A POOLING METHODOLOGY FOR COEFFICIENT OF VARIATION

Transcription

1 Sankhyā : The Indian Journal of Statistics 1995, Volume 57, Series B, Pt. 1, pp A POOLING METHODOLOGY FOR COEFFICIENT OF VARIATION By S.E. AHMED University of Regina SUMMARY. The problem of estimating the coefficient of variation is considered when it is a priori suspected that two coefficients of variations are the same. It is advantageous to pool the data for making inferences about population coefficient of variation. The estimators based on pre-test and the shrinkage principle are proposed. The expressions for the asymptotic bias and asymptotic mean-square error (AMSE) of the proposed estimators are derived and compared with the parallel expressions for the unrestricted and pooled estimators. Interestingly, the proposed estimator dominates the unrestricted estimator in a wider range. Not only that, the size for the preliminary test is also appropriate. Furthermore, an optimal rule for the choice of the level of significance (α) for the preliminary test is presented. Tables for the optimum selection of α is also provided for the use of the shrinkage preliminary test estimator. 1. Introduction The ratio of the standard deviation to the population mean is known as the coefficient of variation (C.V.). For convenience of notation, C.V. will be denoted by greek letter λ. Furthermore, it is worth noting that the population coefficient of variation λ is a pure number free from units of any measure. This has the advantage of being able to compare the variability of two different populations directly using the coefficient of variation. If X is a random variable with mean µ and variance σ, then the coefficient of variation is defined as λ = σ µ, µ 0. It is sometimes a more informative quantity than σ. For instance, a value of 0 for σ has a little meaning unless we can compare it with µ. Ifσ is known to be 0 and µ is known to be 8000, then amount of the variation is small relative Paper received. October 199; revised October AMS (1990) subject classification. 6F10. Key words and phrases. Shrinkage restricted estimator; shrinkage preliminary test estimator; asymptotic biases and risks; asymptotic efficiency.

2 58 s.e. ahmed to the mean. For example, in the stock market the volatility or the level of activity of a stock is often measured by the coefficient of variation of the stock. Thus, it is possible to directly compare the volatility of one stock with another or against that of a known index such as the Dow Jones Industrial Average (DJIA). Also, in the study of precision of a measuring instrument, engineers are typically more interested in estimating λ. Let X 11,X 1,,X 1n1 be independent and identically distributed random variables with mean µ 1 and variance σ1 with λ 1 = σ 1 /µ 1. Also let X 1,X,, X n be independent and identically distributed random variables with mean µ and variance σ with λ = σ /µ. Suppose that this pair of independent random samples are obtained either from similar types of characteristics or at different times. It is natural to suspect that λ 1 may be equal to λ, since the population coefficient of variation is fairly stable over time and over similar types of characteristics. We are interested in estimating λ 1 when it is a priori suspected that both coefficients of variation may be equal and we wish to pool the information from two sources. It is advantageous to utilize information provided by the second sample if both population coefficients of variation are equal. However, in many experimental situations, it is not certain whether the coefficients of variation of the two populations are equal or not. This type of problem may be classified as a statistical estimation problem with uncertain prior information (not in the form of prior distribution). To tackle this uncertainty, it is desirable to perform a preliminary test on the validity of the prior information and then make a choice between restricted and classical statistical inference procedures. This may be regarded as a compromise between two extremes. For some account of the parametric theory on the subject we refer to Judge and Bock (1978) and Ahmed and Saleh (1990) among others. Two bibliographies in this area of research are provided by Bancroft and Han (1977), and Han et al. (1988). In addition, the asymptotic theory of the preliminary test estimation for some discrete models is discussed by Ahmed (1991a, 1991b) among others. In this paper the parameter of interest is λ 1 and the problem is to estimate it on the basis of two samples. Five estimation strategies are developed for the estimation of parameter λ 1. We use the mean-square error (MSE) criterion to appraise the performance of the estimators under the following squared loss function L(ˆλ 1,λ 1 )=(ˆλ 1 λ 1 ),...(1.1) where ˆλ 1 is a suitable estimator of λ 1. Then, the MSE of ˆλ 1 is given by MSE(ˆλ 1,λ 1 )=E(ˆλ 1 λ 1 )....(1.) Further, ˆλ 1 will be termed an inadmissible estimator of λ 1 if there exists an alternative estimator ˆλ 1 such that

