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 [1]. 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 [2] to analyze the uncertainty of fault trees. Gong and Li [3] assessed the strength of ceramics by using the coefficient of variation. Faber and Korn [4] 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 [5]. Billings et al. [6] 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 [7]. Furthermore, Pyne et al. [8] 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 [9], Thompson [10], and Steiger [11]). There are several approaches available for constructing the confidence interval for κ. McKay [12] proposed a confidence interval for κ based on the chi-square distribution; this confidence interval works well when κ<0.33 [13–17]. Later, Vangel [18] 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 [19] modified McKay’s confidence interval by replacing the sample coefficient of variation with the maximum likelihood estimator for a normal distribution. Sharma and Krishna [20] 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 [21] 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 [22].
Mahmoudvand and Hassani [23] 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. [24] and Verrill [25]. Buntao and Niwitpong [26] also introduced an interval estimating the difference of the coefficient of variation for lognormal and delta-lognormal distributions. Curto and Pinto [27] constructed the confidence interval for κ when random variables are not independently and identically distributed. Recent work of Gulhar et al. [28] 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-1∑i=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-1∑i=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=1n∑i=1n(Xi-μ0)2=σ2n∑i=1nZi2,
where Zi=(Xi-μ0/σ)~N(0,1).
Thus,
(5)nS02σ2~χn2.
Let S2=(n-1)-1∑i=1n(Xi-X-)2 and S*2=n-1∑i=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+1σμ0=cn+1κ0,(11)var(κ^0)=(1-cn+12μ02)σ2=(1-cn+12)κ02.
Note that cn+1→1 as n→∞. Therefore, it follows that
(12)limn→∞E(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)limn→∞var(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+12→1/(2n-1). The asymptotic expansion of the gamma function ratio is [32]
(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+12⟶12n-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+12⟶N(0,1).
Therefore, the 100(1-α)% confidence interval for κ0 based on (22) is
(23)κ^0cn+1+z1-α/21-cn+12≤κ0≤κ^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(a≤nS02σ2≤b)=1-α,
we can write
(26)P(κ^0nb≤κ0≤κ^0na)=1-α.
Thus, the 100(1-α)% confidence interval for κ0 based on the pivotal quantity Q is
(27)κ^0nb≤κ0≤κ^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 [34–36].
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^0nχn,1-α/22≤κ0≤κ^0nχn,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 [34–36] 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^=#(L≤κ≤U)M,Length^=∑j=1M(Uj-Lj)M,
where #(L≤κ≤U) 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 [7], McKay’s [12], and Vangel’s [18].
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 [25]:
(36)κ^0≥nχn,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^0≥nχn,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
κ0
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
κ0
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. [37] (cited in Ledolter and Hogg [38], 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.
Shapiro-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^^0≃0.9091. 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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