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Questions tagged [bounds]

Bounds represent the points with which data cannot exceed, such as minima or maxima.

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Concentration inequality for sums of independent gamma random variables

I am dealing with the following problem: Say $X_1, \ldots, X_n$ are independent Gamma random variables, each one having shape and rate parameters $\alpha_i$ and $\beta_i$, respectively. Let $S_n = \...
HeyCool08's user avatar
6 votes
1 answer
289 views

Vintage of this lower bound on skewness for positive data with given mean and sd?

It turns out there is a lower bound on the skewness $g_1$ of any strictly positive set of data having a given mean μ and standard deviation σ: $$ g_1 > \sigma/\mu - \mu/\sigma. $$ Although ...
David C. Norris's user avatar
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8 views

Nonlinear Optimization of Noisy Functions w/ Bound Constraints via SciPy

Can we use scipy.optimize.minimize to find the best parameters $\mathbf{w} \in \Omega^k$, $\Omega \subset \mathbb{R}$, of a function $g = g(f(\mathbf{x}), \mathbf{w}...
Sanjar Adilov's user avatar
1 vote
2 answers
34 views

Calculating variance between paired samples, where one sample is constrained to always be the lower bound?

I'm sure this is a solved question, but I haven't been able to hit on the right search terms. Suppose I have paired samples A and B. A represents a derived variable (say distance "as-the-crow-...
Danielle McCool's user avatar
7 votes
1 answer
218 views

Bound Product of Independent Gaussians

I'm interested in obtaining upper bounds on $$ \Pr[\prod_{i\in[n]}|G_i| > x] $$ where $G_i\sim\mathcal{N}(0,1)$ i.i.d, and $[n] := \{0,1,\dots,n-1\}$. The most naive bound is to note that each $G_i$...
Mark Schultz-Wu's user avatar
6 votes
1 answer
111 views

Bound on Rademacher complexity using polynomial discrimination

This is lemma 4.14 in Wainwright's textbook on High-Dimensional Statistics, it states that given a class of function $\mathcal{F}$ has polynomial discrimination of order $v$, then for all integer $n$ ...
Mondayisgood's user avatar
3 votes
1 answer
93 views

likelihood ratio tests on bounded parameters

I am confused by the likelihood ratio test's boundary condition limitation. A commonly stated is that it causes problem for variance parameter because it is bounded below by 0. Can these models ...
quibble's user avatar
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2 votes
1 answer
51 views

Different versions of additive chernoff

The additive Chernoff Bound says for $X_i \in \{0,1\}$ that satisfies $\mathbb{E}[X_i] = p,$ $$ \mathbb P\left(\sum\limits_{i}^nX_i \geq np+n\epsilon \right) \leq \exp\left(-\frac{(n\epsilon)^2}{2(np+\...
D. S.'s user avatar
  • 69
11 votes
4 answers
229 views

Upper bound for 1-Wasserstein distance between standard uniform and other distribution on $[0,1]$

I want to use the following metric to measure the distance between the standard uniform distribution and any other probability distribution on $[0,1]$. $$\int_0^1 |F(x) - x| dx$$ $F(x)$ is the cdf of ...
spencergw's user avatar
  • 161
1 vote
1 answer
41 views

Using multiple instruments to construct bounds

Suppose I have three candidate instruments $Z_1, Z_2, Z_3$ for the same endogenous variable $T$. I have no clear preference for which one the exclusion restriction is actually valid. Can I combine the ...
Papayapap's user avatar
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Seeking Lower Bound for Partition Probability in Random Variable Analysis

I am reaching out to seek assistance with a probability problem involving random variables. For each $p$ in $[1,\infty)$, consider positive random variables $X_{1,p}, X_{2,p}, \ldots, X_{n,p}$ such ...
Diego Fonseca's user avatar
3 votes
1 answer
229 views

What are bounded distributions? and can a bounded distribution hold the normality assumption?

