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".
27
questions
1
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0answers
73 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.
...
3
votes
1answer
78 views
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
0answers
64 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
0answers
601 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 ...
5
votes
1answer
265 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 $$\int_a^b\...
8
votes
1answer
544 views
Sum of linear combination of product of exponentials is exponential
This problem has arisen in my research: suppose that $V_i \sim \text{ED}$ are iid exponential distributions (ED) with mean $1$ and let $\lambda$ be a nonnegative number. Is it true that
$$
\sum_{k=0}^...
1
vote
0answers
59 views
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 ...
3
votes
1answer
254 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 (...
1
vote
1answer
110 views
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 $n=1$...
1
vote
2answers
2k 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
2answers
2k 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 ...
5
votes
0answers
257 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 ...
46
votes
3answers
8k 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 ...
6
votes
1answer
694 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 ...
1
vote
1answer
709 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 ...
5
votes
2answers
350 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 ...
10
votes
4answers
3k 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
2answers
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+...
23
votes
1answer
2k views
Inverting the Fourier Transform for a Fisher distribution
The characteristic function of Fisher $\mathcal{F}(1,\alpha)$ distribution is:
$$C(t)=\frac{\Gamma \left(\frac{\alpha +1}{2}\right) U\left(\frac{1}{2},1-\frac{\alpha }{2},-i t \alpha \right)}{\Gamma \...
46
votes
5answers
24k views
Generic sum of Gamma random variables
I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ...
22
votes
1answer
11k views
sum of noncentral Chi-square random variables
I need to find the distribution of the random variable
$$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 ...
4
votes
1answer
580 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 ...
2
votes
3answers
227 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 ...
59
votes
9answers
23k 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: ...
14
votes
4answers
728 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 ...
11
votes
3answers
274 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 ...
15
votes
2answers
3k 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 ...
