Questions tagged [bounds]
The bounds tag has no usage guidance.
2
votes
2answers
33 views
Estimator with variance equal to Cramér-Rao lower bound in $N(x_i\theta,1)$-distribution
Let $Y_1,\ldots, Y_n$ be independent and $N(x_i\theta,1)$ distributed, with for each $Y_i$ a mean of $x_i\theta$ for known $x_1,\ldots,x_n$. In a previous section of this exercise I found that the ...
0
votes
0answers
17 views
Prove that $T(\textbf{X}) = \hat{\sigma}^{2}$ reaches the Cramer-Rao bound
Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample whose distribution is given by $\mathcal{N}(\mu,\sigma^{2})$, where both parameters are unknown.
(a) Prove the normal probability density function ...
1
vote
0answers
25 views
Concentration inequality for max component of a multivariate Gaussian in the general case
I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
0
votes
1answer
13 views
Bound for density of random variable with finite second moment
Let $\mathbf{X}$ be a vector-valued random variable with finite second moment and density $\rho$. Assume that $\rho$ is bounded and continuous. As $\mathbf{X}$ has finite second moment, I hope to find ...
0
votes
1answer
36 views
Dealing with measurements falling outside of the theoretical range/boundaries of the data
Imagine I am measuring a bounded variable (with a maximum possible value above which the data doesn't make sense) and I end up with the following dataset with my measurements and measurement errors as ...
1
vote
0answers
51 views
Regression with bounded non-normal dependent variable
I'm wondering what a suitable regression model would be to predict a bounded, continuous, non-normally distributed dependent variable from a binary explanatory variable with partially crossed data.
I'...
0
votes
0answers
5 views
References for generalization bounds?
I'm looking for references (books, papers, lecture notes etc) on generalization bounds and their proofs. Specifically, I'm looking to fully understand the technique of defining a hypothesis class (or ...
1
vote
2answers
36 views
Upper bound of normal cdf
Random variable $X\sim N(0,1)$. Show that, $P(X\geq c) \leq e^{-ct+ \frac{t^{2}}{2}}$ for $c>0$ and for all $t$ in $R$.
I found that $P(X\geq c) = \Phi(-c)$ where $\Phi(x)=\int_{-\infty}^{x}\phi(u)...
0
votes
0answers
11 views
Manipulation of asymptotic bounds for distance between estimators
Suppose I know some asymptotic bounds:
$$\mathbb{E}(|D(a,\hat{a})|) \lesssim O(n^{-1/2}),$$
where $D$ is some distance between probability measures, and $a$ is a probability measure while $\hat{a}$ ...
0
votes
1answer
25 views
Getting From Concentration Inequality to Interval Length
I've seen this used some times and I would like to ask what steps are taken on the way to getting there:
E.g. assuming bounded variance, we can use Chebyshev concentration inequality: for any $t>0$...
0
votes
0answers
6 views
On the practical usage of generalisation error bounds
(This questions is based on a question that I've posted previously here, but I would like it to get more exposure)
In many practical scenarios, one would like to answer how much more data is needed ...
11
votes
6answers
1k views
Linear regression when Y is bounded and discrete
The question is straightforward: Is it appropriate to use linear regression when Y is bounded and discrete (e.g. the test score 1~100, some pre-defined ranking 1~17)? In this case, is it "not good" to ...
3
votes
2answers
38 views
Bounds on Expectation $E[A(B-C)^2]$
[This question has been edited for more given conditions].
Given possibly correlated random variables $A,B,C$, I want to find the best upper bound for $E[A(B-C)^2]$ given the following:
$E[A(B-C)]$
$...
2
votes
0answers
19 views
Bound for type of correlation measure
Assume you have a finite, discrete probability distribution for a joint random variable and such that $P(X=i,Y=j) = p_{i,j}$ for $i \in \{1, \dots, |X|\},j \in \{1, \dots, |Y|\}$. The marginal ...
1
vote
0answers
24 views
Bounds on quantiles of the minimum of summations of (possibly dependent) random variables
Suppose I have $2N$ continuous random variables $X_1, \ldots, X_N, Y_1, \ldots, Y_N$ and that I can evaluate the quantiles of the respective distributions. Given a value $w \in [0, 1]$ I would like to ...
1
vote
1answer
33 views
Is there an upper bound on number of logistic regression responses that yield infinite estimates
Suppose a logistic regression problem has N observations of {0, 1} and that there are p parameters. Also assume the design matrix, X, is full rank with p < N. We know that there will be certain ...
0
votes
0answers
30 views
Approximation of the upper bound on the expectation of log sum of exponentials
I am having some trouble replicating the results in Guillaume Bouchard's paper, Efficient Bounds for the Softmax Function and Applications to Approximate Inference in Hybrid Models, where he discusses ...
