Questions tagged [bounds]

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Sampling a proposed value with a limited range target when running MCMC [duplicate]

I want to do an MCMC algorithm and need to sample a proposed value from a proposed distribution. In the Metropolis algorithm, people usually use a normal distribution as proposal. But if the prior ...
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1answer
16 views

Is a range of values from an exponential distribution still exponentially distributed?

I have to generate numbers of two different exponential distribution ($e_1, e_2$) with parameters respectively $\lambda_1$ and $\lambda_2 = k \lambda_1$, with $0<k<1$. But I also want to ...
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8 views

Rate of convergence of variance of kernel averages

I'm reading Hansen's (2008, p. 729) Theorem 1 where he bounds the variance of averages of the form $$\hat\Psi(x)=\frac{1}{Th}\sum_{t=1}^T Y_t K\bigg(\frac{x-X_t}{h}\bigg)$$ given that $\{(Y_t,X_t)\}_{...
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Bounding variance of noise in a noisy voting scheme

I am looking at the accuracy of a method of human yes/no voting. Essentially, I have the vote totals for a number of binomial processes, which represent different "elections" this method was ran on. ...
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Markov inequality and Boundness in probability

Let $\{X_n\}$ and $\{a_n\}$ be sequences of random variables and real numbers, respectively. Say that $X_n=O_P(a_n)$ iff $\forall\epsilon>0$, $\exists N,M>0$ such that for all $n>N$, we ...
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Calculate bounds of the sum of means of a normal distribution [closed]

I am not sure whether this question really belongs on this StackExchange or whether I should post it on a different one. Please indicate if this is the case. I am programming an algorithm (A1) and I ...
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1answer
24 views

Definitions of VaR (Value at Risk)

Here is the definition of VaR (Value at Risk) taken from McNeil, Alexander J., Rüdiger Frey and Paul Embrechts (2015), Quantitative risk management: Concepts, techniques and tools: $$ \textrm{VaR}_{\...
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1answer
81 views

Bounds on $P(Y, X)$ with $P(Y)$ and $P(X)$ known, as well as $X \geq Y$

Suppose you know the marginal distribution of two random variables, $P(Y)$ and $P(X)$. There are well-known bounds on the joint distribution $P(X, Y)$ that use this information. However, suppose you ...
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1answer
33 views

Bounding data by two parallel lines with minimum distance between them

I have a set of data samples that approximately follow a straight line in 2D. I need to find two parallel lines that are spaced as close as possible such that all of the samples lie between the lines. ...
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1answer
27 views

Model/link function to deal with dependent variable in range [-1,1]?

My dependent variable, $Y$, contains values anywhere from -1 to 1 (i.e. it is bounded continuously on the range $[-1,1]$). I know that a regular OLS regression on such a variable would sometimes ...
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1answer
27 views

Is there a equivalence test for beta coefficients in regression analysis?

There are established ways to rule out medium/high effects like TOST for two-groups. But is there a way to rule out medium/high effects in one multiple regression? Maybe using eta-squared? What ...
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1answer
51 views

Significance test with non-normal, bounded data?

I am attempting to do a one-sample significance test to determine whether a set of data differs from a given value (0 in this case). The issues I have with these data: Non-normally distributed data, ...
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2answers
114 views

Estimator with variance equal to Cramér-Rao lower bound in $N(x_i\theta,1)$-distribution

Let $Y_1,\ldots, Y_n$ be independent and $N(x_i\theta,1)$ distributed, with for each $Y_i$ a mean of $x_i\theta$ for known $x_1,\ldots,x_n$. In a previous section of this exercise I found that the ...
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Prove that $T(\textbf{X}) = \hat{\sigma}^{2}$ reaches the Cramer-Rao bound

Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\mathcal{N}(\mu,\sigma^{2})$, where both parameters are unknown. (a) Prove the normal probability density function ...
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Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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1answer
21 views

Bound for density of random variable with finite second moment

Let $\mathbf{X}$ be a vector-valued random variable with finite second moment and density $\rho$. Assume that $\rho$ is bounded and continuous. As $\mathbf{X}$ has finite second moment, I hope to find ...
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1answer
39 views

Dealing with measurements falling outside of the theoretical range/boundaries of the data

Imagine I am measuring a bounded variable (with a maximum possible value above which the data doesn't make sense) and I end up with the following dataset with my measurements and measurement errors as ...
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87 views

Regression with bounded non-normal dependent variable

I'm wondering what a suitable regression model would be to predict a bounded, continuous, non-normally distributed dependent variable from a binary explanatory variable with partially crossed data. I'...
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References for generalization bounds?

