The bounds tag has no wiki summary.
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How to prove $H(X-Y)\le \log(2\pi eD)$?
In rate distortion theory, difference error entropy $H(X-Y)\le \log(2\pi eD)$, how can we prove this?
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Bound on the variance for [0,1] RVs as a function of the mean
I noticed that if $X$ is a RV in $[0,1]$ then $V[X] \leq E[X](1-E[X])$, which also implies that the bernoulli distribution maximizes variance (one of many solutions).
For interest's sake consider ...
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50 views
Bounded response variable [-1;1] - Should I transform it?
I am planning to use two response variables. One is bounded between 0 and 1, and I guess I can use a binomial (or related) error structure. The second variable is bounded between -1 and 1. I am not ...
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58 views
Can we find bounds on R-squared?
We know that as the number of independent variables increases, the coefficient of determination $R^2$ will increase but the adjusted $R^2$ may or may not increase. In the following question for the ...
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3answers
229 views
Reference for $\text{Var}(X)\le (b-a)^2/4$
I am not a statistician, but am working a proof for the upper bound of an expression which contains the variance of a variable which obtains its values from a closed interval, [0,1].
I have seen in ...
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50 views
Bounds for the population variance?
Suppose we have i.i.d. samples $x_1$, $\ldots$, $x_n$ for a (potentially non-normal) random variable $X$ with finite moments. We can use these samples to construct an unbiased estimates of the ...
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1answer
145 views
Dealing with regression of unusually bounded response variable
I am attempting to model a response variable that is theoretically bounded between -225 and +225. The variable is the total score that subjects got when playing a game. Although theoretically it is ...
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1answer
66 views
Relationship between number of training set and classification performance
Are there any research/paper on the relationship between the number of documents for training and the classification performance using support vector machine?
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About tail distribution of a sum
Do we know anything about the tail distribution of sum of squares of a limited number of i.i.d exponentially distributed random variables? I'm looking for a good bound.
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Upper bounds for the copula density?
The Fréchet–Hoeffding upper bound applies to the copula distribution function and it is given by
$$C(u_1,...,u_d)\leq \min\{u_1,..,u_d\}.$$
Is there a similar (in the sense that it depends on the ...
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0answers
22 views
Generalization error for classification with a nonconvex loss function
I've been working my way through Vapnik's 1998 Statistical Learning Theory book and one thing that I'm still unsure of is if his risk bounds hold for nonconvex loss functions -- i.e., when we can't be ...
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1answer
148 views
Tail bounds on a function of normally distributed variables
I am looking for tail bounds (both at $0$ and at $\infty$) for
$$ Z:=\exp \left(\frac{\alpha}{4}(X-Y)^2+\frac{\alpha}{2}(X+Y)\right)$$
where $\alpha$ is a positive real and $X,Y$ are i.i.d. normal ...
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The product distribution: how fast does dissimilarity increase as a function of number of samples?
If $\mathcal{D}$ is a distribution, let $\mathcal{D}^n$ denote the $n$-fold Cartesian product of $\mathcal{D}$. In other words, $\mathcal{D}^n$ is the distribution of $n$-tuples $(x_1,\dots,x_n)$ ...
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1answer
41 views
Is it correct to compute LR stat after maximising likelihood with bounds?
I use grid search with bounds for example lb=[ 1 1 1 1 1 1]'/1000; ub=[10 10 10 10 10 20]' but it is computationally difficult so it checks 2 points only. Thus i obtain boundary solution consisting ...
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176 views
How can we bound the probability that a random variable is maximal?
Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_n$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$. I am looking for ...
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2answers
938 views
How to set limits using constrOptim in R?
I am using constrOptim to minimize a log likelihood function for maximum likelihood estimation of parameters.
I wish to set the bounds on my parameters, but to not understand the constrOptim ...
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2answers
146 views
Probability of Unique Minimum (Discrete)
This is a discrete problem concerning integers.
If there are $n$ independent random variables $X_1,...,X_n$ that each take on a value from $\{1,...,x\}$ uniformly at random ($x$ distinct values), ...
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3answers
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Bounding the difference between square roots
I want to compute the value of $\frac{1}{\sqrt{a + b + c}}$. Say I can observe a and b, but not c. Instead, I can observe d which is a good approximation for c in the sense that $P( |c-d| \leq 0.001 ...
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1answer
76 views
How does one express the decrease in minimal type II error bound for each observation added?
Problem: I have a "classifier" that uses some arbitrary hypothesis test on observations from one of two known probability distributions:
$P_0$ (null hypothesis $H_0$) is a zero-mean Gaussian ...
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2answers
932 views
Hypothesis testing and total variation distance vs. Kullback-Leibler divergence
In my research I have run into the following general problem: I have two distributions $P$ and $Q$ over the same domain, and a large (but finite) number of samples from those distributions. Samples ...
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1answer
177 views
Bound on variance of sum of variables
Suppose I have two finite sets of data A and B, with equal length n.
What's the best upper ...
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2answers
664 views
What is the variance of the maximum of a sample?
I'm looking for bounds on the variance of the maximum of a set of random variables. In other words, I'm looking for closed-form formulas for $B$, such that
$$
\mbox{Var}(\max_i X_i) \leq B \enspace,
...
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204 views
Lower bound for tail of hypergeometric distribution
There are several simple and widely used upper bounds on the tail of the hypergeometric distribution, including $P(X > E[X]+tn) <= e^{-2t^{2}n}$, where X is hypergeometric with parameters N, M, ...
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Exponential upper bound for $\sum_{k=1}^{N-1} \alpha^k \beta^{\frac{1}{N-k}}$
I will be too happy help me find the exact value or a very tight exponential upper bound for:
$$\sum_{k=1}^{N-1} \alpha^k \beta^{\frac{1}{N-k}}$$
where $0 \leq \alpha < 1$, $\beta= \exp(-N^2\zeta)$ ...
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1answer
344 views
Upper/lower bound and initial domain for lognormal distribution
I'm trying to implement logNormal distribution into my java program because lognormal dist doesn't exist into apache commons math library.
I have no problem to re-write density and cumulative ...
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124 views
Lower bound over a concave function [closed]
I will be to grateful if help me find a lower bound $g(x)$ over the following concave function:
$$f(x) = \sqrt{1+4x} -1 + \log(\sqrt{1+4x}-1) - \log(2x) \geq g(x),$$
where $x \geq 0$.
The taylor ...
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1answer
271 views
Are there bounds on the Spearman correlation of a sum of two variables?
Given $n$-vectors $x, y_1, y_2$ such that the Spearman correlation coefficient of $x$ and $y_i$ is $\rho_i = \rho(x,y_i)$, are there known bounds on the Spearman coefficient of $x$ with $y_1 + y_2$, ...
