Saddlepoint approximation method

The saddlepoint approximation method, initially proposed by Daniels (1954) is a specific example of the mathematical saddlepoint technique applied to statistics, in particular to the distribution of the sum of $N$ independent 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).

Definition

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

{\hat{f}}_{X} (x) = \frac{1}{\sqrt{2 π K^{″} (\hat{s})}} \exp (K (\hat{s}) - \hat{s} x) (1 + ℛ)

where $ℛ$ contains higher order terms to refine the approximation and the saddlepoint approximation to the CDF is defined as:

{\hat{F}}_{X} (x) = {\begin{matrix} Φ (\hat{w}) + ϕ (\hat{w}) (\frac{1}{\hat{w}} - \frac{1}{\hat{u}}) & for x \neq μ \\ \frac{1}{2} + \frac{K^{‴} (0)}{6 \sqrt{2 π} K^{″} (0)^{3 / 2}} & for x = μ \end{matrix}

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

$μ ≜ E [X] = \int_{- \infty}^{+ \infty} x f_{X} (x) d x = \sum_{i = 1}^{N} E [X_{i}] = \sum_{i = 1}^{N} \int_{- \infty}^{+ \infty} x_{i} f_{X_{i}} (x_{i}) d x_{i}$ .

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.

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

[: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.