A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Jismi Ma hew1,∗and Milin K. Anil2
1Depa men o S a is ics, Vimala College (Au onomous), Th issu , Ke ala, India;
[email p o ec ed]
2Depa men o S a is ics, Vimala College (Au onomous), Th issu , Ke ala, India;
[email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : In ecen yea s, Neu osophic s a is ics has eme ged as a powe ul amewo k o
handling unce ain y, inde e minacy, and imp ecision in da a. This e iew pape p esen s a
comp ehensi e e ospec i e analysis o Neu osophic dis ibu ions, acing hei heo e ical
ounda ions, his o ical e olu ion, and me hodological ad ancemen s. The s udy sys ema ically
compa es Neu osophic dis ibu ions wi h classical p obabili y dis ibu ions, emphasizing hei
unique abili y o explici ly model inde e minacy h ough he inco po a ion o u h, inde e -
minacy, and alsi y componen s. Key Neu osophic dis ibu ions in oduced in he li e a u e
a e discussed alongside hei ela ed e ms, wi h a abula p esen a ion o aid cla i y. The
me hodologies employed in exis ing s udies a e examined in de ail. Pa icula a en ion is
gi en o hei applica ions in biomedical esea ch, whe e unce ain y is a c i ical ac o in
decision-making and da a in e p e a ion. The ad an ages, limi a ions, and challenges associa ed
wi h Neu osophic models a e also analyzed. Finally, u u e esea ch di ec ions a e p oposed,
including he de elopmen o new dis ibu ions, imp o ed compu a ional ools, and b oade
in e disciplina y applica ions. This e iew unde sco es he g owing signi icance o Neu osophic
dis ibu ions as a gene aliza ion o classical models, o e ing a mo e ealis ic app oach o da a
analysis in complex, unce ain en i onmen s.
Keywo ds: Neu osophic Dis ibu ions, Unce ain y Modeling, Compa ison wi h Classical
Dis ibu ions
1 In oduc ion
The analysis o da a in eal-wo ld scena ios o en encoun e s he challenge o unce ain y. T adi-
ional s a is ical me hods, while powe ul in many con ex s, ope a e unde he assump ion o
p ecise and well-de ined da a. Howe e , eal-wo ld da a is equen ly augh wi h agueness,
ambigui y, incomple eness, and e en con adic ions. This inhe en unce ain y poses limi a ions
o classical s a is ical app oaches, necessi a ing he de elopmen o me hodologies capable o e -
ec i ely modeling and analyzing such complex in o ma ion. Neu osophic s a is ics has eme ged
as a aluable amewo k o add essing hese limi a ions. I is oo ed in neu osophic logic, which
ex ends uzzy logic by inco po a ing he concep o inde e minacy alongside he adi ional
no ions o u h and alsi y [
15
]. This ex ension allows o a mo e nuanced ep esen a ion o
eal-wo ld sys ems whe e in o ma ion migh no be de ini i ely ue o alse, bu a he con ain
elemen s o he unknown o unclea . Neu osophic s a is ics, he e o e, se es as a gene aliza ion
o classical s a is ics, speci ically designed o handle inde e mina e da a and in e ence me hods
[
4
]. I s abili y o accommoda e a ying deg ees o u h, alsi y, and inde e minacy makes i
pa icula ly sui able o analyzing he messy and complex na u e o eal-wo ld phenomena [
10
].
Fu he mo e, Neu osophic S a is ics is buil upon he ounda ion o Se Analysis, which posi-
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026
Uni e si y o New Mexico
ions In e al S a is ics as a speci ic ins ance wi hin he b oade Neu osophic amewo k. The
in oduc ion o inde e minacy as a undamen al componen ma ks a signi ican depa u e om
adi ional s a is ical pa adigms, o e ing a new pe spec i e on how unce ain y is app oached
in da a analysis. While classical s a is ics p ima ily models andomness using p obabili y, and
uzzy s a is ics ex ends his o handle agueness h ough deg ees o membe ship, neu osophic
s a is ics b oadens he scope u he by explici ly add essing si ua ions whe e in o ma ion is
no jus ague bu also genuinely unknown o con adic o y. This h ee dimensional app oach,
encompassing u h, alsi y, and inde e minacy, allows o a mo e comp ehensi e ep esen a ion
o he complexi ies inhe en in eal-wo ld da a.
This pape aims o p o ide a comp ehensi e e ospec i e analysis o he ield o Neu osophic
dis ibu ions. The objec i e is o explo e he his o ical de elopmen and e olu ion o hese
dis ibu ions, in es iga e hei applica ions ac oss a ious domains, compa e hem wi h o he
ypes o s a is ical dis ibu ions, iden i y hei ad an ages and limi a ions, analyze he me hod-
ologies used in hei s udy, and ul ima ely syn hesize he indings o iden i y key insigh s, ends,
and po en ial u u e esea ch di ec ions. This e ospec i e s udy seeks o o e a holis ic iew
o Neu osophic dis ibu ions, highligh ing hei impac and e ec i eness in add essing he
challenges posed by unce ain y in da a analysis.
In Sec ion 2 i led Fundamen al Concep s o Neu osophic Dis ibu ions we explain he key
concep s o Neu osophy, he s uc u e o Neu osophic se s, and how hey ex end classical and
uzzy logic sys ems. In Sec ion 3, His o ical E olu ion o Neu osophic Dis ibu ions p o ides a
b ie o e iew o he ounda ional concep s and key miles ones in he de elopmen o Neu o-
sophic dis ibu ions. In Sec ion 4, O e iew o Key Neu osophic Dis ibu ions, we p esen a
comp ehensi e able lis ing a ious Neu osophic dis ibu ions and he associa ed e minologies
o ease o e e ence. Sec ion 5 i led Compa ison wi h Classical and Neu osophic Dis ibu ions,
highligh s how Neu osophic dis ibu ions ex end classical amewo ks by inco po a ing inde-
e minacy explici ly. In Sec ion 6, Ad an ages and Challenges o Neu osophic Dis ibu ions,
explo es he bene i s o using Neu osophic models, such as imp o ed lexibili y in modeling
unce ain y, while also add essing hei compu a ional and heo e ical challenges. Sec ion 7
Me hodologies used in Re ospec i e Neu osophic S udies, ou lines key me hodologies used
in e ospec i e s udies, including de i a ions, simula ions, eal da a analysis, and in e ence
echniques. Sec ion 8 i led Neu osophic Dis ibu ions in Biomedical Resea ch highligh s he
applica ions o Neu osophic dis ibu ions in biomedical esea ch. In Sec ion 9, Conclusion
and Fu u e Di ec ions, we summa ize key indings and p opose di ec ions o u u e esea ch,
including new dis ibu ion de elopmen , so wa e implemen a ion, and expanded applica ions.
2 Fundamen al Concep s o Neu osophic Dis ibu ions
Neu osophic dis ibu ions o m he backbone o Neu osophic s a is ics, o e ing a obus ame-
wo k o analyzing da a wi h unce ain y, agueness, and con adic ion. Unlike classical me hods
ha assume p ecise alues, Neu osophic app oaches explici ly inco po a e inde e minacy in o
s a is ical modeling. This sec ion in oduces he essen ial concep s ha de ine he s uc u e and
beha io o Neu osophic dis ibu ions.
2.1 Neu osophic Se s and Logic
A he hea o Neu osophic dis ibu ions lie he undamen al concep s o Neu osophic se s,
logic, and p obabili y. A Neu osophic se is cha ac e ized by he ac ha each elemen wi hin
i possesses a deg ee o membe ship de ined by h ee independen componen s: u h (T),
inde e minacy (I), and alsi y (F) [
15
]. These componen s ep esen he ex en o which an
elemen belongs o he se , he deg ee o which i s membe ship is unknown o unclea , and he
deg ee o which i does no belong o he se , espec i ely. Ope a ing on hese u h alues
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 394
is Neu osophic logic, which se es as he unde lying sys em o easoning abou p oposi ions
in he p esence o inde e minacy. I allows o deg ees o u h, alsehood, and inde e minacy,
o e ing a mo e lexible amewo k han classical bina y logic.
