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 ...
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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 ...
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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. ...
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67 views

Functional relationship between two quantities

Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by $$ \alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{...
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Integral of the deconvolution kernel density estimator

Let $Y_i = m(X_i) + \eta_i$, $W_j = X_j + U_j$, $E[\eta_i | X_i] = 0$ with $X_i \sim f_X$, $U_i \sim f_U$ be an errors-in-variable problem and $K_{U}(x) = \dfrac{1}{2 \pi} \int \mathrm{e}^{-itx} \...
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1 vote
1 answer
41 views

Estimation of Lévy measure of ID distribution

Suppose that the positive random variable $X$ is infinitely divisible and supported on $\mathbb R_+$. Due to Lévy-Khintchine, its moment generating function then writes : $$M(t) = \mathbb E\left(e^{tX}...
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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 ...
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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 ...
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4 votes
2 answers
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. ...
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3 votes
2 answers
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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/...
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1 vote
1 answer
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Non-independent Sub-gaussian variables and concentration

Let $g \in R^{d}$ have $iid$ Gaussian components. Let $a \in R^{d}$, and let $b \in R^{d}$. be arbitrary vectors. Consider the random variable $Y_{g,g}:= \frac{1}{n}\langle g,a \rangle \langle g, b \...
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2 votes
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50 views

Approximate logarithmic bound on expected maximum via central limit theorem

If $Z_i$ are standard normal, possibly dependent, one can show that $$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$ I'm looking for a similar (asymptotic) bound for asymptotically normal ...
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36 views

Create relative correlation [closed]

The problem I am facing is whether I can combine two correlations to one. More specifically, I have a correlation between x and y, but also a correlation between x and y1 and x and y2. I need to ...
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Moments related to a likelihood function

It is well known that, in the $n\rightarrow\infty$ limit, $$Q:=2\ln{L(\hat{\theta};\mathbf{X})}-2\ln{L(\theta;\mathbf{X})}$$ has the $\chi^2_1$ distribution, where $\theta$ is a parameter of a regular ...
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How to recalculate the weights for an event that happens multiple times and requires all outcumes to be unique?

I think it's easiest to explain with an example. I have a weighted probability list A : 0.15 B : 0.15 C : 0.15 D : 0.1 E : 0.1 F : 0.1 G : 0.1 H : 0.075 I : 0.075 ...
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1 vote
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The exact distribution of the conditional distribution of the OLS estimator [migrated]

This is the problem that I have tried figuring it out for a while, and I still need some advice because there is no explicit derivation in the textbook that I have seen so far. The problem looks easy ...
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54 views

Asymptotic distribution of a specific ML estimator

Consider a random independent sample of size $n$ from a distribution defined by the following probability density function$$f(x)=\frac{3}{4\theta}\cdot\left(1-\frac{x^2}{\theta ^2}\right)$$ when $-\...
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4 votes
1 answer
69 views

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 \...
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3 votes
1 answer
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 ...
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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 ...
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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 ...
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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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0 votes
1 answer
29 views

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 $\mathbb R$...
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15 votes
5 answers
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 ...
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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, $f_1,f_2 \in F$, ...
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1 vote
1 answer
160 views

Bound error in approximating $E_x [H(f(x))]$ with random $(1/n) \sum_{i=1}^n \Phi(f(x_i)/h)$ where $H$ is Heaviside function and $\Phi$ is normal CDF

Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently smooth" function. For simplicity, we may consider $f$ to be an affine function, i.e $f(x) \equiv b-x^\top w$, for some $(w,b) \in \mathbb ...
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2 votes
0 answers
59 views

Harish-Chandra–Itzykson–Zuber integral with two terms

We know $$ \int \mathcal{D}U \exp(\mathrm{Tr}(AUBU^*)) $$ can be computed by Harish-Chandra–Itzykson–Zuber(HCIZ) integral. I am wondering whether it is possible to compute $$ I=\int \mathcal{D}U \exp(\...
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1 vote
0 answers
32 views

$L_1$ convergence rates for multivariate kernel density estimation

Let $X$ be a random variable on $\mathbb R^d$ with probability density function $f$, and let $X_1,\ldots,X_n$ of $X$ be $n$ iid copies of $X$. Given a bandwidth parameter $h=h_n > 0$ and a kernel $...
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4 votes
0 answers
128 views

Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)

Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
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5 votes
2 answers
132 views

Placing pins on a Galton board to approximate an arbitrary distribution

Inspired by this reddit post: https://old.reddit.com/r/math/comments/tv3cbg/how_do_you_unbell_curve_a_galtonplinko_board/ The Nth Galton Board, G(N), is a triangular lattice of pegs of height N-1. ...
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2 votes
0 answers
101 views

Consistent approximation of weighted Radon transform of smooth probability density, using kernel density estimation

Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, ...
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2 votes
1 answer
241 views

Does taking minimum preserve density monotonicity?

