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0
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0answers
23 views

How to compare orders of magnitude?

In Fan and Li's paper "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties", they provided a proof to Theorem 1. The very last part of the proof is as follows. Some ...
1
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1answer
27 views

Margin bound for binary classification and Rademacher complexity

In the slides (slide 29) of Mohri a margin bound for binary classification is derived: $$R(h) \leq \hat{R}_\rho(h) + \frac{2}{\rho} \hat{R}_S(H) + 3\sqrt{\frac{\log \frac{2}{\delta}}{2m}}$$ Here ...
4
votes
1answer
70 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
0
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0answers
13 views

Univariate KDE with boundaries [duplicate]

I have to estimate density of bounded variables and I do not know exactly how to deal with it. I know that there are methods as renormalization, reflection or linear combination but I have not found ...
0
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0answers
13 views

largest space in n uniform samples

Suppose I draw n samples from a uniform distribution, and consider the largest space between two adjacent samples (or between the smallest sample and 0, and likewise at the upper end). Is there a ...
1
vote
1answer
49 views

Lower bound on the mean for an exponential distribution at a confidence level of 95%

for the life of me I simply cannot wrap my head around confidence intervals and upper/lower bounds on parameters. E.g $$ f(t;\tau) = \frac{1}{\tau} e^{-t/\tau}, \qquad t \geq0 $$ If $t=1$ (a single ...
0
votes
0answers
78 views

Bounding the bias of standard deviation estimate for stratified sampling (MC)

I cannot find an answer to this issue: in Monte Carlo runs, if one uses stratified sampling then the unknown bias of the variance estimator ( $\bar{\sigma}^2=\frac{1}{N}\sum{(y_i-\bar\mu_y)}$ where ...
9
votes
2answers
169 views

Regression results have unexpected upper bound

I try to predict a balance score and tried several different regression methods. One thing I noticed is that the predicted values seem to have some kind of upper bound. That is, the actual balance is ...
0
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0answers
18 views

Model a continuous response that is bounded by another random variable

I am trying to model a payment amount. The payment amount is bounded by the debt amount. Are there special modeling considerations when a continuous response is bounded by another random variable. ...
5
votes
1answer
54 views

Upper Bound on $E[\frac{1}{1-X}]$ where $E[X]=a$ and $0<a<1$

$X$ is a discrete random variable that can take values from $(0,1)$. Since $\varphi(x)=1/x$ is a convex function, we can use Jensen's inequality to derive a lower bound: $$ ...
0
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0answers
15 views

Getting a “friendly” tailbound from a closed-form description of the probability density (the case of the n-th order statistic)

Suppose I have a probability distribution of an $n$-th order statistic $X_n$ with mean $\mu$ and density $f_n(x)$, where $n$ scales to infinity. If one wants a concrete example, the one I care about ...
0
votes
1answer
23 views

Best lower bound

How could I find the best lower bound of $E[u((1-a)-max(X-a,0))*1_{X>a}]$, where $u$ is an increasing and concave real function with $u(0)=0$ and $a>0$, knowing only the mean and variance of ...
3
votes
0answers
48 views

Asymmetry of the Kullback-Leibler distance in hypothesis testing

My question is related to the asymmetry of the Kullback-Leibler distance. I'm using the discrete definition of the Kullback-Leibler dinstance, so we have: $KL(p,q) = \sum_{s \in S} p(s) ...
1
vote
0answers
38 views

Find expectation or lower bound of log erf

I need to find the expectation of $\log \Phi(x)=\log \left(\int_{-\infty}^x\frac{1}{2\pi}\exp(-\frac{1}{2}s^2)ds\right)$. (I realise this isn't quite the error function, but not sure what to call it). ...
2
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0answers
103 views

rejection region for correlated bivariate normal

STATEMENT OF PROBLEM: Suppose $ \left( \begin{array}{ccc} \ Z_1 \\ Z_2 \end{array} \right)$ follows a Bivariate standard normal with covariance $ \rho $ $ \left( \begin{array}{ccc} \ Z_1 \\ Z_2 ...
0
votes
0answers
162 views

Finding Expectation of a Random Variable Using Its Joint Marginal Density

If X and Y have joint density function $$ f(x,y) = \frac{1}{y} \,\mathbb{I}_{0<x<y<1}, $$ how do I find the expectation of X or Y? Since E[X] requires us to know the PDF of X, I tried to ...
1
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0answers
41 views

Kullback Leibler divergence “efficient” upper bound

For a distribution of N values, how can I efficiently upper-bound the largest divergence between all non-negative distributions over the same random field? For example, for all distributions of a ...
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4answers
4k views

Can mean plus one standard deviation exceed maximum value?

I have mean 74.10 and standard deviation 33.44 for a sample that has minimum 0 and maximum 94.33. My professor asks me how can mean plus one standard deviation exceed the maximum. I showed her ...
1
vote
0answers
30 views

Upper bound for sampling fraction for CLT to hold

What is the upper bound for the sampling fraction for the Central limit theorem to hold when sampling without replacement? Context The context for my question is that it is regularly argued (see e.g. ...
0
votes
0answers
33 views

To model an unknown bounded probability density function by a Gaussian mixture

I have points in dimension 10 coming from an unknown probability distribution. The nature of data strongly suggests that this distribution is bounded. But the boundaries are not precisely known and ...
0
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0answers
49 views

L1 distance between empirical and true distribution for discrete distributions

I have a distribution over the discrete set $\mathcal{A} = \{1, \ldots, d\}$ where the pmf is $p(.)$. That is, $p(i)$ is the probability of obtaining $i$ from $\mathcal{A}$. Given a dataset with $n$ ...
0
votes
1answer
33 views

Data transformation

I was writing with a question regarding a time-varying state space model of the form: \begin{align} y(t) &= \mu_1(t) + A(t)x(t) + v(t); &v(t) &\sim (0, R(t)) \\ x(t) &= ...
6
votes
2answers
104 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 ...
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5answers
237 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
122 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 ...
5
votes
1answer
129 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
125 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
74 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
69 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
102 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
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1answer
45 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
128 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
54 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
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0answers
80 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
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0answers
41 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
38 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
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0answers
109 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
191 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
34 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
190 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
667 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
46 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 $t\in[a,b]$, ...
10
votes
2answers
207 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
95 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
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0answers
42 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
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0answers
303 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
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0answers
225 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
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0answers
538 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
221 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 ...
7
votes
1answer
1k 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 ...