JPS Journal of Probability and Statistics 1687-9538 1687-952X Hindawi Publishing Corporation 324940 10.1155/2013/324940 324940 Research Article Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean Panichkitkosolkul Wararit Chow Shein-chung Department of Mathematics and Statistics Faculty of Science and Technology Thammasat University Phathum Thani 12120 Thailand tu.ac.th 2013 21 11 2013 2013 23 07 2013 25 09 2013 2013 Copyright © 2013 Wararit Panichkitkosolkul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents three confidence intervals for the coefficient of variation in a normal distribution with a known population mean. One of the proposed confidence intervals is based on the normal approximation. The other proposed confidence intervals are the shortest-length confidence interval and the equal-tailed confidence interval. A Monte Carlo simulation study was conducted to compare the performance of the proposed confidence intervals with the existing confidence intervals. Simulation results have shown that all three proposed confidence intervals perform well in terms of coverage probability and expected length.

1. Introduction

The coefficient of variation of a distribution is a dimensionless number that quantifies the degree of variability relative to the mean . It is a statistical measure for comparing the dispersion of several variables obtained by different units. The population coefficient of variation is defined as a ratio of the population standard deviation (σ) to the population mean (μ) given by κ=σ/μ. The typical sample estimate of κ is given as (1)κ^=SX-, where S is the sample standard deviation, the square root of the unbiased estimator of population variance, and X- is the sample mean.

The coefficient of variation has been widely used in many areas such as science, medicine, engineering, economics, and others. For example, the coefficient of variation has also been employed by Ahn  to analyze the uncertainty of fault trees. Gong and Li  assessed the strength of ceramics by using the coefficient of variation. Faber and Korn  applied the coefficient of variation as a way of including a measure of variation in the mean synaptic response of the central nervous system. The coefficient of variation has also been used to assess the homogeneity of bone test samples to help determine the effect of external treatments on the properties of bones . Billings et al.  used the coefficient of variation to study the impact of socioeconomic status on hospital use in New York City. In finance and actuarial science, the coefficient of variation can be used as a measure of relative risk and a test of the equality of the coefficients of variation for two stocks . Furthermore, Pyne et al.  studied the variability of the competitive performance of Olympic swimmers by using the coefficient of variation.

Although the point estimator of the population coefficient of variation shown in (1) can be a useful statistical measure, its confidence interval is more useful than the point estimator. A confidence interval provides much more information about the population characteristic of interest than does a point estimate (e.g., Smithson , Thompson , and Steiger ). There are several approaches available for constructing the confidence interval for κ. McKay  proposed a confidence interval for κ based on the chi-square distribution; this confidence interval works well when κ<0.33 . Later, Vangel  proposed a new confidence interval for κ, which is called a modified McKay’s confidence interval. His confidence interval is based on an analysis of the distribution of a class of approximate pivotal quantities for the normal coefficient of variation. In addition, modified McKay’s confidence interval is closely related to McKay’s confidence interval but it is usually more accurate and nearly exact under normality. Panichkitkosolkul  modified McKay’s confidence interval by replacing the sample coefficient of variation with the maximum likelihood estimator for a normal distribution. Sharma and Krishna  introduced the asymptotic distribution and confidence interval of the reciprocal of the coefficient of variation which does not require any assumptions about the population distribution to be made. Miller  discussed the approximate distribution of κ^ and proposed the approximate confidence interval for κ in the case of a normal distribution. The performance of many confidence intervals for κ obtained by McKay’s, Miller’s, and Sharma-Krishna’s methods was compared under the same simulation conditions by Ng .

Mahmoudvand and Hassani  proposed an approximately unbiased estimator for κ in a normal distribution and also used this estimator for constructing two approximate confidence intervals for the coefficient of variation. The confidence intervals for κ in normal and lognormal were proposed by Koopmans et al.  and Verrill . Buntao and Niwitpong  also introduced an interval estimating the difference of the coefficient of variation for lognormal and delta-lognormal distributions. Curto and Pinto  constructed the confidence interval for κ when random variables are not independently and identically distributed. Recent work of Gulhar et al.  has compared several confidence intervals for estimating the population coefficient of variation based on parametric, nonparametric, and modified methods.

However, the population mean may be known in several phenomena. The confidence intervals of the aforementioned authors have not been used for estimating the population coefficient of variation for the normal distribution with a known population mean. Therefore, our main aim in this paper is to propose three confidence intervals for κ in a normal distribution with a known population mean.

The organization of this paper is as follows. In Section 2, the theoretical background of the proposed confidence intervals is discussed. The investigations of the performance of the proposed confidence interval through a Monte Carlo simulation study are presented in Section 3. A comparison of the confidence intervals is also illustrated by using an empirical application in Section 4. Conclusions are provided in the final section.

2. Theoretical Results

In this section, the mean and variance of the estimator of the coefficient of variation in a normal distribution with a known population mean are considered. In addition, we will introduce an unbiased estimator for the coefficient of variation, obtain its variance, and finally construct three confidence intervals: normal approximation, shortest-length, and equal-tailed confidence intervals.

If the population mean is known to be μ0, then the population coefficient of variation is given by κ0=σ/μ0. The sample estimate of κ0 is (2)κ^0=S0μ0, where S02=n-1i=1n(Xi-μ0)2. To find the expectation of (2), we have to prove the following lemma.

Lemma 1.

Let X1,X2,,Xn be a random sample from normal distribution with known mean μ0 and variance σ2 and let S02=n-1i=1n(Xi-μ0)2. Then (3)E(S0)=cn+1σ,var(S0)=(1-cn+12)σ2, where cn+1=2/n(Γ((n+1)/2)/Γ(n/2)).

Proof of Lemma <xref ref-type="statement" rid="lem1">1</xref>.

By definition, (4)S02=1ni=1n(Xi-μ0)2=σ2ni=1nZi2, where Zi=(Xi-μ0/σ)~N(0,1).

Thus, (5)nS02σ2~χn2.

Let S2=(n-1)-1i=1n(Xi-X-)2 and S*2=n-1i=1n+1(Xi-X-)2. From Theorem B of Rice [29, page 197], the distribution of (n-1)S2/σ2 is central chi-square distribution with n-1 degrees of freedom. Similarly, the distribution of nS*2/σ2 is central chi-square distribution with n degrees of freedom; that is, (6)nS*2σ2~χn2.

One can see that [30, page 181] (7)E(S)=cnσ, where cn=2/(n-1)(Γ(n/2)/Γ((n-1)/2)).

Similarly, (8)E(S*)=cn+1σ, where cn+1=2/n(Γ((n+1)/2)/Γ(n/2)).

Equations (5) and (6) are equivalent. Thus, we obtain E(S0)=E(S*)=cn+1σ. Next, we will find the variance of S0: (9)var(S0)=E(S02)-[E(S0)]2=σ2-cn+12σ2=(1-cn+12)σ2.

By using Lemma 1, we can show that the mean and variance of κ^0 are (10)E(κ^0)=cn+1  σμ0=cn+1κ0,(11)var(κ^0)=(1-cn+12μ02)σ2=(1-cn+12)κ02. Note that cn+11 as n. Therefore, it follows that (12)limnE(κ^0)=  κ0. It means that κ^0 is asymptotically unbiased and asymptotically consistent for κ0. From (10), the unbiased estimator of κ0 is (13)κ^^0=κ^0cn+1. Using Lemma 1, the mean and variance of κ^^0 are given by (14)E(κ^^0)=E(κ^0cn+1)=κ0,(15)var(κ^^0)=var(κ^0cn+1)=1cn+12var(S0μ0)=1cn+12μ02(1-cn+12)σ2=(1-cn+12cn+12)κ02. Thus, (16)limnvar(κ^^0)=0. Hence, κ^^0 is also asymptotically consistent for κ0. Next, we examine the accuracy of κ^^0 from another point view. Let us first consider the following theorem.

Theorem 2.

