A NOTE ON SKEW-NORMAL DISTRIBUTION APPROXIMATION TO THE NEGATIVE BINOMAL DISTRIBUTION

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1 A NOTE ON SKEW-NORMAL DISTRIBUTION APPROXIMATION TO THE NEGATIVE BINOMAL DISTRIBUTION JYH-JIUAN LIN 1, CHING-HUI CHANG * AND ROSEMARY JOU 1 Deartment of Statistics Tamkang University 151 Ying-Chuan Road, Tamsui, Taiei County 51, Taiwan @mail.tku.edu.tw Deartment of Alied Statistics and Information Science Ming Chuan University 5 De-Ming Road, Gui-Shan District, Taoyuan, Taiwan chchang@mcu.edu.tw Graduate Institute of Management Sciences, Tamkang University 151 Ying-Chuan Road, Tamsui, Taiei County 51, Taiwan @s9.tku.edu.tw Abstract: This article revisits the roblem of aroximating the negative binomial distribution. Its goal is to show that the skew-normal distribution, another alternative methodology, rovides a much better aroximation to negative binomial robabilities than the usual normal aroximation. Key-Words: Central Limit Theorem, cumulative distribution function (cdf), skew-normal distribution, skew arameter. * Corresonding author ISBN: ISSN:

2 1. INTRODUCTION This article revisits the roblem of the negative binomial distribution aroximation. Though the most commonly used solution of aroximating the negative binomial distribution is by normal distribution, Peizer and Pratt (1968) has investigated this roblem thoroughly and roosed a different method which outerforms the normal aroximation. However, the methodology used in Peizer and Pratt (1968) is rather comlicated and difficult. With different motive, we roose another alternative, the skew-normal distribution, and show that it rovides a much better aroximation to negative binomial robabilities than the usual normal aroximation. The skew-normal distribution is first introduced by O Hagan and Leonard (1976), which is a generalization of the normal distribution. Its definition is as follows. A r.v. X is said to follow a skew-normal distribution with location arameter μ, scale arameter σ and skew arameter λ (the SN( σ, λ) distribution) if the df of X is given as f ( x σ, λ) = ( / σ ) φ(( x μ) / σ ) Φ( λ( x μ) / σ ), (1.1) where φ and Φ denote the standard normal df and cdf resectively. The arameter sace is: μ (, ), σ > 0 and λ (, ). Besides, the distribution SN ( 0,1, λ) with df f (x 0, 1, λ) = φ (x) Φ( λx) is called the standard skew-normal distribution. When λ = 0, the above df (1.1) boils down to the N(σ ) df. Also arameter λ > 0 (< 0) imlies ositively (negatively) skewed shae of f ( x σ, λ) above. For more skew-normal distribution s roerties, one can see Azzalini (1985), Guta, Nguyen and Sanqui (004), Arnold and Lin (004) or Chang et al. (008). It seems that the skew-normal distribution is a better choice than the normal distribution to aroximate the discrete distribution. The reason is that the skew-normal distribution can take negatively skewed, symmetric or ositively skewed shae through its skew arameter and hence is more versatile than the normal distribution. Comaring to the normal distribution, Chang et al. (008) has shown that the extra (skew) arameter λ would rovide a better aroximation of the binomial distribution by the SN( σ, λ) distribution. Besides, Chang et al. (008) has also mentioned that it should also work for negative binomial distribution aroximation. However, a lot of details of the numerical results are not reorted in Chang et al. (008). Therefore, we will show that the extra (skew) arameter λ would rovide a better aroximation of the negative binomial distribution by the SN (σ, λ) distribution than the normal distribution in this aer along with the details. The skew-normal distribution and its usefulness to aroximate the negative binomial distribution is discussed in the next section. The usual normal aroximation method is also introduced in Section and numerical results and comarisons are rovided in Section.. APPROXIMATION BY THE NORMAL AND SKEW-NORMAL DISTRIBUTIONS In this section we ll see how the SN( σ, λ) distribution can be used to aroximate a negative binomial distribution..1. Normal Aroximation by the CLT It is a common ractice to aroximate the samling distribution of the samle mean by the fact of the Central Limit Theorem (CLT). The CLT is a very owerful theorem and is alying to various areas. For the alications of the CLT, lease see Wang and Wei (008), Giribon, Revetria and Oliva (007) and Mceachen et al (005). By the Central Limit Theorem (CLT), the cdf of NB(r, ) (negative binomial distribution where Y reresents the number of ISBN: ISSN:

3 failures before the r th success) at k, k { 0, 1,, K}, F r,, i.e., P( Y k), is aroximated by the normal cdf as F r where q = 1., Φ(( ( k + 0.5) rq) / rq ).. Skew-Normal Aroximation For a given NB(r, ) distribution, the arameters of the aroximating (or, matching) SN( σ, λ) distribution are found by equating the first three central moments, i.e., (a) E ( Y ) = rq / = E( X ) (b) (c ) = μ + σ ( / π )( λ / 1+ λ ); E( Y E( Y )) = rq / = E( X E( X )) = σ {1 ( / π ) λ /(1 + λ )}; E ( Y E( Y )) = rq(1 + q) / = E( X E( X )) = σ ( / π )( λ / 1+ λ ) ((4 / π ) 1). Solving the above three equations is straight forward. We obtain the values of λ, σ and μ as follows. (A) Given (r, ), the above last two equations yield λ = {( rq ( / π )((4 / π ) 1) /(1 + q)) / + ( / π ) 1} (B) Given ( r, ) and after obtaining above, σ is found as σ = ( rq / ) /{1 (/ π ) λ /(1+ λ )} 1/ 1/ λ as (C) Finally, after obtaining λ and σ as above, get μ as μ = ( rq / ) σ ( / π ) λ / 1+ λ.. After obtaining λ, σ and μ as given above, the negative binomial cdf is aroximated as F k r, f ( x μ, σ, λ) c dx = Ψλ (( k μ) / σ ), where Ψ ( c ) = φ ( z) Φ( λz) dz is the λ standard skew-normal cdf... Restriction A mild restriction is ut on r and, which is ( ) / r(1 ) < , in order to make the skew-normal aroximation work, i.e., F r, Ψλ (( k μ) / σ ). Since the skewness of the skew-normal distribution lies in ( , ), the skew-normal aroximation is alied only when the skewness of NB(r, ), ( ) / r(1 ), lies in this interval. However, the negative binomial distribution is always ositively skewed. Hence, strictly seaking, the restriction is that the skewness of NB(r, ) lies in (0, )..4. Evaluation Criterion Chang et. al. (008) uses the maximal absolute error (MABS) of the skew-normal aroximation to investigate the binomial aroximation B(n, ) as in Schader and Schmid (1989) (though their work is related to the normal aroximation to the binomial distribution). MABS can be substantial when is near 0 or 1. In this aer we will also adot the MABS criterion for this negative binomial aroximation roblem and it is defined as * MABS r, ) = max F F r ( ) (.1) ( r,, k k R Y * where F r, = Φ(( ( k + 0.5) rq) / rq ) is for the normal case and * F r, = Ψλ (( k μ) / σ ) is for the ISBN: ISSN:

4 skew-normal case. Y ~ N( r(1 )/ = , r(1 )/ = ) (.). EXAMPLES AND NUMERICAL RESULTS As a demonstration of the imroved aroximation of the negative binomial distribution by the skew-normal distribution, we first show the matching skew-normal (as in Eq. (.1) and normal (as in Eq. (.)) dfs for the negative binomial distribution ( Y ~ NB( r = 0, = 0.75) ) in the Figure.1 and the errors are reorted in the Table.1. Here ENA and ESA stand for Error in Normal Aroximation and Error in Skew-Normal Aroximation, and they are defined as in (.) and (.4) resectively. It can be seen that the skew-normal curve follows the asymmetric negative binomial distribution closely. Y ~ SN( μ =.4107, σ = (4.4148), λ =.44) (.1) ENA = Fr, Φ(( ( k + 0.5) rq) / rq ) (.) ESA = Fr, Ψλ (( k μ) / σ ) (.4) Next, we calculate the MABS (see Eq. (.1)) for the skew-normal aroximations as well as the normal aroximations (henceforth called MABS(SN) and MABS(N), resectively) for various r and. The results are reorted in Table.. On the other hand, Figure. shows the lots of MABS(SN) and MABS(N) as a function of (for fixed r = 0) and r (for fixed = 0.75). Plots for other values of r showed almost identical atterns. The results and lots both clearly show that the skew-normal aroximation is much suerior to the usual normal aroximation Distribution Negative Binomial Normal Skew-Normal Probability y Figure.1. NB(0, 0.75) mf with matching normal df and skew-normal df. Table.1. Errors in aroximating the NB(r, ) cdf for r = 0 and = k F r, Φ (( ( k + 0.5) rq) / rq) Ψλ (( k μ) / σ ) ENA ESA ISBN: ISSN:

5 Table.. MABS of Normal (N) and Skew-normal (SN) aroximations. r = 10 r = 0 r = 0 r = 40 N SN N SN N SN N SN (r, ) doesn t satisfy the inequality ( 1+ q ) / q < r / ( ). Take k = 0, 1,, 10 5 while MABS is calculated. Table.. (cont.) MABS of Normal (N) and Skew-normal (SN) aroximations. r = 50 r = 100 r = 150 r = 00 N SN N SN N SN N SN Normal Skew-Normal 0.04 r = = Normal Skew-Normal MABS MABS Figure.. Plots of MABS(SN) and MABS(N) r Concluding Remark: In this aer we have shown that the skew-normal distribution can rovide a far better aroximation than the normal distribution, and we believe that it can also be used to aroximate discrete distributions other than the negative binomial distribution. Acknowledgement : The second author s research has been suorted artially by a ISBN: ISSN:

6 research grant from the National Science Council ( M ). References [1] Arnold, B. C. and Lin, G. D. (004). Characterizations of the Skew-Normal and Generalized Chi Distributions. Sankhy a: The Indian Journal of Statistics, Vol. 66, Part - 4, [] Azzalini, A. (1985). A Class of Distributions Which Includes the Normal Ones. Scandinavian Journal of Statistics, 1, [] Chang, C. H., Lin, J. J., Pal, N. and Chiang, M. C. (008). A Note on Imroved Aroximation of the Binomial Distribution by the Skew-Normal Distribution. The American Statistician, 6(), [4] Giribone, P., Revetria, R. and Oliva, F. (007). A Stochastic Simulation Model for Reresenting a Clinical Pathology Laboratory Workout. 6th WSEAS International Conference on System Science and Simulation in Engineering, Venice, Italy, [5] Guta, A. K., Nguyen, T. T. and Sanqui, J. A. T. (004). Characterization of the Skew-Normal Distribution. Annals of the Institute of Statistical Mathematics, 56(), [6] Mceachen, J. C., Zachary J. M., Wang, J. and Cheng, K. W. (005). Conversation Exchange Dynamics: A New Signal Primitive For Visualizing Network Intrusion Detection. 4th WSEAS International Conference On Electronics, Control And Signal Processing, Miami, Florida, USA, [7] O'Hagan, A. and Leonard, T. (1976). Bayes Estimation Subject to Uncertainty about Parameter Constraints. Biometrika, 6, [8] Peizer, D. B. and Pratt, J. W. (1968). A Normal Aroximation for Binomial, F, Beta, and Other Common, Related Tail Probabilities, I, Journal of the American Statistical Association, 6(4), [9] Schader, M. and Schmid, F. (1989). Two Rules of Thumb for the Aroximation of the Binomial Distribution by the Normal Distribution, The American Statistician, 4(1), - 4. [10] Wang J. and Wei C. (008). The asymtotical Behavior of Probability Measures for the Fluctuations of Stochastic Models, WSEAS Transactions On Mathematics, Issue 5, Volume 7, 7-8. ISBN: ISSN: