Stat The ISI’s Journal for the Rapid Dissemination of Statistics Research
(wileyonlinelibrary.com) DOI: 10.1002/sta4.116
Revisiting inference of coefficient of variation: nuisances parameters Saeid Amiri Received 11 July 2016; Accepted 5 August 2016 This paper studies the problem of the statistical inference of coefficient of variation from a different point of view. While the coefficient of variation is well recognized as being pertinent to the interpretation of variability, its application in statistical inference is nevertheless difficult. This work shows that using the estimates of kurtosis and skewness can draw more accurate inference, even under the condition of known distribution. This work investigates a new type of estimator based on an indirect estimate; it is shown when the estimate of the mean in the denominator approaches zero the plug-in estimate overestimates in instances where the coefficient of variation is larger than one, but the indirect estimate provides a more accurate estimate. Copyright © 2016 John Wiley & Sons, Ltd. Keywords: bootstrap plug-in; coefficient of variation; kurtosis; skewness
1
Introduction
The coefficient of variation (CV) is the ratio of the standard deviation to the mean, sometimes expressed as a percentage. It is a dimensionless measurement of dispersion found to be very useful in many situations, denoted by , and widely calculated and interpreted in the study of dispersion. Inferences concerning the population require assumptions concerning the shape and parameter of the distribution that are difficult to infer and are related to the variance of b being expressed in terms of kurtosis, skewness and itself. The inferences reported in the literature are generally based on a parametric model and focus, in particular, on the of a normal population where the values of kurtosis and skewness are three and zero, respectively. However, the exact distribution of the sample b is difficult to obtain, even for a normal distribution. Therefore, a vast amount of literature has been written on the sample b up to the present in efforts to exploit it on various levels of difficulty. The primary work on the statistical inference of b was carried out in McKay (1932), where it was shown that the approximation of the distribution of a function of the sample b has chi-squared distribution, if < 13 . In a more recent work, Forkman & Verrill (2008) shows that McKay’s chi-squared approximation for the CV is a type II non-central beta distribution. An extensive amount of literature exists on the CV. Among many authors, Ahmed (1995) compared the performance of several b s under normality distribution; Vangel (1996) outlined a method based on an analysis of the distribution of a class of approximate pivotal quantities for the normal CV. Wong & Wu (2002) proposed a simple and accurate method to approximate confidence intervals for the CV for normal, Gamma and Weibull models. Nairy & Rao (2003) studied tests of k normal populations under the condition of the normality of observations. Lehmann & Romano (2005) discussed an exact method for approximating the confidence interval for based on the non-central t distribution. Banik & Kibria (2011) considered several confidence intervals for estimating the population CV using simulation; more recent works on statistical inference under the normality of the data are Mahmoudvand & Hassani Department of Natural and Applied Sciences, University of Wisconsin-Green Bay, Green Bay, WI 54311, USA
Email:
[email protected]
Stat 2016; 5: 234–241
Copyright © 2016 John Wiley & Sons, Ltd
Stat
Nuisances parameters in coefficient of variation (wileyonlinelibrary.com) DOI: 10.1002/sta4.116
The ISI’s Journal for the Rapid Dissemination of Statistics Research
(2009), Forkman (2009), Krishnamoorthy & Lee (2014), Jafari & Kazemi (2013) and Jafari (2015), which study the statistical inference under the normality of population. Under the normality of distribution, the value of kurtosis and skewness are three and zero, respectively; unfortunately, these parameters (which go unnoticed) may affect the inference of ; these values often violate real data sets. Therefore, it makes sense to use estimates in the inference. In this paper, we take another important step in developing an inference of b . Our contribution provides new insight into the inference and estimate of the CV. The variance of the CV depends on certain parameters, and the plug-in estimate is a non-trivial estimate of variance. It is shown that the plug-in estimate, b , overestimates when > 1 and b approaches zero. We proposed an indirect estimate that provides an effective way to avoid this problem in situations when the sample size is small. The remainder of this paper is organized as follows. The next section discusses the asymptotic distribution of the CV, and we argue the importance of kurtosis and skewness in the inference of the CV; it is shown that, even under the assumption of knowing the underlying distribution of data, it is better to use estimates of kurtosis and skewness in the estimate of variance. In Section 3, we study several estimators of the variance V.b /. Section 4 considers the indirect estimation of CV; we found that it provides an accurate estimate when > 1 and b approaches zero.
