Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN)^{[1][2][3][4][5][6][7][8]} is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple families, including the half-normal distribution, truncated normal distribution, gamma distribution, and square root of the gamma distribution, all of which are special cases of the MHN distribution. Therefore, it is a flexible probability model for analyzing real-valued positive data. The name of the distribution is motivated by the similarities of its density function with that of the half-normal distribution.

Modified half-normal distribution

Notation	${\displaystyle {\text{MHN}}\left(\alpha ,\beta ,\gamma \right)}$
Parameters	${\displaystyle \alpha >0,\beta >0,{\text{ and }}\gamma \in \mathbb {R} }$
Support	${\displaystyle x\geq 0}$
PDF	${\displaystyle f_{_{\text{MHN}}}(x)={\frac {2\beta ^{\alpha /2}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}$
CDF	${\displaystyle {\begin{aligned}F_{_{\text{MHN}}}(x\mid \alpha ,\beta ,\gamma )={}&{\frac {2\beta ^{\alpha /2}}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}\\[4pt]&{}\times \sum _{i=0}^{\infty }{\frac {\gamma ^{i}}{2i!}}\beta ^{-(\alpha +i)/2}\gamma \left({\frac {\alpha +i}{2}},\beta x^{2}\right),\end{aligned}}}$ where ${\displaystyle \gamma (s,y)}$ denotes the lower incomplete gamma function.
Mean	${\displaystyle E(X)={\frac {\Psi \left({\frac {\alpha +1}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta ^{1/2}\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}$
Mode	${\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}{\text{ if }}\alpha >1}$
Variance	${\displaystyle \operatorname {Var} (X)={\frac {\Psi \left({\frac {\alpha +2}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta \Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}-\left[{\frac {\Psi \left({\frac {\alpha +1}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta ^{1/2}\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}\right]^{2}}$

In addition to being used as a probability model, MHN distribution also appears in Markov chain Monte Carlo (MCMC)-based Bayesian procedures, including Bayesian modeling of the directional data,^[4] Bayesian binary regression, and Bayesian graphical modeling.

In Bayesian analysis, new distributions often appear as a conditional posterior distribution; usage for many such probability distributions are too contextual, and they may not carry significance in a broader perspective. Additionally, many such distributions lack a tractable representation of its distributional aspects, such as the known functional form of the normalizing constant. However, the MHN distribution occurs in diverse areas of research, signifying its relevance to contemporary Bayesian statistical modeling and the associated computation.^{[clarification needed]}

The moments (including variance and skewness) of the MHN distribution can be represented via the Fox–Wright Psi functions. There exists a recursive relation between the three consecutive moments of the distribution; this is helpful in developing an efficient approximation for the mean of the distribution, as well as constructing a moment-based estimation of its parameters.

Definitions

The probability density function of the modified half-normal distribution is

$${\displaystyle f(x)={\frac {2\beta ^{\alpha /2}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}{\text{ for }}x>0}$$

 

where ${\displaystyle \Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)={}_{1}\Psi _{1}\left[{\begin{matrix}({\frac {\alpha }{2}},{\frac {1}{2}})\\(1,0)\end{matrix}};{\frac {\gamma }{\sqrt {\beta }}}\right]}$  denotes the Fox–Wright Psi function.^[9][10][11] The connection between the normalizing constant of the distribution and the Fox–Wright function in provided in Sun, Kong, Pal.^[1]

The cumulative distribution function (CDF) is

$${\displaystyle F_{_{\text{MHN}}}(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\alpha /2}}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}\sum _{i=0}^{\infty }{\frac {\gamma ^{i}}{2i!}}\beta ^{-(\alpha +i)/2}\gamma \left({\frac {\alpha +i}{2}},\beta x^{2}\right){\text{ for }}x\geq 0,}$$

 

where ${\displaystyle \gamma (s,y)=\int _{0}^{y}t^{s-1}e^{-t}\,dt}$  denotes the lower incomplete gamma function.

Properties

The modified half-normal distribution is an exponential family of distributions, and thus inherits the properties of exponential families.