3 pooling methodology for coefficient of variation 59 MSE(ˆλ 1) MSE(ˆλ 1) for all λ 1,...(1.3) with strict inequality for some λ 1. If instead of (1.3) holding for every n we use the asymptotic MSE (AMSE) then we require lim MSE(ˆλ 1) lim MSE(ˆλ 1) for all λ 1,...(1.4) n n with strict inequality for some λ 1 and ˆλ 1 is termed an asymptotically inadmissible estimator of λ 1. The proposed estimators are presented in section. In section 3, the expressions for the asymptotic bias and AMSE of the estimators are presented under local alternatives. The AMSE analysis is provided in section 4. Section 5 discusses how to use the estimators and an example is also provided. Section 6 summarizes the findings.. Estimation strategies and preliminaries If µ i and σi,i=1,, are unknown, then the unrestricted maximum likelihood estimator of (µ i,σi )is( µ i, σ i ), where µ i = 1 n i x ij and σ i = 1 n i (x ij µ i ),j=1,,,n i....(.1) n i n i j=1 j=1 The unrestricted estimator (UE)ofthei-th coefficient of variation is defined as λ i = σ i µ i. Alternatively, if λ 1 = λ, then ˆλ 1 = n 1 λ 1 +n λ n 1+n and is called the pooled or restricted estimator (RE) ofλ 1. The pooled estimator ˆλ 1 performs better than the unrestricted estimator λ 1 when λ 1 = λ but as λ moves away from λ 1, ˆλ 1 may be considerably biased and inefficient. However, the performance of λ 1 remains steady, over such departures. In an effort to increase the precision of estimators, it is often desirable to develop an estimator which is a combination of λ 1 and ˆλ 1 by incorporating a preliminary test on the null hypothesis H o : λ 1 = λ....(.) As a result, when H o is rather suspicious, it is desirable to have a compromised estimator using a preliminary test on H o in (.) and then choose between ˆλ 1 and λ 1 depending on the outcome of the test. This estimator, denoted by ˆλ P 1 is called the preliminary test estimator (PE) of λ 1 which is a convex combination of λ 1 and ˆλ 1 via a test-statistic for testing H o. Thus,

4 60 s.e. ahmed ˆλ P 1 = ˆλ 1 I(D <d α )+ λ 1 I(D d α ),...(.3) where I(A) is an indicator function of the set A and D is a test statistic for H o in (.). For a given level of significance α (0 <α<1), let d α be the upper 100%α critical value using the distribution of D under H o. We develop D in the next section. However, PE may not be perfect in the whole parameter space. Moreover, its use may be limited due to the large level of significance. On the other hand, the use of such a large significance level helps to maximize the minimum efficiency of ˆλ P 1 relative to λ 1 (Ahmed, 1991b). In order to overcome this shortcoming of PE, we propose the shrinkage preliminary test estimator (E). The E may be viewed as an improved version of ˆλ P 1. First, we propose a shrinkage restricted estimator (SRE) of λ 1, ˆλ S 1 =(1 π) λ 1 + πˆλ 1, π [0, 1],...(.4) as a modification of the restricted estimator of λ 1 and π may be defined as coefficient reflecting a degree of confidence in the prior information. The SRE, like UE, yields a smaller AMSE at or near the null hypothesis at the expense of poor performance in the rest of the parameter space. However, the SRE provides a wider range than the restricted estimator in which it dominates the unrestricted estimator. This motivates replacing the restricted estimator RE by SRE in the PE. As a result, the proposed E yields a wider interval in the parameter space in which it dominates the UE. More importantly, it also provides meaningful size for the preliminary test. Finally, we define E by replacing the restricted estimator by SRE in the PE as: ˆλ 1 = {πˆλ 1 +(1 π) λ 1 }I(D <d α )+ λ 1 I(D d α )....(.5) The primary objective here is to focus on the asymptotic properties of E and to compare these with those of PE and UE. An optimal rule for the choice of the level of significance α for the preliminary test is provided. Tables for the optimum selection of α is also presented for the use of the shrinkage preliminary test estimator. p p First, we note that since µ i µi and σ i σi,wehave σ p i µ i σ i µ i, where p means convergence in probability. For the limiting distribution, noting that ( σi ni σ ) i µ i µ i it can be shown that = ni µ i ( σ i σ i )+ σ i µ i µ i { n i (µ i µ i )}, ni ( µ i µ i ) d N(0,σ i ), ni ( σ i σ i ) d N ( 0, (κ 4 1)σi ), 4

