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# 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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79 views

### Comparing nonlinear models using the Chi-squared asymptotic approximation

I am attempting to identify the best model fit for a nonlinear mixed effects model using the asymptotic approximation to the Chi-squared test. When calculating the appropriate number of degrees of ...
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### When do Taylor series approximations to expectations of (entire) functions converge?

Take an expectation of the form $E(f(X))$ for some univariate random variable $X$ and an entire function $f(\cdot)$ (i.e., the interval of convergence is the whole real line) I have a moment ...
498 views

### Low rank approximation of binary valued matrix

How does one get the low rank approximation of binary matrix? Is the low rank approximation also a binary matrix? Note - Here binary matrix just means that any entry of the matrix can either be 0 or ...
248 views

### Classification and regression tree (CART) on large data set

I am trying to approximate a multivariate function $y = f(x_1, ...x_n)$, which I have reason to believe will be well approximated by a classification and regression tree. Some of the variables are ...
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### Approximating the distribution of a linear combination of beta-distributed independent random variables

This question is related with these other two questions in Cross Validated, which has been already answered: Approximate the distribution of the sum of ind. Beta r.v Central limit theorem when the ...
838 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 ...
823 views

### Number of Gaussian mixture components needed to approximate any distribution

I remember reading an actual proven number of components, that can approximate any distribution. Somehow I think it was 18. Can someone point me to a book/article stating something of the sort? Might'...
216 views

### Dvoretzky–Kiefer–Wolfowitz inequality hold for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
141 views

### bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \stackrel{iid}{\sim} P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one ...
4k views

### Difference of two i.i.d. lognormal random variables

Let $X_1$ and $X_2$ be 2 i.i.d. r.v.'s where $\log(X_1),\log(X_2) \sim N(\mu,\sigma)$. I'd like to know the distribution for $X_1 - X_2$. The best I can do is to take the Taylor series of both and ...
413 views

### Arbitrary function approximation in one dimension

Suppose we have some arbitrary function $f: X \mapsto Y, X \in \mathbb{R}, Y \in [0, 1]$. It may be smooth but it may not. I am looking for some way to approximate this function given samples drawn ...
116 views

### Cox PH Modelling: How important are ties to the end result?

I am working on create a survival model using Cox PH, but am concerned with the limitation of the continuous time requirement. In my dataset there are often ties. For example, out of 35000 records, ...
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### Likelihood Function for Complicated Transformations

Suppose that data X have a Normal distribution with some mean $\mu$ and some variance $\sigma^2$. However, you don't get to see X. Instead, you see $Y = g(X)$ where $g$ is a known function. Assume ...
157 views

### expectation of multivariate discrete distribution

The expectation of a function $f(x)$ over a probability distribution $p(x)$ where $x=(x_1,\dots,x_n)$ and $x_i \in \left\{1,\dots,K\right\}$ requires a summation over all possible $K^n$ combinations. ...
186 views

### deterministic sampling for multivariate discrete distributions

Unscented transform can approximate expectations well via deterministic sampling. if $f(x) = N(\mu,\Sigma)$ and $x\in \mathbb{R}^d$ then $\int f(x) h(x) \approx \frac{1}{2d}\sum_{i=1}^{2d} h(x_k)$ ...
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### Approximations to Normality

I have heard of Wilson-Hilferty approximation of Chi-Square Distribution to Normal Distribution. 1) Can this approximation be used with regard to testing of parameters of Chi-Square Distribution? 2)...
102 views

### help with this gradient computation in Expectation Propagation

I am trying to use Expectation propagation (EP) for approximating a posterior distribution in the Gaussian family. In this case, it is done by finding the Gaussian distribution with the same first and ...
4k views

### When is a second hidden layer needed in feed forward neural networks?

I'm using a feed forward neural network to approximate a function with 24 inputs, and 3 outputs. Most of the literature suggests that a single layer neural network with a sufficient number of hidden ...
515 views

### computing KL divergence: M projections for arbitrary distributions

Background I have a generative model for a process that can be described as follows: $$y = t(x, w) + e$$ where $x$ and $y$ observations of a set of random variables which are related by a non-...