I heard that normal distribution should be unbounded, but I want clarification about that, aren't most distributions in the real world bounded, I mean they won't go to infinity they have minimum and ...
NEA's user avatar
  • 33
3 votes
1 answer
83 views

Dependent variables are count variable with an upper bound

I need to test some hypotheses for a social sciences dissertation. In my description below, I refer to the independent as the Xs and the dependent variables as the Ys. I am expecting a straight linear ...
NutellaMonster's user avatar
6 votes
3 answers
203 views

Upper Bound on $\mathbb{E}[\frac{1}{1 + X}]$ where $\mathbb{E}[X] = a$ and $0<𝑎<1$

$𝑋$ is a positive random variable (potentially unbounded) with $0 \le \mathbb{E}[X] = a < 1$. Since $\phi(x) = \frac{1}{x}$ is a convex function, we can use Jensen's inequality to derive a lower ...
Otmane's user avatar
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1 vote
0 answers
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Chernoff bound on probability of binomial random variables is the maximum of its tail

I have this problem that I found on the paper I'm reading. In that paper, it is given that random variable $X = \sum_{i=1}^n X_i$, where each $X_i$ is Bernoulli with parameter $p=1/6$ and they are i.i....
JasonWS 's user avatar
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1 answer
70 views

Regression with bounded dependent variable

I'm using interrupted time series analysis to estimate the impact of an intervention in the same group. However, my target variable is a satisfaction index that goes from -1 to 1. How can I model a ...
Carolina's user avatar
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0 answers
63 views

Concentration inequality for hypergeometric distribution

Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
Dotman's user avatar
  • 71
2 votes
1 answer
17 views

How to interpret this model diagnostics?

A model was fit as below: m1 <- lmer(log (ld50) ~ var * strain * time + (1|rep) + (1|rep:var) + (1|strain:env), dt) The response ld50 ranges from 0.15 (lower ...
Rabin KC's user avatar
1 vote
0 answers
71 views

Why is the multivariate normal distribution is $(\Sigma, C)$ sub-gaussian?

The definition of sub-gaussian from a book I work with is: $X\in\mathbb{R}^n$ is $(\Sigma,C)$ sub-gaussian if $$\mathbb{P}(\lvert X^\top u\rvert>t)<Ce^{-t^2/(2u^\top\Sigma u)}, \qquad u\in\...
Torben I's user avatar
0 votes
0 answers
20 views

Validity of Bootstrap Inference for Bounds

I am facing the following problem: I have access to iid $X_1,\dots ,X_{N_X}$ and $Y_1, \dots, Y_{N_Y}$ from $F_X$ and $F_Y$, respectively, where $X$ stochastically dominates $Y$. My goal is to conduct ...
Tommo's user avatar
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3 votes
2 answers
137 views

Upper bounds on $\mathbb{P}[X \leq k]$ when $k > \mathbb{E}[X]$, for binomial rand. variable $X$

Let $X$ be a binomial random variable, $X \sim \mathcal{B}(n,p)$. When $k > \mathbb{E}[X] = np$, are there no Hoeffding-like bounds on the probability $\mathbb{P}[X \leq k]$? When $k \leq \mathbb{E}...
MikeEVMM's user avatar
1 vote
0 answers
75 views

Conditional Expectation in Uniform Case

Let $X$ and $Y$ be independent random variables where $X \sim uniform[\underline{x}, \bar x]$ and $Y \sim uniform[\underline{y}, \bar y]$. What is the conditional expectation of $X$ given $z = X + Y$? ...
cat123's user avatar
  • 11
4 votes
1 answer
80 views

Which regression best suits double bounded outcomes that aren't binary?

I'm considering some projects in the future which require modeling literacy outcomes as composites (such as word reading scores), which will naturally never have negative values (it's impossible to ...
Shawn Hemelstrand's user avatar
0 votes
1 answer
36 views

Tests that Quantify Deviation from Null Hypotheses

I have been delving into non-parametric tests recently, and I've come to realize that most of these tests offer only a partial perspective. For example, lets say the underlying distribution is $\theta$...
Student's user avatar
  • 235
3 votes
3 answers
122 views

Is the pairwise independence gap bounded to $\left[-\frac{1}{4},\frac{1}{4}\right]$? What about for n variables?

