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-2
votes
0answers
27 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
-2
votes
0answers
22 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
-3
votes
0answers
29 views

Clarifications on proof of Doob's Forward Convergence Theorem, warning related to it and proof of a corollary

From Williams' Probability with Martingales: $X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$ --> Is this supposed to be stronger than $\lim X_n$ does not exist because it's ...
0
votes
0answers
14 views

Upper bound for most likely category in multinomial random vector NOT being max count realized

This is a repost from a question posed at Math stack exchange: http://math.stackexchange.com/questions/1750148/upper-bound-for-most-likely-category-in-multinomial-random-vector-not-being-max Let ...
0
votes
0answers
17 views

online learning- winnow algorithm and mistake bound

I came across an interesting question and I must say I am struggling to figure out how it suppose to work. So we consider the winnow algorithm that learns non-monotone disjunctions. Could someone ...
0
votes
0answers
17 views

Bound on ratio of norms

Let $\mathbf{A}\in\mathbb{R}^{m\times n}$ be fixed and $\mathbf{x}\in\mathbb{R}^{n}$ be a vector of mean zero, independent random variables. I'd like to know if there exists a bound of the form: ...
0
votes
0answers
24 views

Bounds of integration query

My question is for personal research. When finding the distribution of say Z=X+Y where X and Y are both exponentially distributed r.v's where X,Y > 0, I use the convolution method. I do the same for ...
0
votes
0answers
11 views

Empirical Bernstein Bound for distributions outside range(0,1)

I'm working on using empirical Bernstein bounds to estimate the mean difference between 2 variables from different distribution. The algorithm samples from each variable until it detects a ...
25
votes
3answers
837 views

Statistical methods for data where only a minimum/maximum value is known

Is there a branch of statistics that deals with data for which exact values are not known, but for each individual, we know either a maximum or minimum bound to the value? I suspect that my problem ...
0
votes
1answer
21 views

Most likely event in a multinomial distribution setting

I'm looking at the following scenario: $k$ categories, distributed by a multinomial ($p_1,\dots,p_k$) such that $p_1 \ge \dots \ge p_k$. Draw $n$ samples. I'm interested in estimators/lower bounds ...
1
vote
0answers
27 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
votes
0answers
22 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
votes
0answers
47 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, ...
2
votes
2answers
67 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
vote
0answers
35 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
vote
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
votes
0answers
20 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
vote
2answers
55 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
37 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
votes
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
votes
0answers
77 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
79 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
votes
0answers
33 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
vote
1answer
81 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
199 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
votes
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 ...
0
votes
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
62 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
130 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
282 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 ...
6
votes
1answer
74 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
votes
0answers
62 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
53 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
137 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
345 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
90 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 ...
12
votes
4answers
9k 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
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
votes
0answers
44 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
98 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
38 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
111 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
238 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
148 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
149 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
147 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
105 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
82 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
151 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 ...