This tag is used for the saddlepoint approximation to density functions, probability mass functions, cumulative distribution functions, and so on. See Ronald W Butler: "Saddlepoint approximations with applications".

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### P-values from Empirical SaddlePoint Approximation

I try to implement an empirical saddlepoint approximation to compute p-values. Indeed, I want to compute p-values for a bunch of Score Statistics, asymptotically following a Chi-Square distribution of ...
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My question will be very practical. Actually, I work in statistical genetics and I have 1000 Score statistics for which I want to compute the corresponding p-values. To be more specific, I suppose a ...
1 vote
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### Saddle point method used to calculate the inverse Fourier transform

Here I want to find the asymptotic behavior of the following integral $$f(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp(-ikx)*\exp(t(1-\exp(-|k|^\beta)))dk,~~~~~~~Eq~1$$ where $x$ goes to infinity. ...
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### Saddlepoint approximation for Exponential family

I read the following in a book: The saddlepoint approximation of an exponential family density function is $$\tilde P(y;\mu,\phi) = \frac{1}{\sqrt{2\pi \phi V(y)}}exp(-\frac{d(y, \mu)}{2\phi})$$ Where ...
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### Saddlepoint approximation of the generalized chi-square distribution

Following the discussion found here and here, I have been trying to derive the saddlepoint approximation for the generalized chi-square distribution, with the moment generating function defined in ...
1 vote
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### How cost function for simple linear regression behaves under different settings with batch gradient descent? [closed]

In the linear regression problem, using a simple linear model with 1 variable & with 2 model parameters, performing batch Gradient Descent(GD) & assuming I am using Mean Square error as my ...
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### Saddlepoint approximation with weibull distribution

I have some trouble with this computation, I have the moment generating function of a random variable $S$ by: $$M_S(t)=\frac{\beta\mu t}{1+(1+\beta)\mu t-M_X(t)}$$ According to the text that I am ...
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### Variance of angle to $(X,Y)$ where both $X-\mu_X$ and $Y-\mu_Y \sim N(0,\sigma^2)$ independently

$X$ and $Y$ arise from observations contaminated by i.i.d. additive Gaussian noise $\sigma$. I seek the approximate variance of the angle from the origin to $(X,Y)$. What I've tried: The answer (...
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1 vote
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### What is the probability distribution for the squared distance between random points in an $n$-dimensional hypercube?

I choose random points $X,\,Y$ in $[0,\,1]^n$ (I assume all $2n$ Cartesian coordinates are $U(0,\,1)$ iids). What is the probability distribution of $\left\Vert X-Y\right\Vert _{2}^{2}$? Even the $n=1$...
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### "Weighted" Poisson binomial distribution

I have stones of different weights. For each stone, I flip the same fair coin. If it's heads, I add the stone's weight to a running total. Given the weights, can I find the distribution for the total ...
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### Finding the distribution of iid variables X, Y given distribution of X-Y

Say I know the distribution of $X-Y$, but I do not know the distributino of $X$ (or $Y$), but I know that they are statistically independent, and I know they have the same distribution. Is the problem ...
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### Asymptotic distribution of a weighted sum of chi squared variables beyond CLT? [duplicate]

I have a sum $$S = \sum_{i=1}^{n} d_i X_i^2,$$ where $X_i$ are independent standard normals, and $d_i > 0$ are fixed real numbers, for example $d_i = i$. The asymptotic distribution of this sum ...
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### How does saddlepoint approximation work?

How does saddlepoint approximation work? What sort of problem is it good for? (Feel free to use a particular example or examples by way of illustration) Are there any drawbacks, difficulties, things ...
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### Bound for weighted sum of Poisson random variables

Suppose I have some independent Poisson-distributed random variables $X_1 \ldots X_N$ with parameters $\lambda_1 \ldots \lambda_N$. These can be thought of as processes where each arrival/event ...
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### Simulation of PDF of sum of correlated Gamma random variables (in R)

My question is very related to the general sum of Gamma RVs question found in the following link: [The sum of two independent gamma random variables There is some helpful R code there for generating ...
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### Sum of random variables without central limit theorem

I know that using central limit theorem we approximate sum of random variables into Gaussian distribution. Is the any other approximation method available for finding the probability distribution ...
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### Expected number of dice rolls require to make a sum greater than or equal to K?

A 6 sided die is rolled iteratively. What is the expected number of rolls required to make a sum greater than or equal to K? Before Edit ...
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Suppose I have $n$ independent normal random variables $$X_1 \sim \mathrm{N}(\mu_1, \sigma_1^2)\\X_2 \sim \mathrm{N}(\mu_2, \sigma_2^2)\\\vdots\\X_n \sim \mathrm{N}(\mu_n, \sigma_n^2)$$ and $Y=X_1+... • 191 23 votes 1 answer 2k views ### Inverting the Fourier Transform for a Fisher distribution The characteristic function of Fisher$\mathcal{F}(1,\alpha)$distribution is: $$C(t)=\frac{\Gamma \left(\frac{\alpha +1}{2}\right) U\left(\frac{1}{2},1-\frac{\alpha }{2},-i t \alpha \right)}{\Gamma \... • 621 53 votes 5 answers 32k views ### Generic sum of Gamma random variables I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ... • 1,107 24 votes 1 answer 13k views ### sum of noncentral Chi-square random variables I need to find the distribution of the random variable$$Y=\sum_{i=1}^{n}(X_i)^2$$where$X_i\sim{\cal{N}}(\mu_i,\sigma^2_i)$and all$X_i$s are independent. I know that it is possible to first find ... • 421 4 votes 1 answer 630 views ### False-error rate in a Pearson test, when approximation by a$\chi^2$distribution is invalid? The question arises in a cryptographic context involving a regulatory test of a physical source or random bits, with null hypothesis that they are independent and unbiased.$n$samples of 4 bits are ... • 191 2 votes 3 answers 271 views ### Generating random nos based on 'k' moments How do I generate random nos based on say k moments? (no other constraints on support) When k = 2, we generate random nos. from a normal distribution defined by the 2 moments. Can we generalize this ... • 1,862 67 votes 9 answers 28k views ### Advanced statistics books recommendation There are several threads on this site for book recommendations on introductory statistics and machine learning but I am looking for a text on advanced statistics including, in order of priority: ... 15 votes 4 answers 876 views ### Do third order asymptotics exist? Most asymptotic results in statistics prove that as$n \rightarrow \infty$an estimator (such as the MLE) converges to a normal distribution based on a second-order taylor expansion of the likelihood ... • 801 11 votes 3 answers 284 views ### Approximating$Pr[n \leq X \leq m]$for a discrete distribution What's the best way to approximate$Pr[n \leq X \leq m]$for two given integers$m,n$when you know the mean$\mu$, variance$\sigma^2$, skewness$\gamma_1$and excess kurtosis$\gamma_2\$ of a ... 