5 pooling methodology for coefficient of variation 61 σ i µ i µ i P σ i µ i, 1 P 1 µ i µ i where d means convergence in distribution and κ 4 = µ4 i is the kurtosis of the σi 4 distribution. Thus, we have the following lemma: Lemma.1. If X i1,x i,,x ini is a random sample of size n i from a normal distribution with mean µ i (µ i 0)and variance σi, then the following results hold: (i) ( ) ) σ n i d N i µ i σi σ µ i (0, i + σ4 µ i. µ i 4 i (ii) We develop a large sample test statistic for testing H o, ( λ D n = λ 1 ) ( ), ˆτ 1 n n where ˆτ = 1 ˆλ 1 + ˆλ 4 1. For large n 1,n, the distribution D n approaches the central chi-square distribution with 1 degree of freedom. Thus, the critical value d α of D n may be approximated by χ 1,α, the upper 100α% critical value of the chi-square distribution with 1 degree of freedom. To avoid the asymptotic degeneracy, we specify a sequence of local alternatives. This local alternative setting was also used in Kulperger and Ahmed (199) among others. Here the local alternative setting is reasonable since estimators based on the preliminary test principle are only useful in the cases where λ 1 and λ are close. For this reason, a sequence {K n } of local alternatives is considered. That is given a sample of size n = n 1 + n, replace λ by λ 1 + ξ n : where ξ is a fixed real number. K n : λ = λ 1 + ξ n,...(.6) 3. Bias and mean squared error of the estimators In this section, expressions for bias and mean squared errors of the estimators are obtained. First, the asymptotic bias of the proposed estimators are given in theorem 3.1. Theorem 3.1. Under local alternatives, as n in such a way that the relative sample sizes converge, n 1 /n γ (0, 1). By direct computation and using the results from Judge and Bock (1978) Appendix B, we arrive at the following bias expressions of the estimators: B 1 = asymptotic bias of{ γn( λ 1 λ 1 )} =0 B = asymptotic bias of { γn(ˆλ 1 λ 1 )} = τξ (1 γ)

6 6 s.e. ahmed B 3 = asymptotic bias of { γn(ˆλ S 1 λ 1 )} = πτξ (1 γ) B 4 = asymptotic bias of{ γn(ˆλ P 1 λ 1 )} = τξ (1 γ)g 3 (χ 1,α; δ ) B 5 = asymptotic bias of{ γn(ˆλ 1 λ 1 )} = πτξ (1 γ)g 3 (χ 1,α; δ ), where δ γ(1 γ)ξ = τ, τ = 1 λ 1 + λ 4 1 and G m (,δ ) is the cumulative distribution of a noncentral chi-square distribution with m degrees of freedom and noncentrality parameter δ.

7 pooling methodology for coefficient of variation 63 It can easily be seen that B 3 = πb and B 5 = πb 4. Thus, for π (0, 1), B 3 <B and B 5 <B 4. Hence, E and SRE are superior to PE and RE respectively. Thus the shrinkage technique may also be viewed as a bias reduction technique from the point of view of asymptotic bias. Figure 1 displays behavior of the bias function of E and SRE in term of δ for various values of π at selected values of α with γ =0.5. The large values of α is used simply to exhibit its effect on the magnitude of the bias functions (Han and Bancroft, 1968). Since the bias is a component of the AMSE and the control of the AMSE would control both bias and variance, we shall only compare the AMSE from this point onwards. Direct computation provides expressions for the asymptotic mean squared errors (AMSE) of the estimators presented in theorem 3.. Theorem 3.. Under local alternatives and using the results from Judge and Bock (1978) Appendix B, we arrive at the following expressions of the AMSE of the estimators. AMSE 1 (δ ) = asymptotic MSE of{ γn( λ 1 λ 1 )} = τ AMSE (δ ) = asymptotic MSE of { γn(ˆλ 1 λ 1 )} = τ +(1 γ)τ δ (1 γ)τ AMSE 3 (δ ) = asymptotic MSE of { γn(ˆλ S 1 λ 1 )} = τ + π (1 γ)τ δ π( π)(1 γ)τ AMSE 4 (δ ) = asymptotic MSE of { γn(ˆλ P 1 λ 1 } = τ +(1 γ)τ δ {G 3 (χ 1,α; δ ) G 5 (χ 1,α; δ )} (1 γ)τ G 3 (χ 1,α; δ ) AMSE 5 (δ ) = asymptotic MSE of { γn(ˆλ 1 λ 1 } = τ +(1 γ)τ δ {πg 3 (χ 1,α; δ ) π( π)g 5 (χ 1,α; δ )} π( π)(1 γ)τ G 3 (χ 1,α; δ ). In the following section, asymptotic properties of the estimators are studied. 4. Asymptotic analysis We note that the AMSE of λ 1 is a constant line, while the AMSE of ˆλ 1 is a straight line in terms of δ which intersects the AMSE of λ 1 at δ =1. Using the AMSE criterion to appraise performance, if the restriction λ 1 = λ is correct, then the AMSE of ˆλ 1 is less than the AMSE of λ 1. Furthermore, AMSE AMSE 1 if 0 δ 1. Hence, for δ [0, 1], ˆλ 1 performs better than λ 1. However, beyond this interval the AMSE of ˆλ 1 increases without bound. The characteristics of the AMSE function of ˆλ S 1 are similar to that of ˆλ 1.Itis

