Saddlepoint approximation method - Revision history

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2025-01-09T06:21:43Z

v2.05b - Bot T20 CW#61 - Fix errors for CW project (Reference before punctuation - Missing whitespace before a link)

← Previous revision		Revision as of 06:21, 9 January 2025
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	The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954)<ref name=":0">{{Cite journal \|last=Daniels \|first=H. E. \|date=December 1954\|title=Saddlepoint Approximations in Statistics \|url=http://projecteuclid.org/euclid.aoms/1177728652 \|journal=The Annals of Mathematical Statistics \|language=en \|volume=25 \|issue=4 \|pages=631–650 \|doi=10.1214/aoms/1177728652 \|issn=0003-4851}}</ref> is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]], in particular to the distribution of the sum of <math>N</math> independent random variables. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the[[Moment-~~generating_function~~ \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980)<ref>{{Cite journal \|last=Lugannani \|first=Robert \|last2=Rice \|first2=Stephen \|date=June 1980 \|title=Saddle point approximation for the distribution of the sum of independent random variables \|url=https://www.cambridge.org/core/journals/advances-in-applied-probability/article/saddle-point-approximation-for-the-distribution-of-the-sum-of-independent-random-variables/70A031DB905980CA675021C6D9BFFD21 \|journal=Advances in Applied Probability \|language=en \|volume=12 \|issue=2 \|pages=475–490 \|doi=10.2307/1426607 \|issn=0001-8678}}</ref>.		The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954)<ref name=":0">{{Cite journal \|last=Daniels \|first=H. E. \|date=December 1954\|title=Saddlepoint Approximations in Statistics \|url=http://projecteuclid.org/euclid.aoms/1177728652 \|journal=The Annals of Mathematical Statistics \|language=en \|volume=25 \|issue=4 \|pages=631–650 \|doi=10.1214/aoms/1177728652 \|issn=0003-4851}}</ref> is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]], in particular to the distribution of the sum of <math>N</math> independent random variables. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the [[Moment-generating function\|moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980).<ref>{{Cite journal \|last=Lugannani \|first=Robert \|last2=Rice \|first2=Stephen \|date=June 1980 \|title=Saddle point approximation for the distribution of the sum of independent random variables \|url=https://www.cambridge.org/core/journals/advances-in-applied-probability/article/saddle-point-approximation-for-the-distribution-of-the-sum-of-independent-random-variables/70A031DB905980CA675021C6D9BFFD21 \|journal=Advances in Applied Probability \|language=en \|volume=12 \|issue=2 \|pages=475–490 \|doi=10.2307/1426607 \|issn=0001-8678}}</ref>

	== Definition ==		== Definition ==

	If the moment generating function of a random variable <math>X = \sum_{i=1}^{N} X_i</math> is written as <math>M(t)=E\left[e^{tX}\right] = E\left[e^{t\sum_{i=1}^{N}X_i}\right]</math> and the [[cumulant generating function]] as <math>K(t) = \log(M(t)) = \sum_{i=1}^{N}\log E\left[e^{tX_i}\right]</math> then the saddlepoint approximation to the [[Probability density function\|PDF]] of the distribution <math>X</math> is defined as<ref name=":0" />:		If the moment generating function of a random variable <math>X = \sum_{i=1}^{N} X_i</math> is written as <math>M(t)=E\left[e^{tX}\right] = E\left[e^{t\sum_{i=1}^{N}X_i}\right]</math> and the [[cumulant generating function]] as <math>K(t) = \log(M(t)) = \sum_{i=1}^{N}\log E\left[e^{tX_i}\right]</math> then the saddlepoint approximation to the [[Probability density function\|PDF]] of the distribution <math>X</math> is defined as:<ref name=":0" />
	:<math>\hat{f}_X (x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) \,\left(1+\mathcal{R}\right) </math>		:<math>\hat{f}_X (x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) \,\left(1+\mathcal{R}\right) </math>
	where <math>\mathcal{R}</math> contains higher order terms to refine the approximation<ref name=":0" /> and the saddlepoint approximation to the CDF is defined as<ref name=":0" />:		where <math>\mathcal{R}</math> contains higher order terms to refine the approximation<ref name=":0" /> and the saddlepoint approximation to the CDF is defined as:<ref name=":0" />

