Asymptotic sampling distribution of inverse coefficient of variation and its applications: revisited

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1 International Journal of Advanced Statitic and Probability, () (04) 5-0 Science Publihing Corporation doi: 0.449/ijap.vi.475 Short Communication Aymptotic ampling ditribution of invere coefficient of variation and it application: reviited Ahmed N. Albatineh *, B.M. Golam Kibria, Bahar Zogheib 3 Department of biotatitic, Florida international univerity Department of mathematic and tatitic, Florida international univerity 3 Department of mathematic and natural cience, American univerity of Kuwait *Correponding author aalbatin@fiu.edu Copyright 04 Ahmed N. Albatineh et al. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. Abtract Sharma and Krihna [6] derived mathematically an appealing aymptotic confidence interval for the population Signal-to-Noie Ratio (SNR). In thi paper, an evaluation of the performance of thi interval uing Monte Carlo imulation uing randomly generated data from normal, log-normal, χ, Gamma, and Weibull ditribution three of which are dicued in Sharma and Krihna [6]. Simulation revealed that it performance, a meaured by coverage probability, i totally dependent on the amount of noie introduced. A propoal for uing ranked et ampling (RSS) intead of imple random ampling (SRS) improved it performance. It i recommended againt uing thi confidence interval for data from a log-normal ditribution. Moreover, thi interval perform poorly in all other ditribution unle the SNR i around one. Keyword: Signal-to-noie ratio, coefficient of variation, ampling ditribution, confidence interval, ranked et ample, imple random ample.. Introduction The population coefficient of variation (CV) i a unit-free meaure of variability relative to the mean. Thi meaure can be ued to make comparion acro everal population that have different unit of meaurement. The population CV i defined a a ratio of the population tandard deviation (σ) to the population mean (μ), namely CV = σ/μ. In real life cae the population parameter σ and μ are etimated by the ample etimator and, repectively. The CV i widely ued in health cience in decriptive and inferential manner ee Kelley [6] and Gulhar et al. [4] for a dicuion. Some application include meauring the variation in the mean ynaptic repone of the central nervou ytem and meauring the variability in ocioeconomic tatu and prevalence of moking among tobacco control environment, ee Faber and Korn [] and Bernatm et al., []. Becaue the CV i a unit free meaure, the variation of two or more different meaurement method can be compared to each other. In public health etting, the CV i ued in aeing the overall health of an individual where the CV may be ueful in comparing the variability in blood preure meaurement (mmhg) and choleterol meaurement (mg/dl). If variance σ were ued, rather than the CV, then thee two meaure would not be comparable a their unit of meaurement would differ. In analog and digital communication, the reciprocal of the CV i called Signal-to-Noie Ratio (SNR) which i a meaure of ignal trength relative to background noie, while in quality control; the SNR repreent the magnitude of the mean of a proce compared to it variation. The SNR meaure how much ignal ha been corrupted by noie ee McGibney and Smith [0] for a dicuion. The SNR i commonly ued in image proceing, where the SNR of an image i uually calculated a the ratio of the mean piel value over a given neighborhood. For a population with mean μ and tandard deviation σ, the SNR i defined a the ratio of the population mean to the population tandard deviation where the parameter μ and σ are etimated by and, repectively. Hence a ample etimate of the population SNR i given by = /. It i of great interet to find confidence interval etimate for SNR. Confidence interval etimation allow the reearcher to have an idea about the preciion of the point etimate rather than only a p value for rejection or no rejection of a pecified null hypothei. Confidence interval for the SNR are limited in the literature. Sharma and Krihna [6]

2 6 International Journal of Advanced Statitic and Probability derived mathematically an appealing aymptotic confidence interval for the population SNR. The author indicated that uch derivation olved a problem concerning inference about population CV and SNR in addition to making inference (point & interval etimation, and hypothei teting) about the hape parameter of three popular life-teting ditribution: Gamma, Weibull, and log-normal, when the cale parameter i unknown. George and Kibria [3] performed a imulation tudy that compare everal confidence interval etimate for etimating SNR by inverting confidence interval for etimating CV. A powerful alternative ampling technique to imple random ampling (SRS) i ranked et ampling (RSS) which i ued to obtain etimate of the μ and σ in ituation where the variable of interet i too epenive or can t be eaily meaured, but can be eaily ranked. McIntyre [9] wa the firt to ugget uing RSS to etimate the population mean intead of SRS and the idea wa later developed by Takahai and Wakimoto [8] uing mathematical theory to upport their claim. Takahai and Wakimoto [8] proved that the ample mean obtained uing RSS i unbiaed and ha maller variance compared to that obtained uing SRS uing the ame ample ize, ee Samawi and Muttlak [4] and Samawi [5] for dicuion. Stoke [7] propoed an etimator for the variance of a ranked et ample data and howed that the etimator i aymptotically unbiaed and aymptotically more efficient than the ample variance of a imple random ample of a the ame number of obervation. MacEachern et al. [7] propoed an alternative etimator for the variance which i unbiaed and more efficient than Stoke etimator even when the underlying ditribution i not normal and the ranking of the element i not perfect. MacEachern et al. [7] etimator of σ perform well for mall to moderate ample ize and i aymptotically equivalent to Stoke [7] etimator. Stoke [7] ued RSS to derive an aymptotically unbiaed etimate of the variance regardle of preence of error in ranking. MacEachern et al. [7] provided an etimator for the variance which i unbiaed and more efficient than the one propoed by Stoke [7] etimator, even for non-normal underlying ditribution and where judgment ranking are not perfect. Terptra and Nelon [9] ued unbalanced RSS to compare maimum likelihood etimator (MLE) and weighted average (WA) etimate for the population proportion. Later Terptra and Wang [0] compared everal confidence interval for etimating the population proportion uing RSS, which i by far the only work found about implementation and dicuing correponding propertie of RSS in interval etimation for the population proportion. On the other hand, Samawi and Muttlak [4] compared RSS to SRS in etimating ratio and proved that the efficiency of the etimator ha increaed when uing RSS relative to SRS. In thi paper, rather than uing the uual SRS technique, the more powerful RSS technique will be ued to generate the ample. Thi make the current paper very unique in uing RSS to etimate SNR rather than SRS. Thi provided the motivation and ground to implement RSS in etimating a confidence interval for the SNR. In thi paper, an evaluation of the performance of Sharma and Krihna [6] confidence interval for etimating the SNR uing imulated data from everal ditribution. Alo RSS i propoed for interval etimation of the SNR intead of SRS with comparion baed on coverage probability. Table : Balanced RSS with m cycle and et ize k Cycle X [] X [] X []3... X []k Cycle X [] X [] X []3... X []k Cycle 3 X [3] X [3] X [3]3... X [3]k..... Cycle m X [m] X [m] X [m]3... X [m]k Thi paper i organized a follow: in ection an unbiaed etimate μ and σ uing RSS are introduced. Section 3 provide a decription of the aymptotic ditribution of the SNR. Section 4 decribe the imulation tudy and imulation reult, while a concluion i preented in ection 5.. Unbiaed etimate of μ and σ uing RSS Suppoe that we are intereted in obtaining a RSS of ize k from a population. Firt, a SRS of ize k obervation are elected and rank ordered on an attribute of interet. The obervation that i determined to be the mallet i the firt element of the RSS and i denoted X [] and the remaining k- unit are dicarded. A econd SRS of ize k i elected from the population and ranked the ame way and the econd mallet obervation i elected and denoted X []. In a imilar fahion, X [3], X [4] X [k] are elected hence X [], X [] X [k] repreent our firt balanced RSS of ize k. To obtain a balanced RSS of ize n=km, the proce i repeated m independent cycle yielding the balanced RSS of ize n hown in Table. The complete balanced RSS with et ize k and m cycle i given by {X [r]i : r =,, m; i =,,, k}. The term X [r]i i called the rth judgment order tatitic from the ith cycle. It i the obervation that i judged to be the rth order tatitic from the k et in the ith cycle, ee MacEachern et al. [7] for dicuion. Aume that the underlying ditribution ha finite mean μ and variance σ. Stoke [7] propoed an etimator of σ baed on RSS given by n m ˆ m, where ˆ X [ ] n m = ( X ˆ [ r ] i - ) nm - i = r = m () = nm i = r = r i

