# 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.

1,640
questions

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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 ...

2
votes

1
answer

76
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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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0
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16
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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.
...

0
votes

1
answer

67
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### 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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0
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9
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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} \...

1
vote

1
answer

41
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### 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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0
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15
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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 ...

-1
votes

0
answers

55
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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 ...

4
votes

2
answers

93
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### $O(n^{2-\epsilon})$ 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. ...

3
votes

2
answers

124
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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/...

1
vote

1
answer

55
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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 \...

2
votes

0
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50
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### 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 ...

-1
votes

0
answers

36
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### 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 ...

0
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0
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30
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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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29
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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
...

1
vote

0
answers

40
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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 ...

0
votes

0
answers

54
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### 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 $-\...

4
votes

1
answer

69
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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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votes

0
answers

28
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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
\...

3
votes

1
answer

69
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### 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
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0
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34
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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 ...

0
votes

0
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67
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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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0
answers

33
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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 ...

0
votes

1
answer

29
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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 $\mathbb R$...

15
votes

5
answers

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### 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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0
answers

52
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### 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

1
answer

160
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### 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 ...

2
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0
answers

59
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### 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(\...

1
vote

0
answers

32
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### $L_1$ 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 $...

4
votes

0
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128
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### 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 $f$ ...

5
votes

2
answers

132
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### 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.
...

2
votes

0
answers

101
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### 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 $b \in \mathbb R$, ...

2
votes

1
answer

241
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### 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-...

0
votes

1
answer

44
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### 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 ...

2
votes

1
answer

64
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### 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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votes

1
answer

48
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### 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 ...

4
votes

1
answer

120
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### 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, ...

4
votes

1
answer

95
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### 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 $\mathbb R^m$ ...

2
votes

1
answer

67
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### 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). ...

0
votes

1
answer

102
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### 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 ...

1
vote

1
answer

120
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### 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 $\...

1
vote

2
answers

117
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### 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 $...

2
votes

1
answer

128
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### 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 $d(p,q)=\arccos(pq^T)$...

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,\...

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 $W=\sqrt{\frac{\mu}{n}}\sigma\sigma^\top+Z$, ...

1
vote

1
answer

124
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### 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 ...

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 (...

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 ...

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 $\Sigma$....

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 ...