The independence gap is defined as $$\phi_{X_1, \ldots, X_n}(x_1, \ldots, x_n) \triangleq F_{X_1, \ldots, X_n}(x_1, \ldots, x_n) - \prod_{j=1}^n F_{X_j}(x_j)$$ where $F_{X_1, \ldots, X_n}(x_1, \ldots, ...
Galen's user avatar
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2 votes
1 answer
125 views

Upper bound for covariance of Hortvitz-Thompson Estimators

I need to bound on a covariance quantity that has come up in a sampling problem. $\widehat{Y}$ and $\widehat{T}$ are Horvitz-Thompson estimators of population totals, $Y=\sum_{i=1}^N y_i$ and $T=\sum_{...
Eaman's user avatar
  • 41
0 votes
0 answers
33 views

Explicit bounds for logistic regression parameters

To simplify things, I will ask my question in the case of simple logistic regression but I am also interested in the case with multiple explanatory variables. Let $\vec{x} \in \mathbb{R}^N$ be the ...
Steven Gubkin's user avatar
3 votes
1 answer
200 views

Expectation of first of moment of symmetric r.v. in terms of variance

Let $X$ be a symmetric random variable with bounded moments and standard deviation $\sigma$. I want to lower-bound $\mathbb E[|X|]$ in terms of $\sigma$. Here is the formal conjecture; I wonder if ...
AvidLearner's user avatar
6 votes
1 answer
243 views

What is the adequate regression model for bounded, continuous but poisson-like data?

I am trying to compare the lodging resistance scores of different wheat cultivars in an agronomic trial. Lodging is the phenomenon in which wheat plant can bend and lean closer to the ground as a ...
Dangerbuee's user avatar
1 vote
1 answer
28 views

Hypothesis test for a parameter when only the upper and lower bounds of the parameter are estimable

Consider a null hypothesis: \begin{align*} H_0:\;\beta=0 \end{align*} Here, we can estimate only the upper and lower bounds of $\beta$. To be clear, let the upper and lower bounds of $\beta$ be $\...
MinChul Park's user avatar
0 votes
0 answers
57 views

How to bound neural network output?

I have a NN with a single output scalar. I want this scalar to tend towards positive infinity if some of the inputs take on certain values. How can I guarantee this without adding training data?
interatomic's user avatar
1 vote
0 answers
106 views

Number of samples for Hoeffding's Bound with Gaussian R.V

I am trying to obtain the required number of sample $n$ for a given confidence interval $\alpha$ and $X_1 ... X_n$ which are Gaussian rv with $\mu$ mean and $\sigma^2$ variance. I know that \begin{...
victoria's user avatar
0 votes
0 answers
61 views

What can be concluded when standard deviation plus mean exceeds largest value?

The sum of the mean and standard deviation of a non-normal distribution can exceed the value of the largest sample. For a good explanation of why, see Can mean plus one standard deviation exceed ...
jsbox's user avatar
  • 101
1 vote
0 answers
12 views

What is the CRLB of the hyper-parameters of a Gaussian process kernel by using marginal likelihood

I want to derive the CRLB of the hyper-parameters contained in a covariance kernel of a Gaussian process. My kernel looks like the following. $$ K(t, t^{\prime}) = \exp(-\sigma^2/2 (t - t^{\prime})^2) ...
CfourPiO's user avatar
  • 235
0 votes
0 answers
110 views

Why unbounded above activation function is important for training

One of the desirable properties of activation functions is to be unbounded above and bounded below. I guess part of the reasons why it should be unbounded above is to avoid vanishing gradient problems ...
Avv's user avatar
  • 249
1 vote
0 answers
87 views

Asymmetries in the DKW bound

Suppose I have $n$ i.i.d. samples $X_1,...,X_n$ drawn from a distribution with CDF $F$. We use the samples to form the empirical CDF: $$F_n(x)=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{X_i\leq x}$$ The ...
Bill Bradley's user avatar
1 vote
1 answer
139 views

If a random variable is bounded by a constant with high probability, is its expectation also bounded by the same constant with the same probability?

Suppose $X$ is a random variable that is bounded with high probability, i.e., $|X| < M$ for some $M \in \mathbb{R}^+$ with probability $1-p$. Is it correct to say that $\mathbb{E}(|X|)<M$ with ...
user370354's user avatar
4 votes
1 answer
455 views

Upper bound for m.g.f

$X$ is a discrete random variable from power series family (e.g., binomial, poisson etc.). is it possible to find an upper bound for the m.g.f of $X$? N.B: from stack exchange I obtained the following ...
BTM's user avatar
  • 175
3 votes
0 answers
111 views

Relation between generalization bounds of Kernel Ridge Regression and largest eigenvalue of the kernel Gram matrix

Consider a positive-definite, symmetric function $k(x_1, x_2)$ which is used, given the dataset $\{(x_i, y_i)\}_{i=1}^m$, to construct the Gram matrix $K = [k(x_i, x_j)]_{i,j \in 1, ..., m}$. What is ...
incud's user avatar
  • 41
2 votes
1 answer
48 views

Can we give high-probability exponential bounds on the slope of the linear regression function?