Let X1,X2,,Xn be a random sample from a probability density function f(x), which has unknown parameter θ. If θ^ is an unbiased estimator of θ, it can be shown under very general conditions that the variance of θ^ must satisfy the inequality (17)var(θ^)1nE(-2/θ2lnf(x))=1nI(θ), where I(θ) is the Fisher information. This is known as the Cramér-Rao inequality. If var(θ^)=1/(nI(θ)), the estimator θ^ is said to be efficient.

Proof of Theorem <xref ref-type="statement" rid="thm1">2</xref>.

See [31, pages 377–379].

By setting θ=κ0=σ/μ0 in Theorem 2, it is easy to show that (18)var(κ˘0)κ022n, where κ˘0 is any unbiased estimator of κ0. This means that the variance for the efficient estimator of κ0 is κ02/2n.

From (15), we will show that (1-cn+12)/cn+121/(2n-1). The asymptotic expansion of the gamma function ratio is  (19)Γ(j+(1/2))Γ(j)=j(1-18j+1128j2+). Now, if j=n/2 in (19), we have (20)cn+1=2nΓ((n+1)/2)Γ(n/2)=2n[n2(1-14n+132n2+)]=1-14n+ο(1n3/2). Thus, we obtain (21)cn+12=1-12n+ο(1n2),1-cn+12cn+1212n-1. Therefore, var(κ^^0)κ02/(2n-1). This means that κ^^0 is asymptotically efficient (see (18)). In the following section, three confidence intervals for κ0 are proposed.

2.1. Normal Approximation Confidence Interval

Using the normal approximate, we have (22)z=κ^^0-κ0var(κ^^0)=κ^0/cn+1-κ0(1-cn+12)κ02/cn+12=κ^0-cn+1κ0κ01-cn+12N(0,1). Therefore, the 100(1-α)% confidence interval for κ0 based on (22) is (23)κ^0cn+1+z1-α/21-cn+12κ0κ^0cn+1-z1-α/21-cn+12, where z1-α/2 is the 100(1-α/2) percentile of the standard normal distribution.

2.2. Shortest-Length Confidence Interval

A pivotal quantity for σ2 is (24)Q=nS02σ2~χn2. Converting the statement (25)P(anS02σ2b)=1-α, we can write (26)P(κ^0nbκ0κ^0na)=1-α. Thus, the 100(1-α)% confidence interval for κ0 based on the pivotal quantity Q is (27)κ^0nbκ0κ^0na, where a,b>0, a<b, and the length of confidence interval for κ0 is defined as (28)L=κ^0n(1a-1b). In order to find the shortest-length confidence interval for κ0, the following problem has to be solved: (29)goal:mina,bκ^0n(1a-1b)constraint:abfQ(q)dq=1-α, where fQ is the probability density function of central chi-square distribution with n degrees of freedom. From Casella and Berger [33, pages 443-444], the 100(1-α)% shortest-length confidence interval for κ0 based on the pivotal quantity Q is determined by the value of a and b satisfying (30)a3/2fQ(a)=b3/2fQ(b),abfQ(q)dq=1-α.

Table 1 is constructed for the numerical solutions of these equations by using the R statistical software .

The values of a and b for the shortest-length confidence interval for κ0.

