Questions tagged [bounds]

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Equivalence testing: Is it appropriate to set the equivalence bound such that I can reject H0 at alpha=0.05?

I have conducted a survey. One sample answered a binary question (answer A or B), once with and once without treatment. Now there does not seem to be a treatment effect as the proportions of answers ...
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17 views

Causal AR Model?

This questions is about necessary conditions (in form of inequality on coefficients) for the causality of autoregressive models. For instance, $|\phi_1| < 1$ is a necessary condition for an AR(1) ...
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15 views

How to derive this MAE error bound on the central limit theorem?

Is this derived from Chebyshev's inequality or a tail bound theorem? If not, how was it derived? Does this require the existence of the third moment? Does this bound suggest the normal approximation ...
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17 views

Factorial moment bound for discrete Binomial distribution

I need to compute the upped bound for the tail (survivor) probability $P(X \ge t)$ for the discrete Binomial random variable $X$. I could use Chernoff bounds, however according to this paper [1] the ...
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10 views

Theoretical lower limit of area contained in 1 sigma interval of a unimodal distribution

It is known that in case of a normal distribution, the interval of one standard deviation around the mean, $\mu \pm 1\sigma$, contains about $68\%$ of the data. When considering an arbitrary ...
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7 views

Occam Bound using Relative Chernoff Bound

I'm having a bit of a trouble trying to understand one step in the proof of an Occam Bound (Theorem 1) in the paper "A PAC-Bayesian Tutorial with A Dropout Bound" (https://arxiv.org/pdf/1307.2118.pdf) ...
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20 views

Choosing constants for probabilistic bounds

I am studying probabilistic bounds and I have a question regarding how to choose constants from complexity classes. Specifically, consider a biased coin which has the probability of one side $p = \...
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12 views

Time series ARDL-model interpretation

Anyone knows how the interpretation of these ARDL models? Bounds test & ECT are easy to interpretation. But I want to say something about the short run and long run effects of variables. Long ...
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1answer
27 views

How to using the Markov Inequality to find the upper bound for $\mathbb{P}(X > 2)$ given I only have information about $X^4$?

Let $X$ be a nonnegative random variable that satisfies $\mathbb{E}[X^{4}]=4$ . How should I calculate an estimate for the $\mathbb{P}(X \geq 2)$ using the Markov Inequality? I tried to find a ...
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ARDL-model diagnostics, what if the model is unstable CUSUMSQ?

What if your model is unstable after performing diagnostics like the CUSUM and CUSUMSQ? Only cusumsq does not seem to be like it should be.. Can I still give an interpretation of the coefficients? ...
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43 views

Which Distribution functions with increasing hazard rate has x(1-F(x)) tending to 0 when x tends to infinity?

Let $F(x)$ be a cumulated distribution function and $f(x)$ the probability density function with an increasing failure rate (IFR or hazard rate), ie $h(z)=f(x)/(1-F(x))$ is increasing. Which ...
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1answer
38 views

Bounding the structural-risk-minimization (using Hoeffding's inequality twice)

tl;dr: The main question is if I use an inequality that is true with a certain probability (confidence) twice, do I get the same confidence? Original: I've got the following exercise: Where $e_p(h)...
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10 views

Bounded Model Prediction Error

I have a predictive model (not ML based, uses first principles from a science textbook) and I would like to have a confident bound on on the error of the predictions. I am able to collect many ...
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23 views

Rigid regression - show $||w||_2$ is $O(\lambda ^ -1)$

Relevant question: Ridge regression formulation as constrained versus penalized: How are they equivalent? I've got an assignment to show that in rigid regression the coefficients vector $L_2 $ norm, $...
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1answer
62 views

Can you bound the third moment from the second moment?

Suppose $X$ is a random real variable with zero mean and finite second moment $\langle X^2\rangle$. Under what conditions can we give a bound (upper/lower) for the third moment $\langle X^3\rangle$?
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Are the following terminologies error/risk/marmgin/regret bounds related?

I recently come across papers with titles resembling "Error/Risk/Margin/Regret Bounds" and I can't help but wondering if there is any fundamental (mathematical) difference between these terminologies? ...
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29 views

How to compare the distributions of censored data?

Is there a way to test if the distributions of the two samples of censored data? As the data is not defined exactly, Kolmogorov-Smirnov test does not seem to be directly applicable. Generally ...
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markov's inequality generalizability

Let $X : \Omega \rightarrow \mathbb{R}$ be a non-negative random variable on probability space $(\Omega, \mathscr{A}, P)$ and let $c > 0$. Then: $$\mathrm{P}[X > c] \leq \frac{\mathbb{E}(X)}{...
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1answer
55 views

the approximation of the variance of MLE (Cramer-Rai Lower Bound)

This is in In Casella's Statistical Inference,page 473, the approximation of the variance of MLE (Cramer-Rao Lower Bound). I really confused with the conclusion: $Var_{\hat{\theta}}h(\hat{\theta})$ ...
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Lower bound for regression task

I have a dataset $\mathbf{X} = \{ \mathbf{x_1},\mathbf{x_2},...,\mathbf{x_n} \}, x\in\mathcal{R}$ with length $n$ and dimension $d$ along with corresponding labels $\mathbf{y}, y \in \mathcal{R}^+$. ...
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1answer
38 views

Upper Bound on the Wasserstein Distance

I'm interested to know if it's possible to construct an upper bound on the Wasserstein distance in terms of the Kolgomorov distance. The Wasserstein distance can we written as $$W_{1}\left(F, G\...
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1answer
46 views

Fourth moment bound for unit-variance distribution

Given that a random real variable $X$ has zero mean and variance equal to 1, can we bound its fourth moment $\langle X^4\rangle$ (assuming it exists)?
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Comparing numerical stability and computing bounds on the condition number of learned weights

I have an empirical risk minimization problem with two equivalent losses that solves it, $f_1(x; \theta_1)$ and $f_2(x ; \theta_2)$, where $x$ is the data and $\theta$ are the model parameters (in ...
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Exact inference in an approximate model as opposed to approximate inference in an exact model?

I remember hearing a while ago that it was more rigorous to perform approximate inference in an exact model as opposed to exact inference in an approximate model. I can’t now remember the reasoning ...
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1answer
40 views

Changing only one point of a discrete distribution to maximize variance augmentation

X has a discrete distribution with support $x1, x2, ...$ in $ {]}0,1{[}$. You have the right to change only one of the $xi$ to lead to the highest increase in variance (or, at least, a systematic ...
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Lower bound/ upper bound of standard deviation

The attached screenshot is from the 2 page of following publication Dissolution test results. The upper bound for standard deviation is calculated in cell B7. The formula for the calculation is also ...
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Bound on sample size- Hoeffdings inequality

Studying for my upcoming statistics exam I tried to solve the following: In some population, each individual likes exactly one out of 30 possible music genres. In some survey, n people are drawn ...
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19 views

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
18 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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13 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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25 views

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
28 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
107 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
43 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
34 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
53 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
77 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
246 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
25 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
41 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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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
305 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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1answer
32 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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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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2answers
45 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: $E[A(B-C)]$ $...