Questions tagged [approximation]
Approximations to distributions, functions, or other mathematical objects. To approximate something means to find some representation of it which is simpler in some respect, but not exact.
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Subtraction of Monte Carlo integrals - Catastrophic cancellation
I am attempting to estimate a quantity $Q$ which is given by the difference between two functions of Monte Carlo integrals over some set of points $\{x_i\}_{i=1}^N$, call the estimator $\hat{Q}$:
$$ \...
- 404
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Approximation of the expected value of the $i$-th standard normal order statistic in a sample of size n
For random variables $X_1, \cdots, X_n$, we denote the order statistics by
\begin{align}
X_{(1)} & = \min (X_1,\ldots, X_n) \\[6pt]
X_{(2)} & = \text{second-smallest of } X_1,\ldots, X_n \\
&...
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Normal approximation for posterior distribution
I am reading the example 4.3.3 of "The Bayesian Choice" by Christian P. Robert and I was wondering if it is possible to obtain a normal approximation in this case to estimate the posterior. ...
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Continuity correction in a 2 proportion test, with different sample sizes
In a test of 2 proportions (binomial -> Normal), when the sample sizes are different, what does a continuity correction look like?
Usually, in a 1 sample test, we would divide by $n$ (sample size) ...
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Characterize conditions in which Taylor moment approximation is good
I am working with the Projected Gaussian, or Angular Gaussian distribution, which is given by $z = \frac{x}{||x||}$, where $x \sim \mathcal{N}(\mu, \Sigma)$. This is a distribution on the sphere in $\...
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Why is taking the mean RMSE sometimes so far off overall RMSE?
I'm working with a multi-threaded program, which splits a dataset into N chunks, and evaluates some regression model's performance, predicting a score for each item in each chunk.
I'm using RMSD as ...
- 65
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Monte Carlo Approximation on integral of Gaussian pdf on Convex Domain
I have hard time on estimating the following integral on convex domain ($\mathcal D$) using Monte-Carlo approximation.
$$a = \int_{\mathcal D} dx f(x;\mu,\Sigma) $$
where $x \in \mathbb R^d$ and $f$ ...
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taylor approximation multivariate OLS coefficient
Say we have the following multivariate regression model:
$ y = \beta_1 x_1 + \beta_2 x_2 + \varepsilon $
The OLS formula for the first coefficient looks like this
$ \hat{\beta}_1 = \frac{Cov(\tilde{y}...
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votes
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Universal approximation theorem for neural networks reference
On Wikipedia, a nice theorem is given:
However, I can not find the stated theorem in the given references. So where is the stated theorem from?
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Getting extremely poor accuracy while doing function approximation using a neural networks in PyTorch [duplicate]
I have been given a task to approximate the function 5x^3 - 10x^2 - 5x - 9 using a neural network in pytorch. The training data is the set of integers in the range [-100,100] and I have to test the ...
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Approximation for a correlation matrix
I have a cross-correlation matrix of some parameter for each time period. E.g. expected economy growth for each months in the future, i.e. growth for Apr 2014, May 2014, ...., Dec 2018, and ...
3
votes
1
answer
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What paper did Hall suggest the queuing rule of thumb $s \geq \max ( 1, \rho + \sqrt{\rho})$?
According to this site:
Hall (1991) cited an argument of his previous paper that operation research profession could and should be more scientific and less mathematical. In fact, Hall also suggested ...
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Montecarlo Confidence Interval of T distribution
Suppose:
\begin{equation}
x|\sigma^2 \sim \mathcal{N}(x; \mu, \sigma^2) \; \; st. \; \; \sigma^2 \sim \mathcal{X}^{-2}(\sigma^2; \psi, v)
\end{equation}
where $\mathcal{X}^{-2}$ is the inverse ...
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Taylor approximation for function of a random variable [closed]
There is a function $f$ whose domain is the space of CDFs on $\mathbb{R}_+$ and whose range is $[0,1]$, e.g. $f$ maps a CDF on to a real number. Further, $f$ is continuous, increasing with respect to ...
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Moments and PDF of solution to random quadratic equation
Consider the following random quadratic equation,
$$
x^2 + Z x + Y = 0,
$$
where,
$$
\begin{gathered}
Z \sim \mathcal{N}(\mu_Z,\sigma_Z),
\qquad
Y \sim \mathcal{N}(\mu_Y,\sigma_Y).
\end{gathered}
$$
...
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