Jou nal o Enginee ing Science Vol. XXXII, no. 2 (2025), pp. 26 - 34
Fascicle Elec onics and Compu e Science ISSN 2587-3474
Topic Elec onics and Communica ion eISSN 2587-3482
Jou nal o Enginee ing Science June, 2025, Vol. XXXII (2)
h ps://doi.o g/10.52326/jes.u m.2025.32(2).02
UDC 519.2:621.3.076.52
THE LIKELIHOOD FUNCTION BASED ON UNCENSORED/ CENSORED
STATISTICAL DATA FOR MIN-PSD(MAX-PSD) AND MAX-PSD(MIN-PSD) AS
LIFETIME DISTRIBUTIONS IN NETWORK RELIABILITY
Alexei Leahu *, ORCID: 0000-0002-1670-0111,
Ve onica And ie schi-Bag in, ORCID: 0000-0001-8364-9873,
Ma ia Ro a u, ORCID: 0000-0002-3198-0297
Technical Uni e si y o Moldo a, 168 S e an cel Ma e Bl d., Chisinau, Republic o Moldo a
* Co esponding au ho : Alexei Leahu, alexei.leahu@a i.u m.md
Recei ed: 03. 11. 2025
Accep ed: 04. 18. 2025
Abs ac . In his pape gene al o mulas o he likelihood unc ion ha e been de i ed in he
case when uncenso ed/censo ed s a is ical da a e e o he li e ime o se ial-pa allel and
pa allel-se ial ype ne wo ks when he li e imes o he sys em uni s a e independen ,
iden ically dis ibu ed andom a iables, he numbe o subsys ems and he numbe o uni s
in each subsys em a e andom a iables wi h powe se ies ype dis ibu ion. The o mulas
can be applied o ob ain maximum likelihood es ima o s o he pa ame e s o he li e ime
dis ibu ion o he men ioned ne wo ks. The esul s a e illus a ed by examples o conc e e
p obabilis ic models.
Keywo ds: li e ime dis ibu ions, Powe Se ies Dis ibu ion, se ial-pa allel and pa allel-se ial
ne wo ks, likelihood unc ion, maximum likelihood es ima o .
Rezuma . În luc a e au os deduce o mule gene ale pen u uncția de e osimili a e în cazul
în ca e da ele s a is ice necenzu a e/cenzu a e se e e ă la du a a de iață a ețelelo de ip
se ial-pa alel și pa alel-se ial, când du a ele de iață ale uni ățilo sis emului sun a iabile
alia oa e independen e, dis ibui e iden ic, numă ul de subsis eme și numă ul de uni ăți din
ieca e subsis em sun a iabile alia oa e cu dis ibuție de ip se ie de pu e i. Fo mulele po
i aplica e pen u a obține es ima o i de e osimili a e maximă pen u pa ame ii dis ibuției
du a elo de iață ale ețelelo menționa e. Rezul a ele sun ilus a e p in exemple de
modele p obabilis ice conc e e.
Cu in e cheie: dis ibuția du a ei de iață, Dis ibuție de ip Se ie de Pu e i, ețele se ial-pa alelele
și pa alel-se iale, uncția de e osimili a e, es ima o de e osimili a e maximă.
1. In oduc ion
The p oblem o ob aining maximum likelihood es ima o s o he li e ime dis ibu ion
pa ame e s o se ial-pa allel and pa allel-se ial ne wo ks i s equi es knowledge o he
likelihood unc ion based on bo h uncenso ed and censo ed s a is ical da a. Since dynamic
p obabilis ic models ha e al eady been launched and esea ched o he men ioned ne wo ks,
A. Leahu, V. And ie schi-Bag in, M. Ro a u 27
Jou nal o Enginee ing Science June, 2025, Vol. XXXII (2)
ollowing which he mos gene al analy ical o mulas we e ob ained [1], i is na u al ha hey
ha e a simila con inui y in he case o he likelihood unc ion.
2. Auxilia y no ions and esul s
He e a e he esul s om [1] ha we will con inue o ely on. I is abou he wo
ne wo ks ype A, se ial-pa allel and ype B, pa allel-se ial [1], [2], acco ding o Figu e 1.
