Harmonic Unbiased Estimator: Some Properties

Partners Universal Innovative Research Publication (PUIRP) Volume: 03 Issue: 05 | September-October 2025 | ISSN: 3048-586X | www.puirp.com © 2025, PUIRP | PU Publications | DOI:10.5281/zenodo.17536909 Page | 92 Harmonic Unbiased Estimator: Some Properties Dhritikesh Chakrabarty Independent Researcher, Guwahati, Assam, India -------------------------------------------------------------------------------------- Abstract - In an earlier study, concept of harmonic unbiased estimator was introduced and defined based on harmonic expectation. Attempt has here been made to identify some important properties/facts/results of harmonic unbiased estimator. This article is based on the information on this unbiased estimator obtained in the attempt. Keywords: Estimation, Harmonic Unbiasedness, HUE, Some Properties. 1. INTRODUCTION Unbiasedness, in the literature of statistical estimation, is regarded as a desirable property/quality/criterion of an estimator [1 , 8 , 12, 13]. Originally, the concept of unbiasedness [8 , 9] was explained on the basis of the mathematical expectation [2 , 11 , 14], more specifically the arithmetic expectation, of the estimator concerned and accordingly unbiased estimator was defined [1 , 12 , 13]. This definition later was termed as arithmetic unbiased estimator [6]. Recently, concepts of harmonic unbiased estimator [6] was introduced and defined based on harmonic expectation [3 , 4 , 5 , 7]. Attempt has here been made on identifying some important properties/facts/results of harmonic unbiased estimator. This article is based on the information on this unbiased estimator obtained in the attempt. 2. HARMONIC UNBIASED ESTIMATOR Suppose, , , ………. , is a random sample drawn from a population of a non-zero real valued random variable X which follows a probability distribution having parameter θ & T = T( , , ………. , ) is an estimator of θ. Then T can be regarded as harmonic unbiased estimator of parameter θ if EH(T) = θ where EH(T) is the harmonic expectation of T. Let us abbreviate harmonic unbiased estimator by HUE. Note: HUE exists in the case of non-zero real valued estimator. The corresponding parameter θ, in this case, is an unknown non-zero real number. Partners Universal Innovative Research Publication (PUIRP) Volume: 03 Issue: 05 | September-October 2025 | ISSN: 3048-586X | www.puirp.com © 2025, PUIRP | PU Publications | DOI:10.5281/zenodo.17536909 Page | 93 3. ALGEBRAIC PROPERTIES OF HARMONIC UNBIASED IN ESTIMATOR Property (1): If T is HUE of parameter θ then c.T is HUE of parameter c.θ. Proof: This follows from the fact that EH(c.T) = c.EH(T) = c.θ Corollary: If T is HUE of parameter θ then −T is HUE of parameter −θ. This follows from the fact that EH(−T) EH(−1.T) = = −1.EH(T) = −θ Property (2): Harmonic mean (HM) of a finite number of HUEs of a parameter θ is also HUE of the parameter θ. Proof: Suppose, T and S are two HUEs of a parameter θ. Then EH (HM of T and S) = EH ( 2 1 𝑇+1 𝑆 ) = 2EH {(1 𝑇 +1 𝑆)−1} = 2{EA(1 𝑇 +1 𝑆)}−1 , (where EA(T) is the arithmetic expectation of T) = 2{EA(1 𝑇)+ EA(1 𝑆)}−1 = 2[{EH(𝑇)}−1+ {EH(𝑆) }−1]−1 = 2(𝜃 −1+𝜃 −1 )−1 = θ Therefore, HM of T and S is HUE of a parameter θ. Now suppose, 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟 are r HUEs of a parameter θ. Then HM of 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟 = 𝑟 1 𝑇1+1 𝑇2 +⋯……..+ 1 𝑇𝑟 Proceeding similarly as in the earlier case, one can obtain that EH (𝑟 1 𝑇1+1 𝑇2 +⋯……..+ 1 𝑇𝑟 ) = θ Therefore, HM of 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟 is HUE of θ. Thus, Property (2) has been proved for a finite number of estimators. Hence, Property (2) has been established. Partners Universal Innovative Research Publication (PUIRP) Volume: 03 Issue: 05 | September-October 2025 | ISSN: 3048-586X | www.puirp.com © 2025, PUIRP | PU Publications | DOI:10.5281/zenodo.17536909 Page | 94 Property (3): There may exists more than one HUE of a parameter. Proof: Let us consider a population following the uniform discrete distribution [10] described by the probability mass function P(X = ) = , ( = 1 , 2 , ……… , K) with population harmonic mean μH where μH = ( 1 1 𝐾 ∑ 1 𝐾 𝑛 𝑖 = 1 ) Suppose, , , ………. , is a random sample drawn from this population. Then each element of the sample assumes the values 1 , 2 , …….. , k with equal probability , so that by the definition of harmonic expectation, EH ( ) = (1 1 𝐾 ∑ 1 𝐾 𝑛 𝑖 = 1 ) = μH , for each (i = 1 , 2 , …….. , k) This implies each is a HUE of μH . By Property (3), HM of any two elements of the sample is HUE of μH . Similarly, HM of any three elements of the sample is also HUE of μH , HM of any four elements of the sample is also HUE of μH and so on. Thus, Property (3) has been established. Property (4): There may not exists HUE of a parameter. Proof: Let us consider a population following binomial distribution [10] having parameters R (number of trials) and p (probability of success). Suppose , , ………. , is a random sample drawn from this population. For this distribution, HUE of the binomial parameter p does not exist. Thus, Property (4) has been established. Partners Universal Innovative Research Publication (PUIRP) Volume: 03 Issue: 05 | September-October 2025 | ISSN: 3048-586X | www.puirp.com © 2025, PUIRP | PU Publications | DOI:10.5281/zenodo.17536909 Page | 95 Property (5): If T is HUE of parameter θ and S is HUE of parameter φ then HM of T and S is HUE of the HM of θ and φ. In general, if 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟 are r HUEs of the respective parameters 𝜃1 , 𝜃2 , ……….. , 𝜃𝑟. , then the HM of 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟 , is HUE of the HM of 𝜃1 , 𝜃2 , ……….. , 𝜃𝑟 . Proof: This follows from the fact that We EH (HM of T and S) = EH ( 2 1 𝑇+1 𝑆 ) = 2EH {(1 𝑇 +1 𝑆)−1} = 2{EA(1 𝑇 +1 𝑆)}−1 , (where EA(T) is the arithmetic expectation of 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 of θ and φ Proceeding similarly as in the earlier case, one can obtain that EH (HM of 𝑇1 , 𝑇2 , ……….. , 𝑇𝑟) = HM of 𝜃1 , 𝜃2 , ……….. , 𝜃𝑟 Hence, Property (5) has been established. 4. CONCLUSION Concept of geometric unbiasedness is likely to be useful and/or helpful in finding unbiased estimator of a parameter in the situation where the associated data are of ratio type or other allied types. The properties of geometric unbiased estimator, obtained here, are likely to be useful and/or helpful in finding unbiased estimator of a function of parameter in the similar situations. Moreover, the properties of geometric unbiased estimator are likely to be important and useful in enriching the theory of statistical estimation. 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