Performance of three methods of interval estimation of the coefficient of variation

Transcription

1 Performance of three methods of interval estimation of the coefficient of variation C.K.Ng Department of Management Sciences City University of Hong Kong Abstract : Three methods for constructing confidence intervals for the coefficient of variation from normal populations are compared through simulation study. Key Words and Phrases : Coefficient of variation, inverse coefficient of variation, confidence interval, normal population 1. Introduction The coefficient of variation (CV), which is the ratio of the standard deviation to the mean, is a dimensionless measure of dispersion found to be very useful in many situations. In chemical experiments, the CV is often used as a yardstick of precision of measurements; two measurement methods may be compared on the basis of their respective CVs. In finance, the CV can be used as a measure of relative risks (Miller & Karson (1977)) and a test of the equality of the CVs for two stocks, can help to determine if the two stocks possess the same risk or not. Hamer et.al.(1995) used the CV to assess homogeneity of bone test samples produced from a particular method to help assess the effect of external treatments, such as irradiation, on the properties of bones. Ahn (1995) used the CV in the analysis of fault trees. The CV has also been employed by Gong & Li (1999) in assessing the strength of ceramics. Sometimes, it might be easier to work with the reciprocal of the CV, denoted ICV. The ICV has special applications in parametric inference problems for some important lifetime distributions. 1

2 . Background The CV is often estimated by the ratio of the sample standard deviation to the sample mean, called the sample CV. To do inference on the CV, one needs to make assumption about the shape of the population. One also needs to know the distribution of the sample CV. Early in the thirties, Hendricks & Robey (1936) studied the distribution of the sample CV when sample is drawn from a normal population. Koopmans et al (1964) reviewed the relevant literature and obtain confidence intervals for the CV for normal and lognormal distributions. The exact distribution of the sample CV for normal populations was given by Iglewicz (1967). However, this exact sampling distribution is not useful for inferential purposes as one either has to assume that the chance of obtaining a non-positive sample mean is negligible or resort to approximations of this distribution. McKay (193) gave an approximation of the distribution of a statistic derived from the sample CV based on the chi-squared distribution. Studies by Pearson (193), Fieller (193), Iglewicz (1967), Iglewicz & Myers (1970) and Umphrey (1983) have shown that McKay s approximation is very accurate if CV Further studies by Miller (1991) indicated that this approximation is also reasonably accurate when 0.33 CV Except for normal populations, the exact distribution of the sample CV is quite difficult to obtain and is thus difficult to apply to inferences about the population CV. Study on the inference of the CV or ICV for non-normal populations largely remains ignored until Sharma & Krishna (1994) developed the asymptotic distribution of the sample ICV without making an assumption about the population distribution. In particular, the sampling distribution they obtained is claimed to solve inferential problems on the shape parameter of three distributions commonly used in life-testing situations, namely Gamma, Weibull and lognormal distributions.

3 3. Interval Estimation of the CV or ICV Graf et al (1987) derived an approximate confidence interval for the CV by using normal approximation to the exact distribution of the sample CV. The confidence limits can be calculated with considerable accuracy by determining the noncentrality parameter of a certain noncentral t-distribution, say, by Deutler s (1984) series expansion. With Deutler s algorithm, the confidence limits can be determined very accurately even with small sample sizes. To implement Deutler s algorithm, Reh & Scheffler (1996) has written a FORTRAN77 program that calls IMSL routines. In spite of the accuracy, the use of their method is not widespread because their program is not available in any statistical software package. Procedures exist that alleviate the task of having to struggle with FORTRAN programming. The first method, given in Miller (1991), makes use of the asymptotic distribution of the sample CV which can reasonably be assumed normal if the parent population is normal. The second method, given in Sharma & Krishna (1994), involves first finding the confidence interval for the ICV which can then be inverted to give that of the CV. As remarked before, this has the advantage of relieving the normality assumption. The third method, due to McKay (193), employs a chi-squared approximation to a certain statistic derived from the sample CV. Various ways of assessing the performance of confidence intervals have appeared in the literature. Actual coverage is one factor. If different procedures have comparable coverages, then expected interval width and the variance of the interval width would be of concern. Kang & Schmeiser (1986) introduced a graphical assessment technique that takes all three factors into account. Their technique, however, does not gain popularity although its performance is found to be satisfactory (Subramaniam & Leemis (1988)). It is the purpose of this paper to compare the performance of these three methods. Comparison is confined to the cases where the population CV is less than or equal to This is never too restrictive, as Miller (1991) 3