The gene al p obabilis ic model, in he case o bo h ne wo ks, assumes he ollowing:
- he li e imes o he ne wo k uni s a e non-nega i e, independen , iden ically
dis ibu ed andom a iables (i.i.d. . .) wi h he cumula i e dis ibu ion unc ion (c.d. .) F (x)=
FX (x, λ), whe e pa ame e λ ∈⁄ ⊆𝐑𝐑k;
- he numbe M o subne s is a . . wi h possible alues om he se o na u al numbe s,
o he Powe Se ies Dis ibu ions class (PSD) [3], [4} wi h he powe se ies unc ion 𝐵𝐵(𝜔𝜔)=
∑𝑏𝑏𝑚𝑚𝜔𝜔𝑚𝑚
𝑚𝑚≥1 , wi h he adius o con e gence, τ > 0, i.e., 𝑃𝑃(𝑀𝑀=𝑚𝑚)=𝑏𝑏𝑚𝑚𝜔𝜔𝑚𝑚
𝐵𝐵(𝜔𝜔),𝑏𝑏𝑚𝑚≥0, 𝑚𝑚=
1,2, … , 𝜔𝜔 ∈(0, 𝜏𝜏);
- he numbe s Nk o uni s in he subne s k = 1, 2, . . . , M a e 0- unca ed , i.i.d. . ., o
class PSD, wi h powe se ies unc ion, 𝐴𝐴(𝜃𝜃)=∑𝑎𝑎𝑛𝑛𝜃𝜃𝑛𝑛
𝑛𝑛≥1 , wi h he adius o con e gence, τ >
0, i.e., 𝑃𝑃(𝑁𝑁𝑘𝑘=𝑛𝑛)=𝑎𝑎𝑛𝑛𝜃𝜃𝑛𝑛
𝐴𝐴(𝜃𝜃),𝑎𝑎𝑛𝑛≥0, 𝑛𝑛= 1,2, … , 𝜃𝜃 ∈(0, τ);
- he li e imes o he ne wo k uni s and he numbe s M, Nk, k =1, 2, . . . , M a e comple ely
independen . .
a)
b)
Figu e 1. Schema ic ep esen a ion o se ial- pa allel and pa allel-se ial ne wo ks: a) Se ial-
Pa allel Ne wo k scheme; b) Pa allel-Se ial Ne wo k scheme [1].
F om [1] we will call on he ollowing:
P oposi on. The cumula i e dis ibu ion unc ions 𝑈𝑈(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔) ) and 𝑉𝑉(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔) ) o
he li e imes o he ne wo ks, espec i ely, o se ial-pa allel ype and pa allel-se ial ype, can be
calcula ed acco ding o he o mulas:
𝑼𝑼(𝒙𝒙; 𝝀𝝀,∝,𝝎𝝎 )=𝟏𝟏−𝑩𝑩(𝝎𝝎(𝟏𝟏− 𝑨𝑨(𝜽𝜽𝜽𝜽(𝐱𝐱;𝛌𝛌)) / 𝑨𝑨(𝜽𝜽) )
𝑩𝑩(𝝎𝝎) (1)
𝑽𝑽(𝒙𝒙; 𝝀𝝀,∝,𝝎𝝎 )=𝑩𝑩(𝝎𝝎(𝟏𝟏− 𝑨𝑨(𝜽𝜽(𝟏𝟏−𝜽𝜽(𝒙𝒙;𝝀𝝀))) / 𝑨𝑨(𝜽𝜽) )
𝑩𝑩(𝝎𝝎) (2)
Rema k. Because c.d. . 𝑈𝑈(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔) ), as he li e ime dis ibu ion o se ial-pa allel
ne wo k,
coincides wi h c.d. . o he . . 𝑚𝑚𝑚𝑚𝑛𝑛(𝑚𝑚𝑎𝑎𝑥𝑥
1≤𝑖𝑖≤𝑁𝑁1𝑋𝑋𝑖𝑖1,𝑚𝑚𝑎𝑎𝑥𝑥
1≤𝑖𝑖≤𝑁𝑁2𝑋𝑋𝑖𝑖2, … , 𝑚𝑚𝑎𝑎𝑥𝑥
1≤𝑖𝑖≤𝑁𝑁𝑀𝑀𝑋𝑋𝑖𝑖𝑖𝑖) and 𝑉𝑉(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔) ),
as he li e ime dis ibu ion o he pa allel-se ial ne wo k, coincides wi h c.d. . o he . .
𝑚𝑚𝑎𝑎𝑥𝑥(𝑚𝑚𝑚𝑚𝑛𝑛
1≤𝑖𝑖≤𝑁𝑁1𝑋𝑋𝑖𝑖1,𝑚𝑚𝑚𝑚𝑛𝑛
1≤𝑖𝑖≤𝑁𝑁2𝑋𝑋𝑖𝑖2, … , 𝑚𝑚𝑚𝑚𝑛𝑛
1≤𝑖𝑖≤𝑁𝑁𝑀𝑀𝑋𝑋𝑖𝑖𝑖𝑖) , whe e Xij , i=1, ..., Nj , j=1, ..., M a e li e imes o all
uni s, . . M
∈
PSD wi h powe se ies unc ion B(ω) and N1, N2, ..., NM a e i.i.d. . . as . . N
∈
PSD
28 The likelihood unc ion based on uncenso ed/ censo ed s a is ical da a o Min-PSD(Max-PSD) and…
Jou nal o Enginee ing Science June 2025, Vol. XXXII (2)
wi h powe se ies unc ion A(θ), he dis ibu ions 𝑈𝑈(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔) ) and 𝑉𝑉(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔) ) will be
called, espec i ely, Min-PSD(Max-PSD) and Max-PSD(Min-PSD) li e ime dis ibu ions in
ne wo k eliabili y [1].
These gene al o mulas allow us o de e mine, in p ac ice, he likelihood unc ion o
any conc e e se ial-pa allel o pa allel-se ial ype ne wo k model, conside ed as a pa icula
case o he models desc ibed abo e. We no e, ha om he Ec. (1)-(2) we ha e he ollowing
esul [1].