1
vote
0answers
54 views
Finding Chernoff bounds maximum estimators
I am currently trying to resolve the following exercise about Chernoff bounds:
Let $X_{1}, X_{2}, \dots, X_{n}$ be independent, identically distributed (i.i.d) random variables with distribution $N(0,...
0
votes
0answers
21 views
Reference request: Least Squares Concentration Bounds
I've recently become interested in parameter estimation for regression and time-series type models and I often encounter, and indeed need to understand, results using concentration of measure-type ...
1
vote
0answers
53 views
Tight upper bound on the expectation of a concave function
N is a random variable whose sample space is [0,$\infty$). I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The ...
0
votes
1answer
30 views
Bounded functions when x tends to infinity
Please help me understand the below:
The notation $g(x) = O(f(x))$ denotes that $\left|\frac{f(x)}{g(x)}\right|$ is bounded as $x \to \infty$.
For instance if $g(x)=3x^2 + 2$, then $g(x) = O(x^2)$
...
0
votes
1answer
60 views
Find bayesian credible Interval
I am taking this example from here. They have given the steps but i cannot understand them so a little dumbing down of answer is necessary or you can just explain the answer given there. its on page 5
...
1
vote
0answers
112 views
Is it true that normalizing the output of a ReLu feedforward Neural Network that its Rademacher Complexity becomes a constant?
I was trying to understand what happened with the Rademacher Complexity:
$$ R_S(F) = \frac{1}{m} \mathbb E_{\sigma} [\sup_{f \in F}\sum^m_{i=1} \sigma_i f(z_i)] $$
or
$$ R_{P,m} = \mathbb E_{s \...
0
votes
1answer
230 views
ELBO interpretation in Variational Autoencoder (VAE) for anomaly detection
How to interpret different ELBO values when checking anomaly detection possibilities of VAE model on different "testing" datasets?
The higher the ELBO value of the model when testing it on different ...
0
votes
0answers
12 views
Upper bound on single frequency error based on SSE of the full signal
I have an empirical signal:
$y(t) = A_0 + \sum_{i=1}^{M}A_i \cos(\omega_i t + \phi_i) + \epsilon(t)$
The signal is tidal, and so dominated by a small number of frequencies and in particular there is ...
4
votes
1answer
123 views
Fréchet Hoeffding bounds for symmetric random variables
(Edited to clarify the question).
The Hakan & Demirtas (2012 doi: 10.1198/tast.2011.10090) approach to approximating Pearson correlation bounds uses the concept of the Fréchet-Hoeffding bounds by ...
2
votes
0answers
60 views
What's the reasoning behind non-seasonal ARIMA model lag-order bounds?
I read some papers on the non-seasonal ARIMA model, and the consensus I've seen is that for ARIMA(p, d, q), p and q should not be greater than 3, maybe 5. What's the reasoning for that? Is it for ...
7
votes
1answer
215 views
Bounds on the difference of correlated random variables
Given two highly correlated random variables $X$ and $Y$, I'd like to bound the probability that the difference $ |X - Y| $ exceeds some amount:
$$ P( |X - Y| > K) < \delta $$
Assume for ...
0
votes
0answers
38 views
Is max. Eigenvalue of k-sparse PCA always $\leq$ max. Eigenvalue of normal PCA on same dataset?
Is max. Eigenvalue of k-sparse PCA always less than or equal to the max. Eigenvalue of normal PCA on same dataset? K refers to the number of non zero eigenvalues when the dataset is of dimension n <...
0
votes
0answers
27 views
Expected value of (F(x)/x(1-F(x)) where F(x) is CDF?
I was solving a problem and encountered the following:
\sum (f(x)/x) (F(x)/(1-F(x))
Here f(x) is pmf and F(x) is CDF. Is expected value of F(x)/x(1-F(x)) a known function? I found out that f(x)/(1-F(...
3
votes
0answers
58 views
Concentration inequalities for weighted sums of gaussians
Suppose that $x \sim \cal{N}(0,I_d)$ be a $d$-dimensional standard Gaussian vector and let $x_1,\ldots,x_n$ denote $n$ i.i.d. samples drawn from the same distribution.
For some fixed vector $\theta \...
2
votes
2answers
118 views
Understanding the concept of “Bounded in probability”
My statistics book defines the concept of "bounded in probability" in the followng way:
..But doesn't this mean that any sequence of R.V.'s that does not include any R.V.'s with a pdf with infinite ...
4
votes
0answers
44 views
Bounds on probability that a random variable is neither the maximum nor the minimum of a set of random variables
Suppose I have $n$ independent random variables $X_1,\dotsc,X_n$ which are Poisson distributed with $X_i \sim Poi(\lambda_i)$. Without loss of generality, an additional condition $\lambda_1\le\...
1
vote
0answers
38 views
Comparing bound between lasso and OLS
Assume fixed design regression model $Y = \mathbf{X} \beta + \epsilon$ with common assumptions. Lasso estimator $\hat{\beta}_\lambda$ can be shown to have the following bound.
