Pa ne s Uni e sal Inno a i e Resea ch Publica ion (PUIRP)
Volume: 03 Issue: 05 | Sep embe -Oc obe 2025 | ISSN: 3048-586X | www.pui p.com
© 2025, PUIRP | PU Publica ions | DOI:10.5281/zenodo.17536909 Page | 92
Ha monic Unbiased Es ima o : Some P ope ies
Dh i ikesh Chak aba y
Independen Resea che , Guwaha i, Assam, India
--------------------------------------------------------------------------------------
Abs ac - In an ea lie s udy, concep o ha monic unbiased es ima o was in oduced and de ined
based on ha monic expec a ion. A emp has he e been made o iden i y some impo an
p ope ies/ ac s/ esul s o ha monic unbiased es ima o . This a icle is based on he in o ma ion on his
unbiased es ima o ob ained in he a emp .
Keywo ds: Es ima ion, Ha monic Unbiasedness, HUE, Some P ope ies.
1. INTRODUCTION
Unbiasedness, in he li e a u e o s a is ical es ima ion, is ega ded as a desi able p ope y/quali y/c i e ion
o an es ima o [1 , 8 , 12, 13]. O iginally, he concep o unbiasedness [8 , 9] was explained on he basis o
he ma hema ical expec a ion [2 , 11 , 14], mo e speci ically he a i hme ic expec a ion, o he es ima o
conce ned and acco dingly unbiased es ima o was de ined [1 , 12 , 13]. This de ini ion la e was e med as
a i hme ic unbiased es ima o [6]. Recen ly, concep s o ha monic unbiased es ima o [6] was in oduced
and de ined based on ha monic expec a ion [3 , 4 , 5 , 7]. A emp has he e been made on iden i ying some
impo an p ope ies/ ac s/ esul s o ha monic unbiased es ima o . This a icle is based on he in o ma ion
on his unbiased es ima o ob ained in he a emp .
2. HARMONIC UNBIASED ESTIMATOR
Suppose,
, , ………. ,
is a andom sample d awn om a popula ion o a non-ze o eal alued andom a iable X which ollows a
p obabili y dis ibu ion ha ing pa ame e θ
& T = T( , , ………. , )
is an es ima o o θ.
Then T can be ega ded as ha monic unbiased es ima o o pa ame e θ i
EH(T) = θ
whe e EH(T) is he ha monic expec a ion o T.
Le us abb e ia e ha monic unbiased es ima o by HUE.
No e:
HUE exis s in he case o non-ze o eal alued es ima o . The co esponding pa ame e θ, in his case, is an
unknown non-ze o eal numbe .
Pa ne s Uni e sal Inno a i e Resea ch Publica ion (PUIRP)
Volume: 03 Issue: 05 | Sep embe -Oc obe 2025 | ISSN: 3048-586X | www.pui p.com
© 2025, PUIRP | PU Publica ions | DOI:10.5281/zenodo.17536909 Page | 93
3. ALGEBRAIC PROPERTIES OF HARMONIC UNBIASED IN ESTIMATOR
P ope y (1): I T is HUE o pa ame e θ hen c.T is HUE o pa ame e c.θ.
P oo : This ollows om he ac ha
EH(c.T) = c.EH(T) = c.θ
Co olla y: I T is HUE o pa ame e θ hen −T is HUE o pa ame e −θ.
This ollows om he ac ha
EH(−T) EH(−1.T) = = −1.EH(T) = −θ
P ope y (2): Ha monic mean (HM) o a ini e numbe o HUEs o a pa ame e θ is also HUE o he pa ame e
θ.
P oo : Suppose, T and S a e wo HUEs o a pa ame e θ.
Then
EH (HM o T and S) = EH ( 2
1
𝑇+1
𝑆
)
= 2EH {(1
𝑇 +1
𝑆)−1}
= 2{EA(1
𝑇 +1
𝑆)}−1 , (whe e EA(T) is he a i hme ic expec a ion o T)
= 2{EA(1
𝑇)+ EA(1
𝑆)}−1
= 2[{EH(𝑇)}−1+ {EH(𝑆) }−1]−1
= 2(𝜃 −1+𝜃 −1 )−1
= θ
The e o e, HM o T and S is HUE o a pa ame e θ.
Now suppose,
𝑇1 , 𝑇2 , ……….. , 𝑇𝑟
a e HUEs o a pa ame e θ.