4 remarked that normal populations met in practice usually have a CV 0.33 which essentially means there is a negligible probability of the random variable concerned being negative. Taking the CV to be less than 0.67 already extends far beyond the 0.33 upper limit. 4. Notations and distributional results Let x 1,, x n represent a random sample of size n from a normal population with mean μ and variance σ. Let x be the sample mean and s be the sample standard deviation. Suppose s x is used to estimate σ μ, the population CV. Miller (1991) derived the following expansion for s x : s x [ 0.5( σ μ) Y ( σ μ) Y ] + O ( ) = σ μ + m 1 p m where Y 1 and Y are independent standard normal random variables and m = n 1. From this, it can be seen that ( [ ]) s x is AN σ μ, m ( σ μ) ( σ μ) where AN denotes asymptotic normality. Confidence limits for the CV is then self-suggestive : s x ± Z [ ] ( σ μ) 0. ( σ μ) α m 5 + which is approximated by [ ] ( s x) 0. ( s x) s x ± Z + α m 5 where Z α is the upper (1 α) percentile of the standard normal distribution. We label this as Miller s confidence interval. Another 100(1 α)% confidence interval for the CV, due to Sharma & Krishna (1994) can be obtained by inverting the following confidence interval for the ICV : 4

5 P [ x s + Φ ( α ) n < ICV < x s Φ ( α ) n] = 1 α where Φ( ) is the cumulative standard normal distribution. We call this Sharma s confidence interval. The third method is due to McKay (193) who gave the following approximation : [ ] χ [ ( μ σ )]. nv ( 1+ V ) 1+ n where V is the sample CV and χ n is the chi-squared distribution with (n) degrees of freedom. The lower and upper confidence limits for the CV would then be given by : Lower limit : 1 ( x) {[ + ] χ n } 1 S n Upper limit : 1 ( x) α {[ + ] χ n } 1 S n 1 α where S n = ( n ) S n and χ α is the 100α-th percentile of a chi-squared distribution with (n-1) degrees of freedom. 5. Simulation results Here, we basically follow the simulation study conducted by Miller(1991). There, n = 5, 10, 15, 5, 50, 100 and CV = 0.05, 0.1, 0., 0.33, 0.5, Ten thousand simulations were done for each combination of n and CV. The experiment is repeated with confidence levels 0.95 and Without loss of generality, normal populations from which samples are drawn are assumed to have unit standard deviation. So one only needs to adjust the population mean to get the required CV. Tables 1 and summarizes the results produced by a Visual Fortran program running on a Pentium III PC. It can be seen that Sharma s method is consistently inferior to the other two methods in terms of coverage percentage, i.e. the percentage of times the CV is included in the confidence interval. For n 5, McKay s procedure is closer to the nominal level than Miller s, albeit with a slightly longer average interval width. For n=15 and CV 0.33, McKay s procedure is still 5

