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6
votes
2answers
90 views

$E(\frac{1}{1+x^2})$ under a Gaussian

This question is leading on from the following question. http://math.stackexchange.com/questions/360275/e1-1x2-under-a-normal-distribution Basically what is the $E\left(\frac{1}{1+x^2}\right)$ under ...
7
votes
5answers
218 views

If two time series $X$ and $Z$ follow $0 \leq Z \leq X$, can we say that $\text{var}(Z) \leq \text{var}(X)$?

Now I see it can't hold. Thank you for the counter examples... You guys rule! Thank you very much for your comments! I added, however, some observations that were missing. Most importantly is the ...
4
votes
1answer
79 views

Using extreme value theory to estimate bounds

Suppose I have I have a random variable $X$ that I know is doubly bounded on support $[0,\theta]$ but I dont know $\theta$ (we don't know anything on the distribution of $X$, but assume it is not ...
4
votes
1answer
72 views

Cramer-Rao bound for $\chi^2$ distribution parameter estimates

I've struck an unpleasant problem with the noncentral $\chi^2$ distribution. I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ ...
4
votes
2answers
89 views

How to statistically test upper bound

Suppose a theory claims that a random variable $R$ (of unknown distribution $F$) must satisfy a certian upper bound $R < c$ (where $c$ is known constant). Suppose I perform a set of measurements ...
5
votes
1answer
46 views

Treatment Effect Bounds

My supervisor and I have run a randomized experiment in a developing country. Due to administrative problems there we unfortunately have the problem of non-response. This non-response is also not ...
0
votes
1answer
41 views

Big O notation preserved under convex functions?

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$. Does this imply that the random variable $\max\{0,X_T \}$ is $O_p(1)$?
1
vote
1answer
57 views

What are the consequences of knowing (or not knowing) the range of possible answers from a test

Lets say we are interested in some unknown variable x. We poll a x at different intervals and get a set of values that x has been. However we know that if we poll x enough times we will eventually get ...
0
votes
0answers
12 views

Implication of uniform stochastic boundedness?

Let $\theta \in \Theta \subseteq \mathbb{R}^d$ be a parameter vector. Let $Q: \Theta \rightarrow \mathbb{R}$ be a function mapping from the parameter space to the real numbers. Let $Z_T$ be a a ...
0
votes
1answer
28 views

tight bound of bernoulli sums with unknown dependency

Consider n random variables $X_1, \ldots, X_n$ all follow same bernoulli distribution of mean $p$. But the dependency of these variables are unknown (i.e., cannot assume that they are independent). ...
2
votes
1answer
99 views

Way to correct sample selection bias with unknown selection?

I would greatly appreciate some advise on a statistical problem that haunts me. Suppose you wish to estimate the effect of $x$ on $y$, but the probability to observe $\{y_i, x_i\}$ also depends on ...
1
vote
1answer
35 views

Limit multiclassification SVM - ANN

I have some questions on the limits of SVM and ANN for multiclass problem. I know about "one vs all" and "all vs all" strategies but I only want to know the limit of a unique SVM and ANN. Is there a ...
1
vote
0answers
44 views

How to estimate the upper bound of y?

How can you estimate the upper bound of $y$ in this situation? Given: a function $y=f(x_1,x_2,x_3,x_4,x_5)$ with 5 parameters ($y=f(\cdot,\cdot,\cdot,\cdot,\cdot)$ can be any function). for each ...
3
votes
0answers
40 views

Bounds (or model) for estimating probability from generating process

I was curious if there was any bounds or approaches to getting good estimates for a probability from the following generating process. Suppose we have 2 sets of objects $A$ and $B$ where both sets ...
0
votes
0answers
30 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 ...
0
votes
0answers
71 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), ...
5
votes
1answer
136 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 ...
0
votes
0answers
30 views

bounds of wavelet coefficients of function

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 ...
10
votes
1answer
170 views

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 ...
0
votes
1answer
242 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 ...
4
votes
0answers
35 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 $t\in[a,b]$, ...
9
votes
2answers
150 views

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 ...
1
vote
1answer
69 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 ...
2
votes
0answers
36 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 ...
1
vote
0answers
174 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 ...
2
votes
0answers
113 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 ...
11
votes
0answers
532 views

Reference for $\text{Var}(X)\le (b-a)^2/4$ [duplicate]

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 ...
6
votes
1answer
165 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 ...
6
votes
1answer
769 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 ...
1
vote
1answer
87 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?
4
votes
0answers
115 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.
6
votes
0answers
148 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 ...
2
votes
0answers
44 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 ...
3
votes
1answer
219 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 ...
2
votes
0answers
39 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 $(x_1,\dots,x_n)$ ...
1
vote
1answer
45 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 ...
6
votes
0answers
228 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 ...
0
votes
2answers
2k 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 ...
3
votes
2answers
190 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), ...
1
vote
3answers
98 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 ...
1
vote
1answer
111 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 ...
7
votes
2answers
2k 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 ...
0
votes
1answer
275 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 ...
7
votes
2answers
1k 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, ...
1
vote
0answers
305 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, ...
2
votes
0answers
86 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 $0 \leq \alpha < 1$, $\beta= \exp(-N^2\zeta)$ ...
1
vote
1answer
497 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 ...
7
votes
1answer
306 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 $y_1 + y_2$, ...