2
Coefficient of variation
Let b be the plug-in estimate of . We begin by discussing the general asymptotic distribution of b and its inverse. After this discussion, we specify the essential step of considering extra estimates of the parameters in the statistical inferences. The asymptotic distribution of the plug-in estimate is as follows: p 2 2 n1 3 4 C , n.b / AN 0, 4 n
(1)
where n1 ..n 1/K .n 3//, n2 E.X /4 KD , .E.X /2 /2 E.X /3 D , .E.X /2 /3=2
2 D
where K and are the kurtosis and the skewness, respectively. Expression (1) can be proved as follows: 2 NX AN 0, , n and also L
s2 2 ! N.0, 2 /, where 2 D
.n 1/ 4 4 2 ..n 1/K C .n 3// D , n3 n
where K is the kurtosis. Using the Taylor expansion,
Stat 2016; 5: 234–241
235
Copyright © 2016 John Wiley & Sons, Ltd
Stat
S. Amiri (wileyonlinelibrary.com) DOI: 10.1002/sta4.116
s D C NX
The ISI’s Journal for the Rapid Dissemination of Statistics Research
1 1 N 2 2 .s / 2 .X / C Op . 2 n
Denote s2 2 , XN Z2 D p . n Z1 D
N D Using COV.s2 , X/
n1 3 n2
(Cramér, 1945) s 1 Dp NX n
2 1 Z1 2 Z2 C Op , 2 n
where d
Z1 D Z2 N.0, 1/,
n1 . n This well-formulated expression (1) indicates that the inference of CV can be problematic because it includes three parameters; ignoring them may not be appropriate. For instance, the skewness may be affected in the presence of outliers in the data, which are frequent in practical situations. However, the power of parameter can be reduced using logarithmic transformation, and applying the delta method provides the following result: p 2 n 1 (2) n.log.b / log. // AN 0, C 2 , 4 n COV.Z1 , Z2 / D
where E.Y/ D . > 0/ and E.Y 2 / < 1. The next section shows that the inference using log.b / might provide a more accurate inference. Denote the inverse of by D ; in the same way, one can establish the asymptotic distribution of b . Let b be the plug-in estimate of ; p 2 n1 C 1 . n.b / AN 0, 2 4 n Clearly, the variance of b is in terms of itself, the kurtosis and the skewness. This connection is helpful to understand the inherent difficulty in determining the inference of the CV described earlier. Darlington (1970) shows that kurtosis is defined as the expectation of the fourth power of the standardized variable, which simplifies to K D V.Z2 / C 1,
(3)
where Z D .X/=. High kurtosis can arise when (i) the concentration of the probability mass is near (corresponding to a peaked unimodal distribution) or (ii) the concentration of the probability mass is in the tail of the distribution. Formula (3) shows that kurtosis is more strongly affected by the tail behavior of the distribution than by the center of the distribution. High kurtosis has the potential to have outliers in one or both tails of the distribution (Wuensch, 2005); hence, these situations can affect the variance of b , as well. If X N., 2 /, then K D 3; however, in reality, b K > 3 or b K < 3 are more likely to occur. Cramér (1945) shows that 6 E.b K/ D nC1 C 3, which means that if the distribution is normal, then b K < 3 is more likely to be observed. Finding the
Copyright © 2016 John Wiley & Sons, Ltd
236
Stat 2016; 5: 234–241
Stat
Nuisances parameters in coefficient of variation (wileyonlinelibrary.com) DOI: 10.1002/sta4.116
The ISI’s Journal for the Rapid Dissemination of Statistics Research
E.b K/ in other distributions is rather difficult. Cramér (1945, p. 356) proved that the estimation of kurtosis converges with O.n1 /.