df Confidence levels
0.90 0.95 0.99
a b a b a b
2 0.2065 12.5208 0.1015 15.1194 0.0200 20.8264
3 0.5654 13.1532 0.3449 15.5897 0.1140 20.9856
4 1.0200 14.1800 0.6918 16.5735 0.2937 21.8371
5 1.5352 15.3498 1.1092 17.7432 0.5461 22.9867
6 2.0930 16.5807 1.5776 18.9954 0.8567 24.2618
7 2.6828 17.8391 2.0851 20.2863 1.2143 25.6017
8 3.2981 19.1099 2.6235 21.5953 1.6107 26.9749
9 3.9343 20.3848 3.1874 22.9118 2.0394 28.3643
10 4.5883 21.6598 3.7729 24.2303 2.4958 29.7602
11 5.2573 22.9325 4.3768 25.5476 2.9760 31.1580
12 5.9397 24.2016 4.9967 26.8618 3.4771 32.5543
13 6.6337 25.4666 5.6308 28.1717 3.9968 33.9474
14 7.3382 26.7269 6.2776 29.4769 4.5329 35.3358
15 8.0521 27.9825 6.9357 30.7770 5.0840 36.7192
16 8.7745 29.2334 7.6042 32.0720 5.6487 38.0968
17 9.5047 30.4796 8.2820 33.3619 6.2256 39.4688
18 10.2421 31.7212 8.9685 34.6467 6.8139 40.8347
19 10.9861 32.9585 9.6629 35.9266 7.4126 42.1952
20 11.7362 34.1915 10.3647 37.2016 8.0209 43.5498
21 12.4919 35.4205 11.0733 38.4720 8.6383 44.8989
22 13.2530 36.6455 11.7882 39.7379 9.2640 46.2426
23 14.0191 37.8668 12.5092 40.9995 9.8976 47.5810
24 14.7899 39.0844 13.2357 42.2570 10.5385 48.9144
25 15.5650 40.2986 13.9675 43.5105 11.1864 50.2428
26 16.3443 41.5095 14.7043 44.7601 11.8408 51.5665
27 17.1275 42.7171 15.4458 46.0060 12.5014 52.8856
28 17.9144 43.9217 16.1917 47.2483 13.1678 54.2002
29 18.7049 45.1234 16.9419 48.4872 13.8397 55.5107
30 19.4987 46.3222 17.6961 49.7229 14.5170 56.8169
40 27.5919 58.1755 25.4233 61.9217 21.5331 69.6808
50 35.9012 69.8342 33.4085 73.8920 28.8879 82.2534
60 44.3661 81.3479 41.5794 85.6914 36.4863 94.6063
70 52.9501 92.7487 49.8923 97.3573 44.2711 106.7867
80 61.6290 104.0584 58.3183 108.9153 52.2044 118.8272
90 70.3860 115.2925 66.8374 120.3839 60.2597 130.7514
100 79.2086 126.4628 75.4347 131.7767 68.4177 142.5771
150 124.0372 181.6128 119.2737 187.9079 110.3262 200.6194
200 169.6646 235.9748 164.0642 243.1025 153.4834 257.4375
250 215.8057 289.8273 209.4667 297.6910 197.4440 313.4620
300 262.3132 343.3155 255.3057 351.8461 241.9776 368.9185
2.3. Equal-Tailed Confidence Interval

The 100(1-α)% equal-tailed confidence interval for κ0 based on the pivotal quantity Q is (31)κ^0n  χn,1-α/22κ0κ^0n  χn,α/22, where χn,α/22 and χn,1-α/22 are the 100(α/2) and 100(1-α/2) percentiles of the central chi-square distribution with n degrees of freedom, respectively.

3. Simulation Study

A Monte Carlo simulation was conducted using the R statistical software  version 3.0.1 to investigate the estimated coverage probabilities and expected lengths of three proposed confidence intervals and to compare them to the existing confidence intervals. The estimated coverage probability and the expected length (based on M replicates) are given by (32)1-α^=#(LκU)M,Length^=j=1M(Uj-Lj)M, where #(LκU) denotes the number of simulation runs for which the population coefficient of variation κ lies within the confidence interval. The data were generated from a normal distribution with a known population mean μ0=10 and κ0 = 0.05, 0.10, 0.20, 0.33, 0.50, and 0.67 and sample sizes (n) of 5, 10, 15, 25, 50, and 100. The number of simulation runs (M) is equal to 50,000 and the nominal confidence levels 1-α are fixed at 0.90 and 0.95. Three existing confidence intervals are considered, namely, Miller’s , McKay’s , and Vangel’s .

Miller: (33)κ0(κ^0-z1-α/2κ^02n-1(12+κ^02),κ^0+z1-α/2κ^02n-1(12+κ^02)),

McKay: (34)κ0(κ^0[(χn,1-α/22n-1)κ^02+χn,1-α/22n-1]-1/2,κ^0[(χn,α/22n-1)κ^02+χn,α/22n-1]-1/2),

Vangel: (35)κ0(κ^0[(χn,1-α/22+2n-1)κ^02+χn,1-α/22n-1]-1/2,κ^0[(χn,α/22+2n-1)κ^02+χn,α/22n-1]-1/2). The upper McKay’s limit will have to be set to under the following condition : (36)κ^0nχn,α/22(n-1)(n-χn,α/22), and the upper Vangel’s limit will have to be set to under the following condition: (37)κ^0nχn,α/22(n-1)(n-χn,α/22-2). As can be seen from Tables 2 and 3, the three proposed confidence intervals have estimated coverage probabilities close to the nominal confidence level in all cases. On the other hand, the Miller’s, McKay’s, and Vangel’s confidence intervals provide estimated coverage probabilities much different from the nominal confidence level, especially when the population coefficient of variation κ0 is large. In other words, the estimated coverage probabilities of existing confidence intervals tend to be too high. Additionally, the estimated coverage probabilities of existing confidence intervals increase as the values of κ0 get larger (i.e., for 95% McKay’s confidence interval, n=10, 0.9522 for κ0 = 0.05; 0.9539 for κ0 = 0.10; 0.9856 for κ0 = 0.67). However, Figure 1 shows that the estimated coverage probabilities of the three proposed confidence intervals do not increase or decrease according to the values of κ0.

The estimated coverage probabilities and expected lengths of 90% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

n κ 0 Coverage probabilities Expected lengths
Miller McKay Vangel Approx. Shortest Equal-tailed Miller McKay Vangel Approx. Shortest Equal-tailed
5 0.05 0.8499 0.9066 0.8858 0.9016 0.9023 0.8979 0.0555 0.0607 0.0582 0.0741 0.0587 0.0675
0.10 0.8518 0.9069 0.8866 0.9024 0.9008 0.8988 0.1120 0.1237 0.1177 0.1482 0.1174 0.1349
0.20 0.8524 0.9130 0.8960 0.9036 0.8990 0.9006 0.2315 0.2689 0.2457 0.2963 0.2347 0.2696
0.33 0.8572 0.9258 0.9136 0.9038 0.8999 0.9001 0.4099 0.5872 0.4528 0.4895 0.3878 0.4453
0.50 0.8664 0.9430 0.9321 0.9036 0.8994 0.9001 0.6959 1.2123 0.9360 0.7409 0.5869 0.6741
0.67 0.8773 0.9578 0.9428 0.9031 0.9000 0.8992 1.0603 1.5764 1.6394 0.9947 0.7880 0.9050

10 0.05 0.8747 0.9031 0.8870 0.9020 0.8996 0.9006 0.0379 0.0396 0.0382 0.0431 0.0388 0.0416
0.10 0.8792 0.9052 0.8899 0.9024 0.9001 0.9014 0.0765 0.0804 0.0773 0.0864 0.0778 0.0833
0.20 0.8802 0.9135 0.9002 0.9013 0.8993 0.9001 0.1576 0.1686 0.1603 0.1726 0.1553 0.1664
0.33 0.8899 0.9304 0.9202 0.9017 0.9021 0.9015 0.2778 0.3140 0.2893 0.2853 0.2566 0.2750
0.50 0.8999 0.9527 0.9451 0.9007 0.9004 0.8995 0.4709 0.6575 0.5323 0.4329 0.3895 0.4174
0.67 0.9129 0.9694 0.9600 0.9018 0.8999 0.8992 0.7128 1.4205 1.0257 0.5801 0.5218 0.5593

15 0.05 0.8846 0.9010 0.8870 0.9000 0.8989 0.8988 0.0307 0.0316 0.0306 0.0333 0.0311 0.0326
0.10 0.8866 0.9065 0.8925 0.9011 0.9013 0.9001 0.0618 0.0638 0.0617 0.0666 0.0622 0.0652
0.20 0.8913 0.9127 0.9007 0.9006 0.8993 0.8987 0.1271 0.1328 0.1275 0.1330 0.1242 0.1301
0.33 0.9046 0.9308 0.9218 0.9012 0.9022 0.9013 0.2239 0.2418 0.2286 0.2200 0.2054 0.2151
0.50 0.9150 0.9544 0.9477 0.9000 0.9004 0.8991 0.3787 0.4522 0.4087 0.3338 0.3116 0.3264
0.67 0.9280 0.9725 0.9661 0.9010 0.9002 0.8999 0.5713 0.8900 0.7010 0.4466 0.4170 0.4367

