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1
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1answer
41 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
35 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 ...
8
votes
2answers
136 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
votes
0answers
9 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
53 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
votes
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
42 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
32 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). ...
0
votes
0answers
144 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
vote
0answers
34 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 ...
9
votes
4answers
3k 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
28 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
31 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
votes
0answers
36 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
102 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
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
110 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
127 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
117 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
67 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
66 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
87 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
1answer
41 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
126 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
48 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
74 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
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
37 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
101 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
181 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
33 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
189 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
646 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
43 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
206 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
84 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
40 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
276 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
211 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
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
211 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
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 ...
1
vote
1answer
93 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
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0answers
130 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
192 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
53 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
236 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
47 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 $(x_1,\dots,x_n)$ ...