3
Study of variance
In this section, the variance of b , which plays an important role in determining inference, is studied. Four ways to estimate the variance are considered: (I) Asymptotic variance (AV): The estimate using the plug-in estimates of the CV, kurtosis and skewness in expression (1) is 2 1 b 2 n1 b 3 4 b V.b / D b b Cb . n 4 n (II) Naive AV (NAV): Most of the existing literature considered holding the normal distribution as the parent distribution where K D 3 and D 0; hence, the estimate is 1 9 2 4 b V.b / D b Cb . n 4 (III) Jackknife variance (JV): Let b D b n D .Y1 , : : : , Yn / be an estimator of an unknown parameter of . Let .j/ D .Y1 , : : : , Yj1 , YjC1 , : : : , Yn / be the given statistics based on the observations ¹Y1 , : : : , Yj1 , YjC1 , : : : , Yn º, j D 1, : : : , n. Tukey (1958) defined the jackknife pseudo-values as Qj D nb .n 1/b .j/ , j D 1, : : : , n. The pseudo-value Qj might be treated as independent and as an identically distributed random variable; then, Qj p has the same variance as n n ; it is estimated using the jackknife method: 2 X X 1 1 Xb b
D Qj NQ D .n 1/ .j/ .k/ .n 1/ n n
n
n
jD1
jD1
kD1
2
!2 , (4)
p N n. Q /= AN.0, 1/, P where NQ D n1 njD1 Qj . The idea of jackknife for estimating the variance of CV is considered in the study of Chakrabarty & Rao (1968). (IV) Bootstrap variance (BV): In order to estimate variance, one should generate several samples and calculate b b D b b , b D 1, : : : , B. The variance can be obtained by 2 1 X b b N . B B
B2 D
bD1
To compare the discussed methods, we use the mean squared error (MSE) of estimator. Denote & D V.b /, and the MSE is defined as MSE.b & / D .E.b & / &/2 C V.b & /.
Stat 2016; 5: 234–241
237
Copyright © 2016 John Wiley & Sons, Ltd
Stat
S. Amiri (wileyonlinelibrary.com) DOI: 10.1002/sta4.116
The ISI’s Journal for the Rapid Dissemination of Statistics Research
Tables I and II study the MSE of the proposed estimator under N.1, 1/ and N.1,
p
1.5/, where 1.
Clearly, the AV provides a better estimate of variance than does NAV, and almost all papers concerning the inference of b used NAV for the estimate of the variance of b . Hence, it is necessary to use the estimate of the kurtosis and the skewness in variance. Among the proposed computer-intensive methods, the JV provides an appropriate one to estimate the variance.
Table I. The MSE of b V.b / and b V.log.b // of the proposed estimators under N.1, 1/. n Method
10
30
50
100
0.0004 3.4261 0.0008 0.0004
0.0000 2.7667 0.0000 0.0000
0.0001 0.0001 0.0002 0.0001
0.0000 0.0000 0.0000 0.0000
b AV NAV BV JV
0.1708 22.886 143799 0.1532
0.0022 4.5048 0.0087 0.0019
AV NAV BV JV
0.0116 0.0089 0.0355 0.0201
0.0004 0.0003 0.0009 0.007
log.b /
MSE, mean squared error; AV, asymptotic variance; NAV, naive AV; BV, bootstrap variance; JV, jackknife variance
Table II. The MSE of b V.b / and b V.log.b // of the proposed p estimators under N.1, 1.5). n 10
30 b
50
100
AV NAV BV JV
408.05 41640 > 106 49174
0.16845 224.08 35988 0.14583
0.02240 107.96 2.2459 0.01983
0.00225 63.730 0.00494 0.00208
AV NAV BV JV
0.42238 0.41340 0.21995 0.41773
0.00336 0.00299 0.01004 0.00712
0.00069 0.00061 0.00170 0.00156
0.00009 0.00008 0.00023 0.00024
Method
log.b /
MSE, mean squared error; AV, asymptotic variance; NAV, naive AV; BV, bootstrap variance; JV, jackknife variance.
