Questions tagged [bounds]
Bounds represent the points with which data cannot exceed, such as minima or maxima.
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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, ...
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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 ...
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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 $$\mathbb{P}(\lvert X^\top u\rvert>t)<Ce^{-t^2/(2u^\top\Sigma u)}, \qquad u\in\...
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Validity of Bootstrap Inference for Bounds
I am facing the following problem: I have access to iid $X_1,\dots ,X_{N_X}$ and $Y_1, \dots, Y_{N_Y}$ from $F_X$ and $F_Y$, respectively, where $X$ stochastically dominates $Y$.
My goal is to conduct ...
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Upper bounds on $\mathbb{P}[X \leq k]$ when $k > \mathbb{E}[X]$, for binomial rand. variable $X$
Let $X$ be a binomial random variable, $X \sim \mathcal{B}(n,p)$.
When $k > \mathbb{E}[X] = np$, are there no Hoeffding-like bounds on the probability $\mathbb{P}[X \leq k]$?
When $k \leq \mathbb{E}...
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Conditional Expectation in Uniform Case
Let $X$ and $Y$ be independent random variables where $X \sim uniform[\underline{x}, \bar x]$ and $Y \sim uniform[\underline{y}, \bar y]$.
What is the conditional expectation of $X$ given $z = X + Y$?
...
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Which regression best suits double bounded outcomes that aren't binary?
I'm considering some projects in the future which require modeling literacy outcomes as composites (such as word reading scores), which will naturally never have negative values (it's impossible to ...
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Tests that Quantify Deviation from Null Hypotheses
I have been delving into non-parametric tests recently, and I've come to realize that most of these tests offer only a partial perspective.
For example, lets say the underlying distribution is $\theta$...
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Is the pairwise independence gap bounded to $\left[-\frac{1}{4},\frac{1}{4}\right]$? What about for n variables?
The independence gap is defined as
$$\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, ...
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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_{...
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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 ...
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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 ...
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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 ...
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Generalization bound on out of distribution data
Assume we have two sets of data $X_1$ and $X_2$ drawn from two different distributions. Are the loss of the empirical risk minimizer of $X_1$ on $X_2$: $l_{X_2}(f_{X_1})$ the same as the loss of the ...
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Using Dudley Integral to estimate maximum singular value of Gaussian random matrices [duplicate]
On Exercise 5.14 of Wainwright, it provides a way to estimate maximum singular value of Gaussian random matrices using the one-step discretization bound and Gaussian comparison inequality.
Can we use ...
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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 $\...
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General Non-Uniform Berry-Essen
Let $f_n(x)$ by the probability distribution function of a continuous r.v. $X_n$. $X_n$ converges in distribution to $X$, i.e. $|P(X_n < x) - P(X < x)| \rightarrow 0$. On the top of that, $E[|...
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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?
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How to bound neural network output? [duplicate]
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?
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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{...
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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 ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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] \...
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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 ...
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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 ...
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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 $y_A$ ...
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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'...
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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
$$\forall \lambda \in \mathbb{R} \quad \mathbb{E}[e^{\lambda X}]\...
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Why is the lower bound of the confidence interval of a model's error relatively constant compared to the upper bound? [closed]
I am interested in studying the effect of increasing data samples for a regression model on train error and test error. For this I have used 95% confidence intervals for different values of a sample ...
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What is the definition and upper bound on the variable "m" in the definition of the multivariate normal Fisher Information?
Multivariate normal distribution [edit] The FIM for a $N$-variate
multivariate normal distribution, $X \sim N(\mu(\theta),
\Sigma(\theta))$ has a special form. Let the $K$-dimensional vector of
...
user318514
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How to compute a non-trivial lower bound on $P \left[|X| > \frac{|\alpha|}{2} \right]$?
Let $X$ be a random variable such that $E[X] = \alpha$, $\alpha \in \mathbb{R}$ and $E[X^2] = \beta$. The problem is to find a lower bound on the following probability
$$
P \left[|X| > \frac{|\...
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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 ...
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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....
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Predicting limits of bounded dependent variable in Random Forest
I am new to machine learning and trying to use Random Forest to predict a bounded dependent variables (percentage from 0 - 100). The majority of the training data points (~80%) are at the limits of ...
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How to find the bound of a time dependent variable?
I am working on modeling my problem statistically and I need to Know the bound or range of my variable. It is a time series variable position(t) where
position(t+1) = position(t) + a.X - (1-a).Y
a is ...
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Bounding sum of quartic deviations from sample mean
[Cross-posted here with no answers for a few days]
I came - to the very best of my knowledge from reading the source - across the following statement in The Jackknife and Bootstrap, Shao and Tu, p. 87:...
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Can we find upper bound for loss functions?
Is it easy to find upper bound for loss functions like 0-1 loss and hinge loss ?!. I always find this sentence, which is "hinge loss is an upper bound of 0-1 loss", Can we compute the upper ...
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Interpretation of upper bound on the Wasserstein Distance
I am trying to interpret the 2-Wasserstein distance and the upper bound on it. Let's say I have 2-Wasserstein distance between two distributions to be $x$, and I have an upper bound on it which gives ...
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What code lengths can optimal prefix codes assign to the symbols in a given probability distribution?
(Notation) Consider a finite alplhabet $\Sigma\equiv \{x_1,...,x_n\}$, corresponding to a probability distribution $\{p_1,...,p_n\}$. I want to encode this using a uniquely decodable binary code.
Let $...
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How can one show that $\bar{X}$ is the best unbiased estimator for $\lambda$ without using the Cramèr-Rao lower bound?
Assume we have the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\sigma^2 < \infty$. We have that $E[S^2] = \sigma^2$, where $S^2 = \sum_{i = 1}^n \dfrac{(X_i - \bar{X})^2}{n - 1}$ ...
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Upper bound for variance of $\hat{\beta}$ in multiple linear regression
The variance of the beta estimator in an ordinary-least-squares multiple linear regression to express $Y$ as a (linear) function of $X$, $\hat{\beta}$, can be expressed as (knowing $X$ and $\sigma^2$ ...
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How to generate samples of ARIMA(p,d,q) model within an interval?
I am want to generate samples from an ARIMA(p,d,q) or ARMA(p,q) model. There is a Python Package to generate ARMA samples. The problem is that I want to generate scenarios for demand which should be ...
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What Cramer-Rao bound should I use?
I have been researching about the Cramer-Rao bound and I have found two inequalities:
$$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\...
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Show $(E|X|^2)/(E|X^2|) \leq P(X \not =0)$
I'm looking to show this inequality is true, and in turn use it to conclude the second moment method's bound.
Show that $\frac{E|X|^2}{E|X^2|} \leq P(X \not =0)$.
Again, I'm not supposed to use second ...
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Upper Bound for 2nd Raw Moment of Positive Random Variable
Let $X$ be a random variable with support $(0,\infty)$. All I know about $X$ is the support, finite higher moments, and $\mathbb{E}(X)=\mu$. I am trying to come up with a more tractable upper bound ...
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