Suppose $(X,Y), (X_1,Y_1),(X_2,Y_2),\dots$ is a $\mathbb{P}$-i.i.d. sequence of pairs of real-valued random variables such that the support of $\mathbb{P}_{(X,Y)}$ is contained in the square $[-1,1] \...
Bob's user avatar
  • 193
2 votes
0 answers
111 views

Bound on the expectation of a function of random variable having a strictly log-concave probability density

let $\theta \in \mathbb{R}^d$ be a random variable having a strictly log-concave probability density function, i.e \begin{equation} p(\theta) = e^{-\phi(\theta)} \end{equation} where $\phi(\theta)$ is ...
zsheeba's user avatar
  • 21
0 votes
0 answers
37 views

Bound over sample probability

I have a discrete random variable $x \sim \text{Cat}(\textbf{p})$. What I'm trying to compute is the probability that any sample $x$ has an associated probability of at least $\alpha$. I would compute ...
robertc's user avatar
2 votes
1 answer
72 views

Is this derivation in Manski (1990) correct?

Consider the following setting. There are two treatments, $A,B$. Individuals in the population are described by a tuple $(y_A,y_B,z)$ where $z \in \{A,B\}$ denotes the treatment received. Only $y_A$ ...
stats_model's user avatar
  • 2,465
3 votes
3 answers
623 views

Given a correlation between A and B, what are the best bounds on the product of the AC and BC correlations?

Let's say I have three vectors (or random variables) $A, B,$ and $C.$ I can of course calculate the correlation between any of them (and have these numbers). However what I'm interested in is if there'...
user2551700's user avatar
1 vote
1 answer
174 views

Is there an example of a Lipschitz function of a Gaussian vector for which $f(Z)-\mathbb{E}[f(Z)]$ is not sub-Gaussian

Definitions: A random variable $X$ is called sub-Gaussian with parameter $\sigma^2$ if there exists $\sigma \in \mathbb{R}$ such that $$\forall \lambda \in \mathbb{R} \quad \mathbb{E}[e^{\lambda X}]\...
Pablo Ortega's user avatar
1 vote
1 answer
147 views

Why is the lower bound of the confidence interval of a model's error relatively constant compared to the upper bound? [closed]

I am interested in studying the effect of increasing data samples for a regression model on train error and test error. For this I have used 95% confidence intervals for different values of a sample ...
user481031's user avatar
0 votes
1 answer
107 views

What is the definition and upper bound on the variable "m" in the definition of the multivariate normal Fisher Information?

Multivariate normal distribution [edit] The FIM for a $N$-variate multivariate normal distribution, $X \sim N(\mu(\theta), \Sigma(\theta))$ has a special form. Let the $K$-dimensional vector of ...
user avatar
2 votes
1 answer
98 views

How to compute a non-trivial lower bound on $P \left[|X| > \frac{|\alpha|}{2} \right]$?

Let $X$ be a random variable such that $E[X] = \alpha$, $\alpha \in \mathbb{R}$ and $E[X^2] = \beta$. The problem is to find a lower bound on the following probability $$ P \left[|X| > \frac{|\...
Bhisham's user avatar
  • 319
2 votes
1 answer
92 views

What is the probability that every variable of combination of random variables is greater than a specific value?

Suppose there are $N$ positive random variables. Each variable follows an exponential distribution with parameter $\lambda_i$. Now, we choose $n$ variables among the $N$ variables. What is the ...
Cong's user avatar
  • 21
1 vote
0 answers
23 views

Lower bounding the sum of product of two sub-Gaussian variables where one follows an AR(1) process

Suppose we have the sum \begin{equation} \sum_{t=2}^{n}\epsilon_{t-1}u_t \end{equation} where $\epsilon_t$ and $u_t$ are both sub-Gaussian variables. Further suppose that while $u_2,\cdots,u_n$ are i....
Carl's user avatar
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