2.2 Neu osophic P obabili y
Neu osophic p obabili y eme ges as a gene aliza ion o classical p obabili y. In his amewo k,
he p obabili y o an e en is exp essed no as a single alue, bu as a iple ( , i, ), whe e
’ ’ ep esen s he chance o he e en being ue, ’i’ ep esen s he inde e mina e chance ( he
p obabili y o he e en occu ing o no occu ing is unknown), and ’ ’ ep esen s he chance o
he e en being alse [
18
]. A key dis inc ion om classical p obabili y is ha in he neu osophic
ealm, he sum o all space p obabili ies equals 3, e lec ing he inclusion o inde e minacy,
whe eas in classical p obabili y, his sum is 1. The shi om a bina y pe spec i e o ue/ alse
o e en he uzzy pe spec i e o a deg ee o u h o a icho omy o u h, inde e minacy, and
alsi y ep esen s a undamen al concep ual di e ence. This allows Neu osophic me hods o
model a signi ican ly wide ange o eal-wo ld scena ios whe e unce ain y is no simply a ma e
o deg ee bu can also in ol e genuine unknowns o con adic ions.
2.3 Neu osophic Random Va iables
A Neu osophic andom a iable is de ined as a a iable whose possible alues a e Neu osophic
numbe s. These numbe s comp ise wo pa s: a de e mina e pa , which ep esen s he known
o p ecise aspec , and an inde e mina e pa , which cap u es he unce ain y o agueness
[
4
]. A common ep esen a ion o a Neu osophic numbe is XN = a + bI, whe e ’a’ is he
de e mina e componen and ’bI’ is he inde e mina e componen . Consequen ly, a Neu osophic
dis ibu ion is a p obabili y dis ibu ion ha is de ined o a Neu osophic andom a iable.
This dis ibu ion can be ep esen ed in a ious ways, such as by h ee sepa a e unc ions o
in e als co esponding o u h, inde e minacy, and alsi y, o mo e commonly, by pa ame e s o
a classical dis ibu ion ha a e hemsel es Neu osophic numbe s. The unc ion ha models he
Neu osophic P obabili y o a andom a iable x is deno ed as NP(x) = (T(x), I(x), F(x)). The
use o Neu osophic numbe s o de ine bo h andom a iables and hei associa ed dis ibu ions
allows o he di ec and inhe en inco po a ion o unce ain y and agueness in o he s a is ical
model i sel , a he han ea ing unce ain y as an ex e nal ac o o be add essed a e da a
collec ion.
2.4 Rep esen a ion o Inde e minancy
The ep esen a ion o inde e minacy in Neu osophic dis ibu ions is a c ucial aspec . Inde e mi-
nacy is o en exp essed using an in e al o by a pa ame e ha is mul iplied by an inde e mina e
ac o ’I’, whe e ’I’ ypically anges wi hin he in e al o some o he speci ied in e al [
4
]. This
inde e mina e pa se es o e lec he unknown, ague, o e en con adic o y in o ma ion ha
is associa ed wi h he a iable o he pa ame e s o he dis ibu ion. I is impo an o no e ha
Neu osophic s a is ics possesses he capabili y o educe inde e minacy h ough ma hema ical
ope a ions in ce ain cases, a ea u e ha dis inguishes i om in e al s a is ics, whe e such
ope a ions migh lead o an inc ease in inde e minacy. The in e al-based ep esen a ion o inde-
e minacy p o ides a p ac ical and quan i iable way o wo k wi h unce ain y wi hin s a is ical
models. Ins ead o elying on a single p ecise alue, his app oach acknowledges he po en ial
ange o alues ha a pa ame e o obse a ion migh ake, hus o e ing a mo e ealis ic and
lexible amewo k o analysis in he ace o impe ec in o ma ion.
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 395
3 His o ical E olu ion o Neu osophic Dis ibu ions
The genesis o Neu osophic dis ibu ions can be aced o he in oduc ion o Neu osophy by
Flo en in Sma andache in 1995 [
8
]. Concei ed as a gene aliza ion o uzzy logic, Neu osophy
p o ided a philosophical amewo k o add essing neu ali y and inde e minacy, bo h common
in eal-wo ld si ua ions. Following his ounda ion, Neu osophic s a is ics eme ged as a dis-
inc discipline ocused on analyzing da a cha ac e ized by imp ecision and inde e minacy [
7
].
