Questions tagged [bounds]
Bounds represent the points with which data cannot exceed, such as minima or maxima.
297 questions
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Interpreting lower- vs upper-boundary hits in phylogenetic hidden-state models (corHMM): model inadequacy or weak estimability?
I am fitting hidden-rate discrete-trait models (HMMs) for phylogenetic comparative analysis using corHMM (v2.8). The models include two rate regimes (“slow” and “fast”), with regime-specific ...
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How does uniform weak convergence of an empirical process carry to probability bounds at an estimated parameter?
Let $ \mathcal{F} = \{ f_\theta : \theta \in \Theta \} $ be a class of functions indexed by $ \theta $. If $ \mathcal{F} $ is a Donsker class, then the empirical process $ \mathbb{G}_n(f_\theta) = \...
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theilslopes upper and lower bounds
How are the lower and upper bands computed in scipy.stats.theilslopes?
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.theilslopes.html
I looked at the main reference
P.K. Sen, “...
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Tail bounds for bivariate binomial distribution
I'm interested in estimating the joint upper tail probability of two correlated binomial random variables, say:
$$
X \sim \mathrm{Bin}(n, p_1), \quad Y \sim \mathrm{Bin}(n, p_2),
$$
such that $corr_{...
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How to calculate confidence intervals for single side bounded proportion
I have aggregated data on disease occurence for two time intervals, time A and time B say, in different countries.
I am interested in the relative change from time A to time B relative to time A. (...
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Cramér-Rao Lower Bounds for angles and other bounded estimators
The Cramér-Rao lower bound gives a lower bound on the variance of unbiased estimators. However, I have only ever seen it used on variables that can take on any value in $\mathbb{R}$. Is there a ...
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Upper bound for expected value of product of two random variables [closed]
Given two non-negative random variables $X$ and $Y$. We have that $X(\omega)<c$ for some constant $c$ and $\forall \omega \in \Omega$. Is the following statement correct:
$$
E[XY] \leq c E[Y]
$$
Do ...
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Markov chain that does random walk
Let y(t) be a Markov chain with a discrete state space, starting from t = 0 up until t = d (a fixed value) it behaves like a proper Markov chain. However, starting from t = d+1, the chain tosses a ...
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Upper Bound for a Permutation-Based Sum of Ordered Differences
I have a problem that I have been thinking about for many days but without any useful results. I am sharing it with you to seek help.
Suppose we have $X = (X_1, \dots, X_{2n})$, where $X_i \in \...
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Probability of false positive error in an all-or-nothing multiple hypothesis testing
Given I have $m$ hypothesis pairs $(H^j_0,H_j)$ where $j=1,\dots,m$. I do a Bonferroni correction and reject each null hypothesis at a significant level of $\alpha/m$. I am looking at a procedure that ...
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Concentration of softmax transform of i.i.d. gaussians
Consider $n$ i.i.d. standard Gaussian random variables, denoted by $X_1, \ldots, X_n$. I am looking to characterise the concentration of functions like $\sum_{i=1}^n X_i e^{-X_i/\tau}$ and $\sum_{i=1}^...
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Suppose $Z_i$ are i.i.d. $N(0, 1).$ And let $M_n = \max\{Z_1, \ldots, Z_n\}$, show that $P(M_n > t) \leq n(1 - \Phi(t))$
Show that $P(M_n > t) \leq n(1 - \Phi(t))$
My work:
\begin{align}
& P(M_n > t) \leq P\left(\bigcup_{i=1}^n (Z_i > t) \right) \\ \leq {} & \sum_{i=1}^n P(Z_i > t)= n(1 - \Phi(t))
\...
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Ratio of Normal pdf to cdf
I want to show that
$$\Bigl\lvert \frac{\phi(a)}{\Phi(a)} - \frac{\phi(b)}{\Phi(b)} \Bigr\rvert \leq |a-b|$$
where $\phi$ is the standard normal pdf, and $\Phi$ is the standard normal cdf.
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Conditions for a Random Variable to Satisfy a Probability Bound on Boundary Points
Let $X$ be a random variable supported on $\mathcal{X}\subset\mathbb{R}^{d}$, and let $\mathcal{X}$ be compact. Consider $ f $ as the probability density of $ X $.
My question is: What conditions ...
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