Saddlepoint approximation method

Last updated January 04, 2025

The saddlepoint approximation method, initially proposed by Daniels (1954)^[1] is a specific example of the mathematical saddlepoint technique applied to statistics, in particular to the distribution of the sum of $N$ indipendent random variables. It provides a highly accurate approximation formula for any PDF or probability mass function of a distribution, based on the moment generating function. There is also a formula for the CDF of the distribution, proposed by Lugannani and Rice (1980)^[2].

Definition

If the moment generating function of a random variable $X=\sum _{i=1}^{N}X_{i}$ is written as $M(t)=E\left[e^{tX}\right]=E\left[e^{t\sum _{i=1}^{N}X_{i}}\right]$ and the cumulant generating function as $K(t)=\log(M(t))=\sum _{i=1}^{N}\log E\left[e^{tX_{i}}\right]$ then the saddlepoint approximation to the PDF of the distribution $X$ is defined as^[1]:

{\hat {f}}_{X}(x)={\frac {1}{\sqrt {2\pi K''({\hat {s}})}}}\exp(K({\hat {s}})-{\hat {s}}x)\,\left(1+{\mathcal {R}}\right)

where ${\mathcal {R}}$ contains higher order terms to refine the approximation^[1] and the saddlepoint approximation to the CDF is defined as^[1]:

{\hat {F}}_{X}(x)={\begin{cases}\Phi ({\hat {w}})+\phi ({\hat {w}})\left({\frac {1}{\hat {w}}}-{\frac {1}{\hat {u}}}\right)&{\text{for }}x\neq \mu \\{\frac {1}{2}}+{\frac {K'''(0)}{6{\sqrt {2\pi }}K''(0)^{3/2}}}&{\text{for }}x=\mu \end{cases}}

where ${\hat {s}}$ is the solution to $K'({\hat {s}})=x$ , ${\hat {w}}=\operatorname {sgn} {\hat {s}}{\sqrt {2({\hat {s}}x-K({\hat {s}}))}}$ , ${\hat {u}}={\hat {s}}{\sqrt {K''({\hat {s}})}}$ , and $\Phi (t)$ and $\phi (t)$ are the cumulative distribution function and the probability density function of a normal distribution, respectively, and $\mu$ is the mean of the random variable $X$ :

$\mu \triangleq E\left[X\right]=\int _{-\infty }^{+\infty }xf_{X}(x)\,{\text{d}}x=\sum _{i=1}^{N}E\left[X_{i}\right]=\sum _{i=1}^{N}\int _{-\infty }^{+\infty }x_{i}f_{X_{i}}(x_{i})\,{\text{d}}x_{i}$ .

When the distribution is that of a sample mean, Lugannani and Rice's saddlepoint expansion for the cumulative distribution function $F(x)$ may be differentiated to obtain Daniels' saddlepoint expansion for the probability density function $f(x)$ (Routledge and Tsao, 1997). This result establishes the derivative of a truncated Lugannani and Rice series as an alternative asymptotic approximation for the density function $f(x)$ . Unlike the original saddlepoint approximation for $f(x)$ , this alternative approximation in general does not need to be renormalized.

Related Research Articles

In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

<span class="mw-page-title-main">Central limit theorem</span> Fundamental theorem in probability theory and statistics

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.

<span class="mw-page-title-main">Log-normal distribution</span> Probability distribution

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable $X$ is log-normally distributed, then $Y = ln(X)$ has a normal distribution. Equivalently, if $Y$ has a normal distribution, then the exponential function of $Y$ , $X = exp(Y)$ , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).

In probability theory and statistics, the chi-squared distribution with $degrees of freedom is the distribution of a sum of the squares of independent standard normal random variables.$

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.

In probability theory, the Gram–Charlier A series, and the Edgeworth series are series that approximate a probability distribution in terms of its cumulants. The series are the same; but, the arrangement of terms differ. The key idea of these expansions is to write the characteristic function of the distribution whose probability density function $f$ is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover $f$ through the inverse Fourier transform.

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

In probability theory and statistics, the Lévy distribution, named after Paul Lévy, is a continuous probability distribution for a non-negative random variable. In spectroscopy, this distribution, with frequency as the dependent variable, is known as a van der Waals profile. It is a special case of the inverse-gamma distribution. It is a stable distribution.