For any $\delta > ...
2
votes
0answers
68 views
Bounds on variance and mean of maximum of difference of independent random variables
Suppose $X_1,\dotsc,X_n$ are independent but not necessarily identical random variables. $$Y = \max_{1\le i,j\le n}(X_i-X_j)$$
What upper and lower bounds can be derived for expectation and variance ...
2
votes
0answers
457 views
Upper bound on KL divergence
Is there a maximum (unique?) to the KL divergence between discrete distributions p & q, with the restriction that q is a proper probability distribution?
I know KL is unbounded from above when q ...
2
votes
0answers
39 views
Correlation bounds for uniformly distributed matrix?
For a uniformly or Guassian distributed $M\times N $ matrix. Is there any analytical expression in terms of $M$ and $N$ to estimate the maximum and minimum bounds of correlation between the columns of ...
3
votes
0answers
72 views
Tail bounds for F-distribution (not using $\chi^2$ bounds)
Are there any sharp tail bounds for an $F_{p,q}$ distribution? That is, if $X \sim F_{p,q}$, then for a $t_1,t_2 > 0$, what are the sharpest $\delta_1$ and $\delta_2$ known such that
$$P(X > ...
2
votes
0answers
60 views
Bounds on tail conditional expectation of random variable given variance
Given a random variable $X$ with CDF $F(X)$, mean $E(X)=0$, and variance $Var(X) =\sigma^2$, I would like to bound the tail conditional expectation where $X$ is in the tail with probability $1-p$:
$E(...
3
votes
1answer
58 views
Minimum of Poissons
Let $X_i\sim\text{Pois}(\lambda_i)$ for $i=1,2,\ldots,n$ and $Y = \min X_i$. Can we show that, for example $\mathbb{E}[Y] \leq f(\lambda,n)\min\lambda_i$ for some $f : (\mathbb{R}^n,\mathbb{N}) \to [0,...
0
votes
0answers
32 views
Is there a method to fit a bound to the plot of an linear inequality?
I have a physical dataset that is bounded by several different processes, and thus the plot takes the form of a linear inequality:
I'm specifically interested in studying the upper bound. Is there a ...
3
votes
1answer
142 views
Tightest bounds on sample variance given sample size, mean, minimum, and maximum
For real-valued samples (possibly known to lie in some interval, but without further constraints on them), I am interested in the tightest possible bound on the sample variance $\sigma^2$, given the ...
1
vote
0answers
43 views
How to estimate the bound on a random variable with semi-infinite distribution?
Given samples of a random variable which appears to have a continuous unimodal probability density function that is zero on one side of a bound, what statistical methods are there for estimating that ...
0
votes
0answers
151 views
Relationship between big-Oh and little-Oh
I'm reading Wu, J. (2014), Restoring monotonic power in Wald/LM-type tests.
At a certain point in his derivations, an expresion turns out to be $=O_p(\frac{1}{T^{3/10}h^{5/10}}+T^{2/10}h^{20/10})$, ...
0
votes
1answer
63 views
Price Analytics - Price range for a product
I have data on weekly, total number of units sold, at what price, any discount, at which store, product details.
like below
For over 3 years for the Product details across 10 stores.
I would like to ...
0
votes
1answer
86 views
About the expectation of a posterior [closed]
I am interested in finding an upper bound for
$\mathbb{E}\big(
\int_B
\frac{f(X| \theta)}{f(X| \theta_0)} \pi(\theta) d \theta\big)$,
the $\frac{f(X| \theta)}{f(X| \theta_0)}$ is the likelihood ...
2
votes
0answers
83 views
Constraints on the moments of a bounded probability distribution
Consider a probability distribution with support on $[0,1]$. Suppose the first $n$ raw moments $m_1,...,m_n$ are given. What are the constraints on the $(n+1)^\text{th}$ moment? Obviously we have the ...
6
votes
2answers
584 views
disadvantages variational inference
A lot of methods utilize variational inference for hyperparameter calculation.
What are the advantages and disadvantages of variational inference ?
(ex: does it guarantee a global optimal ? )
0
votes
1answer
87 views
Need to know pdf of “x/z+sqrt(y^2-x^2)/z” , or any idea about its upper/lower bounds
I need to know the pdf of the following equation or any upper/lower bound would help. Let $X, Y, Z \sim N(0, \sigma^2)$. Then what is the distribution of:
$$A=\frac{x+ \sqrt{\mid y^2 - x^2 \mid}}{z}$$...
4
votes
1answer
664 views
Gaussian Multi-Armed Bandits and the UCB Algorithm
I've implemented in MATLAB the UCB algorithm for gaussian bandits with zero mean and unit variance (these means were themselves sampled from a gaussian prior of zero mean and unit variance).
Now I ...