	:<math>\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 \\		:<math>\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 \\

80.41.5.39 at 09:48, 5 January 2025

2025-01-05T09:48:14Z

← Previous revision		Revision as of 09:48, 5 January 2025
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	The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954)<ref name=":0">{{Cite journal \|last=Daniels \|first=H. E. \|date=December 1954\|title=Saddlepoint Approximations in Statistics \|url=http://projecteuclid.org/euclid.aoms/1177728652 \|journal=The Annals of Mathematical Statistics \|language=en \|volume=25 \|issue=4 \|pages=631–650 \|doi=10.1214/aoms/1177728652 \|issn=0003-4851}}</ref> is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]], in particular to the distribution of the sum of <math>N</math> ~~indipendent~~ random variables. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the[[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980)<ref>{{Cite journal \|last=Lugannani \|first=Robert \|last2=Rice \|first2=Stephen \|date=June 1980 \|title=Saddle point approximation for the distribution of the sum of independent random variables \|url=https://www.cambridge.org/core/journals/advances-in-applied-probability/article/saddle-point-approximation-for-the-distribution-of-the-sum-of-independent-random-variables/70A031DB905980CA675021C6D9BFFD21 \|journal=Advances in Applied Probability \|language=en \|volume=12 \|issue=2 \|pages=475–490 \|doi=10.2307/1426607 \|issn=0001-8678}}</ref>.		The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954)<ref name=":0">{{Cite journal \|last=Daniels \|first=H. E. \|date=December 1954\|title=Saddlepoint Approximations in Statistics \|url=http://projecteuclid.org/euclid.aoms/1177728652 \|journal=The Annals of Mathematical Statistics \|language=en \|volume=25 \|issue=4 \|pages=631–650 \|doi=10.1214/aoms/1177728652 \|issn=0003-4851}}</ref> is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]], in particular to the distribution of the sum of <math>N</math> independent random variables. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the[[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980)<ref>{{Cite journal \|last=Lugannani \|first=Robert \|last2=Rice \|first2=Stephen \|date=June 1980 \|title=Saddle point approximation for the distribution of the sum of independent random variables \|url=https://www.cambridge.org/core/journals/advances-in-applied-probability/article/saddle-point-approximation-for-the-distribution-of-the-sum-of-independent-random-variables/70A031DB905980CA675021C6D9BFFD21 \|journal=Advances in Applied Probability \|language=en \|volume=12 \|issue=2 \|pages=475–490 \|doi=10.2307/1426607 \|issn=0001-8678}}</ref>.

	== Definition ==		== Definition ==

95.250.223.208: /* Definition */ minor changes

2025-01-03T11:08:25Z

Definition: minor changes

← Previous revision		Revision as of 11:08, 3 January 2025
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	If the moment generating function of a random variable <math>X = \sum_{i=1}^{N} X_i</math> is written as <math>M(t)=E\left[e^{tX}\right] = E\left[e^{t\sum_{i=1}^{N}X_i}\right]</math> and the [[cumulant generating function]] as <math>K(t) = \log(M(t)) = \sum_{i=1}^{N}\log E\left[e^{tX_i}\right]</math> then the saddlepoint approximation to the [[Probability density function\|PDF]] of the distribution <math>X</math> is defined as<ref name=":0" />:		If the moment generating function of a random variable <math>X = \sum_{i=1}^{N} X_i</math> is written as <math>M(t)=E\left[e^{tX}\right] = E\left[e^{t\sum_{i=1}^{N}X_i}\right]</math> and the [[cumulant generating function]] as <math>K(t) = \log(M(t)) = \sum_{i=1}^{N}\log E\left[e^{tX_i}\right]</math> then the saddlepoint approximation to the [[Probability density function\|PDF]] of the distribution <math>X</math> is defined as<ref name=":0" />:
	:<math>\hat{f}_X (x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) \,\left(1+\mathcal{R}\right) </math>		:<math>\hat{f}_X (x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) \,\left(1+\mathcal{R}\right) </math>
	where <math>\mathcal{R}</math> is a ~~remainder~~ ~~term~~ in the approximation<ref name=":0" /> and the saddlepoint approximation to the CDF is defined as<ref name=":0" />:		where <math>\mathcal{R}</math> contains higher order terms to refine the approximation<ref name=":0" /> and the saddlepoint approximation to the CDF is defined as<ref name=":0" />:

	:<math>\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 \\		:<math>\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 \\

95.250.223.208: /* Definition */ minor modifications to formulas.

