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0answers
16 views

Bound on the variance of a product

Let $Z$ be a positive $\mathbb R$-valued random variable bounded above by $M>0$, and $H$ an $\mathbb R^d$-valued random variable (seen as a column vector) such that $\mathbb E[H_i^2]=1$. Define the ...
0
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0answers
16 views

Perceptron Inseparable Case - Hinge loss

I am trying to understand the case where the perceptron algorithm can't find a perfect seperator. I am trying to understand how we bound the number of mistakes by using the hinge loss. My intuition ...
0
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0answers
20 views

How to check if ARDL model is dynamically stable & bounds test, stata

I'm doing research on Spanish and Dutch GDP: the influence of domestic and foreign variables (industrial production, inflation, employment rate) on GDP. All variables are converted into growth rates, ...
5
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0answers
44 views

Empirical Bernstein bound - proof

I'm working on Stopping algorithms and want to prove the bound derived from the Bernstein inequality as developed by Minh et. al. They begin with Hoeffding's inequality $$\begin{equation} ...
2
votes
2answers
53 views

Cramér-Rao inequality and MLEs

I know that if it exists, a regular, unbiased estimator $T$ for $\tau(\theta)$ attains the Cramér-Rao Lower Bound (next, CRLB) if and only if I can decompose the score function as follows: ...
1
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0answers
27 views

How can I evaluate the upper or lower bound for a function of random variables?

Let $\Gamma_1$,$\Gamma_2$ and $\Gamma_3$ be random variables defined as: \begin{align} \Gamma_1 &= \frac{XY}{X + Y + 1} \\[10pt] \Gamma_2 &= \frac{XY}{aX + bY + c} \\[10pt] \Gamma_3 &= ...
1
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0answers
10 views

Tailbounds for a rational function of the average of chi-squares

Let $\chi^2(n)$ refer to a chi-square random variable with $n$ degrees of freedom. This variable has mean $n$ and there are various results about its fluctuation around its mean. In particular, there ...
0
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0answers
15 views

Rademacher complexity of SVM with kernel in terms of whole Kernel Matrix

http://www.cs.nyu.edu/~mohri/mls/lecture_5.pdf In slide no. 18 here, it is shown that Rademacher complexity of SVM with kernel can be written in terms of trace of the matrix. Are there any other ...
1
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2answers
41 views

Understanding Hazard Function Values Exceeding 1

I keep running into problems in understanding hazard rates. I know, for example, that in a strict sense a hazard rate is not a probability and it is continually mentioned that because of this the ...
3
votes
0answers
31 views

Expectation of convex function of a Beta random variable

Suppose I have two Beta random variables $$ X \sim \text{Beta}(a, b), \\ Y \sim \text{Beta}(ca, cb), $$ with $c > 1$. They both have same mean $\tfrac{a}{a+b}$, but $Y$ is more "peaky" than $X$. My ...
8
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3answers
228 views

How to prove that $\frac{\left|X_i -\bar{X} \right|}{S} \leq\frac{n-1}{\sqrt{n}}$

I have been trying to establish the inequality $$\left| T_i \right|=\frac{\left|X_i -\bar{X} \right|}{S} \leq\frac{n-1}{\sqrt{n}}$$ where $\bar{X}$ is the sample mean and $S$ the sample ...
2
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0answers
60 views

convergence rate of Pearson correlation matrix

I posted recently a related question about the convergence rate of a Pearson correlation coefficient, here. Now, I am interested in the matrix version. Let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be ...
2
votes
1answer
66 views

convergence rate of Pearson correlation

I am looking to find the finest bound possible for $$\mathbb{P}\left( | \hat{\rho}(X,Y) - \rho(X,Y) | \geq t\right) \leq ?$$ where $\rho$ is the Pearson correlation defined as $$\rho(X,Y) = ...
0
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0answers
31 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
68 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 ...
5
votes
1answer
169 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 ...
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0answers
14 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 ...
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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
56 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
118 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
265 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
24 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. ...
6
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1answer
70 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
1answer
28 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
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0answers
59 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
48 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
votes
0answers
126 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
311 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
65 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
7k 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
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0answers
36 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
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0answers
43 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
87 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
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1answer
37 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
109 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
145 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 ...
6
votes
1answer
144 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
144 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
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1answer
96 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
79 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
131 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
50 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
145 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
66 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
92 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
42 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
40 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
116 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
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1answer
246 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 ...