Ea ly esea ch es ablished undamen al Neu osophic s a is ical concep s, pa ing he way o
mo e ad anced ools, including p obabili y dis ibu ions. Key Con ibu o s o Neu osophic
Dis ibu ions:
•
Flo en in Sma andache — O igina o o Neu osophy and a p oli ic con ibu o o Neu o-
sophic s a is ics [7].
•
Muhammad Aslam — De eloped nume ous Neu osophic s a is ical es s and dis ibu ions
[15].
•O he no able esea che s and hei con ibu ions include:
–Pa o Sma andache — Neu osophic Binomial and No mal dis ibu ions [15].
–She wani e al. — Con ibu ions o he Neu osophic Binomial dis ibu ion [15].
–S. Al-Duais — Neu osophic Log-Gamma dis ibu ion [12].
–Khan e al. — Neu osophic Nega i e Binomial dis ibu ion [15].
–Musa — Neu osophic Pa e o dis ibu ion [11].
The ield o Neu osophic dis ibu ions, hough ela i ely young, shows ac i e de elopmen , wi h
g owing ecogni ion o i s po en ial o add ess unce ain y in da a analysis. Miles ones in he
E olu ion o Neu osophic Dis ibu ions:
•
Ea ly de elopmen s ocused on ex ending classical dis ibu ions in o he Neu osophic
amewo k:
–Neu osophic Binomial, No mal, and Mul inomial dis ibu ions [15].
•Subsequen expansions included:
–Neu osophic Nega i e Binomial dis ibu ion [15].
–Neu osophic Log-Gamma dis ibu ion [12].
–Neu osophic In e se Exponen ial dis ibu ion [10].
–Neu osophic Maxwell dis ibu ion [13].
–Neu osophic Lindley dis ibu ion [8].
–Neu osophic Pa e o dis ibu ion [11].
–Neu osophic q-Poisson dis ibu ion [3].
Beyond dis ibu ions, Neu osophic concep s ha e been applied o gene alize s a is ical es s and
me hodologies o hypo hesis es ing and in e ence unde unce ain y [
7
]. This gene aliza ion
s a egy e lec s a sys ema ic e o o expand classical s a is ical ools o be e accommoda e
eal-wo ld da a a ec ed by inde e minacy and imp ecision.O e ime, esea che s s eadily buil
on he ea ly ideas, in oducing new Neu osophic dis ibu ions o be e handle he complex
and unce ain na u e o eal-wo ld da a. This s eady g ow h shows how he ield has e ol ed in
esponse o he need o mo e lexible and ealis ic s a is ical ools
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 396
4 O e iew o Key Neu osophic Dis ibu ions
The ield o Neu osophic s a is ics has wi nessed he de elopmen o se e al key p obabili y
dis ibu ions ha ex end hei classical coun e pa s o handle unce ain y and inde e minacy.
A close look a some o hese dis ibu ions e eals hei unique p ope ies and po en ial
applica ions.
The ollowing able summa izes he neu osophic dis ibu ions and hei co esponding p obabili y
unc ions.
Table 1: Neu osophic Dis ibu ions and P obabili y Func ions
SI no
Neu osophic
Dis ibu ion
P obabili y Func ion
1 Binomial (x) =
n
xpx
N(1 −pN)n−x,i x= 0,1, . . . , n
0,o he wise
2 Poisson (x) =
e−λNλx
N
(x)!,i x= 0,1, . . . , n
0,o he wise
3 Exponen ial (x) =
λNe−xλN,i x≥0
0,o he wise
4 Uni o m (x) =
1
bN−aNi a≤x≤b,
0o he wise.