Directional statistics is the subdiscipline of statistics that deals with directions, axes or rotations in Rⁿ. More generally, directional statistics deals with observations on compact Riemannian manifolds including the Stiefel manifold.

In probability and statistics, a circular distribution or polar distribution is a probability distribution of a random variable whose values are angles, usually taken to be in the range [0, 2π). A circular distribution is often a continuous probability distribution, and hence has a probability density, but such distributions can also be discrete, in which case they are called circular lattice distributions. Circular distributions can be used even when the variables concerned are not explicitly angles: the main consideration is that there is not usually any real distinction between events occurring at the opposite ends of the range, and the division of the range could notionally be made at any point.

<span class="mw-page-title-main">Noncentral chi-squared distribution</span> Noncentral generalization of the chi-squared distribution

In probability theory and statistics, the noncentral chi-squared distribution is a noncentral generalization of the chi-squared distribution. It often arises in the power analysis of statistical tests in which the null distribution is a chi-squared distribution; important examples of such tests are the likelihood-ratio tests.

In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

The noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. Whereas the central probability distribution describes how a test statistic t is distributed when the difference tested is null, the noncentral distribution describes how t is distributed when the null is false. This leads to its use in statistics, especially calculating statistical power. The noncentral t-distribution is also known as the singly noncentral t-distribution, and in addition to its primary use in statistical inference, is also used in robust modeling for data.

In probability theory, the inverse Gaussian distribution is a two-parameter family of continuous probability distributions with support on (0,∞).

A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.

<span class="mw-page-title-main">Q-function</span> Statistics function

In statistics, the Q-function is the tail distribution function of the standard normal distribution. In other words, $is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations. Equivalently, is the probability that a standard normal random variable takes a value larger than .$

In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

<span class="mw-page-title-main">Wrapped normal distribution</span>

In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.

A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product $is a product distribution .$

Distributional data analysis is a branch of nonparametric statistics that is related to functional data analysis. It is concerned with random objects that are probability distributions, i.e., the statistical analysis of samples of random distributions where each atom of a sample is a distribution. One of the main challenges in distributional data analysis is that although the space of probability distributions is a convex space, it is not a vector space.

References

1 2 3 4 Daniels, H. E. (December 1954). "Saddlepoint Approximations in Statistics". The Annals of Mathematical Statistics. 25 (4): 631–650. doi:10.1214/aoms/1177728652. ISSN 0003-4851.
↑ Lugannani, Robert; Rice, Stephen (June 1980). "Saddle point approximation for the distribution of the sum of independent random variables". Advances in Applied Probability. 12 (2): 475–490. doi:10.2307/1426607. ISSN 0001-8678.

Butler, Ronald W. (2007), Saddlepoint approximations with applications, Cambridge: Cambridge University Press, ISBN 9780521872508
Daniels, H. E. (1954), "Saddlepoint Approximations in Statistics", The Annals of Mathematical Statistics, 25 (4): 631–650, doi: 10.1214/aoms/1177728652
Daniels, H. E. (1980), "Exact Saddlepoint Approximations", Biometrika, 67 (1): 59–63, doi:10.1093/biomet/67.1.59, JSTOR 2335316
Lugannani, R.; Rice, S. (1980), "Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables", Advances in Applied Probability, 12 (2): 475–490, doi: 10.2307/1426607 , JSTOR 1426607, S2CID 124484743
Reid, N. (1988), "Saddlepoint Methods and Statistical Inference", Statistical Science, 3 (2): 213–227, doi: 10.1214/ss/1177012906
Routledge, R. D.; Tsao, M. (1997), "On the relationship between two asymptotic expansions for the distribution of sample mean and its applications", Annals of Statistics, 25 (5): 2200–2209, doi: 10.1214/aos/1069362394

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[:0-1] 1 2 3 4 Daniels, H. E. (December 1954). "Saddlepoint Approximations in Statistics". The Annals of Mathematical Statistics. 25 (4): 631–650. doi:10.1214/aoms/1177728652. ISSN 0003-4851.

[2] Lugannani, Robert; Rice, Stephen (June 1980). "Saddle point approximation for the distribution of the sum of independent random variables". Advances in Applied Probability. 12 (2): 475–490. doi:10.2307/1426607. ISSN 0001-8678.

[1]

[2]