2024-12-31T16:03:29Z

Definition: minor modifications to formulas.

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	== Definition ==		== Definition ==

	If the moment generating function of a random variable <math>X</math> is written as <math>M(t)=E\left[e^{tX}\right]</math> and the [[cumulant generating function]] as <math>K(t) = \log(M(t))</math> then the saddlepoint approximation to the [[Probability density function\|PDF]] of a distribution <math>X</math> is defined as<ref name=":0" />:		If the moment generating function of a random variable <math>X = \sum_{i=1}^{N} X_i</math> is written as <math>M(t)=E\left[e^{tX}\right] = E\left[e^{t\sum_{i=1}^{N}X_i}\right]</math> and the [[cumulant generating function]] as <math>K(t) = \log(M(t)) = \sum_{i=1}^{N}\log E\left[e^{tX_i}\right]</math> then the saddlepoint approximation to the [[Probability density function\|PDF]] of the distribution <math>X</math> is defined as<ref name=":0" />:
	:<math>\hat{f}_X (x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) \,\left(1+\mathcal{R}\right) </math>		:<math>\hat{f}_X (x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) \,\left(1+\mathcal{R}\right) </math>
	where <math>\mathcal{R}</math> is a remainder term in the approximation<ref name=":0" /> and the saddlepoint approximation to the CDF is defined as<ref name=":0" />:		where <math>\mathcal{R}</math> is a remainder term in the approximation<ref name=":0" /> and the saddlepoint approximation to the CDF is defined as<ref name=":0" />:
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	\frac{1}{2} + \frac{K'''(0)}{6 \sqrt{2\pi} K''(0)^{3/2}} & \text{for } x = \mu		\frac{1}{2} + \frac{K'''(0)}{6 \sqrt{2\pi} K''(0)^{3/2}} & \text{for } x = \mu
	\end{cases} </math>		\end{cases} </math>
	where <math>\hat{s}</math> is the solution to <math>K'(\hat{s}) = x</math>, <math>\hat{w} = \sgn{\hat{s}}\sqrt{2(\hat{s}x - K(\hat{s}))}</math> ,<math>\hat{u} = \hat{s}\sqrt{K''(\hat{s})}</math>, and <math>\Phi(t)</math> ~~is the [[cumulative distribution function]] of a [[normal distribution]],~~ <math>\phi(t)</math> the [[probability density function]] of a normal distribution and <math>\mu</math> is the mean of the random variable <math>X</math>:		where <math>\hat{s}</math> is the solution to <math>K'(\hat{s}) = x</math>, <math>\hat{w} = \sgn{\hat{s}}\sqrt{2(\hat{s}x - K(\hat{s}))}</math> ,<math>\hat{u} = \hat{s}\sqrt{K''(\hat{s})}</math>, and <math>\Phi(t)</math> and <math>\phi(t)</math> are the [[cumulative distribution function]] and the [[probability density function]] of a [[normal distribution]], respectively, and <math>\mu</math> is the mean of the random variable <math>X</math>:

	<math>\mu \triangleq E \left(X\right)</math>.		<math>\mu \triangleq E \left[X\right] = \int_{-\infty}^{+\infty} x f_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</math>.