Consequence. I he li e ime o each uni is a . . o (absolu ely) con inuous ype, hen he
li e imes o he espec i e ne wo ks will be . . o (absolu e) con inuous ype oo. A he same ime,
i he li e ime o each uni is a . . o disc e e ype, hen he li e imes o he espec i e ne wo ks
will be . . o disc e e ype oo.
3. Likelihood unc ion based on uncenso ed da a
Le us conside he sample o size n o uncenso ed da a (x1, x2, ..., xn) o he li e imes o
a ne wo k o ype A o B. The ac ha he da a a e uncenso ed means ha he alues x1, x2,
..., xn ep esen he esul s o n obse a ions made on he li e ime o he ne wo k om he
s a o i s ope a ion un il i s ailu e. The likelihood unc ion is de ined based on he
Maximum Likelihood P inciple, acco ding o which: i a andom e en has occu ed, hen i
means ha his is he e en wi h he highes p obabili y o occu .
In he assump ion ha c.d. . o he li e ime o each uni o he ne wo k depends on a
pa ame e , le ’s say, λ, i.e., F (x) = F(x; λ). Then Ec. (1) − (2) show ha he dis ibu ions o
ne wo k li e imes, gene ally, depend on 3 pa ame e s, λ, θ, ω.
Case1. The li e ime o each uni s is a . . o (absolu e) con inuous ype
Mos o he p obabilis ic models ha aim a he li e ime o he ne wo ks we app oach
s a om he assump ion ha he li e ime o each uni is a . . o absolu e con inuous ype.
So, acco ding o ou Consequence, he li e ime o he ne wo k will also be a . . o absolu e
con inuous ype. This means ha , ha ing he Ec. (1) − (2) a ou disposal, we can de e mine
he p obabili y densi y unc ion (p.d. .) o he li e imes o se ial-pa allel ne wo ks and se ial-
pa allel ne wo ks, espec i ely, acco ding o he o mulas:
𝑢𝑢(𝑥𝑥; 𝜆𝜆,∝,ω )=𝑑𝑑𝑑𝑑(𝑥𝑥; λ,∝,ω )
𝑑𝑑𝑥𝑥 , 𝑣𝑣(𝑥𝑥; λ,∝,ω )=𝑑𝑑𝑑𝑑(𝑥𝑥; 𝜆𝜆,∝,ω )
𝑑𝑑𝑥𝑥 .
This means ha he espec i e likelihood unc ions will be w i en as ollows:
𝑳𝑳𝑼𝑼(𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐 , . . . , 𝒙𝒙𝒏𝒏; 𝝀𝝀,∝,𝛚𝛚 )=∏𝒖𝒖(𝒙𝒙𝒌𝒌;𝝀𝝀,∝,𝛚𝛚 )
𝒏𝒏
𝒌𝒌=𝟏𝟏 (3)
𝑳𝑳𝑽𝑽(𝒙𝒙𝟏𝟏,𝒙𝒙𝟐𝟐 ,...,𝒙𝒙𝒏𝒏; 𝝀𝝀,∝,𝛚𝛚 )=∏𝒗𝒗(𝒙𝒙𝒌𝒌;𝝀𝝀,∝,𝛚𝛚 )
𝒏𝒏
𝒌𝒌=𝟏𝟏 (4)
Example 1. We will ake as a special case o he models desc ibed by us, he case
when he numbe o subne wo ks M is no andom, bu is cons an , being equal wi h na u al
numbe M , and he numbe o uni s Ni in he subne wo k numbe i = 1, …, M is also cons an
, being known and equal o a na u al numbe N. The e o e, o mally, we can conside ha M
is o he PSD class wi h he powe se ies unc ion B(ω) = ωM , bu also ha N is o he PSD
class wi h he powe se ies unc ion A(θ) = θN . We assume, o example, ha he li e ime o
each uni is exponen ially dis ibu ed . . wi h pa ame e λ > 0, i.e., wi h c.d. . F(x)=(1-e-
λx)I[0,+∞](x).
A. Leahu, V. And ie schi-Bag in, M. Ro a u 29
Jou nal o Enginee ing Science June, 2025, Vol. XXXII (2)
The p oblem a ises o cons uc ing a maximum likelihood es ima o (m.l.e.) o his
pa ame e , ha ing a ailable he expe imen al da a o he li e imes (x1, x2, ..., xn) aimed a he
esul s o he ope a ion o n iden ical se ial -pa allel o pa allel-se ial ne wo ks.
