Questions tagged [bounds]

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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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15 views

Inferring bounds from joint typicality on three variables

Consider the following exercise from Cover and Thomas: And the given solution from the solutions manual: It is reasonably clear that these bounds are valid (one simply follows the counting argument ...
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85 views

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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1answer
36 views

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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1answer
94 views

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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1answer
30 views

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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1answer
30 views

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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9 views

Expected number of hits of decision nodes in decision tree

Given a dataset $X\in \mathbb{R}^{n \times m}, Y \in \{0,1\}^n$. One can fit a decision tree model. Assuming the fitted decision tree have $p$ Decision node. If we a child node is about feature $i$, ...
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1answer
25 views

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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54 views

What does knowing two pairwise copulas tell us about the third

Say we have three random variables, which are all standard uniforms: X ~ U(0,1), Y ~ U(0,1), and Z ~ U(0,1) If we know two of the pairwise copulas, $C_{XY}$ and $C_{YZ}$, what can be said about the ...
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“Z-value” equivalent for sample variance

For a random variable $X$ (mean $\mu$, variance $\sigma^2$, kurtosis $\kappa$), I take $n$ i.i.d. samples $X_1,\dots,X_n$ and find their mean, $\hat \mu^{(n)}$. By linearity of expectation, I know it ...
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Bounding values of a Dirichlet distribution

Consider $k$ random variables $X_1, X_2, \ldots, X_k$ such that $(X_1, X_2, \ldots, X_k)$ follow a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution. For a large enough $k$, I am trying to bound/find ...
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31 views

How to derive Chernoff Bounds for Sample Variance?

I was reading a paper on Bandits where I encountered this: After searching around on the internet I found and understood the first set of bounds quite well. However, I could not find any explanation ...
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Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that?

Background I am taking an introductory course on probability and inference. We recently covered several useful inequalities which I will list below: Markov's Inequality Let $X$ be a non-negative ...
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1answer
46 views

Conjectures regarding EM approximations of mixtures of multivariate normal distributions

Consider $X\in\mathbb{R}^{N\times d}$ containing data for $N$ points in $d$ dimensions drawn from a bimodal multivariate normal distribution, where any row $x$ of $X$ follows the mixed multivariate ...
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1answer
26 views

Measures of correlation / influence for predictors with bounded outcome

I'm doing a systematic review of epidemic models that project "the % reduction in incidence ($Y$) after K years" given a particular simulated intervention. The models include various ...
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58 views

Which regression model distribution or transformation for data bounded between -1 and 1?

It seems quite common in studies of plant interactions to find response variables that are bounded between -1 and 1, such as this relative interaction index (from Armas et al 2004, Ecology 85, https://...
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93 views

How does maximising ELBO for a Gaussian mixture model fit the model to data?

I am following along in Bishop's Pattern Recognition and ML chapters 9 and 10, and I understand that the EM algorithm works by iteratively updating model parameters using equations derived from ...
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35 views

Bounds on distance between two independently variables drawn from the same distribution

Suppose $X_1$ and $X_2$ are iid from an arbitrary distribution with variance $\sigma^2$. How can we derive an upper bound for: $$P(|X_1-X_2|\ge\delta)$$ One simple idea is Chebyshev's Inequality, ...
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Are there supposed to be bounds on parameters in 2PL Item Response Theory models?

Recently I've been studying Item Response Theory (IRT) and have come across some issues with the application side of it. I currently have a dataset of ~200 respondents x 7405 questions (quite ...
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15 views

How to find upper and lower bound

Let $\Sigma \in S_{++}^n$ be a symmteric positive definte matrix with all diagonal entries one. Let $U \in R^{n \times k_1}$, $W \in R^{n \times k_2}$, $\Lambda \in R^{k_1 \times k_1}$ and $T \in R^{...
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28 views

Bounding the norm of the difference between two related probability densities

Suppose we have a continuous random variable $X$ and two continuous functions $f$ and $g$ such that $f(X)$ and $g(X)$ are continuous random variables. Let $p_A$ be the probability density function of ...
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2answers
373 views

Is there a statistical distribution whose values are bounded $[-1,1]$ and sum to 1?

The Dirichlet distribution contains values that are bounded $[0,1]\in \mathbb{R}$ and sum to $1$. Is there a parametric distribution or similar method whose values do the same but reach as low as $-1$?...
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27 views

How to get this bound?

I read the following part in a paper, it is trying to show that the difference between $g(x,\gamma)$ and its linearized version is small. Here $g(z,\gamma)$ depends on two generic functions $\gamma=(\...
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25 views

What is a common-sensical approach to setting the boundaries of an interval?