Moments

Let ${\displaystyle X\sim {\text{MHN}}(\alpha ,\beta ,\gamma )}$ . Choose a real value ${\displaystyle k\geq 0}$  such that ${\displaystyle \alpha +k>0}$ . Then the ${\displaystyle k}$ th moment is

$${\displaystyle E(X^{k})={\frac {\Psi \left({\frac {\alpha +k}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta ^{k/2}\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}.}$$

 

Additionally,

$${\displaystyle E(X^{k+2})={\frac {\alpha +k}{2\beta }}E(X^{k})+{\frac {\gamma }{2\beta }}E(X^{k+1}).}$$

 

The variance of the distribution is ${\displaystyle \operatorname {Var} (X)={\frac {\alpha }{2\beta }}+E(X)\left({\frac {\gamma }{2\beta }}-E(X)\right).}$  The moment generating function of the MHN distribution is given as

$${\displaystyle M_{X}(t)={\frac {\Psi \left({\frac {\alpha }{2}},{\frac {\gamma +t}{\sqrt {\beta }}}\right)}{\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}.}$$

 

Modal characterization

Consider ${\displaystyle {\text{MHN}}(\alpha ,\beta ,\gamma )}$  with ${\displaystyle \alpha >0}$ , ${\displaystyle \beta >0}$ , and ${\displaystyle \gamma \in \mathbb {R} }$ .

• If ${\displaystyle \alpha \geq 1}$ , then the probability density function of the distribution is log-concave.
• If ${\displaystyle \alpha >1}$ , then the mode of the distribution is located at ${\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.}$ 
• If ${\displaystyle \gamma >0}$  and ${\displaystyle 1-{\frac {\gamma ^{2}}{8\beta }}\leq \alpha <1}$ , then the density has a local maximum at ${\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}}$  and a local minimum at ${\displaystyle {\frac {\gamma -{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.}$ 
• The density function is gradually decreasing on ${\displaystyle \mathbb {R} _{+}}$  and mode of the distribution does not exist, if either ${\displaystyle \gamma >0}$ , ${\displaystyle 0<\alpha <1-{\frac {\gamma ^{2}}{8\beta }}}$  or ${\displaystyle \gamma <0,\alpha \leq 1}$ .

Additional properties involving mode and expected values

Let ${\displaystyle X\sim {\text{MHN}}(\alpha ,\beta ,\gamma )}$  for ${\displaystyle \alpha \geq 1}$ , ${\displaystyle \beta >0}$ , and ${\displaystyle \gamma \in \mathbb {R} {}}$ , and let the mode of the distribution be denoted by ${\displaystyle X_{\text{mode}}={\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.}$ 

If ${\displaystyle \alpha >1}$ , then

$${\displaystyle X_{\text{mode}}\leq E(X)\leq {\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta }}}{4\beta }}}$$

 

for all ${\displaystyle \gamma \in \mathbb {R} }$ . As ${\displaystyle \alpha }$  gets larger, the difference between the upper and lower bounds approaches zero. Therefore, this also provides a high precision approximation of ${\displaystyle E(X)}$  when ${\displaystyle \alpha }$  is large.

On the other hand, if ${\displaystyle \gamma >0}$  and ${\displaystyle \alpha \geq 4}$ , then

$${\displaystyle \log(X_{\text{mode}})\leq E(\log(X))\leq \log \left({\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta }}}{4\beta }}\right).}$$

 

For all ${\displaystyle \alpha >0}$ , ${\displaystyle \beta >0}$ , and ${\displaystyle \gamma \in \mathbb {R} }$ , ${\displaystyle {\text{Var}}(X)\leq {\frac {1}{2\beta }}}$ . Also, the condition ${\displaystyle \alpha \geq 4}$  is a sufficient condition for its validity. The fact that ${\displaystyle X_{\text{mode}}\leq E(X)}$  implies the distribution is positively skewed.

Mixture representation

Let ${\displaystyle X\sim \operatorname {MHN} (\alpha ,\beta ,\gamma )}$ . If ${\displaystyle \gamma >0}$ , then there exists a random variable ${\displaystyle V}$  such that ${\displaystyle V\mid X\sim \operatorname {Poisson} (\gamma X)}$  and ${\displaystyle X^{2}\mid V\sim \operatorname {Gamma} \left({\frac {\alpha +V}{2}},\beta \right)}$ . On the contrary, if ${\displaystyle \gamma <0}$  then there exists a random variable ${\displaystyle U}$  such that ${\displaystyle U\mid X\sim {\text{GIG}}\left({\frac {1}{2}},1,\gamma ^{2}X^{2}\right)}$  and ${\displaystyle X^{2}\mid U\sim {\text{Gamma}}\left({\frac {\alpha }{2}},\left(\beta +{\frac {\gamma ^{2}}{U}}\right)\right)}$ , where ${\displaystyle {\text{GIG}}}$  denotes the generalized inverse Gaussian distribution.

References

1. 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.
2. 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.
3. 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.
4. 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.
5. Trangucci, Robert Neale (2023). Bayesian Model Expansion for Selection Bias in Epidemiology (Thesis). doi:10.7302/8573. hdl:2027.42/178116.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.