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Questions tagged [bounds]

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2
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2answers
33 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 ...
0
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0answers
17 views

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 ...
1
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0answers
25 views

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 ...
0
votes
1answer
13 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 ...
0
votes
1answer
36 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 ...
1
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0answers
51 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'...
0
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0answers
5 views

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 ...
1
vote
2answers
36 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)...
0
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0answers
11 views

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 $\hat{a}$ ...
0
votes
1answer
25 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 $t>0$...
0
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0answers
6 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 ...
11
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6answers
1k views

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 ...
3
votes
2answers
38 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: $E[A(B-C)]$ $...
2
votes
0answers
19 views

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 ...
1
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0answers
24 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 ...
1
vote
1answer
33 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 ...
0
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0answers
30 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 ...
1
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0answers
54 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,...
0
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0answers
21 views

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 ...
1
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0answers
53 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 ...
0
votes
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 $g(x) = O(x^2)$ ...
0
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1answer
60 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 ...
1
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0answers
112 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 \...
0
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1answer
230 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 ...
0
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0answers
12 views

Upper bound on single frequency error based on SSE of the full signal

I have an empirical signal: $y(t) = A_0 + \sum_{i=1}^{M}A_i \cos(\omega_i t + \phi_i) + \epsilon(t)$ The signal is tidal, and so dominated by a small number of frequencies and in particular there is ...
4
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1answer
123 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 ...
2
votes
0answers
60 views

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 ...
7
votes
1answer
215 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 ...
0
votes
0answers
38 views

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 <...
0
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0answers
27 views

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(...
3
votes
0answers
58 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 \...
2
votes
2answers
118 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 ...
4
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0answers
44 views

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\...
1
vote
0answers
38 views

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 > ...
2
votes
0answers
68 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 ...
2
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0answers
457 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 ...
2
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0answers
39 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 ...
3
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0answers
72 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 > ...
2
votes
0answers
60 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(...
3
votes
1answer
58 views

Minimum of Poissons

Let $X_i\sim\text{Pois}(\lambda_i)$ for $i=1,2,\ldots,n$ and $Y = \min X_i$. Can we show that, for example $\mathbb{E}[Y] \leq f(\lambda,n)\min\lambda_i$ for some $f : (\mathbb{R}^n,\mathbb{N}) \to [0,...
0
votes
0answers
32 views

Is there a method to fit a bound to the plot of an linear inequality?

I have a physical dataset that is bounded by several different processes, and thus the plot takes the form of a linear inequality: I'm specifically interested in studying the upper bound. Is there a ...
3
votes
1answer
142 views

Tightest bounds on sample variance given sample size, mean, minimum, and maximum

For real-valued samples (possibly known to lie in some interval, but without further constraints on them), I am interested in the tightest possible bound on the sample variance $\sigma^2$, given the ...
1
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0answers
43 views

How to estimate the bound on a random variable with semi-infinite distribution?

Given samples of a random variable which appears to have a continuous unimodal probability density function that is zero on one side of a bound, what statistical methods are there for estimating that ...
0
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0answers
151 views

Relationship between big-Oh and little-Oh

I'm reading Wu, J. (2014), Restoring monotonic power in Wald/LM-type tests. At a certain point in his derivations, an expresion turns out to be $=O_p(\frac{1}{T^{3/10}h^{5/10}}+T^{2/10}h^{20/10})$, ...
0
votes
1answer
63 views

Price Analytics - Price range for a product

I have data on weekly, total number of units sold, at what price, any discount, at which store, product details. like below For over 3 years for the Product details across 10 stores. I would like to ...
0
votes
1answer
86 views

About the expectation of a posterior [closed]

I am interested in finding an upper bound for $\mathbb{E}\big( \int_B \frac{f(X| \theta)}{f(X| \theta_0)} \pi(\theta) d \theta\big)$, the $\frac{f(X| \theta)}{f(X| \theta_0)}$ is the likelihood ...
2
votes
0answers
83 views

Constraints on the moments of a bounded probability distribution

Consider a probability distribution with support on $[0,1]$. Suppose the first $n$ raw moments $m_1,...,m_n$ are given. What are the constraints on the $(n+1)^\text{th}$ moment? Obviously we have the ...
6
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2answers
584 views

disadvantages variational inference

A lot of methods utilize variational inference for hyperparameter calculation. What are the advantages and disadvantages of variational inference ? (ex: does it guarantee a global optimal ? )
0
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1answer
87 views

Need to know pdf of “x/z+sqrt(y^2-x^2)/z” , or any idea about its upper/lower bounds

I need to know the pdf of the following equation or any upper/lower bound would help. Let $X, Y, Z \sim N(0, \sigma^2)$. Then what is the distribution of: $$A=\frac{x+ \sqrt{\mid y^2 - x^2 \mid}}{z}$$...
4
votes
1answer
664 views

Gaussian Multi-Armed Bandits and the UCB Algorithm

I've implemented in MATLAB the UCB algorithm for gaussian bandits with zero mean and unit variance (these means were themselves sampled from a gaussian prior of zero mean and unit variance). Now I ...