Chiang Mai J. Sci. 2016; 43(3) : Contributed Paper

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1 Chiang Mai J Sci 06; 43(3) : Contributed Paper Upper Bounds of Generalized p-values for Testing the Coefficients of Variation of Lognormal Distributions Rada Somkhuean, Suparat Niwitpong and Sa-aat Niwitpong* Department of Applied Statistics, Faculty of Applied Science, King Mongkut s University of Technology North Bangkok, Bangkok 0800, Thailand *Author for correspondence; snw@kmutnbacth Received: 3 February 04 Accepted: 4 April 04 ABSTRACT The coefficient of variation is defined as the ratio of the standard deviation to the mean It has been widely used to compare the variability between groups of observations Recently, there have been many researchers proposed tests and confidence intervals for the coefficients of variation of data from normal distribution However, this paper presents the new generalized p-values for testing the single coefficient of variation and the difference between coefficients of variation for lognormal distribution using the generalized p-value developed by Tsui & Weerahandi [6] Moreover, for each generalized p-value, we derive a closed form expression of its upper bound Numerical computations illustrate the theoretical results Keyword : upper bound, generalized p-value, coefficient of variation INTRODUCTION Statistical inference for lognormal distributions has been widely used in economics, healthcare, and biological and environmental research (Zou et al [], Aitchison et al [], Crow et al [3], Krishnamoorth & Mathew [4]) In particular, inference based upon the difference between lognormal means has been presented by Abdollahnezhad et al [5] The confidence intervals and hypotheses testing for parameters of means of two independent lognormal distributions, have been presented by Krishnamoorthy & Mathew [4], Chen & Zhou [6], Gill [7], and Gupta & Li [8] Pardo & Pardo [9] presented new statistics, based upon Rényi's divergence, in order to test the equality of coefficient variations Forkman [0] proposed a new test statistic F, which is approximately F test Curto & Pinto [] derived a hypothesis test, in order to compare the coefficients of variation based upon asymptotically normal distribution, using the bootstrap method Amirin [] used the bootstrap method to test for the comparisons of coefficients of variation under assumed normal populations Bai et al [3] used the mean-variance ratio (MVR) test in order to test the ratio of coefficients of variation in small samples Niwitpong [4], and Buntao & Niwitpong [5], proposed confidence intervals for the coefficients of variation of lognormal distributions with restricted parameter spaces In this paper, the generalized p-value approach, based upon Tsui

2 67 Chiang Mai J Sci 06; 43(3) and Weerahandi s approach [6], is used to find new generalized p-values, and that is in order to test: ) the single coefficient of variation and ) the difference between coefficients of variation in lognormal distributions Further, we applied a problem described by Tang & Tsui [7], in order to construct the upper boundaries of the mentioned generalized p-values In section, we provide the paths to some basic steps used to construct the generalized p- values The processes of deriving the upper boundaries, as previously mentioned, are then presented in section 3 Numerical results are displayed in section 4 Finally, in section 5, our conclusion is drawn BASIC CONCEPT A GENERALIZED P-VALUES AND PROBABILITY DENSITY FUNCTION FOR LOGNORMAL DISTRIBUTION In this section, we reviewed the probability density function for lognormal distribution, and their mean, variance and coefficient of variation Moreover, we reviewed the generalized p-value to test the hypotheses of the single coefficient of variation and the difference between the coefficient of variation for lognormal distributions Probability Density Function for Longormal Distribution Let X be a random variable having a lognormal distribution, and let and denote Z ln X ~ N, The the mean and variance of ln X respectively, so that probability density function of X is lnx exp ; 0 if x f x,, x () 0 ; otherwise, where and denote the mean and variance of lnx respectively, so that Z ln X ~ N, In particular, the mean, variance and the coefficient of variation for lognormal distribution are given by, see eg, Niwitpong [4], E X EexpZ exp Var( X ) exp exp CV exp, where CV denotes the coefficient of variation of X which is computed from Var X / EX Generalized P-values The concept of the generalized p-value has been introduced by Tsui & Weerahandi [6] and Weerahandi [8] We first briefly review some basic steps to construct the generalized p- value for testing hypothesis problems Let X be a random variable with a density function f X, where,, is the parameter of interest, and is a nuisance parameter