	When the distribution is that of a sample mean, Lugannani and Rice's saddlepoint expansion for the cumulative distribution function <math>F(x)</math> may be differentiated to obtain Daniels' saddlepoint expansion for the probability density function <math>f(x)</math> (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 <math>f(x)</math>. Unlike the original saddlepoint approximation for <math>f(x)</math>, this alternative approximation in general does not need to be renormalized.		When the distribution is that of a sample mean, Lugannani and Rice's saddlepoint expansion for the cumulative distribution function <math>F(x)</math> may be differentiated to obtain Daniels' saddlepoint expansion for the probability density function <math>f(x)</math> (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 <math>f(x)</math>. Unlike the original saddlepoint approximation for <math>f(x)</math>, this alternative approximation in general does not need to be renormalized.

Naraht: repair date

2024-12-30T23:24:55Z

repair date

← Previous revision		Revision as of 23:24, 30 December 2024
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	The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954)<ref name=":0">{{Cite journal \|last=Daniels \|first=H. E. \|date=1954~~-12~~ \|title=Saddlepoint Approximations in Statistics \|url=http://projecteuclid.org/euclid.aoms/1177728652 \|journal=The Annals of Mathematical Statistics \|language=en \|volume=25 \|issue=4 \|pages=631–650 \|doi=10.1214/aoms/1177728652 \|issn=0003-4851}}</ref> is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]], in particular to the distribution of the sum of <math>N</math> indipendent random variables. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the[[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980)<ref>{{Cite journal \|last=Lugannani \|first=Robert \|last2=Rice \|first2=Stephen \|date=1980~~-06~~ \|title=Saddle point approximation for the distribution of the sum of independent random variables \|url=https://www.cambridge.org/core/journals/advances-in-applied-probability/article/saddle-point-approximation-for-the-distribution-of-the-sum-of-independent-random-variables/70A031DB905980CA675021C6D9BFFD21 \|journal=Advances in Applied Probability \|language=en \|volume=12 \|issue=2 \|pages=475–490 \|doi=10.2307/1426607 \|issn=0001-8678}}</ref>.		The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954)<ref name=":0">{{Cite journal \|last=Daniels \|first=H. E. \|date=December 1954\|title=Saddlepoint Approximations in Statistics \|url=http://projecteuclid.org/euclid.aoms/1177728652 \|journal=The Annals of Mathematical Statistics \|language=en \|volume=25 \|issue=4 \|pages=631–650 \|doi=10.1214/aoms/1177728652 \|issn=0003-4851}}</ref> is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]], in particular to the distribution of the sum of <math>N</math> indipendent random variables. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the[[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980)<ref>{{Cite journal \|last=Lugannani \|first=Robert \|last2=Rice \|first2=Stephen \|date=June 1980 \|title=Saddle point approximation for the distribution of the sum of independent random variables \|url=https://www.cambridge.org/core/journals/advances-in-applied-probability/article/saddle-point-approximation-for-the-distribution-of-the-sum-of-independent-random-variables/70A031DB905980CA675021C6D9BFFD21 \|journal=Advances in Applied Probability \|language=en \|volume=12 \|issue=2 \|pages=475–490 \|doi=10.2307/1426607 \|issn=0001-8678}}</ref>.

	== Definition ==		== Definition ==

Naraht: /* References */ add reflist

2024-12-30T23:20:46Z

References: add reflist

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	== References ==		== References ==
			{{reflist}}

	* {{citation \|last1=Butler \|first1=Ronald W. \|title= Saddlepoint approximations with applications \|publisher=Cambridge University Press \|location=Cambridge\|year=2007 \|isbn=9780521872508 }}		* {{citation \|last1=Butler \|first1=Ronald W. \|title= Saddlepoint approximations with applications \|publisher=Cambridge University Press \|location=Cambridge\|year=2007 \|isbn=9780521872508 }}

95.250.223.208: /* Definition */ clarification of certain terms of the approximation

2024-12-30T22:42:24Z

Definition: clarification of certain terms of the approximation

← Previous revision		Revision as of 22:42, 30 December 2024
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	The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954) is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]]. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the [[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980).		The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954)<ref name=":0">{{Cite journal \|last=Daniels \|first=H. E. \|date=1954-12 \|title=Saddlepoint Approximations in Statistics \|url=http://projecteuclid.org/euclid.aoms/1177728652 \|journal=The Annals of Mathematical Statistics \|language=en \|volume=25 \|issue=4 \|pages=631–650 \|doi=10.1214/aoms/1177728652 \|issn=0003-4851}}</ref> is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]], in particular to the distribution of the sum of <math>N</math> indipendent random variables. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the[[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980)<ref>{{Cite journal \|last=Lugannani \|first=Robert \|last2=Rice \|first2=Stephen \|date=1980-06 \|title=Saddle point approximation for the distribution of the sum of independent random variables \|url=https://www.cambridge.org/core/journals/advances-in-applied-probability/article/saddle-point-approximation-for-the-distribution-of-the-sum-of-independent-random-variables/70A031DB905980CA675021C6D9BFFD21 \|journal=Advances in Applied Probability \|language=en \|volume=12 \|issue=2 \|pages=475–490 \|doi=10.2307/1426607 \|issn=0001-8678}}</ref>.

	== Definition ==		== Definition ==

	If the moment generating function of a ~~distribution~~ is written as <math>M(t)</math> and the cumulant generating function as <math>K(t) = \log(M(t))</math> then the saddlepoint approximation to the PDF of a distribution is defined as:		If the moment generating function of a random variable <math>X</math> is written as <math>M(t)=E\left[e^{tX}\right]</math> and the [[cumulant generating function]] as <math>K(t) = \log(M(t))</math> then the saddlepoint approximation to the [[Probability density function\|PDF]] of a distribution <math>X</math> is defined as<ref name=":0" />:
	:<math>\hat{f}(x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) </math>		:<math>\hat{f}_X (x) = \frac{1}{\sqrt{2 \pi K''(\hat{s})}} \exp(K(\hat{s}) - \hat{s}x) \,\left(1+\mathcal{R}\right) </math>
	and the saddlepoint approximation to the CDF is defined as:		where <math>\mathcal{R}</math> is a remainder term in the approximation<ref name=":0" /> and the saddlepoint approximation to the CDF is defined as<ref name=":0" />:

	:<math>\hat{F}(x) = \begin{cases} \Phi(\hat{w}) + \phi(\hat{w})(\frac{1}{\hat{w}} - \frac{1}{\hat{u}}) & \text{for } x \neq \mu \\		:<math>\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		\frac{1}{2} + \frac{K'''(0)}{6 \sqrt{2\pi} K''(0)^{3/2}} & \text{for } x = \mu
	\end{cases} </math>		\end{cases} </math>
	where <math>\hat{s}</math> is the solution to <math>K'(\hat{s}) = x</math>, <math>\hat{w} = \sgn{\hat{s}}\sqrt{2(\hat{s}x - K(\hat{s}))}</math> ~~and~~ <math>\hat{u} = \hat{s}\sqrt{K''(\hat{s})}</math>.		where <math>\hat{s}</math> is the solution to <math>K'(\hat{s}) = x</math>, <math>\hat{w} = \sgn{\hat{s}}\sqrt{2(\hat{s}x - K(\hat{s}))}</math> ,<math>\hat{u} = \hat{s}\sqrt{K''(\hat{s})}</math>, and <math>\Phi(t)</math> is the [[cumulative distribution function]] of a [[normal distribution]], <math>\phi(t)</math> the [[probability density function]] of a normal distribution and <math>\mu</math> is the mean of the random variable <math>X</math>:

			<math>\mu \triangleq E \left(X\right)</math>.

	When the distribution is that of a sample mean, Lugannani and Rice's saddlepoint expansion for the cumulative distribution function <math>F(x)</math> may be differentiated to obtain Daniels' saddlepoint expansion for the probability density function <math>f(x)</math> (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 <math>f(x)</math>. Unlike the original saddlepoint approximation for <math>f(x)</math>, this alternative approximation in general does not need to be renormalized.		When the distribution is that of a sample mean, Lugannani and Rice's saddlepoint expansion for the cumulative distribution function <math>F(x)</math> may be differentiated to obtain Daniels' saddlepoint expansion for the probability density function <math>f(x)</math> (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 <math>f(x)</math>. Unlike the original saddlepoint approximation for <math>f(x)</math>, this alternative approximation in general does not need to be renormalized.