5 No mal (x) =
1
σN√2πexp −(x−µN)2
2σ2
N,i −∞ <x<∞
0,o he wise
6 Weibull (x) =
βN
αβN
N
xβN−1e−(x/αN)βN,i x > 0
0,o he wise
7 Be a (x) =
1
B(αN,βN)xαN−1(1 −x)βN−1,i 0≤x≤1
0,o he wise
whe e B(αN, βN) = Γ(αN)Γ(βN)
Γ(αN+βN)
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 397
8 Gamma (x) =
1
Γ(αN)λαN
N
xαN−1e−(x
λN),i x≥0
0,o he wise
9 Lomax (x) =
αN
βN1 + x
βN−(αN+1)
,i x≥0
0,o he wise
10 Laplace (x) =
1
2βNexp −|x−θN|
βN, o −∞ <x<∞
0,o he wise
11 Geome ic (x) =
pN(1 −pN)x,i x= 0,1, . . . , n
0,o he wise
12 Bu -III (x) =
ckx−c−1(1 + x−c)−k−1, o x > 0, c > 0, k > 0
0,o he wise
13
Gene alized
Pa e o (NGPD)
(x) =
1
βN1 + αNxN
βN−1
αN−1
,i x > 0
0,o he wise
14
Nega i e Bino-
mial
(x) =
N+x−1p N
N(1 −pN)x,i x= 0,1, . . . , n
0,o he wise
15 Kuma aswamy (x) =
αNβNxαN−1(1 −xαN)βN−1,i x∈(0,1)
0,o he wise
16 Rayleigh (x) =
x
θ2
N
e−x2
2θ2
N,i x > 0
0,o he wise
17
Log Gamma
(NLGD)
(x) =
bp
Γ(p)xb−1(log(x))p−1,i x≥1, p, b > 0
0,o he wise
18
In e se Expo-
nen ial (NIE)
(x) =
θN
x2e−θN
x,i x > 0
0,o he wise
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 398
19 Maxwell (x) =
2
πλ3
N
x2e−x2
2λ2
N o x > 0,
0o he wise.
20 Lindley (x) =
ϑ2
(1+ϑ)(1 + x)e−ϑx, o x≥0,
0,o he wise.
21 Pa e o (x) =
αNθαN
N
xαN+1 ,i x>θN, αN>0, θN>0
0,o he wise
The ollowing a e some commonly used Neu osophic p obabili y dis ibu ions along wi h
hei key a eas o applica ion in eal-wo ld unce ain eni onmen s:
•
Neu osophic Binomial: Used in s a is ics and quali y con ol o analyze de ec a es
when success p obabili ies a e unce ain.
•
Neu osophic Poisson: Applied in enginee ing and heal hca e o coun e en s wi h
unp edic able o a iable a es.
•
Neu osophic Exponen ial: Use ul in eliabili y and ope a ions esea ch o model ime
un il ailu e unde unce ain condi ions.
•Neu osophic Uni o m: Helps simula e da a wi hin a known bu imp ecise ange.
•
Neu osophic No mal: Used in inance and enginee ing when da a has an unce ain
mean o a iance.
•
Neu osophic Weibull: Ideal o modeling p oduc li espans and eliabili y in ague o
unce ain en i onmen s.
•
Neu osophic Be a: Commonly used in Bayesian in e ence and s a is ics o model
unce ain success p obabili ies.
•
Neu osophic Gamma: Applied in heal hca e and enginee ing o model ea men
du a ions wi h a ying e ec s.
•
Neu osophic Lomax: Use ul in inance and ac ua ial science o modeling ex eme
losses wi h imp ecise isks.
•
Neu osophic Laplace: Handles da a wi h sha p peaks and unce ain pa ame e s, o en
used in economics and signal p ocessing.
•
Neu osophic Geome ic: Models he numbe o ials un il he i s success, especially
in quali y con ol and compu e science.
•
Neu osophic Bu -III: Used in inance and eliabili y enginee ing o ailu e ime
analysis unde unce ain s ess le els.
•
Neu osophic Gene alized Pa e o: Sui able o en i onmen al science and hyd ology
o model a e ex eme e en s wi h imp ecise da a.
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 399
•
Neu osophic Nega i e Binomial: Applied in epidemiology o model disease sp ead
wi h unce ain ansmission a es.
•
Neu osophic Kuma aswamy: Helps model bounded da a like i e lows wi h imp ecise
measu emen s.
•
Neu osophic Rayleigh: Used in elecommunica ions and elec onics o model de ice
li espans unde luc ua ing condi ions.