3 Chiang Mai J Sci 06; 43(3) 673 Suppose we have the following hypothesis to test: H : vs H : where is a specified quantity Let x be a particular observed sample The generalized test 0 variable, TXx,,, satisfies the following three conditions: (i) For fixed x and,, the distribution,, parameter (ii) t Txx,, obs is free from any unknown parameter T Xx is free from the nuisance (iii) For fixed x and, if TXx,, is either stochastically increasing or decreasing in for any given t Under the above conditions, if TXx,, is a stochastically increasing test variable, then the subset of space an extreme region C consisting of all the samples X that are as extreme as the observed x Usually, C is of the form C X: TXx,, 0 Given the observed sample x, the generalized p-value is defined as p x sup P X C sup P X : T X, x, 0, H0 H0 for further details and for several applications based on the generalized p-value, we refer to the book by Weerahandi [8] Moreover, Tsui & Weerahandi [6] used the generalized p-value px for the Behrens- Fisher problem of testing the difference of two independent normal distribution means with possibly unequal variances Later, Tang & Tusi [7] extended the works of Weerahandi [8], r of the generalized Gamage & Weerahadi [9] to derived the formula of the upper bound p-value px which is in the form of, see also eg, Kabaila & Lloyd [0], P px r r In this paper, we also extend Tang & Tsui [7] s work to find upper bounds of generalized p-values p x for the testing hypotheses of the coefficients of variation for lognormal distributions of the single coefficient of variation and the difference between coefficients of variation 3 UPPER BOUNDS OF GENERALIZED P-VALUES FOR COEFFICIENTS OF VARIATION FOR LOGNORMAL DISTRIBUTIONS Let X X X X and Y Y Y Y,,, n,,, be two independent samples from m a lognormal distributions, and let Z ln X ~ N,, G ln Y ~ N, which the coefficients of variation are k and x k where y, Var X Var Y k, k, E X exp, E Y exp x y E X E Y Var X exp exp, Var Y exp exp and the interest parameters are k x exp and k k x y exp exp

4 674 Chiang Mai J Sci 06; 43(3) for test the null hypotheses, H : vs H : and H : vs H :, 0 0 a a 0 the sufficient statistics of involving in this problem are ZGS,,, S, where n m n m Z Z G G S Z Z S G G n m n m,,, i i i j i j i j The distributions of the underlying random variables are given by n S m S Z ~ N,, ~,, ~, ~, G N n m n m and the randon variables ZGS,, and S are all independent We denote D X, X,, X n and Q X, X,, X, Y, Y,, Y, q x x x y y y (), d x, x,, x,,,,,,,, n n m n m where d, q are respectively the vector of observed sample vectors of D and Q Let z, g, s, s be the observed value of the sufficient statisticzgs,,, S Following Tang & Tsui [7], the repeated sampling, z, g, s, s follows the same probability distributions as () CASE I: The Hypothesis Testing of Single Coefficient of Variation We interested the hypothesis testing of the single coefficient of variation for lognornal distribution: H : vs H : (3) 0 0 a 0 The parameter of the single coefficient of variation for lognormal distribution is A generalized test variable for is exp k x T Xx,, exp s exp S n s exp, where ~ n U It is easy to see that,, The generalized p-value, pd is defined, under the null hypothesis H, to be 0 U (4) T Xx in (4) satisfies conditions (i)-(iii) in section sup,,,,,,,, pd PT Xx T xx PT Xx T xx (5) H0