25 0.05 0.8933 0.9056 0.8932 0.9029 0.9035 0.9021 0.0236 0.0240 0.0233 0.0247 0.0238 0.0244
0.10 0.8948 0.9054 0.8939 0.9010 0.9014 0.9004 0.0475 0.0485 0.0471 0.0495 0.0475 0.0489
0.20 0.9022 0.9146 0.9042 0.9028 0.9008 0.9021 0.0977 0.1003 0.0971 0.0988 0.0949 0.0976
0.33 0.9144 0.9318 0.9238 0.9005 0.9016 0.9004 0.1716 0.1796 0.1727 0.1630 0.1566 0.1610
0.50 0.9285 0.9548 0.9491 0.8976 0.8977 0.8978 0.2893 0.3207 0.3027 0.2471 0.2374 0.2440
0.67 0.9430 0.9768 0.9722 0.9018 0.8999 0.9008 0.4363 0.5418 0.4930 0.3310 0.3179 0.3268

50 0.05 0.8941 0.8992 0.8905 0.8993 0.8977 0.8989 0.0166 0.0167 0.0164 0.0170 0.0166 0.0168
0.10 0.8996 0.9043 0.8949 0.9004 0.9007 0.8997 0.0334 0.0337 0.0330 0.0339 0.0333 0.0337
0.20 0.9061 0.9118 0.9041 0.8994 0.8989 0.8996 0.0688 0.0697 0.0680 0.0678 0.0665 0.0674
0.33 0.9220 0.9314 0.9253 0.8997 0.8996 0.8994 0.1206 0.1236 0.1204 0.1118 0.1096 0.1112
0.50 0.9436 0.9583 0.9539 0.9009 0.9022 0.9010 0.2031 0.2153 0.2084 0.1695 0.1662 0.1685
0.67 0.9588 0.9801 0.9770 0.9010 0.9009 0.9008 0.3062 0.3460 0.3309 0.2271 0.2226 0.2257

100 0.05 0.8998 0.9026 0.8961 0.9019 0.8902 0.9017 0.0117 0.0117 0.0115 0.0118 0.0113 0.0118
0.10 0.9000 0.9031 0.8959 0.8995 0.8878 0.8992 0.0236 0.0237 0.0233 0.0236 0.0227 0.0236
0.20 0.9110 0.9131 0.9072 0.9011 0.8901 0.9008 0.0485 0.0488 0.0480 0.0472 0.0453 0.0471
0.33 0.9277 0.9329 0.9282 0.9015 0.8900 0.9012 0.0850 0.0863 0.0847 0.0779 0.0748 0.0777
0.50 0.9485 0.9589 0.9561 0.9001 0.8885 0.8998 0.1429 0.1486 0.1455 0.1180 0.1133 0.1177
0.67 0.9678 0.9810 0.9790 0.9020 0.8910 0.9021 0.2157 0.2347 0.2289 0.1582 0.1519 0.1578

The estimated coverage probabilities and expected lengths of 95% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

n κ 0 Coverage probabilities Expected lengths
Miller McKay Vangel Approx. Shortest Equal-tailed Miller McKay Vangel Approx. Shortest Equal-tailed
5 0.05 0.8829 0.9533 0.9440 0.9538 0.9504 0.9511 0.0661 0.0785 0.0762 0.1058 0.0758 0.0870
0.10 0.8827 0.9537 0.9457 0.9549 0.9501 0.9506 0.1333 0.1608 0.1544 0.2113 0.1513 0.1737
0.20 0.8847 0.9578 0.9508 0.9548 0.9500 0.9507 0.2756 0.3630 0.3282 0.4226 0.3026 0.3475
0.33 0.8904 0.9647 0.9599 0.9542 0.9501 0.9501 0.4880 0.8954 0.6498 0.6986 0.5001 0.5743
0.50 0.8934 0.9711 0.9656 0.9537 0.9487 0.9491 0.8276 1.3796 1.4333 1.0561 0.7561 0.8683
0.67 0.9042 0.9795 0.9721 0.9548 0.9495 0.9502 1.2550 1.5758 1.9791 1.4140 1.0124 1.1625