•
Neu osophic Log-Gamma: Models indus ial g ow h o economic change wi h unce ain
in luencing ac o s.
•
Neu osophic In e se Exponen ial: Sui able o componen s ha imp o e o e ime
bu s ill ha e unce ain ailu e imes.
•
Neu osophic Maxwell: Applied in physics and chemis y o s udy molecula speeds
unde unce ain measu emen s.
•
Neu osophic Lindley: Models su i al imes in medical s udies whe e da a is ague o
imp ecise.
•
Neu osophic Pa e o: Use ul in economics and sociology o s udy income inequali y
when high-income da a is unce ain.
•
Neu osophic Diagnosis Tes s: Designed o medical science o in e p e unclea o
ague symp oms and es esul s.
5 Compa ison wi h Classical and Neu osophic Dis ibu ions
Neu osophic dis ibu ions ex end classical p obabili y models by inco po a ing inde e minacy,
making hem be e sui ed o analyzing unce ain, ague, o incomple e da a. When he
inde e minacy componen is se o ze o, Neu osophic dis ibu ions educe o hei classical
coun e pa s [
15
]. Classical models assume p ecise da a and pa ame e s, whe eas Neu osophic
models explici ly handle unce ain y, o e ing mo e lexibili y in eal-wo ld applica ions [
10
]. The
key di e ence lies in hei abili y o add ess a ious o ms o unce ain y:
Table 2: Compa ison o Classical and Neu osophic Dis ibu ions
Fea u e Classical Neu osophic
Handles De e minacy Yes Yes
Models Randomness Yes (P ima y Focus) Yes
Models Vagueness Limi ed Yes (Th ough Inde e minacy)
Models Inde e minacy No Yes (Explici Componen )
Models Con adic ion No
Yes (Th ough T u h and Falsi y)
Gene aliza ion o Classical No Yes
Based on Logic Boolean Logic Neu osophic Logic
To highligh he p ac ical signi icance o neu osophic p obabili y dis ibu ions, se e al case
s udies a e p esen ed below. These examples showcase eal-wo ld da ase s om di e se ields
such as heal hca e, eliabili y enginee ing, en i onmen al science, inance, and manu ac u ing.
Each s udy includes a compa ison be ween classical and neu osophic models, demons a ing
how he inco po a ion o inde e minacy imp o es model i and in e p e abili y.
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 400
5.1
Enginee ing Da a Modeling Using he Neu osophic Bu -XII Dis ibu-
ion
To demons a e he e ec i eness o he Neu osophic Bu -XII (NeS-B XII) dis ibu ion, i was
applied o a eal-wo ld da ase (DS) in ol ing ime- o- ailu e da a o 20 elec onic componen s
[
1
]. The ailu e imes we e epo ed as in e als (e.g., (0.001, 0.06), (0.011, 0.15)) due o
measu emen limi a ions, making hem ideal o neu osophic modeling.
Table 3: Desc ip i e neu osophic summa y o DS
Da a N Mean Median
Va iance
Skewness
Ku osis Min Max
DS1 20 2.53 1.87 6.86 2.29 5.77 0.03 12
Table 4: ML es ima es and in o ma ion c i e ia o DS
Dis ibu ion
MLEs (SE) In o ma ion C i e ion
ˆηNeS ˆγNeS AIC CAIC BIC HQIC
NeS-B XII
[0.72,1.57]
([0.16, 0.13])
[1.32,0.81]
([0.13, 0.13])
[69.01,
77.43]
[69.72,
78.13]
[71.00,
79.42]
[69.40,
77.82]
BuXII 1.60 (0.36) 0.70 (0.19) 86.11 86.82 88.10 86.50
Wb 0.83 (0.21) 0.64 (0.08) 68.49 69.19 70.48 68.88
B III 1.27 (0.23) 1.40 (0.32) 85.94 86.65 87.93 86.33
NH 1.08 (0.51) 0.35 (0.27) 81.09 81.79 83.08 81.48
A summa y o he da ase (Table 3) shows mode a e a iabili y and posi i e skewness, wi h
key s a is ics like mean, a iance, and ku osis exp essed as in e als— e lec ing unce ain y
in he da a. The NeS-B XII model was compa ed wi h ou classical models (Bu -XII, Bu -
III, Weibull, Nada ajah–Haghighi) using maximum likelihood es ima ion. Goodness-o - i was
assessed ia AIC, BIC, CAIC, and HQIC (Table 4).