3 International Journal of Advanced Statitic and Probability 7 Stoke [7, page 37] howed that thi etimator i a biaed etimator of σ, but it i aymptotically unbiaed a either n or m approach. Moreover, Stoke [7] indicated that the RSS etimator ha more preciion over the ample mean; ay obtained uing SRS becaue of independence of the order tatitic compoing the ranked et ample. In fact, the author howed that n var ( ) n var ( ). The balanced RSS i ued in thi paper. The etimator of the variance of a RSS propoed by MacEachern et al. [7] will be implemented in the imulation ince it ha been hown to perform very well for mall a well a large ranked et ample, which i an improvement on Stoke [7] etimator. The evaluation of Sharma and Krihna [6] interval will be baed on coverage probability which eem to be the major factor for comparion, ee Mahmoudvand and Haani [8], Terptra and Wang [0], Panichkitkoolkul [,], and Kang and Schmeier [5] for dicuion. 3. Aymptotic ditribution of SNR Let, n be an independently and identically ditributed (iid) random ample of ize n from a population with finite mean μ and finite variance σ. Let be the ample mean and being the ample tandard deviation. Uing the central limit theorem converge in ditribution to tandard normal ditribution and in probability to m, while converge in probability to σ. The following Lemma i from Sharma and Krihna [6]. Lemma : Let [X n, Yn], n =, be a equence of pair of random variable. Then for a contant k 0, If X n L ¾ ¾ X,Y n P ¾ ¾ k, then X n Y n L ¾ ¾ X k Applying Lemma to and to obtain converge in ditribution to the tandard normal with mean and variance, in probability. Sharma and Krihna [6] indicated that the method of moment can be ued to how that i an etimator of and that i an unbiaed and conitent etimator of. Moreover, i aymptotically ditributed a tandard normal. Sharma and Krihna [6] indicated that the aymptotic ditribution of i ueful for teting and etimating population value of m and hence olved an important problem in urvival analyi involving inference on the hape parameter of Gamma, Weibull, and log-normal ditribution. Sharma and Krihna [6, page 63] provided a (-α) 00% confidence interval etimate for the population SNR= m which i given by P ¾ ¾ m ± Z n m Note that the width of thi interval i Z which i a function of the ample ize n only. The width will decreae n a the ample ize n increae. Therefore, for a fied n, the width will be contant for any value of and obtained uing either SRS or RSS. For thi reaon, the width wa not conidered a a criterion for performance comparion. In the following ection, a critical evaluation of the performance of thi interval i performed uing coverage probability a a performance criterion. Data were generated from five ditribution, three of which are ued by Sharma and Krihna [6], with a range of ample ize and value of SNR uing both SRS and RSS technique. 4. Simulation tudy The objective of thi paper i to evaluate the performance of Sharma and Krihna [6] interval etimator of the SNR uing SRS compared to RSS. Thi interval ha an appealing mathematical derivation, but to evaluate it performance a imulation tudy ha been conducted uing R tatitical oftware. Five ditribution are ued in imulation, namely: normal, log-normal, χ, Gamma, Weibull ditribution, three of which have been ued in Sharma and Krihna [6]. The nominal confidence level i et to 95%. 4.. Simulation technique In thee imulation, random ample of ize n=5, 5, 50, 00 are generated with pecific parameter from normal, log-normal, χ, Gamma, and Weibull ditribution. For each combination of n and SNR, 5000 replication were generated. The correponding imulated coverage probabilitie are preented in Table. m m n( - m ) n ()