I'm looking for references (books, papers, lecture notes etc) on generalization bounds and their proofs. Specifically, I'm looking to fully understand the technique of defining a hypothesis class (or ...
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2answers
41 views

Upper bound of normal cdf

Random variable $X\sim N(0,1)$. Show that, $P(X\geq c) \leq e^{-ct+ \frac{t^{2}}{2}}$ for $c>0$ and for all $t$ in $R$. I found that $P(X\geq c) = \Phi(-c)$ where $\Phi(x)=\int_{-\infty}^{x}\phi(u)...
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Manipulation of asymptotic bounds for distance between estimators

Suppose I know some asymptotic bounds: $$\mathbb{E}(|D(a,\hat{a})|) \lesssim O(n^{-1/2}),$$ where $D$ is some distance between probability measures, and $a$ is a probability measure while $\hat{a}$ ...
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1answer
28 views

Getting From Concentration Inequality to Interval Length

I've seen this used some times and I would like to ask what steps are taken on the way to getting there: E.g. assuming bounded variance, we can use Chebyshev concentration inequality: for any $t>0$...
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10 views

On the practical usage of generalisation error bounds

(This questions is based on a question that I've posted previously here, but I would like it to get more exposure) In many practical scenarios, one would like to answer how much more data is needed ...
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Linear regression when Y is bounded and discrete

The question is straightforward: Is it appropriate to use linear regression when Y is bounded and discrete (e.g. the test score 1~100, some pre-defined ranking 1~17)? In this case, is it "not good" to ...
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41 views

Bounds on Expectation $E[A(B-C)^2]$

[This question has been edited for more given conditions]. Given possibly correlated random variables $A,B,C$, I want to find the best upper bound for $E[A(B-C)^2]$ given the following: $E[A(B-C)]$ $...
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Bound for type of correlation measure

Assume you have a finite, discrete probability distribution for a joint random variable and such that $P(X=i,Y=j) = p_{i,j}$ for $i \in \{1, \dots, |X|\},j \in \{1, \dots, |Y|\}$. The marginal ...
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28 views

Bounds on quantiles of the minimum of summations of (possibly dependent) random variables

Suppose I have $2N$ continuous random variables $X_1, \ldots, X_N, Y_1, \ldots, Y_N$ and that I can evaluate the quantiles of the respective distributions. Given a value $w \in [0, 1]$ I would like to ...
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1answer
41 views

Is there an upper bound on number of logistic regression responses that yield infinite estimates

Suppose a logistic regression problem has N observations of {0, 1} and that there are p parameters. Also assume the design matrix, X, is full rank with p < N. We know that there will be certain ...
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37 views

Approximation of the upper bound on the expectation of log sum of exponentials

I am having some trouble replicating the results in Guillaume Bouchard's paper, Efficient Bounds for the Softmax Function and Applications to Approximate Inference in Hybrid Models, where he discusses ...
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1answer
61 views

Finding Chernoff bounds maximum estimators

I am currently trying to resolve the following exercise about Chernoff bounds: Let $X_{1}, X_{2}, \dots, X_{n}$ be independent, identically distributed (i.i.d) random variables with distribution $N(0,...
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Reference request: Least Squares Concentration Bounds

I've recently become interested in parameter estimation for regression and time-series type models and I often encounter, and indeed need to understand, results using concentration of measure-type ...
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82 views

Tight upper bound on the expectation of a concave function

N is a random variable whose sample space is [0,$\infty$). I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The ...
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1answer
30 views

Bounded functions when x tends to infinity

Please help me understand the below: The notation $g(x) = O(f(x))$ denotes that $\left|\frac{f(x)}{g(x)}\right|$ is bounded as $x \to \infty$. For instance if $g(x)=3x^2 + 2$, then $g(x) = O(x^2)$ ...
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1answer
78 views

Find bayesian credible Interval

I am taking this example from here. They have given the steps but i cannot understand them so a little dumbing down of answer is necessary or you can just explain the answer given there. its on page 5 ...
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124 views

Is it true that normalizing the output of a ReLu feedforward Neural Network that its Rademacher Complexity becomes a constant?

