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Modified half-normal distribution
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<h2 id="definitions">Definitions</h2> <p>The probability density function of the modified half-normal distribution is f ( x ) = 2 β α / 2 x α − 1 exp ( − β x 2 + γ x ) Ψ ( α 2 , γ β ) for x > 0 {\displaystyle f(x)={\frac {2\beta ^{\alpha /2}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}{\text{ for }}x>0} where Ψ ( α 2 , γ β ) = 1 Ψ 1 [ ( α 2 , 1 2 ) ( 1 , 0 ) ; γ β ] {\displaystyle \Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)={}_{1}\Psi _{1}\left[{\begin{matrix}({\frac {\alpha }{2}},{\frac {1}{2}})\\(1,0)\end{matrix}};{\frac {\gamma }{\sqrt {\beta }}}\right]} denotes the <a href="/facts/Fox%25E2%2580%2593Wright_function/U197C9hr">Fox–Wright Psi function</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> The connection between the normalizing constant of the distribution and the Fox–Wright function in provided in Sun, Kong, Pal.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p><p>The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> (CDF) is F MHN ( x ∣ α , β , γ ) = 2 β α / 2 Ψ ( α 2 , γ β ) ∑ i = 0 ∞ γ i 2 i ! β − ( α + i ) / 2 γ ( α + i 2 , β x 2 ) for x ≥ 0 , {\displaystyle F_{_{\text{MHN}}}(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\alpha /2}}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}\sum _{i=0}^{\infty }{\frac {\gamma ^{i}}{2i!}}\beta ^{-(\alpha +i)/2}\gamma \left({\frac {\alpha +i}{2}},\beta x^{2}\right){\text{ for }}x\geq 0,} where γ ( s , y ) = ∫ 0 y t s − 1 e − t d t {\displaystyle \gamma (s,y)=\int _{0}^{y}t^{s-1}e^{-t}\,dt} denotes the lower <a href="/facts/Incomplete_gamma_function/vojnYSu4">incomplete gamma function</a>. </p> <h2 id="properties">Properties</h2> <p>The modified half-normal distribution is an <a href="/facts/Exponential_family/1LkkqEIf">exponential family</a> of distributions, and thus inherits the properties of exponential families. </p> <h3>Moments</h3> <p>Let X ∼ MHN ( α , β , γ ) {\displaystyle X\sim {\text{MHN}}(\alpha ,\beta ,\gamma )} . Choose a real value k ≥ 0 {\displaystyle k\geq 0} such that α + k > 0 {\displaystyle \alpha +k>0} . Then the <i> k {\displaystyle k} </i>th <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moment</a> is E ( X k ) = Ψ ( α + k 2 , γ β ) β k / 2 Ψ ( α 2 , γ β ) . {\displaystyle E(X^{k})={\frac {\Psi \left({\frac {\alpha +k}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta ^{k/2}\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}.} Additionally, E ( X k + 2 ) = α + k 2 β E ( X k ) + γ 2 β E ( X k + 1 ) . {\displaystyle E(X^{k+2})={\frac {\alpha +k}{2\beta }}E(X^{k})+{\frac {\gamma }{2\beta }}E(X^{k+1}).} The variance of the distribution is Var ( X ) = α 2 β + E ( X ) ( γ 2 β − E ( X ) ) . {\displaystyle \operatorname {Var} (X)={\frac {\alpha }{2\beta }}+E(X)\left({\frac {\gamma }{2\beta }}-E(X)\right).} The moment generating function of the MHN distribution is given as M X ( t ) = Ψ ( α 2 , γ + t β ) Ψ ( α 2 , γ β ) . {\displaystyle M_{X}(t)={\frac {\Psi \left({\frac {\alpha }{2}},{\frac {\gamma +t}{\sqrt {\beta }}}\right)}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}.} </p> <h3>Modal characterization</h3> <p>Consider MHN ( α , β , γ ) {\displaystyle {\text{MHN}}(\alpha ,\beta ,\gamma )} with α > 0 {\displaystyle \alpha >0} , β > 0 {\displaystyle \beta >0} , and γ ∈ R {\displaystyle \gamma \in \mathbb {R} } . </p> <ul><li>If α ≥ 1 {\displaystyle \alpha \geq 1} , then the probability density function of the distribution is log-concave.