5 Chiang Mai J Sci 06; 43(3) 675 Following (5) the generalized p-value for (3) can be defined as,,,, pd PT Xx T xx ns P exp 0 U n s P exp 0 U ns P ln 0 U ns P U ln 0 n s P U ln 0 n s EU n, (6) ln 0 n S where E U is an expectation operator with respect to U ~ n and n is a cdf of the chi-square distribution with n degrees of freedom Theorem Let f u n u then Proof: We have f u f hu, and u of u Hence h u u f u is a convex function of u be the probability density function f hu f hu huh u huh u huh u since u 0 then hu 0 Hence hu 0 and hu 0 Moreover Hence f hu 0, and h u h u 0 Theorem The upper bound of where 0 r 05, k Proof: From (6) Proof: From (6) ln 0 f u is convex in u pd in (6) takes the form pd k r n n and the inverse function of p d E U n n s ln n 0 U EU n 0 ln U ln 0 E, k Un k n

6 676 Chiang Mai J Sci 06; 43(3) For any 05 r and pd r Hence, by theorem, f U E U n By Jensen s inequality pd E fu feu fn For 0 r 05, we have p d : p d r P d : p d r d d P p d r U n P r n k n P n r k P n r k n s s Pn rk n s PU n rk s E n n rk by Jensen s inequality k r where k n n U is convex in U k r n n E k n n s k, p ln 0 and e 0 (7) CASE II: The Hypothesis Testing of Diference Between Coefficients of Variation The hypothesis testing of the difference between coefficients of variation for lognormal distributions is set in the form: H : vs H : (8) 0 0 a 0 The parameter of the difference between coefficients of variation for lognormal distributions is A generalized test variable for is k k exp exp x y T XYx,,, y, exp exp s s exp exp S S n s m s U V exp exp, where U ~, V ~ (9) n m d

7 Chiang Mai J Sci 06; 43(3) 677 It is easy to see that,,,, T XYx y in (9) satisfies conditions (i)-(iii) in section Without loss of generality, suppose 0 0 under the null hypothesis H, to be 0 The generalized p-value, sup,,,,,,,, p q P T X Y x y T x y x y H0,,,,,,,, p q is defined, P T X Y x y T x y x y (0) Following (0) the generalized p-value for (8) can be defined as p q P T X, Y, x, y, T x, y, x, y, where n s m s P exp exp 0 U V n s m s P exp exp U V n s m s P exp exp U V P U V U n s F n s m s P V m s P E n n s m m s s F n, m s, () E F is an expectation operator with respect to F and n, m n and m degrees of freedom distribution with Theorem 3 If g f f n, m then g f is a convex function of f Proof: We have g f gh f, and f function of f Hence h f f is a cdf of the F- be the probability density g h f g h f h f h f h f h f h f h f since f 0 then h f 0 Hence h f 0 and h f 0 Moreover h f h f 0 Hence f h f 0, and g f is convex in f

8 678 Chiang Mai J Sci 06; 43(3) Theorem 4 The upper bound of where 0 r 05, k m m 3 n, m inverse function of Proof: From () For any 05 p q pq in () takes the form pq k r E E m, n n, m, is distribution of F and m, n s F n, m s F F n, m E Fc c,, F n m n, m the r and pq r Hence by theorem 3, such that E gf gef p q E Fc ce F F n, m c m m 3 For 0 r 05, we have p q : p q r P q : p q r q n, m q n, m F p P p q r q c m P n, m m 3 P c m m 3 n, m r r r 3 3 s s m Pc n, m m m PF r n, m m m 3 s E r m, n n, m m s m 3 s r E m, n n, m m s m 3 r m, n n, m m k m, n n, m r () where k m 3 m

