Questions tagged [saddlepoint-approximation]

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 ...
0 votes
0 answers
41 views

SaddlePoint Approximation Score Statistic

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
0 answers
191 views

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 ...
3 votes
1 answer
155 views

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
0 answers
839 views

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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5 votes
1 answer
293 views

Constant of Laplace approximation

I'm reading Example 3.16 of Robert & Casella's Monte Carlo Statistical Methods. It uses a Laplace approximation for approximating an integral related with the Gamma distribution namely $$\int_a^b\...
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8 votes
1 answer
687 views

Sum of linear combination of product of exponentials is exponential

This problem has arisen in my research: suppose that $V_i \sim \text{ED}$ are iid exponential distributions (ED) with mean $1$ and let $\lambda$ be a nonnegative number. Is it true that $$ \sum_{k=0}^...
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1 vote
0 answers
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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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3 votes
1 answer
411 views

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
1 answer
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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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2 answers
3k views

"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 ...
6 votes
2 answers
3k views

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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5 votes
0 answers
288 views

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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54 votes
3 answers
11k views

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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6 votes
1 answer
880 views

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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1 vote
1 answer
942 views

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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5 votes
2 answers
412 views

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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10 votes
4 answers
4k views

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 ...
8 votes
2 answers
1k views

Sum of normal truncated random variables

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+...
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23 votes
1 answer
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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15 votes
2 answers
4k views

Trigonometric operations on standard deviations

Addition, subtraction, multiplication and division of normal random variables are well defined, but what about trigonometric operations? For instance, let us suppose that I'm trying to find the angle ...
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