Saddlepoint approximation method - Alchetron, the free social encyclopedia

The saddlepoint approximation method, initially proposed by Daniels (1954) is a specific example of the mathematical saddlepoint technique applied to statistics. 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 distribution is written as M ( t ) and the cumulant generating function as K ( t ) = log ⁡ ( M ( t ) ) then the saddlepoint approximation to the PDF of a distribution is defined as:

f ^ ( x ) = 1 2 π K ″ ( s ^ ) exp ⁡ ( K ( s ^ ) − s ^ x )

and the saddlepoint approximation to the CDF is defined as:

F ^ ( x ) = { Φ ( w ^ ) + ϕ ( w ^ ) ( 1 w ^ − 1 u ^ ) for x ≠ μ 1 2 + K ‴ ( 0 ) 6 2 π K ″ ( 0 ) 3 / 2 for x = μ

where s ^ is the solution to K ′ ( s ^ ) = x , w ^ = sgn ⁡ s ^ 2 ( s ^ x − K ( s ^ ) ) and u ^ = s ^ K ″ ( s ^ )

References

Saddlepoint approximation method Wikipedia

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