6 superior to Miller s but deteriorates abruptly when CV=0.67 in both tables. This deterioration is even more pronounced for n= 5,10 when the CV>0.50 when 1-α =0.95. When 1-α = 0.99, this occurs as early as when CV>0.33. On closer examination, the exceptionally poor performance is due to the fact that the expression under the square root sign in McKay s interval becomes negative very frequently and these samples have to be discarded. In summary, McKay s procedure is recommended when n 15 and when CV 0.33, otherwise if CV>0.33, Miller s interval should be used. When n 10, Miller s procedure seems to be the better choice especially if the CV is known, a-priori, to be large. n CV (7.14E-) a 0.895(4.7E-) 0.916(3.77E-) 0.98(.85E-) 0.94(1.99E-) 0.946(1.40E-) (5.84E-3) b 0.131(4.10E-3) 0.140(3.34E-3) 0.139(.58E-3) 0.145(1.83E-3) 0.147(1.9E-3) 0.958(9.85E-) c 0.949(5.33E-) 0.951(4.06E-) 0.95(.97E-) 0.95(.03E-) 0.949(1.41E-) 0.844(0.145) 0.89(9.51E-) 0.909(7.58E-) 0.96(5.76E-) 0.944(4.0E-) 0.947(.8E-) (.39E-) 0.66(1.66E-) 0.67(1.35E-) 0.81(1.04E-) 0.76(7.3E-3) 0.73(5.17E-3) 0.955(0.04) 0.948(0.108) 0.950(8.18E-) 0.949(6.01E-) 0.95(4.11E-) 0.950(.85E-) 0.84(0.305) 0.894(0.198) 0.91(0.157) 0.933(0.119) 0.937(8.8E-) 0.94(5.81E-) (0.107) 0.491(6.97E-) 0.510(5.57E-) 0.506(4.3E-) 0.513(.96E-) 0.51(.08E-) 0.955(0.510) 0.949(0.33) 0.953(0.173) 0.951(0.15) 0.951(8.49E-) 0.948(5.88E-) 0.843(0.563) 0.893(0.359) 0.910(0.84) 0.98(0.13) 0.938(0.147) 0.943(0.103) (0.391) 0.686(0.17) 0.701(0.166) 0.714(0.13) 0.717(8.35E-) 0.77(5.8E-) 0.741(1.306) 0.949(0.469) 0.949(0.39) 0.954(0.31) 0.95(0.153) 0.951(0.105) 0.844(1.005) 0.89(0.66) 0.909(0.484) 0.98(0.363) 0.94(0.47) 0.946(0.173) (11.178) 0.8(0.63) 0.836(0.41) 0.847(0.96) 0.86(0.194) 0.857(0.133) 0.361(1.790) 0.85(1.56) 0.954(0.691) 0.955(0.47) 0.954(0.68) 0.955(0.181) 0.840(9.) 0.887(1.01) 0.905(0.764) 0.93(0.561) 0.939(0.376) 0.94(0.61) (8.138) 0.858(5.070) 0.893(0.954) 0.904(0.584) 0.918(0.359) 0.914(0.4) 0.181(.13) 0.51(1.93) 0.798(1.556) 0.954(0.863) 0.96(0.44) 0.960(0.88) Table 1. Coverage percentage and average width (in brackets) for 1 α = 0.95 a : Miller b : Sharma c : McKay 6

7 n CV (9.18E-) a 0.945(6.09E-) 0.958(4.90E-) 0.970(3.7E-) 0.980(.61E-) 0.987(1.84E-) (5.81E-3) b 0.133(4.10E-3) 0.130(3.34E-3) 0.144(.59E-3) 0.145(1.83E-3) 0.146(1.9E-3) 0.991(0.168) c 0.983(6.77E-) 0.989(5.68E-) 0.991(4.05E-) 0.989(.7E-) 0.991(1.87E-) 0.896(0.186) 0.937(0.1) 0.959(9.88E-) 0.967(7.53E-) 0.981(5.7E-) 0.986(3.71E-) (.396E-) 0.65(1.66E-) 0.73(1.35E-) 0.76(1.04E-) 0.79(7.3E-3) 0.78(5.17E-3) 0.99(0.373) 0.980(0.137) 0.991(0.116) 0.990(8.1E-) 0.990(5.49E-) 0.991(3.78E-) 0.894(0.387) 0.940(0.55) 0.958(0.04) 0.97(0.155) 0.979(0.109) 0.986(7.63E-) (0.107) 0.49(6.96E-) 0.485(5.55E-) 0.506(4.4E-) 0.507(.957) 0.505(.08E-) 0.834(1.00) 0.983(0.300) 0.990(0.48) 0.99(0.17) 0.988(0.114) 0.990(7.81E-) 0.891(0.704) 0.937(0.46) 0.955(0.366) 0.967(0.78) 0.976(0.193) 0.984(0.136) (0.389) 0.69(0.15) 0.707(0.165) 0.71(0.1) 0.719(8.36E-) 0.70(5.83E-) 0.338(1.845) 0.979(0.69) 0.989(0.500) 0.990(0.35) 0.990(0.07) 0.990(0.141) 0.886(1.11) 0.933(0.806) 0.950(0.69) 0.964(0.469) 0.977(0.34) 0.981(0.6) (3.680) 0.818(0.63) 0.836(0.44) 0.849(0.94) 0.857(0.194) 0.859(0.133) 0.117(.01) 0.73(1.631) 0.895(1.35) 0.99(0.657) 0.991(0.37) 0.990(0.43) 0.885(61.75) 0.93(1.73) 0.950(0.981) 0.965(0.71) 0.975(0.490) 0.984(0.34) (11.38) 0.858(.793) 0.895(0.95) 0.907(0.579) 0.916(0.360) 0.9(0.4) 4.77E-(.94) 0.36(.195) 0.50(.33) 0.889(1.554) 0.99(0.654) 0.993(0.396) Table. Coverage percentage and average width (in brackets) for 1 α = 0.99 a : Miller b : Sharma c : McKay 7

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