Then
HM o 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟 = 𝑟
1
𝑇1+1
𝑇2 +⋯……..+ 1
𝑇𝑟
P oceeding simila ly as in he ea lie case, one can ob ain ha
EH (𝑟
1
𝑇1+1
𝑇2 +⋯……..+ 1
𝑇𝑟 ) = θ
The e o e, HM o 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟 is HUE o θ.
Thus, P ope y (2) has been p o ed o a ini e numbe o es ima o s.
Hence, P ope y (2) has been es ablished.
Pa ne s Uni e sal Inno a i e Resea ch Publica ion (PUIRP)
Volume: 03 Issue: 05 | Sep embe -Oc obe 2025 | ISSN: 3048-586X | www.pui p.com
© 2025, PUIRP | PU Publica ions | DOI:10.5281/zenodo.17536909 Page | 94
P ope y (3): The e may exis s mo e han one HUE o a pa ame e .
P oo : Le us conside a popula ion ollowing he uni o m disc e e dis ibu ion [10] desc ibed by he
p obabili y mass unc ion
P(X = ) = , ( = 1 , 2 , ……… , K)
wi h popula ion ha monic mean μH whe e
μH = ( 1
1
𝐾 ∑ 1
𝐾
𝑛
𝑖 = 1 )
Suppose,
, , ………. ,
is a andom sample d awn om his popula ion.
Then each elemen o he sample assumes he alues
1 , 2 , …….. , k
wi h equal p obabili y ,
so ha by he de ini ion o ha monic expec a ion,
EH ( ) = (1
1
𝐾 ∑ 1
𝐾
𝑛
𝑖 = 1 ) = μH , o each (i = 1 , 2 , …….. , k)
This implies each is a HUE o μH .
By P ope y (3),
HM o any wo elemen s o he sample is HUE o μH .
Simila ly, HM o any h ee elemen s o he sample is also HUE o μH ,
HM o any ou elemen s o he sample is also HUE o μH
and so on.
Thus, P ope y (3) has been es ablished.
P ope y (4): The e may no exis s HUE o a pa ame e .
P oo : Le us conside a popula ion ollowing binomial dis ibu ion [10] ha ing pa ame e s R (numbe o
ials) and p (p obabili y o success).
Suppose
, , ………. ,
is a andom sample d awn om his popula ion.
Fo his dis ibu ion, HUE o he binomial pa ame e p does no exis .
Thus, P ope y (4) has been es ablished.
Pa ne s Uni e sal Inno a i e Resea ch Publica ion (PUIRP)
Volume: 03 Issue: 05 | Sep embe -Oc obe 2025 | ISSN: 3048-586X | www.pui p.com
© 2025, PUIRP | PU Publica ions | DOI:10.5281/zenodo.17536909 Page | 95
P ope y (5): I T is HUE o pa ame e θ and S is HUE o pa ame e φ hen HM o T and S is HUE o he HM o θ
and φ.
In gene al, i
𝑇1 , 𝑇2 , ……….. , 𝑇𝑟
a e HUEs o he espec i e pa ame e s
𝜃1 , 𝜃2 , ……….. , 𝜃𝑟. ,
hen he HM o 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟 , is HUE o he HM o 𝜃1 , 𝜃2 , ……….. , 𝜃𝑟 .
P oo : This ollows om he ac ha We
EH (HM o T and S) = EH ( 2
1
𝑇+1
𝑆
)
= 2EH {(1
𝑇 +1
𝑆)−1}
= 2{EA(1
𝑇 +1
𝑆)}−1 , (whe e EA(T) is he a i hme ic expec a ion o T)
= 2{EA(1
𝑇)+ EA(1
𝑆)}−1
= 2[{EH(𝑇)}−1+ {EH(𝑆) }−1]−1
= 2(𝜃 −1+ 𝜑−1)−1
= HM HM o θ and φ
P oceeding simila ly as in he ea lie case, one can ob ain ha
EH (HM o 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟) = HM o 𝜃1 , 𝜃2 , ……….. , 𝜃𝑟
Hence, P ope y (5) has been es ablished.
4. CONCLUSION
Concep o geome ic unbiasedness is likely o be use ul and/o help ul in inding unbiased es ima o o a
pa ame e in he si ua ion whe e he associa ed da a a e o a io ype o o he allied ypes.
The p ope ies o geome ic unbiased es ima o , ob ained he e, a e likely o be use ul and/o help ul in
inding unbiased es ima o o a unc ion o pa ame e in he simila si ua ions.