Solu ion. Subs i u ing in he Ec. (1)-(2) he conc e e exp essions o he unc ions F(x),
A(θ) and B(θ), we deduce ha in his case he li e ime c.d. . o he se ial-pa allel and pa allel-
se ial ne wo k a e, espec i ely,
𝑼𝑼(𝒙𝒙;𝛌𝛌,𝐍𝐍,𝐌𝐌) = 𝟏𝟏 – �𝟏𝟏 – �𝜽𝜽 (𝒙𝒙)�𝑵𝑵�𝑴𝑴=𝟏𝟏−�𝟏𝟏 – �𝟏𝟏−𝒆𝒆− 𝝀𝝀 𝒙𝒙�𝑵𝑵�𝑴𝑴𝑰𝑰[𝟎𝟎,+∞)(𝒙𝒙) (5)
𝑉𝑉 (𝑥𝑥; 𝜆𝜆,𝑁𝑁 , 𝑀𝑀) = �1 − �1 − 𝐹𝐹 (𝑥𝑥)�𝑁𝑁�𝑖𝑖= (1 – 𝑒𝑒− 𝜆𝜆𝑁𝑁𝑥𝑥)𝑖𝑖𝐼𝐼[0,+∞)(𝑥𝑥) (6)
Acco ding o he abo e Consequence, because he li e ime o each uni is (absolu ely)
con inuous . ., we deduce ha he li e imes o ou ne wo ks a e (absolu ely) con inuous . .
wi h, espec i ely, p obabili y densi y unc ions
𝑢𝑢(𝑥𝑥;𝜆𝜆,𝑁𝑁 , 𝑀𝑀)=𝑑𝑑𝑑𝑑
𝑑𝑑𝑥𝑥 = (MNλ)𝑒𝑒−𝜆𝜆𝑥𝑥(1 −𝑒𝑒−𝜆𝜆 𝑥𝑥)𝑁𝑁−1 (1 – (1 −𝑒𝑒−𝜆𝜆 𝑥𝑥)𝑁𝑁)𝑖𝑖−1𝐼𝐼[0,+∞](𝑥𝑥)
𝑣𝑣(𝑥𝑥;𝜆𝜆,𝑁𝑁 , 𝑀𝑀)=𝑑𝑑𝑉𝑉
𝑑𝑑𝑥𝑥 == MNλ𝑒𝑒−𝜆𝜆𝑁𝑁𝑥𝑥�1 −𝑒𝑒−𝜆𝜆𝑁𝑁𝑥𝑥 �M −1𝐼𝐼[0,+∞](𝑥𝑥)
So, Likelihood unc ion, co esponding o he con inuous da a, o se ial-pa allel
ne wo k is
𝐿𝐿𝑑𝑑(𝑥𝑥1,𝑥𝑥2 , . . . , 𝑥𝑥𝑛𝑛; 𝜆𝜆,𝑁𝑁 , 𝑀𝑀)=�𝑢𝑢(𝑥𝑥𝑘𝑘;𝜆𝜆,𝑁𝑁 , 𝑀𝑀)=
𝑛𝑛
𝑘𝑘=1
(𝑀𝑀𝑁𝑁𝜆𝜆)𝑛𝑛𝑒𝑒−𝜆𝜆𝑁𝑁∑𝑥𝑥𝑘𝑘
𝑛𝑛
𝑘𝑘=1 ∗� �1−𝑒𝑒−𝜆𝜆 𝑥𝑥𝑘𝑘�𝑁𝑁−1
𝑛𝑛
𝑘𝑘=1 ((1 – (1 −𝑒𝑒−𝜆𝜆 𝑥𝑥𝑘𝑘)𝑁𝑁)𝑖𝑖−1
and o pa allel-se ial ne wo k is
𝐿𝐿𝑑𝑑(𝑥𝑥1,𝑥𝑥2 ,...,𝑥𝑥𝑛𝑛; 𝜆𝜆,𝑁𝑁 , 𝑀𝑀) = �𝑣𝑣(𝑥𝑥𝑘𝑘;𝜆𝜆,𝑁𝑁 , 𝑀𝑀)
𝑛𝑛
𝑘𝑘=1 =
(𝑀𝑀𝑁𝑁𝜆𝜆)𝑛𝑛𝑒𝑒−𝜆𝜆𝑁𝑁∑𝑥𝑥𝑘𝑘
𝑛𝑛
𝑘𝑘=1 � �1−𝑒𝑒−𝜆𝜆 𝑁𝑁𝑥𝑥𝑘𝑘�𝑖𝑖−1
𝑛𝑛
𝑘𝑘=1
By he de ini ion, he maximum likelihood es ima o (m.l.e.) o he pa ame e λ,
pa ame e s M, N being known, ep esen s ha alue 𝜆𝜆
o which he likelihood unc ion akes
i s maximum alue (see Maximum Likelihood P inciple) [5]. Fo M > 1, N>1 , ou likelihood
unc ions can be maximized using nume ical me hods only, bu o M = 1, o he wise, when
he pa allel-se ial ne wo k is always mo e eliable han he se ial-pa allel ne wo k, his
p oblem can be explici ly sol ed o he pa allel-se ial ne wo k. Indeed, in his case, he
maximum likelihood es ima o (m.l.e.) 𝜆𝜆
o he pa ame e λ, ou likelihood unc ion can be
maximized, sol ing likelihood equa ion:
𝑑𝑑
𝑑𝑑𝜆𝜆𝑙𝑙𝑛𝑛𝐿𝐿𝑑𝑑(𝑥𝑥1,𝑥𝑥2 ,...,𝑥𝑥𝑛𝑛; 𝑁𝑁,λ)=0
i.e., equa ion 𝑑𝑑
𝑑𝑑𝜆𝜆[𝑛𝑛(𝑙𝑙𝑛𝑛𝑁𝑁+𝑙𝑙𝑛𝑛𝜆𝜆)−∑𝜆𝜆𝑁𝑁𝑥𝑥𝑘𝑘
𝑛𝑛
𝑘𝑘=1 ]= 0 i.e.,
𝑛𝑛
𝜆𝜆−𝑁𝑁�𝑥𝑥𝑘𝑘
𝑛𝑛
𝑘𝑘=1 = 0
In his way, o pa alel-se ial ne wo k, we ind ha
𝑚𝑚.𝑙𝑙.𝑒𝑒.𝜆𝜆
=𝑛𝑛
𝑁𝑁∑𝑥𝑥𝑘𝑘
𝑛𝑛
𝑘𝑘=1
30 The likelihood unc ion based on uncenso ed/ censo ed s a is ical da a o Min-PSD(Max-PSD) and…
Jou nal o Enginee ing Science June 2025, Vol. XXXII (2)
Case 2. The li e ime o each uni s is a . . o disc e e ype
