O the Variability Estimatio of Logormal Distributio Based o Sample Harmoic ad Arithmetic Meas Edward Y. Ji * ad Bria L. Ji ** Abstract: For the logormal distributio, a ubiased estimator of the squared coefficiet of ariatio is deried from the relatie ratio of sample arithmetic to harmoic meas. Aalytical proofs ad simulatio results are preseted. Methods of estimatig the parameters of logormal distributio are summarized by Aitchiso ad Brow 1, Crow ad Shimizu, ad others. Fiey s method (1941) leads to the geeral solutios of uiformly miimum ariace ubiased estimators (UMVUE) 3, which are expressed by the ifiite series or geeralized hypergeometric fuctios. O the other had, for the harmoic mea estimatio method i umerous egieerig applicatios, some iestigators preferred other estimators to UMVUE for issues such as the egatie alues i the series ad computatioal simplicity 4. Here we show a ubiased estimator of the squared coefficiet of ariatio based o the relatie ratio of sample arithmetic to harmoic meas. This work is ispired by the ew semicoductor itegrated circuit architecture where the harmoic ad arithmetic meas are collectiely measured 5. While our results were origially obtaied first by the empirical fittigs to Mote Carlo simulatios, here we start with the aalytical proofs ad followed by the simulatio results. The two-parameter log-ormal (LN) distributio of a positie radom ariable X is deoted as X~LL(μ Y, σ Y ), with the probability desity fuctio, f LL (x; μ Y, σ Y ) = 1 xσ Y π eee (ll(x) μ Y ) σ Y, x > 0 * Mout Kisco, NY 10549. ** College of Naoscale Sciece ad Egieerig, SUNY Polytechic Istitute, Albay, NY 103. Email: bji@suycse.com 1
where μ Y ad σ Y are the mea ad ariace i the logarithmic space, Y = ll(x). I this paper, the populatio parameters of X i the real space are deoted as: the arithmetic mea α E[X], the ariace β E[(X α) ], the coefficiet of ariatio C β α, ad the harmoic mea h 1 E 1. X Furthermore, defie a ratio ω α h, ad a relatie ratio k ω 1 (α h) 1. For the logormal distributio, it is kow that 1 α = exp μ Y + 1 σ Y, h = exp μ Y 1 σ Y, β = eee(μ Y + σ Y )(eee(σ Y ) 1), C = exp(σ Y ) 1, so that ω = eee(σ Y ) ad k = eee(σ Y ) 1 = C 0. Let,,, X,, be idepedet ad idetically distributed (i.i.d.) radom ariables haig LL(μ Y, σ Y ). The sample arithmetic mea is A X (1 ) X i. The sample harmoic mea is H X H i=1(1 X i ). The sample relatie ratio is defied as K A 1. H i=1 Propositio 1. For LL(μ Y, σ Y ), the expected alue of the sample relatie ratio K is, E(K ) = 1 1 k = C (1) Proof. From the multiplicatie properties 1,, if X a ~LL(μ a, σ a ) ad X b ~LL(μ b, σ b )are idepedet, the their product is a logormal ariable, X a X b ~LL(μ a + μ b, σ a + σ b ). Sice 1/X i ~LL( μ Y, σ Y ), / ~LL(0, σ Y ), so that E( / ) = eee(σ Y ) = k + 1. Thus, E(K ) = E 1 X i 1 X j 1 = 1 E 1 + X i X j 1 = 1 + ( 1)E 1 i=1 j=1 i=j=1 = 1 1 1 + ( 1)(k + 1) 1 = k = C i j Propositio. Let k = K 1 = 1 A 1, the k H is a ubiased estimator of k ad C, that is, Ek = k = C () Proof. Equatio () is straightforward from equatio (1).
