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 ...
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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 ...
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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 ...
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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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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 $$\int_a^b\...
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Sum of linear combination of product of exponentials is exponential
This problem has arisen in my research: suppose that $V_i \sim \text{ED}$ are iid exponential distributions (ED) with mean $1$ and let $\lambda$ be a nonnegative number. Is it true that
$$
\sum_{k=0}^...
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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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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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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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Inverting the Fourier Transform for a Fisher distribution
The characteristic function of Fisher $\mathcal{F}(1,\alpha)$ distribution is:
$$C(t)=\frac{\Gamma \left(\frac{\alpha +1}{2}\right) U\left(\frac{1}{2},1-\frac{\alpha }{2},-i t \alpha \right)}{\Gamma \...
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Generic sum of Gamma random variables
I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ...
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sum of noncentral Chi-square random variables
I need to find the distribution of the random variable
$$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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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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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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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: ...
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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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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 ...
A. N. Other
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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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