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Journal of Biometrics & Biostatistics
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Estimator and Tests for Coefficient of Variation in Uniform Distribution

Jamal Hoseini1* and Anwar Mohammadi2

1Clinical Research Developmental Center, Kermanshah University of Medical Sciences, Kermanshah, Iran

2Mathematical Sciences Department, Tarbiat Modares University, Tehran, Iran

*Corresponding Author:
J. Hoseini
Clinical Research Developmental Center
Kermanshah University of Medical Sciences, Kermanshah, Iran
Tel: +98 831 4276299
Fax: +98 831 4276299
E-mail: [email protected]

Received date: May 16, 2012; Accepted date: June 25, 2012; Published date: June 25, 2012

Citation: Hoseini J, Mohammadi A (2012) Estimator and Tests for Coefficient of Variation in Uniform Distribution. J Biomet Biostat 3:149. doi:10.4172/2155-6180.1000149

Copyright: © 2012 Hoseini J, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Coefficient of variation has been widely used as a measure of precision of measurement equipment in particular for medical laboratories. The objective of this paper is to Inference for the coefficient of variation in uniform distributions. By using the two approaches of central limit theorem and generalized variable, confidence interval for coefficient of variation are considered. The coverage properties of the proposed confidence intervals for proposed two methods are assessment by simulation. An example using data of the approved metrological company from Iran Standard and Industrial Researches Organization is provided.

Keywords

Coefficient of variation; Uniform distribution; Generalized variable; Central limit theorem

Introduction

Evaluation and monitoring the precision of medical laboratory equipment are very important. Ignoring the monitoring and control of measurement errors in laboratory equipment may reduce the accuracy of experiment results. In addition, before calibrating the equipment, we must check the precision of measurements [1]. Measurement errors are divided in two parts: systematic and random. Precision depends only on the distribution of random errors [2]. According to ISO/IEC GUIDE 99, precision is the closeness of agreement between indications or measured quantity values obtained by replicate measurements on the same or similar objects under specified conditions and usually expressed numerically by measures of imprecision, such as standard deviation or coefficient of variation (CV) under the specified conditions of measurement [3]. Then, coefficient of variation has been widely used as a measure of precision of measurement equipment in particular for medical laboratories [2-4].

The coefficient of variation of a population is defined as the ratio of population standard deviation to the population mean, and then the coefficient of variation of X is given by:

                                 equation

In order to draw inferences concerning the population CV, it is necessary to can estimate CV by equation and then use the delta method to find the asymptotic distribution of the centeredand scaled estimator. An exact method for inferences about the coefficient of variation from a normal population based on the non-central t distribution is available, but it is computationally cumbersome [5]. The first published approximation method for the construction of a CI for normal coefficient of variation was by McKay and David [6,7]. Vangel [8] provided a modification to McKay’s method which was based on analysis of the distribution of a class of approximate pivotal quantities [8]. Koopmans et al. [9] inferred the confidence intervals of the CV for normal and lognormal distributions [9]. Sharma and Krishna [10] developed the asymptotic distribution of the reciprocal of CV, which is called ICV, without making any assumptions about the distribution. Wong and Wu [11] developed small sample asymptotic method to obtain approximate confidence intervals for the coefficient of variation for both normal and no normal distributions. Pang et al. [12] proposed a simulation based Bayesian approach for constructing the interval estimation of the CV for Weibull, lognormal and gamma distributions [12]. Forkman and Verrill [13] shown that McKay’s approximation is type II noncentral beta distributed. Liang [14] studied the empirical Bayes estimation for the coefficient of variation in shifted exponential distributions. Mahmoudvand and Hassani [15] introduce two new estimators for the population coefficient of variation, in a normal distribution. Banik and Kibria [16] attempt has been made to estimating the population coefficient of variation by confidence intervals. But, the statistical tests in the literature not reported inferences about the coefficient of variation in uniform distribution. Therefore, it is of practical and theoretical importance to develop procedures for interval estimation and hypothesis testing for the coefficient of variation based on independent uniform samples.

In this paper, we introduce an unbiased estimator for the population coefficient of variation, in a uniform distribution.