4 8 International Journal of Advanced Statitic and Probability 4... Normal Ditribution Let, n be an iid random ample from a normal ditribution with finite mean μ and variance σ. For a random variable X uch that X ~ N (μ, σ ), the population SNR= μ/σ. Thu, for σ = 0, 5, 3, and μ = 0 the correponding population value of SNR are,, 3.33, and 0 are obtained Log-Normal Ditribution Let, n be iid random ample from a log-normal ditribution ln N(μ,σ ) where the hape parameter σ > 0 and the cale parameter μ a real number. The mean and variance of a log normal random variable are given by m = e m+ m + And = ( -)( e ) e. Therefore, the population SNR i given by SNR = ( e - ) - (3) Clearly the value of SNR in equation (3) depend only on the hape parameter σ. For imulation purpoe, et μ = 0 (arbitrary), and in order to get SNR= 0, 3.33,,, we mut et σ = , 0.938, 0.474, 0.836, repectively χ ditribution Let, n be an iid random ample from a χ (v) ditribution with v degree of freedom. Then the mean and variance of a χ random variable are given by μ = v and σ = v, repectively. Therefore, the population SNR for a χ random variable i given by v SNR = (4) For imulation purpoe, and in order to get SNR= 0, 3.33,,, we mut et v = 00,, 8,, repectively Gamma Ditribution Let, n be an iid random ample from a Gamma ditribution with hape and cale parameter α and β, repectively. The mean and variance of a random variable from Gamma (α, β) ditribution are μ = αβ and σ = αβ. Therefore, the value of the population SNR i given by SNR = (5) Clearly, SNR in equation (5) i free of the cale parameter β. For imulation purpoe, we et β = (arbitrary) and in order to get SNR= 0, 3.33,,, we mut et α = 00,.0889, 4,, repectively Weibull ditribution Let, n be an iid random ample from a Weibull ditribution with cale parameter λ > 0 and hape parameter k > 0. Then the mean and variance of a random variable which ha a Weibull ditribution are m = G (+ k ) and = G ( + ) - m. Thu, the value of SNR i given by k G ( + ) G ( + ) SNR = k = k G ( + ) - m G ( + ) - G ( + ) k k k For imulation purpoe, and in order to get SNR= 0, 3.33,,, we et λ = 4 (arbitrary) and k =.6, 3.7,.05,, repectively. 4. Simulation reult To evaluate the performance of Sharma and Krihna [6] confidence interval, 5000 random ample were generated for each combination of n and SNR. The random ample were generated from each of the following ditribution: normal, log-normal, χ, Gamma, and Weibull ditribution. Coverage probabilitie were calculated uing SRS and RSS technique. Table preent the imulation reult which reveal that for normal data and ample ize greater than or equal to 5, coverage probabilitie uing RSS were cloer to the 0.95 nominal value only when SNR=. For log-normal (9)

5 International Journal of Advanced Statitic and Probability 9 data the interval performed the wore for all ample ize and SNR value, with the highet coverage attained of on the other hand, for χ, Gamma, and Weibull ditribution, the interval attained coverage probability cloe to the nominal value for all ample ize, uing both RSS and SRS, only when SNR=. A the SNR value increae from to 0, the coverage probability got wore. Baed on thee imulation, thi interval etimate of the SNR hould not be ued in interval etimation of the SNR unle the value of the SNR i cloe to one, with a preference of RSS over SRS epecially when the data i normally ditributed. In real life etting, it could be difficult for SNR to equal one for thi interval to perform well. Thi i a evere limitation of the performance of thi interval, which could not been identified without etenive imulation, given the appealing mathematical formation of thi interval. Table : Etimated coverage probabilitie uing normal (I), log-normal (II), χ (III), Gamma (IV), and Weibull (V) ditribution for ample ize 5,5,50,00 and SNR of,, 3.33, and 0 Ditribution I II III IV V n=5 SNR= RSS SRS SNR= RSS SRS RSS SRS SNR=0 RSS SRS n=5 SNR= RSS SRS SNR= RSS SRS RSS SRS SNR=0 RSS SRS n=50 SNR= RSS SRS SNR= RSS SRS RSS SRS SNR=0 RSS SRS n=00 SNR= RSS SRS SNR= RSS SRS RSS SRS SNR=0 RSS SRS Concluding remark In thi paper, a imulation tudy i conducted to evaluate the performance of Sharma and Krihna [6] confidence interval of SNR. Data were imulated from normal, log-normal, χ, Gamma, and Weibull ditribution with ample ize 5, 5, 50, and 00 with SNR value of,, 3.33, and 0. A ranked et ampling trategy i propoed intead of imple random ampling which improved the performance a meaured by coverage probability. Even though thi confidence