8 64 s.e. ahmed worth noting that ˆλ S 1 dominates λ 1 when 0 δ π 1 ( π). Thus,the range in which AMSE 3 AMSE 1 is wider than the range in which AMSE AMSE 1. In an effort to identify some important characteristics of the shrinkage preliminary test estimator, first note that G 3 (χ 1,α; δ ) G 1 (χ 1,α; δ ) G 1 (χ 1,α;0)=1 α,... (4.1) for α (0, 1) and δ 0. The left hand side of (4.1) converges to 0 as δ approaches infinity. Now, we compare the AMSE of E with UE. AMSE 5 AMSE 1 according as δ > ( π)g 3 (χ 1,α; δ ) { G 3 (χ 1,α; δ ) ( π)g 5 (χ 1,α; δ ) } 1....(4.) Thus, we notice that ˆλ 1 dominates λ 1 whenever δ ( π)g 3 (χ 1,α; δ ) { G 3 (χ 1,α; δ ) ( π)g 5 (χ 1,α; δ ) } 1....(4.3) It is evident from (4.3) that the AMSE of E is less than the AMSE of UE when δ is equal to or near 0. Moreover, as α, the level of the statistical significance, approaches one, AMSE 5 tends to AMSE 1. Also, when δ increases and tends to infinity, the AMSE of E approaches the AMSE of UE. Further, for larger values of δ, the value of the AMSE of E increases, reaches its maximum after crossing the AMSE of UE and then monotonically decreases and approaches the AMSE of UE. Therefore, there are points in the parameter space where E has a larger AMSE than UE and a sufficient condition for this result to occur is that (4.) holds. Now, we examine the value of δ when α 0 in the equation (4.3). Here 0 δ ( π)....(4.4) π It is evident that the AMSE of PE will be less than that of UE as long as 0 δ 1 when α 0, whereas the AMSE of E will be less than AMSE of UE as long as (4.4) holds. Thus, the range for which AMSE 5 AMSE 1 is greater than the range for which AMSE 4 AMSE 1. Hence the shrinkage preliminary test estimator provides a wider range than ˆλ P 1 in which its AMSE is smaller than the AMSE of λ 1. This clearly indicates the superiority of E over PE. Next, investigating the dominance range for ˆλ 1 and ˆλ 1. Note that under the null hypothesis AMSE 5 - AMSE = τ (1 γ) [ 1 { 1 (1 π) G 3 (χ 1,α;0) }] > 0, for α (0, 1). Alternatively, when δ deviates from the null hypothesis the AMSE of ˆλ 1 grows and become unbounded whereas the AMSE of ˆλ 1 remains bounded. The departure from H o is fatal to ˆλ 1 whereas ˆλ 1 has good AMSE control.

