Questions tagged [bounds]
Bounds represent the points with which data cannot exceed, such as minima or maxima.
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Concentration inequality for sums of independent gamma random variables
I am dealing with the following problem:
Say $X_1, \ldots, X_n$ are independent Gamma random variables, each one having shape and rate parameters $\alpha_i$ and $\beta_i$, respectively. Let $S_n = \...
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Vintage of this lower bound on skewness for positive data with given mean and sd?
It turns out there is a lower bound on the skewness $g_1$ of any strictly positive set of data having a given mean μ and standard deviation σ:
$$
g_1 > \sigma/\mu - \mu/\sigma.
$$
Although ...
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Nonlinear Optimization of Noisy Functions w/ Bound Constraints via SciPy
Can we use scipy.optimize.minimize to find the best parameters $\mathbf{w} \in \Omega^k$,
$\Omega \subset \mathbb{R}$, of a function
$g = g(f(\mathbf{x}), \mathbf{w}...
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Calculating variance between paired samples, where one sample is constrained to always be the lower bound?
I'm sure this is a solved question, but I haven't been able to hit on the right search terms.
Suppose I have paired samples A and B. A represents a derived variable (say distance "as-the-crow-...
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Bound Product of Independent Gaussians
I'm interested in obtaining upper bounds on
$$
\Pr[\prod_{i\in[n]}|G_i| > x]
$$
where $G_i\sim\mathcal{N}(0,1)$ i.i.d, and $[n] := \{0,1,\dots,n-1\}$.
The most naive bound is to note that each $G_i$...
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Bound on Rademacher complexity using polynomial discrimination
This is lemma 4.14 in Wainwright's textbook on High-Dimensional Statistics, it states that given a class of function $\mathcal{F}$ has polynomial discrimination of order $v$, then for all integer $n$ ...
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likelihood ratio tests on bounded parameters
I am confused by the likelihood ratio test's boundary condition limitation. A commonly stated is that it causes problem for variance parameter because it is bounded below by 0. Can these models ...
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Different versions of additive chernoff
The additive Chernoff Bound says for $X_i \in \{0,1\}$ that satisfies $\mathbb{E}[X_i] = p,$
$$
\mathbb P\left(\sum\limits_{i}^nX_i \geq np+n\epsilon \right) \leq \exp\left(-\frac{(n\epsilon)^2}{2(np+\...
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Upper bound for 1-Wasserstein distance between standard uniform and other distribution on $[0,1]$
I want to use the following metric to measure the distance between the standard uniform distribution and any other probability distribution on $[0,1]$.
$$\int_0^1 |F(x) - x| dx$$
$F(x)$ is the cdf of ...
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Using multiple instruments to construct bounds
Suppose I have three candidate instruments $Z_1, Z_2, Z_3$ for the same endogenous variable $T$. I have no clear preference for which one the exclusion restriction is actually valid.
Can I combine the ...
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Seeking Lower Bound for Partition Probability in Random Variable Analysis
I am reaching out to seek assistance with a probability problem involving random variables.
For each $p$ in $[1,\infty)$, consider positive random variables $X_{1,p}, X_{2,p}, \ldots, X_{n,p}$ such ...
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What are bounded distributions? and can a bounded distribution hold the normality assumption?
I heard that normal distribution should be unbounded, but I want clarification about that, aren't most distributions in the real world bounded, I mean they won't go to infinity they have minimum and ...
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Dependent variables are count variable with an upper bound
I need to test some hypotheses for a social sciences dissertation. In my description below, I refer to the independent as the Xs and the dependent variables as the Ys.
I am expecting a straight linear ...
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Upper Bound on $\mathbb{E}[\frac{1}{1 + X}]$ where $\mathbb{E}[X] = a$ and $0<𝑎<1$
$𝑋$ is a positive random variable (potentially unbounded) with $0 \le \mathbb{E}[X] = a < 1$.
Since $\phi(x) = \frac{1}{x}$ is a convex function, we can use Jensen's inequality to derive a lower ...
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Chernoff bound on probability of binomial random variables is the maximum of its tail
I have this problem that I found on the paper I'm reading. In that paper, it is given that random variable $X = \sum_{i=1}^n X_i$, where each $X_i$ is Bernoulli with parameter $p=1/6$ and they are i.i....
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