6 0 International Journal of Advanced Statitic and Probability interval for etimating the SNR ha an appealing tatitical reaoning, it wa found that the performance of thi interval i totally dependent on the value of the SNR, with bet performance when SNR i one. Thi interval hould not be ued when the data follow a log-normal ditribution due to low coverage probability. For data from χ, Gamma, and Weibull, the interval attained the nominal value coverage of 0.95 when SNR= uing either RSS or SRS. For data from normal ditribution, the interval attained nominal coverage value when SNR= and a ample ize greater than or equal to 5 (not urpriing ince thi interval i baed on an aymptotic reult), with RSS performing better than SRS in term of coverage probability. The more noie introduced in the data, the wore the performance of thi interval. Thi wa noticed acro all ditribution dicued. The concluion of thi paper are retricted to the et of imulation condition we have conidered. For definite tatement, one might need more imulation under different condition. All imulation were performed uing R [3] tatitical oftware. Reference [] Bernatm, D.H., Lazovich, D., Forter, J.L., Oake, J.M., Chen, V., Area-level variation in adolecent moking, Preventing Chronic Dieae, 6,, (009) -8. [] Faber, D. S. and Korn, H., Applicability of the coefficient of variation for analyzing ynaptic platicity, Biophyical Society, 60, (99) [3] George. F. and Kibria, B. M. Golam, Confidence interval for etimating the population ignal-to-noie ratio: a imulation tudy, Journal of Applied Statitic, 39, (0) [4] Gulhar, M., Kibria, B. M. G., Albatineh, A. N., and Ahmed, N. U., A Comparion of ome confidence interval for etimating the population coefficient of variation: a imulation tudy, SORT, 36, (0) [5] Kang, K. and Schmeier, B. W., Method for evaluating and comparing confidence interval procedure, Technical Report, Dept. Indutrial Engineering, Univerity of Miami, Florida, U.S.A, (986). [6] Kelly, K., Sample ize planning for the coefficient of variation from the accuracy in parameter etimation approach, Behavior Reearch Method, 39, 4, (007) [7] MacEachern, S., Ozturk, O., and Wolfe, D.A., A New ranked et ample etimator of variance, Journal of the Royal Statitical Society B., 64, Part, (00) [8] Mahmoudvand, R. and Haani, H., Two new confidence interval for the coefficient of variation, Journal of Applied Statitic, 36, (009) [9] McIntyre, G. A., A Method for unbiaed elective ampling uing ranked et, Autralian Journal of Agricultural Reearch, 3, (95) [0] McGibney, G. and Smith, M. R., An unbiaed ignal-to-noie ratio meaure for magnetic reonance image, Medical Phyic, 0, (993) [] Panichkitkoolkul, W., Aymptotic confidence interval for the coefficient of variation of Poion ditribution: a imulation tudy, Maejo International Journal of Science and Technology, 4, (00) - 7. [] Panichkitkoolkul, W., Improved confidence interval for a coefficient of variation of a normal ditribution, Thailand Statitician, 7, (009) [3] R Development Core Team, R: A language and environment for tatitical computing, R foundation for tatitical computing, Vienna, Autria. ISBN , URL (007). [4] Samawi, H. M. and Muttlak, H. A., Etimation of ratio uing ranked et ampling, Biometrical Journal, 6, (996) [5] Samawi, H. M., More efficient Monte Carlo Method obtained by uing ranked et imulated ample, Communication in Statitic - Simulation and Computation, 8, (999) [6] Sharma, K. K. and Krihna, H., Aymptotic ampling ditribution of invere coefficient- of -variation and it application. IEEE Tranaction on Reliability, 43, (994) [7] Stoke, S.L., Etimation of variance uing judgment ordered ranked et ample, Biometric, 36, (980) [8] Takahahi, K. and Wakimoto, K., On unbiaed etimate of the population mean baed on the ample tratified by mean of ordering, Annal of the Intitute of Statitical Mathematic, 0, (968) - 3. [9] Terptra, J. and Nelon, E., Optimal rank et ampling etimate for a population proportion, Journal of Statitical Planning and Inference, 7, (005) [0] Terptra, J. and Wang, P., Confidence interval for a population proportion baed on a ranked et ample, Journal of Statitical Computation and Simulation, 78, (008)