9 pooling methodology for coefficient of variation 65 Now, we wish to compare AMSE behavior of ˆλ 1 and ˆλ S 1 and determine the dominance range of these estimators in terms of the non-centrality parameter δ. First under the null hypothesis, i.e., δ = 0 the AMSE of of ˆλ S 1 is τ {1 π( π)(1 γ))} and AMSE 5 - AMSE 3 > 0, for α (0, 1). Hence, we conclude that under the null hypothesis ˆλ S 1 performs better than ˆλ 1. Furthermore, we find that the AMSE of ˆλ S 1 is smaller than the AMSE of ˆλ 1 when ( 1 G3 δ (χ 1,α; δ ) ) < ( π) ( 1 1 G 3 (χ 1,α ; δ ) ) ( 1 G 5 (χ 1,α ; δ ) ) and for ( 1 G3 δ (χ 1,α; δ ) ) > ( π) ( 1 1 G 3 (χ 1,α ; δ ) ) ( 1 G 5 (χ 1,α ; δ ) ) the opposite conclusion holds. Hence neither ˆλ 1 nor ˆλ S 1 asymptotically dominates the other under local alternatives. We now proceed to compare the AMSE of E and PE and determine the conditions under which E dominates PE. It may be noticed that AMSE 4 AMSE 5 = τ (1 γ)δ { (1 π)g 3 (χ 1,α; δ ) (1 π) G 5 (χ 1,α; δ ) } τ (1 γ)(1 π) G 3 (χ 1,α; δ )....(4.5) It is clear from (4.5) that the AMSE of ˆλ P 1 will be smaller than ˆλ 1 in the neighborhood of δ = 0. However, the AMSE difference may be negligible for larger values of π. On the other hand, as δ increases, the AMSE difference in (4.5) becomes positive and ˆλ 1 dominates ˆλ P 1 uniformly in the rest of the parameter space. Let δπ be a point in the parameter space at which the AMSE of E and PE intersect for a given π. Then, for δ (0,δπ], PE performs better than E, while for δ (δπ, ), E dominates PE uniformly. Further, for large values of π (close to 1), the interval (0,δπ] may be negligible. Nevertheless, PE and E share a common asymptotic property that, as δ, their AMSE converge to a common limit, i.e., to the AMSE of λ 1. In addition, from (4.5) AMSE 4 AMSE 5 according as δ < (1 π)g 3 ((χ 1,α; δ ) { G 3 (χ 1,α; δ ) (1 π)g 5 (χ 1,α; δ ) } 1....(4.6) Thus, E dominates PE whenever δ (1 π)g 3 (χ 1,α; δ ) { G 3 (χ 1,α; δ ) (1 π)g 5 (χ 1,α; δ ) } 1 When α 0 in (4.6), the value of δ is...(4.7) 0 δ (1 π) (1 + π).

10 66 s.e. ahmed [ ] Thus, PE dominates the E only when δ 0, (1 π) (1+π) which suggests that π should be chosen large. For example, if π =0.9, then PE dominates E in a tiny interval [0, 1 19 ]. We have plotted the AMSE of λ 1, ˆλS 1 and ˆλ 1 against δ for π =0.1, 0.3, 0.5, γ =0.5, and for selected values of α. Figure exhibits such properties of these estimators. It appears from figure that for smaller levels of significance, when π is fixed, the variation in the AMSE functions is greater. The large α values are used in figure to observe the variation between the maximum and minimum AMSE of the selected estimators. Moreover, the larger the value of π, the greater is the variation in the AMSE of ˆλ 1. Also, it may be seen that for smaller values of γ,

11 pooling methodology for coefficient of variation 67 when π and α are fixed, the variation in the AMSE functions is greater. Finally, we conclude none of the estimators is inadmissible to the any others under the local alternatives. However, under the null hypothesis, the dominance picture of the estimators is presented in the following theorem. Theorem 4.1. None of the five estimators is inadmissible with respect to the other four. However, at δ =0, the AMSE of the estimators may be ordered according to the magnitude of their AMSE as follows. ˆλ 1 ˆλ S 1 ˆλ P 1 ˆλ 1 λ 1, for a range of π,... (4.8) where denotes domination. In order to compare the performance of the estimators it is conventional to consider the asymptotic relative efficiency (ARE) of the estimators defined by ARE = AMSE( λ 1,λ 1 ) AMSE( λ 1,λ 1), while keeping in mind that a value of ARE greater than 1 signifies improvement of λ 1 over λ 1. The ARE of ˆλ 1 relative to λ 1 is given by where ARE 1 (α, δ,π)= AMSE( λ 1,λ 1 ) AMSE(ˆλ 1,λ 1 ) = 1 1+Ψ(δ ),...(4.9) Ψ(δ ) = δ (1 γ) { πg 3 (χ 1,α; δ ) π( π)g 5 (χ 1,α,δ ) } π( π)(1 γ)g 3 (χ 1,α; δ )]. Similarly, the asymptotic relative efficiency of...(4.10) ˆλ 1 relative to ˆλ 1 is given by ARE (α, δ,π)= AMSE(ˆλ 1,λ 1 ) γ +(1 γ)δ = AMSE(ˆλ 1,λ 1 ) 1+Ψ(δ....(4.11) ) Now, we present the analysis of the asymptotic relative efficiencies. First we consider ˆλ 1 relative to λ 1. The ARE 1 is a function of α, δ and π. This function for α 0 has its maximum at δ = 0 with value E = {1 π( π)(1 γ)g 3 (χ 1,α;0)} 1 ( 1)....(4.1) Moreover, for fixed values of α and π, ARE 1 decreases as δ increases from 0, crossing the line ARE 1 = 1, attains a minimum value at a point δ o and then increases asymptotically to 1. However, for fixed π, E is a decreasing function of α while the minimum efficiency is an increasing function of α. We have plotted ARE(α, π, δ ) against δ for γ =0.5 and α =0.05, 0.10, 0.0, at selected values of π in figure 3.