The Ec. (1) - (2) being alid also when he li e ime o he ne wo k uni s is . . o disc e e
ype, hen, acco ding o ou Consequence, he li e imes o he ne wo ks will o disc e e ype
oo. Mo e p ecisely, i he li e ime X o each uni is his, o example, a . . wi h alues om
he se {0,1,2, ..., k, ...} gi en by he pa ame ic p obabilis ic dis ibu ion 𝑃𝑃𝜆𝜆 (𝑋𝑋=
𝑘𝑘),whe e 𝑃𝑃𝜆𝜆 (𝑋𝑋=𝑘𝑘)≥0, 𝑘𝑘= 0,1,2, . . ., ∑𝑃𝑃𝜆𝜆 (𝑋𝑋=𝑘𝑘)= 1
𝑘𝑘≥𝑜𝑜 , hen he li e imes U and V o
ou ne wo ks will also be . . o disc e e ype oo, wi h he possible alues om he same se .
Because o any in ege alue x om he se o possible alues, o example, o . . U, we
ha e ha c.d. . is equal wi h 𝑈𝑈(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔) = ∑𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑈𝑈=𝑘𝑘),
𝑘𝑘:𝑘𝑘≤𝑥𝑥 i u ns ou ha o he
li e ime U o he se ial-pa allel ne wo k
𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑈𝑈= 0)= U(0; 𝜆𝜆,∝,𝜔𝜔), and
𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑈𝑈=𝑘𝑘)=𝑈𝑈(𝑘𝑘; 𝜆𝜆,∝,𝜔𝜔)−𝑈𝑈(𝑘𝑘−1; 𝜆𝜆,∝,𝜔𝜔),𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘≥1 (7)
Analogously, o he li e ime V o he pa allel-se ial ype ne wo k
𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑉𝑉= 0)=𝑉𝑉(0; 𝜆𝜆,∝,𝜔𝜔), and
𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑉𝑉=𝑘𝑘)=𝑉𝑉(𝑘𝑘; 𝜆𝜆,∝,𝜔𝜔)−𝑉𝑉(𝑘𝑘−1; 𝜆𝜆,∝,𝜔𝜔),𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘≥1 (8)
Acco ding o he Maximum Likelihood P inciple in Case 2, using Ec. (7) - (8), he
espec i e unc ions o se ial-pa allel and pa allel-se ial ne wo ks will be w i en as ollows:
𝐿𝐿𝑑𝑑(𝑥𝑥1,𝑥𝑥2 , . . . , 𝑥𝑥𝑛𝑛; 𝜆𝜆,∝,ω )=∏𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑈𝑈=𝑥𝑥𝑘𝑘)
𝑛𝑛
𝑘𝑘=1 . (9)
𝐿𝐿𝑑𝑑(𝑥𝑥1,𝑥𝑥2 ,...,𝑥𝑥𝑛𝑛; 𝜆𝜆,∝,ω )=∏𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑉𝑉=𝑥𝑥𝑘𝑘)
𝑛𝑛
𝑘𝑘=1 (10)
whe e 𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑈𝑈=𝑥𝑥𝑘𝑘) 𝑎𝑎𝑛𝑛𝑑𝑑 𝑃𝑃𝜆𝜆,∝,𝜔𝜔 (𝑉𝑉=𝑥𝑥𝑘𝑘) a e calcula ed acco ding o he Ec. (7) – (8).
Example 2. We will conside he same model as in Example 1, wi h he di e ence ha
he li e imes a e andom a iables, independen , iden ically dis ibu ed geome ically wi h
he pa ame e λ, 0<λ<1, i.e., li e ime is a . . gi en by dis ibu ion
𝑃𝑃𝜆𝜆 (𝑋𝑋=𝑘𝑘)=λ(1 −𝜆𝜆)𝑘𝑘,𝑘𝑘= 0,1,2, …
So, o each x, x=0,1,2, ..., c.d. .