As I am trying to present my results to a non-expert audience, I am wondering about what the most commonly used boundaries are for intervals. I mean specifically, which of the four versions explained ...
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How to bound a regressor function?

I've seen similar questions on here, but none seem to quite apply to my use case. I want to predict Metacritic scores bases on a number of features. Metacritic scores are bounded to a 0-100 scale, ...
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1answer
218 views

Use Chebyshev's inequality to find a lower bound of a Chi-Square Distribution

I'm trying to solve the following exercise but I'm not sure if what I'm doing is right. "Let $X$ be an r.v. distributed as $\chi_{40}^{2}$. Use Tchebichev’s inequality in order to find a lower ...
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1answer
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Positive or negatively bounded CDFs [closed]

If $X\in\mathbb{R}^n$ is a continuous random variable whose cumulative distribution function is ordinarily $$F_X(x) = \int_{-\infty}^{\infty} f_X(x) dx $$ what is the meaning of $$F_X(x) = \int_{0}^{\...
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1answer
56 views

Does a generalization bound that holds with high probability imply a bound that holds in expectation?

I am interested in generalization bounds, for example PAC bounds (Probably Approximately Correct). In particular, I wonder if a high probability bound implies a bound in expectation (or vice versa). ...
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1answer
34 views

How to deal with training models on data where the examples are highly dependent on each other?

Say you have a dataset of products sold at a store with the special condition that each day there is only one of each product in stock. That is, if there are multiple orders for a given product on a ...
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1answer
51 views

A tail bound for an unknown distribution via sampling

I know that many results exist for making an argument about the tail of a distribution, i.e., for a random variable $X$, one can find a bound $\epsilon$ such that $\Pr[X \geq a]<\epsilon$. Some ...
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1answer
94 views

Is a bounded real-number random variable discrete or continuous?

A discrete random variable is countable (such as integers and natural numbers), whereas a continuous r.v. is not countable (like the real numbers $\mathbb{R}$). If I have a dataset whose observations ...
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4answers
748 views

How to generate random numbers normally distributed in R or any software with limitations (bounds)?

I am working on a project where I need to generate random numbers for a given task time which is normally distributed with mean = 40, and standard deviation = 150. Because of the high SD, I will get ...
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2answers
89 views

On the difference between the main effect in a one-factor and a two-factor regression

Consider a linear regression (based on least squares) on two predictors including an interaction term: $$Y=(b_0+b_1X_1)+(b_2+b_3X_1)X_2$$ $b_2$ here corresponds to the conditional effect of $X_2$ when ...
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1answer
56 views

Cramer-Rao Lower Bound Proof (fuzzy step)

The following is the derivation of the Cramer-Rao lower bound as detailed on p.336 of Casella and Berger's Statistical Inference: $\frac{d}{d\theta}E[W(\bf{X})|\theta] = \int_{\chi}W(\bf{x})\left[\...
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Suppose $max\{a_i\}_{i=1}^{Rn}\overset{p}{\rightarrow} a_0$, where $a_i$ are i.i.d.r.v.. Are there any results on its rate of convergence?

Suppose $max\{a_i\}_{i=1}^{Rn}\overset{p}{\rightarrow} a_0$, where $a_i$ are i.i.d. random variables, $a_0$ is a constant and $R_n\rightarrow\infty$ as $n\rightarrow\infty$. Are there any results on ...
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Symmetrization in Proof of Hoeffding's Lemma

This alternative proof of a slightly weaker version of Hoeffding's Lemma features in Stanford's CS229 course notes. What's notable about this proof is its use of symmetrization. However, I find this ...
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What's a good way to select a bound that's close to zero?

I have a bunch of position data that I transformed into speed data. I'm assuming that I have some noise in my data and that the noise got worse after transforming to speed. I used a Kalman filter to ...
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1answer
23 views

Why $Pr[X-\mu \geq t]= Pr[e^{\lambda(X-\mu)} \geq e^{\lambda t}]$ for all $\lambda> 0$

I hope everyone is having a nice day. I don't know why this inequality holds. $$ Pr[X-\mu \geq t]= Pr[e^{\lambda(X-\mu)} \geq e^{\lambda t}] $$ For $\lambda >0$. I guess it has something to do ...
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1answer
117 views

Does the 1-Wasserstein distance have an upper and a lower bound?

Given $u$ and $v$ two probability distributions and U and V their respective $CDFs$, the $1$-Wasserstein distance is formulated as follows: $l_1(u,v)=\int_{-\infty}^{+\infty}|U-V|$ Does $l_1$ have ...
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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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40 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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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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25 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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1answer
41 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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2answers
186 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
60 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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25 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
249 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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