9 Chiang Mai J Sci 06; 43(3) NUMERICAL RESULTS In this section, we used functions written in the R program [] to compute the values of the upper bounds of the generalized p-values proposed in Theorems and 4 For given values of n, m,, k 0, k, r, we computed the upper bounds of pd and pq, by using the results from Theorems, 4, shown in Tables - As we can see in these tables, all results of the upper bounds of the generalized p-values proposed in Theorems, 4 are depend mainly on a variety of values of n, m,, k, 0 i i,, and r As a result, these upper bounds confirm our proof in Theorems, 4 CONCLUSION In this paper, we proposed two new generalized p-values for testing the hypotheses of a single coefficient of variation and the difference between coefficients of variation for lognormal distributions We also provied new upper bounds for our proposed generalized p- values We note here that the result for these findings for the hypothses case and case, were analogous to the upper bound of the generalized p-value for the Behrens-Fisher problem proposed by Tang & Tsui [7] Numerical results shown in Tables and, confirmed our findings in Theorems and 4 From Figures and, we also found that the proposed upper bounds for the hypothses case and case are increasing up on the parameter values of k and k ACKNOWLEDGEMENTS The authors would like to thank the editor and the referees for their constructive comments, which have led to substantial improvements in this paper The authors also thank the partial financial support from King Mongkut s University of Technology North Bangkok The third author is grateful to the grant number KMUTNB-GOV from King Mongkut s University of Technology North Bangkok REFERENCES [] Zou GY, Huo CY and Taleban J, Environmetrics, 009; 0: 7-80 DOI 000/env99 [] Aitchison J and Brown IAC, The Lognormal Distribution, Cambridge, UK, 957 [3] Crow EL and Shimizu K, Lognormal Distributions: Theory and Application, Environmetrics, New York, USA, 988 [4] Krishnamoorthy K and Mathew T, J Stat Plan Infer, 003; 5: 03- DOI 006/S (0)0053- [5] Abdollahnezhad K, Babanezhad M and Jafari AA, J Stat Econ Meth, 0; : 5-3 [6] Chen YH and Zhou XH, Stat Med, 006; 5: DOI 000/sim504 [7] Gill PS, Biometrics, 004; 60(): DOI 0/j X x [8] Gupta RC and Li X, Comput Stat Data Anal, 006; 50: DOI 006/jcsda [9] Pardo MC and Pardo JA, J Comput Appl Math, 000; 6: DOI 006/S (99)003-X [0] Forkman FJ, Coefficients of Variation an Approximate F-test, PhD Thesis, Saint Louis University, USA, 005 [] Curto JD and Pinto JC, J Appl Stat, 009; 36: -3 DOI 0080/ [] Amiri S, On the Application of Bootstap Coeffication of Variation, Contingency Table, Information Theory and Ranked Set Sampling, PhD Thesis, Uppsala University, Sweden, 0

10 680 Chiang Mai J Sci 06; 43(3) [3] Bai Z, Wang K and Wong W-K, Stat Prob Lett, 0; 8: DOI 006/jspl00035 [4] Niwitpong S, Appl Math Sci, 03; 7: [5] Buntao N and Niwitpong S, Appl Math Sci, 0; 6: [6] Tsui K-W and Weerahandi S, J Am Stat Assoc, 989; 84: DOI 0080/ [7] Tang S and Tsui K-W, Stat Prob Lett, 007; 77: -8 DOI 006/jspl [8] Weerahandi S, Exact Statistical Methods for Data Analysis, Springer, New York, 995 [9] Gamage J and Weerahandi S, Comm Stat, 998; 7(3): DOI 0080/ [0] Kabaila P and Lloyd CJ, J Aus Stat Assoc, 997; 39(): DOI 0/j467-84X997 tb00535x [] The R Development Core Team, An introduction to R, Vienna, 00 Table Upper bounds of generalized p-values for the hypothesis case when σ = n θ 0 k r=00 r=00 r=005 r=00 r= Table Upper bounds of generalized p-values for the hypothesis case n, m k r=00 r=00 r=005 r=00 r=00 5, , , , , ,

11 Chiang Mai J Sci 06; 43(3) 68 r = 00 k=3 k=33 k=34 r = 00 k=3 k=33 k=34 r = 005 k=3 k=33 k=34 r = 00 r = 00 k=3 k=33 k=34 k=3 k=33 k=34 Figure Upper bounds of generalized p-values for the hypothesis case r = 00 k=3 k=33 3 k=34 4 r = 00 k=3 k=3 k=3 r = 005 k=3 k=33 k=34 r = 00 k=3 k=33 k=34 r = 00 k=3 k=33 k=34 Figure Upper bounds of generalized p-values for the hypothesis case