</li> <li>If α > 1 {\displaystyle \alpha >1} , then the mode of the distribution is located at γ + γ 2 + 8 β ( α − 1 ) 4 β . {\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.} </li> <li>If γ > 0 {\displaystyle \gamma >0} and 1 − γ 2 8 β ≤ α < 1 {\displaystyle 1-{\frac {\gamma ^{2}}{8\beta }}\leq \alpha <1} , then the density has a local maximum at γ + γ 2 + 8 β ( α − 1 ) 4 β {\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}} and a local minimum at γ − γ 2 + 8 β ( α − 1 ) 4 β . {\displaystyle {\frac {\gamma -{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.} </li> <li>The density function is gradually decreasing on R + {\displaystyle \mathbb {R} _{+}} and mode of the distribution does not exist, if either γ > 0 {\displaystyle \gamma >0} , 0 < α < 1 − γ 2 8 β {\displaystyle 0<\alpha <1-{\frac {\gamma ^{2}}{8\beta }}} or γ < 0 , α ≤ 1 {\displaystyle \gamma <0,\alpha \leq 1} .</li></ul> <h3>Additional properties involving mode and expected values</h3> <p>Let X ∼ MHN ( α , β , γ ) {\displaystyle X\sim {\text{MHN}}(\alpha ,\beta ,\gamma )} for α ≥ 1 {\displaystyle \alpha \geq 1} , β > 0 {\displaystyle \beta >0} , and γ ∈ R {\displaystyle \gamma \in \mathbb {R} {}} , and let the mode of the distribution be denoted by X mode = γ + γ 2 + 8 β ( α − 1 ) 4 β . {\displaystyle X_{\text{mode}}={\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.} </p><p>If α > 1 {\displaystyle \alpha >1} , then X mode ≤ E ( X ) ≤ γ + γ 2 + 8 α β 4 β {\displaystyle X_{\text{mode}}\leq E(X)\leq {\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta }}}{4\beta }}} for all γ ∈ R {\displaystyle \gamma \in \mathbb {R} } . As α {\displaystyle \alpha } gets larger, the difference between the upper and lower bounds approaches zero. Therefore, this also provides a high precision approximation of E ( X ) {\displaystyle E(X)} when α {\displaystyle \alpha } is large. </p><p>On the other hand, if γ > 0 {\displaystyle \gamma >0} and α ≥ 4 {\displaystyle \alpha \geq 4} , then log ( X mode ) ≤ E ( log ( X ) ) ≤ log ( γ + γ 2 + 8 α β 4 β ) . {\displaystyle \log(X_{\text{mode}})\leq E(\log(X))\leq \log \left({\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta }}}{4\beta }}\right).} For all α > 0 {\displaystyle \alpha >0} , β > 0 {\displaystyle \beta >0} , and γ ∈ R {\displaystyle \gamma \in \mathbb {R} } , Var ( X ) ≤ 1 2 β {\displaystyle {\text{Var}}(X)\leq {\frac {1}{2\beta }}} . Also, the condition α ≥ 4 {\displaystyle \alpha \geq 4} is a sufficient condition for its validity. The fact that X mode ≤ E ( X ) {\displaystyle X_{\text{mode}}\leq E(X)} implies the distribution is positively skewed. </p> <h3>Mixture representation</h3> <p>Let X ∼ MHN ( α , β , γ ) {\displaystyle X\sim \operatorname {MHN} (\alpha ,\beta ,\gamma )} . If γ > 0 {\displaystyle \gamma >0} , then there exists a random variable V {\displaystyle V} such that V ∣ X ∼ Poisson ( γ X ) {\displaystyle V\mid X\sim \operatorname {Poisson} (\gamma X)} and X 2 ∣ V ∼ Gamma ( α + V 2 , β ) {\displaystyle X^{2}\mid V\sim \operatorname {Gamma} \left({\frac {\alpha +V}{2}},\beta \right)} . On the contrary, if γ < 0 {\displaystyle \gamma <0} then there exists a random variable U {\displaystyle U} such that U ∣ X ∼ GIG ( 1 2 , 1 , γ 2 X 2 ) {\displaystyle U\mid X\sim {\text{GIG}}\left({\frac {1}{2}},1,\gamma ^{2}X^{2}\right)} and X 2 ∣ U ∼ Gamma ( α 2 , ( β + γ 2 U ) ) {\displaystyle X^{2}\mid U\sim {\text{Gamma}}\left({\frac {\alpha }{2}},\left(\beta +{\frac {\gamma ^{2}}{U}}\right)\right)} , where GIG {\displaystyle {\text{GIG}}} denotes the <a href="/facts/Generalized_inverse_Gaussian_distribution/7tdENlkE">generalized inverse Gaussian distribution</a>. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics - Theory and Methods. 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Trangucci, Rob; Chen, Yang; Zelner, Jon (18 Aug 2022). "Modeling racial/ethnic differences in COVID-19 incidence with covariates subject to non-random missingnes". arXiv:2206.08161. PPR533225. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Wang, Hai-Bin; Wang, Jian (23 August 2022). "An exact sampler for fully Baysian elastic net". Computational Statistics. doi:10.1007/s00180-022-01275-8. ISSN 1613-9658. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Pal, Subhadip; Gaskins, Jeremy (2 November 2022). "Modified Pólya-Gamma data augmentation for Bayesian analysis of directional data". Journal of Statistical Computation and Simulation. 92 (16): 3430–3451. doi:10.1080/00949655.2022.2067853. ISSN 0094-9655. S2CID 249022546. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Trangucci, Robert Neale (2023). Bayesian Model Expansion for Selection Bias in Epidemiology (Thesis). doi:10.7302/8573. hdl:2027.42/178116. <a href="https://deepblue.lib.umich.edu/handle/2027.42/178116" target="_blank">https://deepblue.lib.umich.edu/handle/2027.42/178116</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Haoran, Xu; Ziyi, Wang (18 May 2023). "Condition Evaluation and Fault Diagnosis of Power Transformer Based on GAN-CNN". Journal of Electrotechnology, Electrical Engineering and Management. 6 (3): 8–16. doi:10.23977/jeeem.2023.060302. S2CID 259048682. <a href="https://www.clausiuspress.com/article/7101.html" target="_blank">https://www.clausiuspress.com/article/7101.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Gao, Fengxin; Wang, Hai-Bin (17 August 2022). "Generating Modified-Half-Normal Random Variates by a Relaxed Transformed Density Rejection Method". www.researchsquare.com. doi:10.21203/rs.3.rs-1948653/v1. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Копаниця, Юрій (5 October 2021). "ПОВІТРЯНИЙ СТОВП НАПІРНОГО ГІДРОЦИКЛОНУ ІЗ ПНЕВМАТИЧНИМ РЕГУЛЯТОРОМ". Проблеми водопостачання, водовідведення та гідравліки (in Ukrainian) (36): 4–10. doi:10.32347/2524-0021.2021.36.4-10. ISSN 2524-0021. S2CID 242771336. <a href="https://doi.org/10.32347%2F2524-0021.2021.36.4-10" target="_blank">https://doi.org/10.32347%2F2524-0021.2021.36.4-10</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Pal, Subhadip; Gaskins, Jeremy (2 November 2022). "Modified Pólya-Gamma data augmentation for Bayesian analysis of directional data". Journal of Statistical Computation and Simulation. 92 (16): 3430–3451. doi:10.1080/00949655.2022.2067853. ISSN 0094-9655. S2CID 249022546. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Wright, E. Maitland (1935). "The Asymptotic Expansion of the Generalized Hypergeometric Function". Journal of the London Mathematical Society. s1-10 (4): 286–293. doi:10.1112/jlms/s1-10.40.286. ISSN 1469-7750. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Fox, C. (1928). "The Asymptotic Expansion of Generalized Hypergeometric Functions". Proceedings of the London Mathematical Society. s2-27 (1): 389–400. doi:10.1112/plms/s2-27.1.389. ISSN 1460-244X. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Mehrez, Khaled; Sitnik, Sergei M. (1 November 2019). "Functional inequalities for the Fox–Wright functions". The Ramanujan Journal. 50 (2): 263–287. arXiv:1708.06611. doi:10.1007/s11139-018-0071-2. ISSN 1572-9303. S2CID 119716471. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics - Theory and Methods. 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> </ol>