12 68 s.e. ahmed It appears from figure 3 that the smaller the value of α, the greater is the variation in the ARE. On the other hand, for any fixed α, the maximum value of ARE is an increasing function of π and the minimum efficiency is a decreasing function of π. The shrinkage factor π may also be viewed as a variation controlling factor among the maximum and minimum efficiencies. Next, we compare ˆλ 1 with ˆλ 1. First, we note that under the null hypothesis γ ARE (α, 0,π)= 1 π( π)(1 γ)g 3 (χ ( γ). 1,α ;0)

13 pooling methodology for coefficient of variation 69 Thus, γ ARE (α, 0,π) 1 ARE 1 (α, 0,π). On the other hand, as δ moves away from the the null hypothesis ARE (α, δ,π) 1 whenever δ 1+(1 γ) 1 Ψ(δ ) ˆλ Thus, except when δ is small, 1 is relatively more efficient than ˆλ 1. Finally, noticing that the ARE 1 of ˆλ 1 depends on α, the size of the preliminary test, which must be determined by the user. One method to determine α is to compute the minimum guaranteed efficiency. We outline this procedure in the following section. 5. Maximum efficiency criterion for selecting the estimators The ARE of ˆλ 1 is a function of δ, α and π and one method to determine α and π is to use a maxmin rule given by Ahmed (199) among others. We preassign a value of the minimum efficiency (E o ) that we are willing to accept to use this rule. Consider the set A = {α, π ARE(α, π, δ ) E o, δ }....(5.1) The estimator is chosen which maximizes ARE (α, π, δ )overallα, π A and δ. Thus, we solve for α and π such that { } sup inf ) = E(α,π )=E o....(5.) α,π A δ For given π = π o, we determine the value of α such that { inf α, π o,δ ) } = E(α,π o )=E o....(5.3) δ sup α A

14 70 s.e. ahmed TABLE 1. MAXIMUM AND MINIMUM ASYMPTOTIC GUARANTEED EFFICIENCIES FOR γ =0.4. α/π E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo

15 pooling methodology for coefficient of variation 71 TABLE. MAXIMUM AND MINIMUM ASYMPTOTIC GUARANTEED EFFICIENCIES FOR γ =0.5. α/π E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo E E δo

16 7 s.e. ahmed TABLE 3. MAXIMUM AND MINIMUM ASYMPTOTIC GUARANTEED EFFICIENCIES FOR γ =0.6. α/π E E δ E E δ E E δ E E δ E E δ E E δ E E δ E E δ E E δ E E δ E E δ E E δ E E δ0