Estimators and Confidence Intervals for CV

Let X be a random variable having a uniform distribution with a probability density

                                 equation (1)

where equation, σ>0, and I(A) is the indicator function of the event A. Given equation , equation Thus, the CV of a uniform distribution is equation. Our goal is to estimate κ.

LetX1 ,...,Xn be a sample of size n(≥ 2) taken from a uniform distribution.

Let X(1) = min (X1 ,...,Xn )and X(n) = max(X1 ,...,Xn ). Then(Y1 ,Yn ) is complete and sufficient for (μ,σ). Given (μ,σ),

equation and equation are the MLEs of μ and σ, respectively. Note that equation, thus, equation is unbiased estimator of μ. But equation,then equation is biased estimator of σ. To generate a unbiased estimator from equation , must use the statistic of equation. Also, equation and equation are the UMVUEs of μ and σ, respectively.

Let equation and equation. Thus, given (μ,σ), (T, R), has a joint probability density

                                 equation (2)

Note that equation is UMVUE of equation[17].

Theorem 1: Let X(1) = min (X1 ,...,Xn ), X(n) = max(X1 ,...,Xn ) and equation.Then                                  equation(3)

Proof: Let equation , by central limit theorem, denote a standardized normal random variable. Then,

                                 equation

where,

                                 equation

Thus,

                                 equation

                                 equation

                                 equation

Hence

                                 equation

Therefore, the 100(1-α)% confidence interval for τ based on Equation

                                 equation (4)

Theorem 2: Let equation and equation. Then

                                 equation (5)

Is generalized pivotal quantity forequation. If equationthenequation is the generalized test variable for testingequation vs equation

Proof: equation

equation

                                 equation

where U1 ~ Beta(n −1,2), and U2 ~ Beta(1,n),U3 ~ Beta(n,1), then Rθ has a distribution free of μ and σ. Obviously, the observed value of Rθ as T = t and R = risθ. Moreover, the distribution of Rθ does not depend on unknown parameters. Therefore, Rθ is a generalized pivot for θ. The generalized pivot for κ =σ / μ is Rκ=1 / Rθ . Note that T κ satisfy the three conditions above: (1) For fixed t and r, the distribution of T κ is free of nuisance parameters; (2) t κ , the observed value of T κ is 0, and hence is free of any unknown parameters; and (3) T κ is stochastically decreasing in κ .

Therefore, T κ is bonafide generalized pivotal variable for constructing confidence interval of and its quantiles may be used to construct confidence limits for κ =σ / μ .

For a given data set xi where i = 1,2,.,n, the generalized confidence intervals and the generalized P-values can be computed by the following steps:

1. Compute t and r.

2. Generate U1 ~ Beta(n −1, 2), U2 ~ Beta(1,n) , U3 ~ Beta(n,1) , then compute Rκ .

3. Repeat Step 2 a total m times and obtain an array of Rκ’s.

4. Rank this array of Rκ’s from small to large.

The 100αth percentile of Rκ’s, Rκ(α ) is an estimate of the lower bound of the one-sided

100(1 − α) percent a confidence interval and Rκ(α/ 2) Rκ; (1- α/ 2) is a two-sided 100(1−α) per cent confidence interval. The percentage that Rκ’s are less than or equal to 0 κ is a Monte Carlo estimate of the generalized P-value for testingκ=κ0vs κ <κ0 .

Comparison of the Methods

Two approximation methods for the construction of CI for k were highlighted in Sec. 2. If a CI for CV is to be constructed, there is no clear guidance as to which of these methods has to be used. In particular, there is no information regarding relative performance of the CI from central limit theorem and generalized variable methods introduced in this article. In view of this, there is a need to compare these approximation methods in order to assess their performance and select one which is appropriate. The two approximate methods are compared in the construction of a CI for CV using simulation and real data sets.