Tassedethe: v2.05 - Repaired 1 link to disambiguation page - (You can help) - Henry Daniels

2024-07-16T20:13:38Z

v2.05 - Repaired 1 link to disambiguation page - (You can help) - Henry Daniels

← Previous revision		Revision as of 20:13, 16 July 2024
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	The saddlepoint approximation method, initially proposed by [[~~Henry_Daniels~~ \| Daniels]] (1954) is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]]. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the [[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980).		The saddlepoint approximation method, initially proposed by [[Henry Daniels (statistician)\|Daniels]] (1954) is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]]. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the [[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980).

	== Definition ==		== Definition ==

Anarchyte: not accurate. see WP:PAREN (depreciated but still invalid tag)

2024-04-14T16:14:06Z

not accurate. see WP:PAREN (depreciated but still invalid tag)

← Previous revision		Revision as of 16:14, 14 April 2024
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	{{No footnotes\|date=April 2023}}
	The saddlepoint approximation method, initially proposed by [[Henry_Daniels \| Daniels]] (1954) is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]]. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the [[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980).		The saddlepoint approximation method, initially proposed by [[Henry_Daniels \| Daniels]] (1954) is a specific example of the mathematical [[method_of_steepest_descent\| saddlepoint ]] technique applied to [[statistics]]. It provides a highly accurate approximation formula for any [[Probability density function\|PDF]] or probability mass function of a distribution, based on the [[Moment-generating_function \| moment generating function]]. There is also a formula for the [[cumulative_distribution_function \| CDF ]] of the distribution, proposed by Lugannani and Rice (1980).

OAbot: Open access bot: doi added to citation with #oabot.

2023-08-12T23:29:09Z

Open access bot: doi added to citation with #oabot.

← Previous revision		Revision as of 23:29, 12 August 2023
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	* {{citation \|last1=Daniels \|first1=H. E. \|title= Saddlepoint Approximations in Statistics \|journal=The Annals of Mathematical Statistics \|volume=25 \|issue=4 \|pages=631–650 \|year=1954 \|doi=10.1214/aoms/1177728652\|doi-access=free }}		* {{citation \|last1=Daniels \|first1=H. E. \|title= Saddlepoint Approximations in Statistics \|journal=The Annals of Mathematical Statistics \|volume=25 \|issue=4 \|pages=631–650 \|year=1954 \|doi=10.1214/aoms/1177728652\|doi-access=free }}
	* {{citation \|last1=Daniels \|first1=H. E. \|title= Exact Saddlepoint Approximations \|journal=Biometrika \|volume=67 \|issue=1 \|pages=59–63 \|year=1980 \|doi=10.1093/biomet/67.1.59\|jstor=2335316 }}		* {{citation \|last1=Daniels \|first1=H. E. \|title= Exact Saddlepoint Approximations \|journal=Biometrika \|volume=67 \|issue=1 \|pages=59–63 \|year=1980 \|doi=10.1093/biomet/67.1.59\|jstor=2335316 }}
	* {{citation \|last1=Lugannani\|first1=R. \| last2=Rice \| first2=S. \|title= Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables \|journal=Advances in Applied Probability \|volume=12 \|issue=2 \|pages=475–490 \|year=1980 \|doi=10.2307/1426607\|jstor=1426607 \|s2cid=124484743 }}		* {{citation \|last1=Lugannani\|first1=R. \| last2=Rice \| first2=S. \|title= Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables \|journal=Advances in Applied Probability \|volume=12 \|issue=2 \|pages=475–490 \|year=1980 \|doi=10.2307/1426607\|jstor=1426607 \|s2cid=124484743 \|doi-access=free }}
	* {{citation \|last1=Reid\|first1=N.\|title= Saddlepoint Methods and Statistical Inference \|journal=Statistical Science \|volume=3 \|issue=2 \|pages=213–227 \|year=1988 \|doi=10.1214/ss/1177012906\|doi-access=free }}		* {{citation \|last1=Reid\|first1=N.\|title= Saddlepoint Methods and Statistical Inference \|journal=Statistical Science \|volume=3 \|issue=2 \|pages=213–227 \|year=1988 \|doi=10.1214/ss/1177012906\|doi-access=free }}
	* {{citation \|last1=Routledge\|first1=R. D. \| last2=Tsao \| first2=M. \|title= On the relationship between two asymptotic expansions for the distribution of sample mean and its applications \|journal=Annals of Statistics \|volume=25 \|issue=5 \|pages=2200–2209 \|year=1997 \|doi=10.1214/aos/1069362394 \|doi-access=free }}		* {{citation \|last1=Routledge\|first1=R. D. \| last2=Tsao \| first2=M. \|title= On the relationship between two asymptotic expansions for the distribution of sample mean and its applications \|journal=Annals of Statistics \|volume=25 \|issue=5 \|pages=2200–2209 \|year=1997 \|doi=10.1214/aos/1069362394 \|doi-access=free }}

