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### 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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### Expected number of times the empirical mean will exceed a value

Given a sequence of i.i.d. random variables, say, $X_i \in [0,1]$ for $i = 1,2,...,n$, I'm trying to bound the expected number of times the empirical mean $\frac{1}{n}\sum_{i=1}^n X_i$ will exceed a ...
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### 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 $y_1 + y_2$, ...
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### Tail bounds on Euclidean norm for uniform distribution on $\{-n,-(n-1),…,n-1,n\}^d$

What are known upper bounds on how often the Euclidean norm of a uniformly chosen element of $\:\{-n,~-(n-1),~...,~n-1,~n\}^d\:$ will be larger than a given threshold? I'm mainly interested in bounds ...
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### 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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### 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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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 $$\mbox{Var}(\max_i X_i) \leq B \enspace, ... 1answer 92 views ### Probability of pairwise difference of samples from distribution with finite support I'd appreciate any help on the following problem: Let X_1, X_2, \dots, X_N be i.i.d. continuous random variables with support [0, 1]. What is a reasonable bound on the probability that some pair ... 0answers 108 views ### Upper bounds for the copula density? The Fréchet–Hoeffding upper bound applies to the copula distribution function and it is given by$$C(u_1,...,u_d)\leq \min\{u_1,..,u_d\}.$$Is there a similar (in the sense that it depends on the ... 0answers 95 views ### About tail distribution of a sum Do we know anything about the tail distribution of sum of squares of a limited number of i.i.d exponentially distributed random variables? I'm looking for a good bound. 2answers 176 views ### Probability of Unique Minimum (Discrete) This is a discrete problem concerning integers. If there are n independent random variables X_1,...,X_n that each take on a value from \{1,...,x\} uniformly at random (x distinct values), ... 1answer 334 views ### Dealing with regression of unusually bounded response variable I am attempting to model a response variable that is theoretically bounded between -225 and +225. The variable is the total score that subjects got when playing a game. Although theoretically it is ... 1answer 187 views ### Tail bounds on a function of normally distributed variables I am looking for tail bounds (both at 0 and at \infty) for$$ Z:=\exp \left(\frac{\alpha}{4}(X-Y)^2+\frac{\alpha}{2}(X+Y)\right)$$where \alpha is a positive real and X,Y are i.i.d. normal ... 0answers 32 views ### Index of dispersion with approximate distribution I have an unknown discrete probability distribution D (D is a probability mass function), defined on an interval [a,b] (a>0) and an estimation \hat{D} such that, for all t\in[a,b], ... 0answers 59 views ### Bounds for the population variance? Suppose we have i.i.d. samples x_1, \ldots, x_n for a (potentially non-normal) random variable X with finite moments. We can use these samples to construct an unbiased estimates of the ... 0answers 33 views ### Bound on the variance for [0,1] RVs as a function of the mean I noticed that if X is a RV in [0,1] then V[X] \leq E[X](1-E[X]), which also implies that the bernoulli distribution maximizes variance (one of many solutions). For interest's sake consider ... 0answers 75 views ### Can we find bounds on R-squared? We know that as the number of independent variables increases, the coefficient of determination R^2 will increase but the adjusted R^2 may or may not increase. In the following question for the ... 0answers 34 views ### Generalization error for classification with a nonconvex loss function I've been working my way through Vapnik's 1998 Statistical Learning Theory book and one thing that I'm still unsure of is if his risk bounds hold for nonconvex loss functions -- i.e., when we can't be ... 0answers 35 views ### The product distribution: how fast does dissimilarity increase as a function of number of samples? If \mathcal{D} is a distribution, let \mathcal{D}^n denote the n-fold Cartesian product of \mathcal{D}. In other words, \mathcal{D}^n is the distribution of n-tuples (x_1,\dots,x_n) ... 0answers 80 views ### Exponential upper bound for \sum_{k=1}^{N-1} \alpha^k \beta^{\frac{1}{N-k}} I will be too happy help me find the exact value or a very tight exponential upper bound for:$$\sum_{k=1}^{N-1} \alpha^k \beta^{\frac{1}{N-k}}$$where 0 \leq \alpha < 1, \beta= \exp(-N^2\zeta) ... 3answers 83 views ### 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 ... 1answer 98 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 ... 1answer 398 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 ... 1answer 56 views ### Theoretical upper bounds of classification accuracy? I'm looking for theoretical upper bounds of classification accuracy. Please let me know if you are familiar with results like the following. The setup below is a general one, but please share results ... 1answer 75 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? 1answer 43 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 ... 0answers 109 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 ... 0answers 262 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, ... 2answers 1k 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 ... 1answer 217 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 ... 1answer 102 views ### Bounded in probability and finite expectation Let x_t = O_p(1), meaning that for all \varepsilon > 0 there exists M_{\varepsilon} < \infty s.t. P(|X_t| > M_{\varepsilon}) < \epsilon for all t \in \mathbb{N}. Does it imply ... 0answers 19 views ### Quadratic lower bound on Gaussian Suppose I have a multivariate Gaussian such that p(y)=\mathcal{N}(\mathbf{0},\Sigma). What would be a quadratic lower bound, f(y) on on p(y). i.e. for what values of k and \Omega will ... 0answers 32 views ### Normalising Constant for exponentiated function What would the normalising constant be of the following, or atleast an approximation? I would like to avoid sampling.$$f(\theta)=\exp(-k_1e^{-k_2\theta^2}-\theta^2)\qquad\theta\in(-\infty,\infty), ...
Let $A_k^p:=\{f:\, \|(ix)^k\hat{f}(x)\|_p\leq 1\}, k \in Z_+, p \in (1, \infty)$. I am wondering what the application of the lower and upper bounds of wavelet coefficients of the function on the ...