17 pooling methodology for coefficient of variation 73 Tables 1 3 provide the maximum relative efficiencies E, minimum relative efficiencies E o and δo at which the minimum occurred for π =0.1(0.1)1.0 and γ =0.4, 0.5, 0.6 respectively. The tables for other values of γ are also prepared but not provided here to save the space. These tables are prepared for those values of α as were used by Han and Bancroft (1968) among others. However, the maximum and minimum efficiency may be readily computed for smaller values of the α. We notice that for smaller values γ, while π and α are held constant, the variation in the maximum ARE and the minimum ARE is greater. Infact, the maximum ARE is a decreasing function of γ, while the minimum ARE is an increasing function of γ. Furthermore, it is evident from the tables that maximum relative efficiency (E ) increases with increase in π and minimum relative efficiency (E o ) decreases. Hence, there does not exist a π satisfying (5.). The value of π can be determined by the researcher according to his prior belief in the uncertain prior information. However, we recommend the following step for selecting the size of the preliminary test. Suppose the experimenter does not know the size of the test but asserts π = π o and wishes to accept an estimator which has relative efficiency not less than E o. Then the maxmin principle determines α = α such that ARE (α, π o,δ )=E o. Therefore, a user who wishes to find a good alternative to the SRE, RE and UE should be able to specify the minimum relative efficiency (E o ). Example 5.1. Suppose times were recorded by the 100 meter and 1000 meter runners of Canadian pre-olympic track team. After viewing both the sample data of running times, one of the coaches commented that the the running times for both races are the same. Since, the standard deviation in this case is not a good measure of variability in the ability of runners, therefore the coaches wish to estimate the coefficient of variation based on the two data set. The sample mean and sample variance based on 30 samples for 100 meter and 1000 meter running times (in minutes) is calculated and then the coefficient of variation of both data computed as and respectively. Thus, λ 1 =0.058 and λ = The calculated value of the test statistic D n is Since both the samples are reasonably large, the significant value for the distance statistic may be determined by using table 1. Furthermore, if the coaches suspect that π =0.4 and are looking for an estimator with a minimum ARE of at least 0.80, then from table 1 α is Such a choice of α would yield an estimator with a maximum efficiency of 1.38 at δ = 0 with a minimum guaranteed efficiency of 0.80 at δo =5.. The critical value based on 1 degree of freedom is 3.84 and the test is not significant. Hence, the estimate of λ 1 is ˆλ 1 = ˆλ S 1 = On the other hand, if the user wishes to rely on data completely and uses π = 1 when H o is accepted, then from table 1 the size of the preliminary test will be approximately 0.0. In addition, the maximum efficiency drops from 1.38 to 1.0. The use of PE may be limited due to the

18 74 s.e. ahmed large size of Œa, pha, the level of significance, as compared to E. The E has a remarkable edge over PE with respect to the size of the preliminary test. 6. Concluding remarks We have presented the shrinkage preliminary test estimator of the coefficient of variation of a normal distribution when additional sample is aorks having the following features: (1) Each network consists of logical nodes (or events) and directed branches (or activities). () A branch has a probability that the activity associated with it will be performed. (3) Other parameters describe the activities represented by the branches. In this paper, however, reference will be made to a sample size parameter only. The sample size n associated with a branch is characterised by the moment generating function (mgf) of the form M n (θ) = exp (nθ)f(n), where f(n) n denotes the density function of n and θ is any real variable. The probability φ that the branch is realised is multiplied by the mgf to yield the W -function such that W (θ) =φm n (θ)...(3.1) The W -function is used to obtain the information on the relationship which exists between the nodes. 4. GERT analysis of the plan The possible states of the C-1 inspection system described in section () can be defined as follows: S 0 : Initial state of the plan. S 1 (k) : State in which k(= 1,..., i) preceding units are found clear of defects during 100% inspection. : Initial state of partial inspection. S : State in which a unit is not inspected (i.e. passed) during sampling inspection.

19 pooling methodology for coefficient of variation 75 A : State in which a unit is found free of defects during partial (sampling) inspection. R : State in which a unit is found defective during partial inspection. S A : State in which current unit is accepted. S R : State in which current unit is rejected. The above states enable us to construct GERT network representation of the inspection system as shown in Fig. (1) and (). Suppose that the process is in statistical control, so that the probability of any incoming unit being defective is (p) and the probability of any unit being non-defective is q =1 p. First of all, we will show that the probability of acceptance and rejection of a unit during sampling inspection [see Fig. (1)] is same as that of its acceptance and rejection during 100% inspection. Now, by applying Mason s (1953) rule in the representation in Fig. (1), the W -functions from the initial node S 0 to the terminal nodes A and R are respectively found as W 1A (θ) = fqi+1 [1 (1 f)e θ ]e θ + f(1 f)q i+1 e θ 1 [(1 q i )+(1 f)e θ ]+(1 f)(1 q i )e θ...(4.1) and W 1R (θ) = fpqi+1 e θ [1 (1 f)e θ ]+fpq i+1 (1 f)e θ 1 [(1 q i )+(1 f)e θ ]+(1 f)(1 q i )e θ...(4.) From the W -functions defined above, we obtain the probability that a unit is accepted and rejected respectively by sampling procedure as [W 1A (θ)] θ=0 = q

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