𝐹𝐹(𝑥𝑥; 𝜆𝜆)=∑𝜆𝜆(1 −𝜆𝜆)𝑘𝑘
𝑥𝑥
𝑘𝑘=0 = 1 − (1 −λ)𝑥𝑥+1
Using Ec. (5) – (6) we ind ha
𝑃𝑃𝜆𝜆,,𝑁𝑁,𝑖𝑖 (𝑈𝑈=𝑘𝑘)= (1 −(1 −(1−𝜆𝜆)𝑘𝑘)𝑁𝑁)𝑖𝑖−(1 −(1 −(1−𝜆𝜆)𝑘𝑘+1)𝑁𝑁)𝑖𝑖,𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘≥0
Replacing hese p obabili ies, espec i ely, in he Ec. (9) – (10), we ob ain he
co esponding Likelihood Func ions o se ial-pa allel and pa allel-se ial ne wo ks. As hey
show, 𝑚𝑚.𝑙𝑙.𝑒𝑒.𝜆𝜆
can only be ound by nume ical me hods.
4. Likelihood unc ion based on censo ed da a
In S a is ics da a a e called censo ed when he alue o an obse a ion is pa ially
known. This is usually he case in su i al/ eliabili y analysis, whe e he ime o a ce ain
e en is o in e es , bu o some s udies, he e en has no ye occu ed a he ime o
analysis. Fo example, i we a e s udying he li e ime o a p oduc , he censo ed da a would
be cases whe e he p oduc is s ill wo king a he end o he s udy pe iod, so we do no ha e
an exac alue o li e ime.
The e a e se e al ypes o censo ship [6,7]:
Righ -censo ing. We don' know wha happens a e a ce ain poin .
A. Leahu, V. And ie schi-Bag in, M. Ro a u 31
Jou nal o Enginee ing Science June, 2025, Vol. XXXII (2)
Le censo ing. Le -censo ed obse a ions occu in li e es applica ions when a
sys em has ailed a he ime o i s i s inspec ion; all ha is known is ha he uni
ailed be o e he inspec ion ime. We ha e no in o ma ion abou wha happened
be o e a ce ain poin .
In e al censo ing. We know ha he e en occu ed wi hin a ce ain in e al, bu we
do no know he exac ime.
Censo ing is impo an because i a ec s how we analyze and in e p e da a. S a is ical
me hods mus be adap ed o accoun o he incomple e in o ma ion p o ided by censo ed
da a. I is impo an o us o know ha in he case o censo ed da a, he Likelihood Func ion
is w i en mo e simply, because ega dless o he ype o da a (disc e e o (absolu ely)
con inuous), we only need he c.d. . o he obse ed a iable.
Rega ding ou case, when he li e ime Y, gi en by c.d. . FY(x), a ge s se ial-pa allel o
pa allel-pa allel ne wo ks, he censo ed da a is ep esen ed by andom e en s o he o m:
{𝑌𝑌≤𝑎𝑎}, i.e., he da a is le censo ed; {𝑎𝑎<𝑌𝑌≤𝑏𝑏}, i.e., he da a is in e al censo ed, {𝑌𝑌>𝑏𝑏},
i.e., he da a is igh censo ed, whe e 0<a<b<+∞. O he wise, since he li e ime is a non-nega i e
. ., we no e ha bo h le censo ing and igh censo ing can be conside ed special cases o
in e al censo ing. Indeed: p obabili ies o hese e en s a e equals o
𝑃𝑃{𝑌𝑌≤𝑎𝑎}=𝐹𝐹𝑌𝑌(𝑎𝑎), 𝑃𝑃{𝑎𝑎<𝑌𝑌≤𝑏𝑏}=𝐹𝐹𝑌𝑌(𝑏𝑏)−𝐹𝐹𝑌𝑌(𝑎𝑎),𝑃𝑃{𝑌𝑌>𝑏𝑏}= 1 −𝐹𝐹𝑌𝑌(𝑏𝑏).
So, we may sol e he p oblem o w i ing he Likelihood Func ion. Fo his, i is
su icien o know he ollowing in o ma ion:
(1) he a and b, as he alues ha de e mine he 3 ypes o da a censo ing;
(2) he c.d. . FY(x) o he li e ime Y, also known as a unc ion, ha depends on 3
pa ame e s, mo e exac ly, FY(x) = 𝐹𝐹𝑌𝑌(x; 𝜆𝜆,∝,ω);
(3) he p obabili ies 𝑃𝑃{𝑌𝑌≤𝑎𝑎},𝑃𝑃{𝑌𝑌>𝑏𝑏},𝑃𝑃{𝑎𝑎<𝑌𝑌≤𝑏𝑏};
(4) he sample size n o da a and he numbe s n1, n2 and n3, espec i ely, o he le ,
in e al and igh censo ed da a, whe e n1+n2 + n3=n.