← Previous revision		Revision as of 23:29, 12 August 2023
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	* {{citation \|last1=Daniels \|first1=H. E. \|title= Saddlepoint Approximations in Statistics \|journal=The Annals of Mathematical Statistics \|volume=25 \|issue=4 \|pages=631–650 \|year=1954 \|doi=10.1214/aoms/1177728652\|doi-access=free }}		* {{citation \|last1=Daniels \|first1=H. E. \|title= Saddlepoint Approximations in Statistics \|journal=The Annals of Mathematical Statistics \|volume=25 \|issue=4 \|pages=631–650 \|year=1954 \|doi=10.1214/aoms/1177728652\|doi-access=free }}
	* {{citation \|last1=Daniels \|first1=H. E. \|title= Exact Saddlepoint Approximations \|journal=Biometrika \|volume=67 \|issue=1 \|pages=59–63 \|year=1980 \|doi=10.1093/biomet/67.1.59\|jstor=2335316 }}		* {{citation \|last1=Daniels \|first1=H. E. \|title= Exact Saddlepoint Approximations \|journal=Biometrika \|volume=67 \|issue=1 \|pages=59–63 \|year=1980 \|doi=10.1093/biomet/67.1.59\|jstor=2335316 }}
	* {{citation \|last1=Lugannani\|first1=R. \| last2=Rice \| first2=S. \|title= Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables \|journal=Advances in Applied Probability \|volume=12 \|issue=2 \|pages=475–490 \|year=1980 \|doi=10.2307/1426607\|jstor=1426607 \|s2cid=124484743 }}		* {{citation \|last1=Lugannani\|first1=R. \| last2=Rice \| first2=S. \|title= Saddle Point Approximation for the Distribution of the Sum of Independent Random Variables \|journal=Advances in Applied Probability \|volume=12 \|issue=2 \|pages=475–490 \|year=1980 \|doi=10.2307/1426607\|jstor=1426607 \|s2cid=124484743 \|doi-access=free }}
	* {{citation \|last1=Reid\|first1=N.\|title= Saddlepoint Methods and Statistical Inference \|journal=Statistical Science \|volume=3 \|issue=2 \|pages=213–227 \|year=1988 \|doi=10.1214/ss/1177012906\|doi-access=free }}		* {{citation \|last1=Reid\|first1=N.\|title= Saddlepoint Methods and Statistical Inference \|journal=Statistical Science \|volume=3 \|issue=2 \|pages=213–227 \|year=1988 \|doi=10.1214/ss/1177012906\|doi-access=free }}
	* {{citation \|last1=Routledge\|first1=R. D. \| last2=Tsao \| first2=M. \|title= On the relationship between two asymptotic expansions for the distribution of sample mean and its applications \|journal=Annals of Statistics \|volume=25 \|issue=5 \|pages=2200–2209 \|year=1997 \|doi=10.1214/aos/1069362394 \|doi-access=free }}		* {{citation \|last1=Routledge\|first1=R. D. \| last2=Tsao \| first2=M. \|title= On the relationship between two asymptotic expansions for the distribution of sample mean and its applications \|journal=Annals of Statistics \|volume=25 \|issue=5 \|pages=2200–2209 \|year=1997 \|doi=10.1214/aos/1069362394 \|doi-access=free }}