Simulation study

Simulation studies are done to compare the length of the 95% CI for CV obtained from central limit theorem and generalized variable methods. Data were generated from Uniform distribution with CV= 0.1, 0.3, 0.6, 1 and 3, and sample size n = 10, 25, 50 and 100. All simulations were performed using programs written in the R statistical software [13-15] with the number of simulation runs M = 10,000 at level of significance α= 0.05 and 0.01.Given the role that m (repeat frequent) in function of constructed confidence intervals by generalized variable method, Comparisons are done between confidence interval from central limit theorem method and confidence intervals from generalized variable methods with m = 5, 10, 30, 50, 100.The simulation results are shown in Tables 1-2, in which the following information, viz. The estimated coverage percentages of the confidence intervals from central limit theorem and generalized variable methods for a Uniform distribution at α= 0.05 and 0.01, is presented respectively.

Sample Size
(n)
  Coverage Percentage
Method (I) Method (II)
m=5 m=10 m=30 m=50 m=100
0.1 10 0.1267 0.6512 0.7906 0.9030 0.9205 0.9427
25 0.2493 0.6482 0.7962 0.9031 0.9247 0.9428
50 0.3820 0.6514 0.7939 0.9051 0.9212 0.9427
100 0.5309 0.6466 0.8005 0.9096 0.9294 0.9480
0.3 10 0.4726 0.6664 0.8187 0.9296 0.9462 0.9650
25 0.6820 0.6638 0.8262 0.9299 0.9446 0.9575
50 0.7907 0.6699 0.8231 0.9330 0.9461 0.9590
100 0.8697 0.6761 0.8300 0.9329 0.9475 0.9611
0.6 10 0.7256 0.6683 0.8242 0.9351 0.9445 0.9560
25 0.8427 0.6690 0.8270 0.9316 0.9431 0.9514
50 0.8977 0.6684 0.8323 0.9354 0.9434 0.9506
100 0.9257 0.6742 0.8360 0.9351 0.9436 0.9535
1 10 0.8166 0.6468 0.7991 0.9132 0.9263 0.9447
25 0.8921 0.6538 0.8020 0.9146 0.9243 0.9388
50 0.9209 0.6576 0.8185 0.9175 0.9304 0.9434
100 0.9395 0.6630 0.8197 0.9200 0.9342 0.9461
3 10 0.8636 0.7420 0.8594 0.9215 0.9263 0.9383
25 0.9206 0.6650 0.8083 0.9232 0.9328 0.9485
50 0.9332 0.6445 0.7941 0.9000 0.9122 0.9318
100 0.9458 0.6498 0.8033 0.9005 0.9143 0.9331

Table 1: Coverage Percentage of 95% confidence Intervals from central limit theorem and generalized variable methods for a Uniform distribution.

Sample Size
(n)
  Coverage Percentage
Method (I) Method (II)
m=5 m=10 m=30 m=50 m=100
0.1 10 0.1823 0.6693 0.8135 0.9357 0.9639 0.9816
25 0.3452 0.6649 0.8210 0.9348 0.9618 0.9802
50 0.5051 0.6711 0.8152 0.9403 0.9603 0.9794
100 0.6747 0.6695 0.8230 0.9418 0.9637 0.9818
0.3 10 0.6013 0.6781 0.8442 0.9499 0.9728 0.9832
25 0.8002 0.6829 0.8462 0.9559 0.9706 0.9839
50 0.9001 0.6903 0.8439 0.9566 0.9732 0.9840
100 0.9484 0.6862 0.8413 0.9527 0.9703 0.9804
0.6 10 0.8245 0.6756 0.8381 0.9506 0.9688 0.9784
25 0.9293 0.6892 0.8457 0.9547 0.9654 0.9760
50 0.9646 0.6836 0.8525 0.9524 0.9663 0.9762
100 0.9796 0.6856 0.8457 0.9476 0.9634 0.9733
1 10 0.9009 0.6645 0.8188 0.9389 0.9571 0.9735
25 0.9583 0.6747 0.8328 0.9401 0.9579 0.9705
50 0.9789 0.6828 0.8419 0.9503 0.9626 0.9755
100 0.9864 0.6788 .8338 0.9436 0.9604 0.9738
3 10 0.9400 0.7501 0.8734 0.9391 0.9522 0.9681
25 0.9754 0.6814 0.8314 0.9507 0.9668 0.9800
50 0.9841 0.6608 0.8209 0.9297 0.9521 0.9722
100 0.9870 0.6668 0.8192 0.9287 0.9524 0.9722

Table 2: Coverage Percentage of 99% confidence Intervals from central limit theorem and generalized variable methods for a Uniform distribution.

