# Questions tagged [taylor-series]

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### Finding C for a PMF of a frequency distribution

N has probability mass function: $p_o = p_1 =0$ and $p_k = c/k!$ for $k=2,3,4,...$ I used exp series $\sum_{n=1}^{\infty} \frac{x^k}{k!} = e^x$ to get $c\sum_{n=1}^{\infty} \frac{1}{k!}$ then $ce=1$ ...
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### Question about delta method and variance-stabilization

The delta method or variance-stabilizing transformation can be applied to make the variance be "nearly constant" (https://en.wikipedia.org/wiki/Variance-stabilizing_transformation). They use ...
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### how Taylor series expansion is applied on the pixel array in python?

I wanted to compute Taylor series expansion of each pixel values from the pixel array. I am wondering how Taylor expansion is going to approximate each pixel values with certain approximation order. ...
20 views

### Analytical Approximation for Conditional Moments

Say I have a function of a latent variable: $F(X_{t+1})$. $F(X_{t+1})=-log(\sum\limits_{\substack{k \neq j}}\alpha^{k}_{j}\frac{S^{k}_{t+1}}{S^{j}_{t+1}})$ The term in brackets is $X_{t+1}$. I know ...
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### How does an influence function-based estimator estimate a target functional for an unknown distribution?

How exactly does a "1-step" influence function-based estimator estimate a target functional (like average treatment effect) for an unknown distribution? As described in Aaron Fisher and Edward H. ...
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### Error Propagation for Unbiased Means

I am reading through https://arxiv.org/pdf/1210.3781.pdf, and do not understand its derivation for propagation of errors with respect to means. According to the text, when trying to estimate a ...
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### How do you approximate the variance for arcsin of a proportion?

I want to calculate the variance of $$\sqrt{n}\arcsin \sqrt{P}$$ and I believe I'm supposed to use the Taylor approximation where $$\sqrt{n}\arcsin \sqrt{P} = Z,$$ where $$Z = g(P).$$ I'm a bit ...
62 views

### What is the general second-order Taylor approximation to $\mathbb{V}(f(X))$?

If $X \sim \text{N}(0, \sigma^2)$ it is well-known that we have the second-order Taylor approximation: $$\mathbb{V}[f(X)] \approx f'(\mu)^2 \cdot \sigma^2 + \frac{f''(\mu)^2}{2} \cdot \sigma^4.$$ ...
42 views

### Taylor expansion of a conditional expectation of a function of a random variable

I saw this post on hazard function and conditional pdf, Why is the Hazard function not a pdf?, and the main outcome there is that the argument of a conditional pdf cannot depend on the conditioning ...
201 views

### Probability distribution function expressed in terms of a divergent series

I'm interested in finding the CDF and PDF of $U_i$ defined as follows, $$U_i=\frac g{d^{\alpha}}$$ where $g$ is a gamma distributed random variable with shape $k$ and scale $\theta$, and $d$ is a ...
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