Then, acco ding o he Maximum Likelihood P inciple, Likelihood Func ion
co esponding o he censo ed da a o li e ime Y is done by he o mula:
𝐿𝐿𝑌𝑌(𝑛𝑛1,𝑛𝑛2,𝑛𝑛3 ;λ,∝,ω)=
[𝐹𝐹𝑌𝑌(𝑎𝑎; λ,∝,ω)]𝑛𝑛1]∗[1 −𝐹𝐹𝑌𝑌(𝑎𝑎; λ,∝,ω)]𝑛𝑛2∗[𝐹𝐹𝑌𝑌(b; λ,∝,ω)−𝐹𝐹𝑌𝑌(a; λ,∝,ω)]𝑛𝑛3.
Replacing in his o mula c.d. . 𝐹𝐹𝑌𝑌(𝑥𝑥; λ,∝,ω) wi h c.d. . 𝑈𝑈(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔 ) o c.d. .
𝑉𝑉(𝑥𝑥; 𝜆𝜆,∝,𝜔𝜔 ), gi en by he Ec. (1)-(2), we ob ain Likelihood Func ions o se ial-pa allel and
pa allel-se ial ne wo ks.
Rema k. The abo e likelihood unc ion ep esen s he case when he da a a e censo ed
on 3 in e als, bu i can be ex ended, simila ly, o he case when he numbe o censo ing
in e als is g ea e han 3. This will be illus a ed in Exemple 3.
Example 3. We will conside he same model as in Example 1, wi h he di e ence ha
we ha e n cenzo ed da a, whe e numbe o le cenzo ed da a wi h he h eshold a is equal
wi h n1, he numbe o censo ed da a by in e al (a,b] is equal wi h n3 and numbe o igh
cenzo ed da a wi h he h eshold b is equal wi h n3, a>0, a<b<+∞, n1+n2 + n3=n. Then, on he
base o Ec. (11), using, espec i ely Ec. (5)-(6), he Likelihood Func ions o se ial-pa allel and
pa allel ne wo ks may by w i e, espec i ely, as
𝐿𝐿𝑑𝑑(𝑛𝑛1,𝑛𝑛2,𝑛𝑛3 ; 𝜆𝜆,𝑁𝑁,𝑀𝑀=
32 The likelihood unc ion based on uncenso ed/ censo ed s a is ical da a o Min-PSD(Max-PSD) and…
Jou nal o Enginee ing Science June 2025, Vol. XXXII (2)
�𝒏𝒏!
𝒏𝒏
𝟏𝟏
!𝒏𝒏
𝟐𝟐
!𝒏𝒏
𝟑𝟑
!� ∗�𝟏𝟏−(𝟏𝟏−�𝟏𝟏−𝒆𝒆− 𝝀𝝀 𝒂𝒂)𝑵𝑵�𝑴𝑴�𝒏𝒏𝟏𝟏�[�𝟏𝟏 – �𝟏𝟏−𝒆𝒆− 𝝀𝝀 𝒂𝒂)𝑵𝑵��𝑴𝑴
−[�𝟏𝟏 – �𝟏𝟏−𝒆𝒆− 𝝀𝝀 𝒃𝒃)𝑵𝑵��𝑴𝑴�𝒏𝒏𝟐𝟐∗ {[�𝟏𝟏 – �𝟏𝟏−𝒆𝒆− 𝝀𝝀 𝒃𝒃)𝑵𝑵��𝑴𝑴}𝒏𝒏𝟑𝟑
(11)
𝑳𝑳𝑽𝑽(𝒏𝒏𝟏𝟏,𝒏𝒏𝟐𝟐,𝒏𝒏𝟑𝟑 ; 𝝀𝝀,𝑵𝑵,𝑴𝑴)=
�𝒏𝒏!
𝒏𝒏
𝟏𝟏
!𝒏𝒏
𝟐𝟐
!𝒏𝒏
𝟑𝟑
!���𝟏𝟏−𝒆𝒆
−𝝀𝝀𝑵𝑵𝒂𝒂
�𝑴𝑴�𝒏𝒏
𝟏𝟏
∗[�𝟏𝟏−𝒆𝒆
−𝝀𝝀𝑵𝑵𝒃𝒃
�𝑴𝑴−�𝟏𝟏−𝒆𝒆
−𝝀𝝀𝑵𝑵𝒂𝒂
�𝑴𝑴]
𝒏𝒏𝟐𝟐
∗
�𝟏𝟏−�𝟏𝟏−𝒆𝒆−𝝀𝝀𝑵𝑵𝒃𝒃�𝑴𝑴�𝒏𝒏𝟑𝟑
(12)
To ind ou he m.l.e. 𝜆𝜆
o he pa ame e λ we will ake he case when he pa allel-
se ial ne wo kis mo e eliable han se ial pa allel, i.e., he case when, acco ding o he wo k
[1], N<M. Fo example: N=2, M=3, and as he uni o li e ime we will ake 1 yea . To es he
algo i hm o ob aining a maximum likelihood es ima e o he pa ame e λ, we simula e
Mon e Ca lo [8]-[15], using Cha GPT 4, alues o he li e ime o ou ne wo k in he case, o
example, when λ =0.2, i.e., when he li e ime . o each uni in he ne wo k has an a e age
alue equal o 1/ λ =5 yea s. He e a e he simula ed alues w i en as a a ia ional s ing in
ascending o de :
(1.23,1.67,1.89, 2.34,2.56,2.78,2.89,3.12,3.45,3.78,4.01,4.23,4.56,5.01,5.34,5.67,5.89,
6.12,6.45,6.78).