Suppose $X$ and $Y$ are continuous random variables with a joint density function $f_{X,Y}$. Both $X$ and $Y$ are supported on $(0,1)$ and have continuous (can be assumed differentiable) and non-...
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0 votes
1 answer
44 views

Bounds on the number of samples needed to learn a real valued function class

Let us see Theorem 6.8 in this book, https://www.cs.huji.ac.il/w~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf It gives us a lowerbound (and also an ...
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  • 505
2 votes
1 answer
64 views

Lower bound on the error of proportion estimation

Let $X \sim \operatorname{Bin}(n,p)$. Suppose we estimate $p$ by $\hat{p}=\frac{X}{n}$. By Hoeffding’s inequality it holds for all $\delta \in (0,1)$ with probability at least $1-\delta$ that, $$\...
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-1 votes
1 answer
48 views

The distribution of the sum of values from a normal and a truncated normal distribution

Using R to extract truncated normal distribution samples and normal distribution samples separately, when they are combined, the image drawn by the hist function is very similar to a normal ...
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4 votes
1 answer
120 views

About non-reversible Metropolis Hastings Markov chain

I am reading a paper about constructing a non-reversible Metropolis Hastings Markov chain from a reversible one as described at a high level in paragraph $3$ of page $1$. But I don't understand how, ...
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4 votes
1 answer
95 views

Consistent empirical estimation of Radon transform of a multivariate density function

Let $P$ be a "nice" distribution on $\mathbb R^m$ (e.g., multivariate Gaussian, etc.), with density $p$. Let $H := \{x \in \mathbb R^m \mid x^\top w = b\}$ be a hyperplane in $\mathbb R^m$ ...
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2 votes
1 answer
67 views

Derive equation for regularized logistic regression with batch updates

I am trying to understand this paper by Chapelle and Li "An Empirical Evaluation of Thompson Sampling" (2011). In particular, I am failing to derive the equations in algorithm 3 (page 6). ...
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0 votes
1 answer
102 views

Sparse linear regression model

Define a set $C_3(S):=\{\Delta\in R^d: \|\Delta_{S^c}\|_1\le 3\|\Delta_S\|_1\}$. Suppose we form a random design matrix $X\in R^{n\times d}$ with rows drawn iid from a $N(0,\Sigma)$ distribution and ...
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1 vote
1 answer
120 views

Lasso of sparse linear regression model

Consider the sparse linear regression model $y=X\theta^*+w$, where $w\sim N(0, \sigma^2 I_{n\times n})$ and $\theta^*\in R^d$ is supported on a subset $S$. Suppose that the sample covariance matrix $\...
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  • 156
1 vote
2 answers
117 views

Central limit theorem of random vectors when the dimension is increasing

This is a question about central limit theorems when the dimension is increasing. Suppose now I have a random vector $X_N = (X_{N1}, \cdots, X_{Np})\in\mathbb{R}^p$. For all $c_p\in\mathbb{R}^p$ with $...
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2 votes
1 answer
128 views

The mean of positive points on a unit $n$-sphere $S^n$

My question is similar to The mean of points on a unit n-sphere $S^n$. I have a unit $n$-sphere $S^n$ and a set $P$ of points lying on its surface. I use geodesic distance metric $d(p,q)=\arccos(pq^T)$...
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0 votes
0 answers
34 views

Confidence Set of Maximum Likelihood Estimator

Question Assuming we want to estimate a conditional distribution $p^*(y | x)$, where $(x,y)\in \mathcal X\times \mathcal Y$ and $\mathcal Y$ is a finite space, i.e., we can write $\mathcal Y = \{1,2,\...
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0 votes
0 answers
60 views

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 $W=\sqrt{\frac{\mu}{n}}\sigma\sigma^\top+Z$, ...
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  • 433
1 vote
1 answer
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 ...
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  • 141
0 votes
0 answers
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 (...
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  • 187
1 vote
1 answer
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 ...
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  • 187
0 votes
0 answers
34 views

Prove the statistical rate lower bound of a given complicated statistics

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 $\Sigma$....
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1 vote
1 answer
76 views

Permute a sparse random matrix to resemble a diagonal matrix as much as possible

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 ...
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