Mo eo e , he p ope ies o geome ic unbiased es ima o a e likely o be impo an and use ul in en iching
he heo y o s a is ical es ima ion.
REFERENCES
[1] Bi nbaum Allan
(1961): “A Uni ied Theo y o Es ima ion, The Annals o Ma hema ical S a is ics. 32(1),
112
–
135.
doi
:
10.1214/aoms/1177705145
.
[2] Cha am elli, R., Shanmugam, R. (2024). “Ma hema ical Expec a ion”, In: Random Va iables o
Scien is s and Enginee s. Syn hesis Lec u es on Enginee ing, Science, and Technology. Sp inge , Cham.
h ps://doi.o g/10.1007/978-3-031-58931-7_1.
Pa ne s Uni e sal Inno a i e Resea ch Publica ion (PUIRP)
Volume: 03 Issue: 05 | Sep embe -Oc obe 2025 | ISSN: 3048-586X | www.pui p.com
© 2025, PUIRP | PU Publica ions | DOI:10.5281/zenodo.17536909 Page | 96
[3] Dh i ikesh Chak aba y (2024): “Idea o A i hme ic, Geome ic and Ha monic Expec a ions”, Pa ne s
Uni e sal In e na ional Inno a ion Jou nal (PUIIJ), 02(01), 119 – 124. www.puiij.com .
DOI:10.5281/zenodo.10680751.
[4] Dh i ikesh Chak aba y (2024): “A i hme ic, Geome ic and Ha monic Expec a ions: Expec ed Rainy
Days in India”, Pa ne s Uni e sal In e na ional Resea ch Jou nal (PUIRJ), (ISSN: 2583-5602), 03(01), 119
– 124. www.pui j.com . DOI:10.5281/zenodo.10825829.
[5] Dh i ikesh Chak aba y (2024): “Rhy hmic Addi i e P ope y o Ha monic Expec a ion”, Pa ne s
Uni e sal In e na ional Inno a ion Jou nal (PUIIJ), 02(05), 37 – 42. www.puiij.com .
DOI:10.5281/zenodo.13995073.
[6] Dh i ikesh Chak aba y (2025): “A i hme ic, Geome ic, Ha monic and Quad a ic Unbiasedness in
Es ima ion”, Pa ne s Uni e sal In e na ional Inno a ion Jou nal (PUIIJ), 24 – 32. www.puiij.com.
DOI: h ps://doi.o g/10.5281/zenodo.17060844.
[7] Dh i ikesh Chak aba y (2025): “Con inuous Random Va iable A i hme ic, Geome ic, Ha monic and
Quad a ic Expec a ions”, Pa ne s Uni e sal Inno a i e Resea ch Publica ion (PUIRP), (ISSN: 3048-
586X), 03(04), 36 – 42. www.pui p.com. DOI:10.5281/zenodo.16996732.
[8] Keeping E. S. (1962): “In oduc ion o S a is ical In e ence”, P ince on, N.J.: D. Van Nos and Co., Inc.
[9] Lehmann E. L.
(1951): “A Gene al Concep o Unbiasedness”, The Annals o Ma hema ical
S a is ics, 22(4),
587 –
592.
doi
:
10.1214/aoms/1177729549
.
JSTOR
2236928
.
[10] Pepelyshe A. & Zhiglja sky A. (2020): “Disc e e Uni o m and Binomial Dis ibu ions wi h In ini e
Suppo ”, So Compu , 24, 17517 – 17524. h ps://doi.o g/10.1007/s00500-020-05190-2 .
[11] P ei e P. E. (1990): “Ma hema ical Expec a ion”, In: P obabili y o Applica ions. Sp inge Tex s in
S a is ics, Sp inge , New Yo k, NY. h ps://doi.o g/10.1007/978-1-4615-7676-1_15.
[12]
Taylo Cou ney (2019): “
Unbiased and Biased Es ima o s”,
Though Co
, Re ie ed 2020-09-12
.
[13]
Voino Vassily G.; Nikulin, Mikhail S. (1993): “Unbiased Es ima o s and Thei Applica ions”, Vol. 1:
Uni a ia e case. Do d ec : Kluwe Academic Publishe s.
ISBN
0-7923-2382-3
.
[14]Yada S. K., Singh S. & Gup a, R. (2019): “Random Va iable and Ma hema ical Expec a ion”, In:
Biomedical S a is ics, Sp inge , Singapo e. h ps://doi.o g/10.1007/978-981-32-9294-9_26.