As can be seen the Tables 1 and 2, all of the confidence intervals, for 95% confidence level only for two location that constructed confidence interval by central limit theorem method and eighteen location that constructed confidence interval by generalized variable method, have the most of coverage percentage. Also, for 99% confidence level only for five locations that constructed confidence interval by central limit theorem method and five teen locations that constructed confidence interval by generalized variable method, have the most of coverage percentage.

Effect of coefficient of variation value on confidence intervals: With a brief look at the results that can be made to the weakness of confidence interval by central limit theorem method for small values of the coefficient of variation. So that in both confidence level for the first three values of parameters of coefficient of variation (0.1, 0.3 and 0.6), the highest of cover percentage to has been the confidence intervals constructed by generalized variable method with m=100. However, increasing the parameter of coefficient of variation, performance of confidence intervals of constructed to central limit theorem method is improved. But in general, performance of confidence intervals of constructed to central limit theorem method for small values of the coefficient of variation parameter is not suitable. But for larger values and confidence intervals of constructed to generalized variable method, its performance is improved.

Effect of sample size on confidence intervals: Note that the central limit theorem support of confidence intervals of constructed to central limit theorem method, Not unexpected that the confidence intervals in the high size of samples to show better performance. This issue carefully in both tables 1 and 2 can be clearly seen. So that for the five-coefficient of variations parameter value with increasing sample size clearly increase to coverage percentages of confidence interval of constructed to central limit theorem method. But it problem for confidence interval of constructed to generalized variable method is not seen. Then, does not appear to increase the sample size may have an effect on the performance improvement. Thus it appears that in practice the use of confidence interval of generalized variable method for a small sample size is much more reasonable.

Efficiency of repeat frequent on constructed confidence intervals by generalized variable method: With attention to results of simulation study the important role of this agent in the performance of confidence intervals is clearly visible. In all positions with increasing m, coverage percentages of confidence interval of constructed to generalized variable method significantly increased.

A real example

The approved metrological company from Iran Standard and Industrial Researches Organization has conducted the study for the purpose of evaluation of precision for medical laboratory micropipettes according to NCCLS EP5-A2 and ISO 8655-6. CV is commonly presented in the report of the measurements. Samples were collect for three kinds of micropipettes which are common in medical diagnosis labs are used and are shown by A, B and C. A lab unit technician sampled the distilled water in a standard lab condition at the beginning of the work time and repeated the sampling two hours later. Overall, there were 40 runs in 28 consecutive days, and in every measurement 10 times sampling was conducted for each three kinds of micropipettes. The summary statistics of data collected from the three kinds of micropipettes of A, B and Care for every run given in Table 3. Initial review of 10 times of sampling in every run for any micropipette by Kolmogorov-Smirnov test shows that data follow a uniform distribution (Table 3). Therefore, we are interested in making inference about the coefficient of variation of every run based data of three kinds of micropipettes. Take 0.006 as a hypothetical value for the criteria of good precision for the measurements of micropipette according to ISO 8655-6. Upper limit of 95% confidence interval for 40 runs in three kinds of micropipettes by using two statistical methods are shown in Table 4.