Figu e 1: Compa ison o in o ma ion c i e ia o DS ac oss i e models
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 401
Neu osophic dis ibu ions is expec ed o expand u he , con ibu ing o he ad ancemen o
e idence-based medicine in unce ain en i onmen s.
9 Conclusion and Fu u e Di ec ions
This e ospec i e s udy p o ides a s uc u ed o e iew o he e olu ion, heo e ical ounda ions,
and eal-wo ld applica ions o Neu osophic p obabili y dis ibu ions. By gene alizing classical
dis ibu ions o inco po a e inde e minacy, hese models o e a powe ul amewo k o analyzing
da a cha ac e ized by unce ain y, agueness, and imp ecision—challenges o en encoun e ed in
ields such as enginee ing, heal hca e, en i onmen al science, and decision-making. A unique
con ibu ion o his wo k is he sys ema ic compa ison be ween classical and Neu osophic
dis ibu ions, which highligh s how Neu osophic models ex end beyond andomness o explic-
i ly cap u e a ious o ms o unce ain y. The s udy also syn hesizes p ac ical applica ions,
demons a ing he supe io pe o mance o Neu osophic models in handling imp ecise and
in e al- alued da a compa ed o classical me hods. Howe e , ce ain limi a ions emain. The
heo e ical de elopmen o Neu osophic dis ibu ions is ongoing, wi h speci ic challenges in
s anda dizing he in e p e a ion and quan i ica ion o inde e minacy. Addi ionally, inc eased
compu a ional complexi y and limi ed a ailabili y o use - iendly so wa e cu en ly es ic
hei widesp ead adop ion.
Fu u e esea ch di ec ions a e c ucial o add ess hese gaps and unlock he ull po en ial o
Neu osophic me hods. Key a eas o u he explo a ion include:
•Ad ancing heo e ical p ope ies o exis ing and new Neu osophic dis ibu ions.
•De eloping e icien compu a ional algo i hms and accessible so wa e ools.
•Ex ending Neu osophic models o mul i a ia e, ime-se ies, and high-dimensional da a.
•
C ea ing obus goodness-o - i es s and model selec ion c i e ia speci ic o Neu osophic
amewo ks.
•
Applying Neu osophic me hods in eme ging ields such as a i icial in elligence, big da a
analy ics, biomedical esea ch, and complex sys em modeling.
•
Conduc ing compa a i e s udies wi h o he unce ain y modeling app oaches, including
uzzy logic and Bayesian s a is ics.
•
In es iga ing highe -o de Neu osophic s a is ics o cap u e mo e complex o ms o
unce ain y.
In conclusion, Neu osophic dis ibu ions o e a p omising ad ancemen in s a is ical modeling
o unce ain en i onmen s. Thei abili y o in eg a e inde e minacy p o ides a iche and mo e
ealis ic analy ical amewo k. Con inued esea ch is essen ial o s eng hen hei heo e ical
unde pinnings, b oaden hei applica ions, and es ablish hem as p ac ical ools o eal-wo ld
decision-making unde unce ain y.
Re e ences
[1]
Al-Essa, L.A., Jamal, F., Sha iq, S., Khan, S., Abbas, Q., Khan She wani, R.A., & Aslam,
M. (2025). P ope ies and Applica ions o Neu osophic Bu XII Dis ibu ion. In e na ional
Jou nal o Compu a ional In elligence Sys ems, 18(1), 10.
[2]
Albassam, M., Ahsan-ul Haq, M., & Aslam, M. (2023). Weibull dis ibu ion unde inde e -
minacy wi h applica ions. AIMS Ma hema ics, 8, 10745–10757.
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 408
[3]
Alsoboh, A., Amou ah, A., Da us, M., & Sha e een, R.I.A. (2023). Applica ions o neu-
osophic q-Poisson dis ibu ion se ies o subclass o analy ic unc ions and bi-uni alen
unc ions. Ma hema ics, 11(4), 868.