Propositio 3. For LL(μ Y, σ Y ), the ariace ad stadard deiatio (the square root of ariace) of the sample relatie ratio K are, respectiely, VVV(K ) = ( 1) k 1 + k + k = ( 1) C 4 1 + C + C 4 (3) ss(k ) = k k ( 1) 1 + k + = C ( 1) 1 + C + C (4) Proof. The ariace of the sum is the sum of the coariaces, VVV( i=1 A i ) = i,j=1 CCCA i, A j. The similar terms i the expressio of the ariace of K as the sum are collected ad calculated with the coariace property, CCC(A, B) = E(AA) E(A)E(B) ad the logormal properties, for example, E = E = ELL(0, 6σ Y ) = eee(3σ Y ) = ω 3. We hae, 4 VVV = CCC, = E E E = eee(4σ X Y ) eee(σ Y ) = ω 4 ω CCC, = E E E = 1 eee(σ X Y ) = 1 ω ; CCC 1 = ω 3 ω ; CCC = ω ω ; CCC, = ω ω ; CCC, = ω 3 ω ; CCC X 4 = 0 For the geeral case of 4: VVV(K ) = VVV 1 X i 1 X j = 1 VVV 1 + X i 4 i=1 j=1 i=j=1 X j i j = 1 4 ( 1)CCC, + ( 1)CCC, + ( 1)( )CCC + ( 1)( )CCC + ( 1)( )CCC, + ( 1)( )CCC, + ( 1)( )( 3)CCC X 4 = 1 4 {( 1)(ω4 ω ) + ( 1)(1 ω ) + ( 1)( )(ω 3 ω ) + ( 1)( )(ω ω ) = + ( 1)( )(ω ω ) + ( 1)( )(ω 3 ω )} ( 1) k 1 + k + k 3
The cases of = ad 3 are straightforward. Example 1. For the case of = : VVV(K ) = 1 4 VVV ( + ) 1 + 1 = 1 4 VVV + + = 1 4 VVV + = 1 4 CCC, + CCC, + CCC, + CCC, = 1 4 {(ω4 ω ) + (1 ω )} = 1 8 (ω 1) = 1 8 k (k + ) = 1 k 1 + k + k 4 Propositio 4. The ariace ad stadard deiatio of k are, respectiely, VVVk = 1 k 1 + k + k = C 1 4 1 + C + C 4 (5) ssk = k k 1 + k + = C 1 1 + C 1 + C 4 (6) Proof. Equatios (5) ad (6) are straightforward from equatios (3) ad (4). Mote Carlo simulatio data of k for arious sample size ad C alues are show i Figure 1, with aalytical fittigs to alidate equatios () ad (6). The simulatios were programmed with R laguage 6. The logormal radom ariables are geerated with mealog = 0 ad arious sdlog or C alues, usig the lorm fuctio. The expected alue ad stadard deiatio i Figure 1 are plotted o the same scale for guidace to the egieerig applicatio desigs. A traditioal way to study the estimator efficiecy is the large-sample ariaces 1. Fiey s UMVUE geerally apply to the parameter form, θ a,b,c = σ Y c exp (aμ Y + bσ Y ), therefore the parameter ω = exp(σ Y ) is a special case of a = 0; b = 1; c = 0. For ω UUUUU, followig the steps show o page 37 of Crow ad Shimizu 1, we obtai the special case solutio, VVV(ω UUUUU ) 1 σ Y 4 exp(σ Y ). Sice k = ω 1, VVVk UUUUU 1 σ Y 4 thus the large-sample efficiecy of k is, exp(σ Y ). From equatio (5), VVVk 1 eσ Y 1 exp(σy ), 4
eee. k = VVVk UUUUU 1 = σ 4 Y VVVk 1 e σ Y 1 Figure shows the large-sample efficiecy of k accordig to (7). (7) The logormal probability desity fuctio ca be alteratiely expressed i terms of g ad k, f LL (x; g, k) = 1 xπππ(1+k) eee ll(x/g) (8) ll(1+k) where g = eee(μ Y ) is the geometric mea. The geometric, arithmetic ad harmoic meas are related by 7 ah = g. It is easy to see that g = A H is a cosistet estimator, that is, asymptotically ubiased whe goes to ifiity. The measuremet cost should be ealuated for statistical samplig. Oe ca defie the measuremet cost as the umber of the eeded measuremets. The coetioal method of ariability estimatio requires the detailed kowledge of each replicate i the sample, thus the measuremet cost umber for sample size is simply. Usig the itegrated circuits of replicatig deices, some sample meas may be measured collectiely by arious laws of physics ad circuits (for example, series ad parallel circuits). I these cases, the sample harmoic mea H or arithmetic mea A ca be obtaied with a measuremet cost umber of 1. The sample estimators k ad g eed a measuremet cost umber of. The -to- measuremet cost umber reductio is obiously sigificat. Ackowledgemet We thak K. Liu ad M. B. Ketche for helpful discussios about the related ad broader topics. Referece [1] Aitchiso, J. & Brow, J. A. C. The Logormal Distributio. (Cambridge Uiersity Press, 1957). [] Crow, E L. & Shimizu, K., eds. Logormal distributios: Theory ad applicatios. (M. Dekker, New York, 1988). 5
[3] Fiey, D. J. O the distributio of a ariate whose logarithm is ormally distributed. J. Roy. Statist. Soc. Suppl., 7, 155-161 (1941). [4] Limbruer, J. F., Vogel, R. M., & Brow, L. C. Estimatio of harmoic mea of a logormal ariable. Joural of Hydrologic Egieerig 5 (1), 59-66 (000). [5] Ji, E. Y., Method ad Apparatus of Itegrated Sesor Systems. U. S. Patet Applicatio (pedig), 014. [6] R Deelopmet Core Team R: A laguage ad eiromet for statistical computig. R Foudatio for Statistical Computig, Viea, Austria. ISBN 3-900051-07-0, URL http://www.r-project.org/ (01). [7] Rossma, L. A. Desig stream flows based o harmoic meas. J. Hydraulic Egieerig 116, 946 950 (1990). Figure 1 Expected alue ad stadard deiatio of k of arious sample size ad arious C alues. Dots: simulatio data. The expected alues ad the stadard deiatios are calculated from Mote Carlo simulatio of 10 7 /(-1) rus. Lies: aalytical predictios by equatios () ad (6). 6
Figure Large-sample efficiecy of k as a fuctio of σ Y. 7