10 0.05 0.9115 0.9522 0.9440 0.9511 0.9502 0.9495 0.0451 0.0490 0.0478 0.0551 0.0480 0.0515
0.10 0.9125 0.9539 0.9460 0.9529 0.9510 0.9505 0.0912 0.0997 0.0968 0.1105 0.0962 0.1031
0.20 0.9156 0.9588 0.9522 0.9521 0.9506 0.9498 0.1881 0.2113 0.2023 0.2209 0.1924 0.2062
0.33 0.9201 0.9663 0.9620 0.9507 0.9499 0.9489 0.3311 0.4052 0.3718 0.3645 0.3174 0.3401
0.50 0.9281 0.9788 0.9751 0.9510 0.9492 0.9500 0.5606 0.9797 0.7415 0.5528 0.4814 0.5159
0.67 0.9372 0.9856 0.9812 0.9504 0.9500 0.9492 0.8470 1.7544 1.5776 0.7398 0.6442 0.6904

15 0.05 0.9244 0.9517 0.9443 0.9507 0.9506 0.9499 0.0366 0.0386 0.0377 0.0415 0.0380 0.0398
0.10 0.9250 0.9523 0.9446 0.9494 0.9501 0.9475 0.0737 0.0781 0.0761 0.0830 0.0760 0.0796
0.20 0.9294 0.9592 0.9537 0.9516 0.9509 0.9507 0.1520 0.1637 0.1584 0.1660 0.1521 0.1593
0.33 0.9324 0.9681 0.9634 0.9502 0.9493 0.9489 0.2669 0.3016 0.2861 0.2737 0.2508 0.2626
0.50 0.9418 0.9811 0.9783 0.9501 0.9490 0.9497 0.4495 0.5917 0.5267 0.4141 0.3794 0.3973
0.67 0.9528 0.9894 0.9862 0.9496 0.9510 0.9493 0.6819 1.3361 1.0306 0.5564 0.5097 0.5338

25 0.05 0.9356 0.9513 0.9458 0.9504 0.9500 0.9505 0.0281 0.0290 0.0284 0.0302 0.0287 0.0295
0.10 0.9338 0.9509 0.9452 0.9491 0.9481 0.9484 0.0566 0.0586 0.0573 0.0604 0.0574 0.0590
0.20 0.9383 0.9580 0.9527 0.9497 0.9495 0.9491 0.1167 0.1219 0.1188 0.1209 0.1149 0.1181
0.33 0.9453 0.9701 0.9664 0.9510 0.9497 0.9505 0.2043 0.2192 0.2118 0.1990 0.1892 0.1945
0.50 0.9575 0.9839 0.9816 0.9520 0.9516 0.9512 0.3456 0.4004 0.3783 0.3024 0.2875 0.2956
0.67 0.9651 0.9920 0.9899 0.9512 0.9503 0.9505 0.5205 0.7126 0.6382 0.4045 0.3845 0.3953

50 0.05 0.9400 0.9504 0.9458 0.9504 0.9488 0.9499 0.0198 0.0201 0.0197 0.0204 0.0199 0.0202
0.10 0.9431 0.9520 0.9473 0.9493 0.9491 0.9492 0.0398 0.0405 0.0398 0.0409 0.0399 0.0405
0.20 0.9479 0.9581 0.9534 0.9496 0.9491 0.9491 0.0819 0.0837 0.0821 0.0817 0.0797 0.0808
0.33 0.9581 0.9695 0.9669 0.9506 0.9510 0.9502 0.1437 0.1490 0.1457 0.1349 0.1316 0.1334
0.50 0.9686 0.9853 0.9834 0.9518 0.9512 0.9514 0.2420 0.2615 0.2538 0.2044 0.1994 0.2022
0.67 0.9776 0.9940 0.9927 0.9507 0.9506 0.9510 0.3652 0.4272 0.4089 0.2740 0.2673 0.2710