Copyright © 2016 John Wiley & Sons, Ltd
238
Stat 2016; 5: 234–241
Stat
Nuisances parameters in coefficient of variation (wileyonlinelibrary.com) DOI: 10.1002/sta4.116
The ISI’s Journal for the Rapid Dissemination of Statistics Research
In Section 2, the following results where given: 2 2 n1 3 4 n.b / AN 0, C , 4 n p 2 n 1 2 C , > 0, n.log.b / log. // AN 0, 4 n
p
V.log.b // is a second power of in comparison to V.b /, which is a power of four; hence, the log transformation decreases variability and might be useful for the inference when > 1. The numerical results presented in Tables I and II demonstrate that determining the inference via log transformation might be more appropriate when is large.
4
Indirect estimate
Statement (1) shows that the variance of the plug-in estimate is very large for > 1. Finding a more robust estimate under > 1 is of interest. Obviously, when approaches zero, the value of increases. Here, we propose a technique that combines a computer-intensive approach and an asymptotic method to solve the problem and refer to this method as indirect estimation. Under > 0, using delta method shows the following: p
N log.// AN.0, 2 /, n.log.Y/
(5)
p N is the square of . Our method provides a new approach to estimate which demonstrates that the variance of n log.Y/ p the CV of populations. Consider k samples with different sizes, nr , r D 1, : : : , k, and denote Yr D nr log.YN r /, r D P 1, : : : , k and YN D 1k krD1 Yr . The CV of these samples can be estimated using the following equation: v u k u 1 X t .Yr YN /2 . k 1 rD1 It seems difficult to use this method for estimating the CV, but it can be carried out using resampling techniques; p let us define .y1 , : : : , yn / D n log.yN /, take a resample of ¹y1 , : : : , yn º and denote it as ¹y1 , : : : , yn º; calculate p .y1 , : : : , yn / D n log.yN / by repeating the procedure B times as follows: b y1 , : : : , yn , b D 1, : : : , B, The CV can be estimated using the following: v u u b Dt
B
2 1 X b y1 , : : : , yn N y1 , : : : , yn . B1
(6)
bD1
Let us refer to this method as indirect estimation. The relation given in (5) is asymptotic; that is, it is held under large n, and it makes sense to run the procedure using n > n, for example, n D 3n or 4n. To compare the plug-in and indirect estimates, we generated 10,000 samples of size n from N.1, 1/, where the parent distribution has a D 1 and calculate b m and b m , m D 1, : : : , 10, 000. Because the is known, the MSE of the estimator is calculated: /. MSE.b / D .E.b / /2 C V.b
Stat 2016; 5: 234–241
239
Copyright © 2016 John Wiley & Sons, Ltd
Stat
S. Amiri (wileyonlinelibrary.com) DOI: 10.1002/sta4.116
The ISI’s Journal for the Rapid Dissemination of Statistics Research
Figure 1. Relative efficiency for the proposed estimators under N.1, 1/ and N.1,
p
1.5/.
The plug-in estimate is considered as a standard method, and the indirect estimate is compared with it using relative accuracy (RA), RA.b ,b / D
MSE.b / , MSE.b /
RA.b ,b / < 1 implies that b outperforms b . We present the result as plots that yield a visual representation. The p plots in Figure 1 show the RA under N.1, 1/ and N.1, 1.5/ for different ns. Under N.1, 1/, the b outperforms the b p for n 18, but for n > 18, the performance is similar. Under N.1, 1.5/, where D 1.5, for n < 40, RA < 1 and the b has very good performance; therefore, the indirect estimate can be nominated as an alternative estimator under small sample sizes and > 1.
