion is ...
67 views

### Verification of a certain computation of VC dimension

Disclaimer: I'm not very familiar with the concept of VC dimensions and how to manipulate such objects. I'd be grateful if expects on the subject (learning theory, probability), could kindly proof ...
33 views

### Problem about providing a good estimator in 2SLS

I am now studying the 2-stages least-squares method and have been curious about the following circumstances. Suppose that I have $Y_i = X^{T}_{i}β +e_{i}$ with $\mathbb{E}(e_{i}X_{i}) ̸\ne 0$, that ...
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### Rademacher complexity of function class $\{(x,y) \mapsto 1[|yf(x)-\alpha| \ge \beta]$ in terms of $\alpha$, $\beta$, and Rademacher complexity of $F$

Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$...
1k views

### Longest increasing subsequence as measure of randomness

Although I am by no means an applied mathematician, I like to occasionally explain applications of the math I teach to real world problems. Right now I am teaching some students about longest ...
52 views

### Upper-bound for bracketing number in terms of VC-dimension

Let $P$ be a probability distribution on a measurable space $\mathcal X$ (e.g;, some euclidean $\mathbb R^m$) and let $F$ be a class of funciton $f:\mathcal X \to \mathbb R$. Given, $f_1,f_2 \in F$, ...
1 vote
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1 vote
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### Exact recovery of spiked Wigner model

We are given a bisection $\sigma\in\{\pm 1\}^n$, i.e. there are half 1s and half -1s in it. However there are some Gaussian noise, i.e. our observation is $W=\sqrt{\frac{\mu}{n}}\sigma\sigma^\top+Z$, ...
1 vote
124 views

### Expected value of a function of normal random variable

Suppose $X\sim \mathcal{N}(0,\sigma^2)$, find the expectation $\mathbb{E}\left[\frac{1}{(1+X^2)^a}\right]$ where $a$ is a fixed positive real number. Is there an explicit formula for the above ...
17 views

### Compute the autocovariance function of a stationary process

Say we have a stationary process, but we observe samples at random times $\{t_n\}$ which itself is a stochastic point process (e.g. Poisson process). The resulting sample is also a stationary process (...
1 vote
107 views

### Weakly dependent central limit theorem

Say I have $N$ random variables $X_1,\cdots,X_i,\cdots,X_N$, with zero mean and finite variance. $X_i$ and $X_j$ are independent iif $|i-j|>m$, and positively correlated otherwise (say the ...
Given a i.i.d. sequence of random variables $\{Z_i\}_{i=1}^n$ who has mean zero. Two i.i.d. sequence of random vectors $\{X_i\}_{i=1}^n$, $\{Y_i\}_{i=1}^n$ who have the same covariance matrix $\Sigma$....
Say we generate an $N \times N$ sparse random matrix $W$, where each element $W_{ij}$ was independently chosen to be $1$ with probability $p=\frac{a}{N}$, and $0$ with probability $1-p$. We are ...