100 0.05 0.9454 0.9502 0.9463 0.9496 0.9496 0.9492 0.0139 0.0140 0.0138 0.0141 0.0140 0.0141
0.10 0.9479 0.9528 0.9494 0.9511 0.9502 0.9507 0.0281 0.0283 0.0279 0.0283 0.0280 0.0282
0.20 0.9545 0.9590 0.9554 0.9500 0.9501 0.9501 0.0578 0.0584 0.0576 0.0566 0.0559 0.0563
0.33 0.9621 0.9697 0.9675 0.9493 0.9489 0.9491 0.1013 0.1034 0.1018 0.0934 0.0923 0.0929
0.50 0.9758 0.9844 0.9834 0.9489 0.9486 0.9488 0.1705 0.1789 0.1757 0.1416 0.1399 0.1408
0.67 0.9849 0.9946 0.9939 0.9495 0.9489 0.9492 0.2570 0.2840 0.2775 0.1896 0.1873 0.1886

The estimated coverage probabilities of 90% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

As can be seen from Figure 2, McKay’s and Vangel’s confidence intervals have longer expected lengths than Miller’s and the proposed confidence intervals. While the expected lengths of the three proposed confidence intervals are shorter than the lengths of the existing ones in almost all cases. Additionally, when the sample sizes increase, the lengths become shorter (i.e., for 95% shortest-length confidence interval, κ0 = 0.20, 0.1553 for n=10; 0.0949 for n = 25; 0.0665 for n = 50).

The expected lengths of 90% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

4. An Empirical Application

To illustrate the application of the confidence intervals proposed in the previous section, we used the weights (in grams) of 61 one-month old infants listed as follows:(38)4960513042605160405052404350436039304410461045504460294041604110441048005130367045504290495052103210403035804360436039204050463037564586533628284172425645944866478445205238432053303836591650104344349641484044519243684180410252104382507050443530The data are taken from the study by Ziegler et al.  (cited in Ledolter and Hogg , page 287). The histogram, density plot, Box-and-Whisker plot, and normal quantile-quantile plot are displayed in Figure 3. Algorithm 1 shows the result of the Shapiro-Wilk normality test.

<bold>Algorithm 1: </bold>Shapiro-Wilk test for normality of the weights of 61 one-month old infants.

S hapiro-Wilk normality test

data: weight

W = 0.978, P-value = 0.3383

(a) Histogram, (b) density plot, (c) Box-and-Whisker plot, and (d) normal quantile-quantile plot of the weights of 61 one-month old infants.

As they appear in Figure 3 and Algorithm 1, we find that the data are in excellent agreement with a normal distribution. From past research, we assume that the population mean of the weight of one-month old infants is about 4400 grams. An unbiased estimator of the coefficient of variation is κ^^00.9091. The 95% of proposed and existing confidence intervals for the coefficient of variation are calculated and reported in Table 4. This result confirms that the three confidence intervals proposed in this paper are more efficient than the existing confidence intervals in terms of length of interval.

The 95% confidence intervals for the coefficient of variation of the weight of one-month old infants.

Methods Confidence intervals Lengths
Lower limit Upper limit
Miller 0.1131 0.1635 0.0504
McKay 0.1163 0.1675 0.0512
Vangel 0.1162 0.1674 0.0511
Normal approx. 0.1179 0.1689 0.0510
Shortest 0.1159 0.1659 0.0500
Equal-tailed 0.1175 0.1681 0.0506
5. Conclusions

The coefficient of variation is the ratio of standard deviation to the mean and provides a widely used unit-free measure of dispersion. It can be useful for comparing the variability between groups of observations. Three confidence intervals for the coefficient of variation in a normal distribution with a known population mean have been developed. The proposed confidence intervals are compared with Miller’s, McKay’s, and Vangel’s confidence intervals through a Monte Carlo simulation study. Normal approximation, shortest-length, and equal-tailed confidence intervals are better than the existing confidence intervals in terms of the expected length and the closeness of the estimated coverage probability to the nominal confidence level.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to Professor Dr. Tonghui Wang, Professor Dr. John J. Borkowski, and anonymous referees for their valuable comments and suggestions, which have significantly enhanced the quality and presentation of this paper.

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