RUN Micropipette A Micropipette B Micropipette C
Minimum Maximum P-Value Minimum Maximum P-Value Minimum Maximum P-Value
   1 0.04951 0.05001 0.508 0.04922 0.04978 0.200 0.04751 0.04800 0.533
2 0.04960 0.05017 0.592 0.04932 0.05029 0.671 0.04706 0.04793 0.089
3 0.04968 0.05018 0.198 0.04942 0.05006 0.422 0.04818 0.04858 0.436
4 0.04983 0.05032 0.855 0.04938 0.05034 0.188 0.04813 0.04873 0.944
5 0.04918 0.05025 0.017 0.04958 0.05008 0.042 0.04711 0.04794 0.472
6 0.04930 0.05030 0.960 0.04963 0.05051 0.347 0.04740 0.04853 0.419
7 0.04986 0.05018 0.919 0.04921 0.05056 0.227 0.04816 0.04979 0.965
8 0.04933 0.05007 0.329 0.04951 0.05007 0.002 0.04909 0.04951 0.311
9 0.04961 0.05012 0.886 0.04953 0.05023 0.230 0.04716 0.04860 0.288
10 0.04955 0.05025 0.597 0.04910 0.04990 0.013 0.04808 0.04922 0.685
11 0.04905 0.05004 0.060 0.04905 0.04985 0.496 0.04805 0.04938 0.213
12 0.04983 0.05048 0.427 0.04910 0.05026 0.448 0.04821 0.04892 0.268
13 0.04943 0.04988 0.115 0.04918 0.04975 0.893 0.04666 0.04765 0.920
14 0.04961 0.05021 0.819 0.04920 0.05036 0.000 0.04794 0.04862 0.949
15 0.04982 0.05017 0.065 0.04944 0.05070 0.690 0.04600 0.04774 0.155
16 0.04978 0.05012 0.949 0.04933 0.05009 0.455 0.04726 0.04810 0.178
17 0.04949 0.05003 0.897 0.04913 0.05003 0.097 0.04744 0.04881 0.012
18 0.04985 0.05091 0.095 0.04891 0.05027 0.919 0.04999 0.05034 0.819
19 0.04925 0.05018 0.267 0.04933 0.05016 0.710 0.04825 0.04924 0.396
20 0.04918 0.04996 0.360 0.04914 0.05040 0.017 0.04810 0.04878 0.832
21 0.04967 0.05033 0.130 0.04943 0.05014 0.998 0.04747 0.04863 0.329
22 0.04975 0.05040 0.661 0.04918 0.05041 0.238 0.04844 0.04992 0.855
23 0.04978 0.05055 0.044 0.04931 0.04972 0.917 0.04819 0.04923 0.530
24 0.04935 0.05017 0.841 0.04934 0.05028 0.253 0.04869 0.04967 0.896
25 0.04999 0.05063 0.709 0.04918 0.05013 0.010 0.04819 0.04991 0.433
26 0.04925 0.05028 0.680 0.04936 0.05087 0.019 0.04819 0.04926 0.056
27 0.04950 0.05004 0.022 0.04924 0.05032 0.143 0.04884 0.04957 0.500
28 0.04947 0.05003 0.958 0.04924 0.04994 0.748 0.04649 0.04920 0.582
29 0.04941 0.05025 0.999 0.04934 0.05136 0.000 0.04779 0.04850 0.431
30 0.04988 0.05058 0.597 0.04911 0.05023 0.396 0.04730 0.04834 0.314
31 0.04947 0.05010 0.673 0.04928 0.04973 0.707 0.04839 0.04908 0.243
32 0.04971 0.05019 0.736 0.04911 0.04979 0.341 0.04715 0.04864 0.173
33 0.04982 0.05025 0.022 0.04935 0.04990 0.536 0.04714 0.04802 0.975
34 0.04970 0.05010 0.172 0.04914 0.05003 0.126 0.04731 0.04821 0.250
35 0.04996 0.05033 0.041 0.04936 0.04984 0.193 0.04846 0.04934 0.786
36 0.04954 0.05001 0.798 0.04940 0.04998 0.131 0.04868 0.04923 0.184
37 0.04961 0.05011 0.413 0.04928 0.05025 0.020 0.04650 0.04854 0.001
38 0.04963 0.04997 0.514 0.04935 0.04993 0.096 0.04902 0.04991 0.456
39 0.04986 0.05056 0.673 0.04968 0.05009 0.044 0.04868 0.04938 0.748
40 0.04989 0.05018 0.947 0.04959 0.05004 0.819 0.04786 0.04832 0.225

Table 3: Summary statistics and P-value of Kolmogorov-Smirnov test for data collected from the three kinds of micropipettes of A, B and C are for every run.