I was trying to understand what happened with the Rademacher Complexity: $$ R_S(F) = \frac{1}{m} \mathbb E_{\sigma} [\sup_{f \in F}\sum^m_{i=1} \sigma_i f(z_i)] $$ or $$ R_{P,m} = \mathbb E_{s \...
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1answer
256 views

ELBO interpretation in Variational Autoencoder (VAE) for anomaly detection

How to interpret different ELBO values when checking anomaly detection possibilities of VAE model on different "testing" datasets? The higher the ELBO value of the model when testing it on different ...
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1answer
165 views

Fréchet Hoeffding bounds for symmetric random variables

(Edited to clarify the question). The Hakan & Demirtas (2012 doi: 10.1198/tast.2011.10090) approach to approximating Pearson correlation bounds uses the concept of the Fréchet-Hoeffding bounds by ...
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What's the reasoning behind non-seasonal ARIMA model lag-order bounds?

I read some papers on the non-seasonal ARIMA model, and the consensus I've seen is that for ARIMA(p, d, q), p and q should not be greater than 3, maybe 5. What's the reasoning for that? Is it for ...
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261 views

Bounds on the difference of correlated random variables

Given two highly correlated random variables $X$ and $Y$, I'd like to bound the probability that the difference $ |X - Y| $ exceeds some amount: $$ P( |X - Y| > K) < \delta $$ Assume for ...
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Is max. Eigenvalue of k-sparse PCA always $\leq$ max. Eigenvalue of normal PCA on same dataset?

Is max. Eigenvalue of k-sparse PCA always less than or equal to the max. Eigenvalue of normal PCA on same dataset? K refers to the number of non zero eigenvalues when the dataset is of dimension n <...
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Expected value of (F(x)/x(1-F(x)) where F(x) is CDF?

I was solving a problem and encountered the following: \sum (f(x)/x) (F(x)/(1-F(x)) Here f(x) is pmf and F(x) is CDF. Is expected value of F(x)/x(1-F(x)) a known function? I found out that f(x)/(1-F(...
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78 views

Concentration inequalities for weighted sums of gaussians

Suppose that $x \sim \cal{N}(0,I_d)$ be a $d$-dimensional standard Gaussian vector and let $x_1,\ldots,x_n$ denote $n$ i.i.d. samples drawn from the same distribution. For some fixed vector $\theta \...
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2answers
528 views

Understanding the concept of “Bounded in probability”

My statistics book defines the concept of "bounded in probability" in the followng way: ..But doesn't this mean that any sequence of R.V.'s that does not include any R.V.'s with a pdf with infinite ...
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Bounds on probability that a random variable is neither the maximum nor the minimum of a set of random variables

Suppose I have $n$ independent random variables $X_1,\dotsc,X_n$ which are Poisson distributed with $X_i \sim Poi(\lambda_i)$. Without loss of generality, an additional condition $\lambda_1\le\...
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Comparing bound between lasso and OLS

Assume fixed design regression model $Y = \mathbf{X} \beta + \epsilon$ with common assumptions. Lasso estimator $\hat{\beta}_\lambda$ can be shown to have the following bound. For any $\delta > ...
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87 views

Bounds on variance and mean of maximum of difference of independent random variables

Suppose $X_1,\dotsc,X_n$ are independent but not necessarily identical random variables. $$Y = \max_{1\le i,j\le n}(X_i-X_j)$$ What upper and lower bounds can be derived for expectation and variance ...
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678 views

Upper bound on KL divergence

Is there a maximum (unique?) to the KL divergence between discrete distributions p & q, with the restriction that q is a proper probability distribution? I know KL is unbounded from above when q ...
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41 views

Correlation bounds for uniformly distributed matrix?

For a uniformly or Guassian distributed $M\times N $ matrix. Is there any analytical expression in terms of $M$ and $N$ to estimate the maximum and minimum bounds of correlation between the columns of ...
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84 views

Tail bounds for F-distribution (not using $\chi^2$ bounds)

Are there any sharp tail bounds for an $F_{p,q}$ distribution? That is, if $X \sim F_{p,q}$, then for a $t_1,t_2 > 0$, what are the sharpest $\delta_1$ and $\delta_2$ known such that $$P(X > ...
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80 views

Bounds on tail conditional expectation of random variable given variance

Given a random variable $X$ with CDF $F(X)$, mean $E(X)=0$, and variance $Var(X) =\sigma^2$, I would like to bound the tail conditional expectation where $X$ is in the tail with probability $1-p$: $E(...