We assume ha hese da a a e censo ed acco ding o he alues a=2 yea s and b=4
yea s. So, we assume ha we ha e a ou disposal, om a o al numbe o n = 20 da a, n1=3
da a censo ed on he le , n2=7 da a censo ed on he in e al (2,4] and n3=10 da a censo ed
on he igh . Then he Likelihood Func ion is
𝐿𝐿𝑑𝑑(3, 7,10; λ, 2, 3)=20!
3! 7! 10!∗((1 −𝑒𝑒−4𝜆𝜆)3)3∗
[�1−𝑒𝑒−8𝜆𝜆�3−�1−𝑒𝑒−4𝜆𝜆�3]7∗[1 −�1−𝑒𝑒−8𝜆𝜆�3]10
By means o he Ma hema ica 14.0 we ind ha alue o λ o which he Likelihood
Func ion akes i s global maximum, ha is, we ind ha m.l.e. 𝜆𝜆
=0.0882487. Compa ing i
wi h he ue alue o λ=0.2, we ind ha he app oxima ion is no so good. So i 's a p oblem
ela ed o he na u e o censo ship.
Now, on he base o he abo e simula ed da a, we assume ha hese da a a e censo ed
acco ding o he alues a=2 yea s, b=4 yea s and c=6 yea s. Tha means we assume ha we
ha e a ou disposal, om a o al numbe o n = 20 da a, n1=3 da a censo ed on he le , n2=7
da a censo ed on he in e al (2,4], n3=7 da a censo ed on he in e al (4,6], and n4=3 da a
censo ed on he igh . I in he p e ious case we had da a censo ed on 3 in e als, now we
will ha e o deal wi h da a censo ed on 4 in e als. The me hod o calcula ing he Likelihood
Func ion being simila , we ind ha
𝐿𝐿𝑑𝑑(3, 7,7,3; λ, 2, 3) = 20!
3!7!7!3!∗((1 −𝑒𝑒−4𝜆𝜆)3)3∗((1 −𝑒𝑒−8𝜆𝜆)3−�1−𝑒𝑒−4𝜆𝜆�3)7∗
∗(1 −�1−𝑒𝑒−12𝜆𝜆�3)3
Now, also by means o Ma hema ica 14.0, we ind ha alue o λ o which he
Likelihood Func ion akes i s global maximum, i.e., we ind ha m.l.e. 𝜆𝜆
=0.214796
sa is ac o ily app oxima es he ue alue o he pa ame e λ=0.2.
A. Leahu, V. And ie schi-Bag in, M. Ro a u 33
Jou nal o Enginee ing Science June, 2025, Vol. XXXII (2)
5. Conclusions
Gene al o mulas in Eq. (1)-(2) o de e mining c.d. . o li e imes is a la ge sou ce o
dynamic p obabilis ic models o se ial-pa allel o pa allel-se ial ne wo ks, bu also a basis
o w i ing he Likelihood Func ion, when he da a a e uncenso ed o censo ed. W i ing he
likelihood unc ion o censo ed da a becomes simple , because i does no depend on he
ype o li e ime as . . (disc e e o con inuous), using only c.d. . Bu he p oblem o inding an
m.l.e. which app oxima es as well as possible he ue alue o he unknown pa ame e is
complica ed, because i is di icul o ma ch he censo ing in e als. The ac ha ma ching
he censo ing in e als is di icul , e en when we ely on simula ion da a, shows us ha in
he case o eal p oblems he use o maximum likelihood es ima o s mus be done wi h g ea
cau ion i he choice o censo ing in e als does no ha e a ma hema ical easoning.
Howe e , his is a p oblem ha dese es o be esea ched mo e deeply. The examples gi en
show ha inding he maximum likelihood es ima o s becomes a maximiza ion p oblem ha
can be sol ed, as a ule, by nume ical me hods.
These esul s we e p esen ed a he In e na ional Con e ence on Elec onics,
Communica ions and Compu ing, ECCO 2024, 17-18 Oc obe , Chisinau, Republic o Moldo a.
Acknowledgmen s: The esul s we e ob ained wi hin he Ins i u ional Resea ch P ojec
020404 concluded wi h he Minis y o Educa ion and Resea ch o he Republic o Moldo a.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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34 The likelihood unc ion based on uncenso ed/ censo ed s a is ical da a o Min-PSD(Max-PSD) and…
Jou nal o Enginee ing Science June 2025, Vol. XXXII (2)
Ci a ion: Leahu, A.; And ie schi-Bag in, V.; Ro a u, M. The likelihood unc ion based on uncenso ed/ censo ed
s a is ical da a o Min-PSD(Max-PSD) and Max-PSD(Min-PSD) as li e ime dis ibu ions in ne wo k eliabili y.
Jou nal o Enginee ing Science. 2025, XXXII (2), pp. 26-34. h ps://doi.o g/10.52326/jes.u m.2025.32(2).02.
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