RUN Micropipette A Micropipette B Micropipette C
CLM GV CLM GV CLM GV
    1 0.002896 0.002914 0.003265 0.003281 0.00296 0.002976
2 0.003291 0.003308 0.005614 0.005639 0.005286 0.005303
3 0.002883 0.0029 0.003701 0.003727 0.002385 0.002395
4 0.00282 0.00284 0.005538 0.005587 0.003569 0.003595
5 0.006166* 0.006232* 0.002888 0.002905 0.005023 0.005057
6 0.00578 0.005828 0.005067 0.005101 0.006792* 0.006837*
7 0.001846 0.001856 0.007802* 0.007853* 0.009573* 0.009659*
8 0.004283 0.004317 0.00325 0.003261 0.002458 0.00247
9 0.002949 0.002961 0.004048 0.004063 0.008624* 0.008706*
10 0.004044 0.004067 0.004674 0.004685 0.006754* 0.006793*
11 0.005736 0.005784 0.004665 0.004683 0.007878* 0.007902*
12 0.003739 0.003755 0.006734* 0.006765* 0.004204 0.004236
13 0.002609 0.002623 0.00332 0.003335 0.006044* 0.006076*
14 0.003465 0.003482 0.006747* 0.006749* 0.004056 0.004079
15 0.002021 0.00203 0.007238* 0.007296* 0.01059* 0.010764*
16 0.001962 0.001976 0.004407 0.004438 0.005086 0.005113
17 0.003129 0.003149 0.005239 0.005267 0.00813* 0.00826*
18 0.006074* 0.006106* 0.007895* 0.007959* 0.002012 0.002025
19 0.005371 0.005426 0.00481 0.004839 0.005851 0.005891
20 0.004527 0.004562 0.00732* 0.007339* 0.004043 0.00407
21 0.003808 0.003826 0.004108 0.004134 0.006927* 0.006997*
22 0.003741 0.003765 0.007121* 0.007165* 0.008654* 0.008729*
23 0.004432 0.00445 0.002387 0.0024 0.006134* 0.00619*
24 0.004741 0.004777 0.005448 0.005471 0.005741 0.005777
25 0.003666 0.003683 0.005483 0.005538 0.010099* 0.010149*
26 0.005951 0.005993 0.00871* 0.008723* 0.006295* 0.006358*
27 0.003121 0.003148 0.006253* 0.006296* 0.00427 0.004305
28 0.003245 0.003258 0.004064 0.004085 0.016176* 0.016395*
29 0.004855 0.004881 0.011642* 0.011616* 0.004242 0.00427
30 0.004019 0.004042 0.00649* 0.00654* 0.006251* 0.006307*
31 0.003642 0.003676 0.002622 0.002641 0.004083 0.004113
32 0.00277 0.002784 0.003954 0.003983 0.008969* 0.009011*
33 0.002481 0.002491 0.003191 0.003213 0.005323 0.005361
34 0.002313 0.002325 0.005178 0.005206 0.005438 0.005466
35 0.002131 0.002137 0.002793 0.002803 0.00519 0.005213
36 0.002724 0.002734 0.003368 0.00338 0.003232 0.003253
37 0.002887 0.002904 0.005593 0.005644 0.012204* 0.012426*
38 0.001968 0.001979 0.003372 0.003386 0.005185 0.005214
39 0.004018 0.004039 0.002373 0.002381 0.004113 0.004136
40 0.001672 0.001679 0.002603 0.002617 0.002761 0.002771

Table 4: Upper limit of 95% confidence interval of CV for micropipettes A, B and C.

Conclusion

In this paper, we propose two approaches for making inferences about the coefficient of variation in uniform distribution. These approaches utilize the concepts of generalized variables and central limit theorem. The performance of generalized variable method, alongside central limit theorem method, was assessed using simulation and real data. A comparison indicates that the coverage percentage of the proposed confidence interval of the generalized variable method is generally satisfactory, but it is computationally time consuming. Moreover, central limit theorem method takes a shorter time to calculate.

Acknowledgements

We are grateful to the Pishgaman Andazeshenasi Daghigh Company (approved metrological company from Iran Standard and Industrial Researches Organization), and especially to Dr. Hamid AlaviMajd, for providing us with precision of micropipettes the data.

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