5
Conclusion
A considerable amount of research has been carried out to elaborate on the statistical inference of CV. This paper explored the asymptotic distribution of b . It is shown that the variance of b is in terms of kurtosis, skewness and . Kurtosis and skewness are qualitative properties of the distribution; hence, the accurate inference of the b requires knowledge of the underlying distribution. Using the estimates in variance instead of using the values in the assumed distribution leads to more efficient and accurate estimation of CV during the process of statistical inference, which has wide-ranging consequences. In this paper, we have described the performance of a number of different approaches for estimating the variance in the CV, such as the AV, NAV, JV, or BV method. The results show that the NAV most widely used in the literature performs quite poorly when > 1 compared with the others. We show that log transformation where > 0 might improve the accuracy of inference. One of the focuses of this paper is the study of > 1. Under the condition of > 1 and a small sample size, the plug-in estimate overestimates the CV and its variance is high. For the problem to be overcomed, an alternative computer-intensive method is presented. As these results show, the proposed procedure can take great advantage of the asymptotic result and the computer-intensive method to provide a more accurate estimate of b .
References Ahmed, SE (1995), ‘A pooling methodology for coefficient of variation’, Sankhya: The Indian Journal of Statistics, Series B, 57(1), 57–75. Banik, S & Kibria, BG (2011), ‘Estimating the population coefficient of variation by confidence intervals’, Communications in Statistics-Simulation and Computation, 40(8), 1236–1261.
Copyright © 2016 John Wiley & Sons, Ltd
240
Stat 2016; 5: 234–241
Stat
Nuisances parameters in coefficient of variation (wileyonlinelibrary.com) DOI: 10.1002/sta4.116
The ISI’s Journal for the Rapid Dissemination of Statistics Research
Chakrabarty, RP & Rao, J (1968), ‘The bias and stability of jackknife variance estimator in ratio estimation’, Journal of the American Statistical Association, 63, 748–749. Cramér, H (1945), Mathematical Methods of Statistics, Almqvist & Wiksells, Uppsala. Darlington, RB (1970), ‘Is kurtosis really peakedness?’ The American Statistician, 24(2), 19–22. Forkman, J & Verrill, S (2008), ‘The distribution of McKay’s approximation for the coefficient of variation’, Statistics & Probability Letters, 78(1), 10–14. Forkman, J (2009), ‘Estimator and tests for common coefficients of variation in normal distributions’, Communications in Statistics-Theory and Methods, 38(2), 233–251. Jafari, AA & Kazemi, MR (2013), ‘A parametric bootstrap approach for the equality of coefficients of variation’, Computational Statistics, 28(6), 2621–2639. Jafari, AA (2015), ‘Inferences on the coefficients of variation in a multivariate normal population’, Communications in Statistics-Theory and Methods, 44(12), 2630–2643. Krishnamoorthy, K & Lee, M (2014), ‘Improved tests for the equality of normal coefficients of variation’, Computational Statistics, 29(1-2), 215–232. Lehmann, EL & Romano, JR (2005), Testing Statistical Hypothesis, 3rd edn., Springer, New York. Mahmoudvand, R & Hassani, H (2009), ‘Two new confidence intervals for the coefficient of variation in a normal distribution’, Journal of Applied Statistics, 36(4), 429–442. McKay, AT (1932), ‘Distribution of the coefficient of variation and the extended t-distribution’, Journal of the Royal Statistical Society, 95(4), 695–698. Nairy, KS & Rao, KA (2003), ‘Tests of coefficients of variation of normal population’, Communications in StatisticsSimulation and Computation, 32(3), 641–661. Tukey, JW (1958), ‘Bias and confidence in not-quite large samples’, Annals of Mathematical Statistics, 29, 614. Vangel, MG (1996), ‘Confidence intervals for a normal coefficient of variation’, The American Statistician, 50(1), 21–26. Wong, ACM & Wu, J (2002), ‘Small sample asymptotic inference for the coefficient of variation: normal and nonnormal models’, Journal of Statistical Planning and Inference, 104(1), 73–82. Wuensch, KL (2005), ‘Kurtosis’, Encyclopedia of Statistics in Behavioral Science in Everitt, BS & Howell, DC (eds), Wiley, Chichester, UK, pp. 1028–1029.
Stat 2016; 5: 234–241
241
Copyright © 2016 John Wiley & Sons, Ltd