[4]
Aslam, M. (2024). The neu osophic nega i e binomial dis ibu ion: algo i hms and p ac ical
applica ion. REVSTAT–S a is ical Jou nal.
[5]
Aslam, M., & Albassam, M. (2024). Neu osophic Geome ic Dis ibu ion: Da a Gene a ion
unde Unce ain y and P ac ical Applica ions. AIMS Ma hema ics, 9, 16436–16452.
[6]
Aslam, M., & Saleem, M. (2023). Neu osophic es o linea i y wi h applica ion. AIMS
Ma hema ics, 8(4), 7981–7989.
[7]
Aslam, M., & Sma andache, F. (2023). Chi-squa e es o imp ecise da a in consis ency
able. F on ie s in Applied Ma hema ics and S a is ics, 9.
[8]
Bashi , D., Masood, B., Shahzadi, I., Al-Husseini, Z., & Aslam, M. (2025). Neu osophic
Lindley Dis ibu ion: Simula ion, Applica ion, and Compa a i e S udy. Con empo a y
Ma hema ics, 6, 551.
[9]
Fa ima, A., A zal, U., Aslam, M., & Sma andache, F. (2025). A Comp ehensi e Re iew o
Neu osophic S a is ics o Da a Analysis in Applied Sciences. Jou nal o Reliabili y and
S a is ical S udies, 18(12), 25–40.
[10]
Mansou , M. (2025). Op imize Decision-Making in he Indus ial Sec o unde Unce -
ain y: A Neu osophic In e se Exponen ial Dis ibu ion App oach. In e na ional Jou nal
o Neu osophic Science, 25, 295–304.
[11]
Musa, A. (2025). De elopmen o Neu osophic Pa e o Dis ibu ion o Su i al Analysis.
In e na ional Jou nal o Neu osophic Science, 25, 305–315.
[12]
Al-Duais, F.S. (2024). Neu osophic log-gamma dis ibu ion and i s applica ions o indus ial
g ow h. Neu osophic Se s and Sys ems, 72(1), 22.
[13]
Shah, F., Aslam, M., Khan, Z., Almazah, M.M., & Alduais, F.S. (2022). [Re ac ed] On
Neu osophic Ex ension o he Maxwell Model: P ope ies and Applica ions. Jou nal o
Func ion Spaces, 2022(1).
[14]
She wani, R.A.K., Iqbal, S., Abbas, S., Aslam, M., & Al-Ma shadi, A.H. (2021a). [Re ac ed]
A New Neu osophic Nega i e Binomial Dis ibu ion: P ope ies and Applica ions. Jou nal
o Ma hema ics, 2021(1).
[15]
She wani, R.A.K., Naeem, M., Aslam, M., Raza, M., Abid, M., & Abbas, S. (2025).
Neu osophic Ex ension o he Nega i e Binomial Dis ibu ion: P ope ies and Applica ions
o In e al-Valued Da a. Ad ances and Applica ions in S a is ics, 92(1), 1–25.
[16]
She wani, R.A.K., Naeem, M., Aslam, M., Raza, M., Abid, M., & Abbas, S. (2021b).
Neu osophic be a dis ibu ion wi h p ope ies and applica ions. In ini e S udy.
[17]
Sma andache, F. (1998). Neu osophy: Neu osophic P obabili y, Se , and Logic: Analy ic
Syn hesis & Syn he ic Analysis. Ame ican Resea ch P ess.
[18]
Sma andache, F. (2014). In oduc ion o Neu osophic Measu e, Neu osophic In eg al, and
Neu osophic P obabili y. In ini e S udy.
[19]
Thaku , R., Malik, S., & Raj, M. (2024). Neu osophic Laplace Dis ibu ion wi h Applica ion
in Financial Da a Analysis. Neu osophic Se s and Sys ems, 57, 224.
Jismi Ma hew and Milin K. Anil, A Re ospec i e S udy on Neu osophic Dis ibu ions and Thei
Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 409
Recei ed: Ap il 18, 2025. Accep ed: Sep 29, 2025
