Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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".

29 questions
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9 answers

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: maximum likelihood, generalized linear models,…
Robert Kubrick
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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 general set of Gamma random variables. I have…
OSE
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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 to watch out for, or traps for the unwary?
Glen_b
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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 the product of all moment generating functions…
pitfall
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Inverting the Fourier Transform for a Fisher distribution

The characteristic function of Fisher $\mathcal{F}(1,\alpha)$ distribution is: $$C(t)=\frac{\Gamma \left(\frac{\alpha +1}{2}\right) U\left(\frac{1}{2},1-\frac{\alpha }{2},-i t \alpha \right)}{\Gamma \left(\frac{\alpha }{2}\right)}$$ where $U$ is the…
Nero
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4 answers

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 function. I believe there's a similar result in…
gabgoh
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2 answers

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 of a triangular wedge (modelled as a right-angle…
Bossykena
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3 answers

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 discrete distribution $X$, and it is clear from the…
A. N. Other
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4 answers

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 P(Sum>=1 in exactly 1 roll)=1 P(Sum>=2 in exactly 1 roll)=5/6 P(Sum>=2 in exactly 2 rolls)=1/6 P(Sum>=3 in…
Usual Suspect
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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}^{\infty} \frac{\lambda^k e^{-\lambda}V_{0}…
Alex
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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 $Y=X_1+X_2+\dotsm+X_n$. How would I characterize the…
Devin
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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 of finding the distribution well-defined, as in…
pkofod
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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 generates a loss of $a_1 \ldots a_N$. What I'm…
B Fuchs
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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…
ZHU
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Asymptotic distribution of a weighted sum of chi squared variables beyond CLT?

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 is given by the central limit theorem as…
Kirill
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