Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
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1 Metodološki zvezki, Vol. 13, No., 016, Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece iterval for the reciprocal of a ormal populatio mea with a kow coefficiet of variatio is proposed. This has applicatios i the area of uclear physics, agriculture ad ecoomic whe the researcher kows the coefficiet of variatio. The proposed cofidece iterval is based o the approximate expectatio ad variace of the estimator by Taylor series expasio. A Mote Carlo simulatio study was coducted to compare the performace of the proposed cofidece iterval with the existig cofidece iterval. Simulatio results show that the proposed cofidece iterval performs as well as the existig oe i terms of coverage probability. However, the approximate cofidece iterval is very easy to calculate compared with the exact cofidece iterval. 1 Itroductio The reciprocal of a ormal mea is applied i the area of uclear physics, agriculture ad ecoomic. For example, Lamaa et al. (1981) studied a charged particle mometum, reciprocal of a ormal mea is defied by p µ 1 where µ is the track curvature of a particle. The θ µ 1, where µ is the populatio mea. May researchers studied the reciprocal of a ormal mea. For istace, Zama (1981) discussed the estimators without momets i the case of the reciprocal of a ormal mea. The maximum likelihood estimate of the reciprocal of a ormal mea with a class of zero-oe loss fuctios was proposed by Zama (1985). Withers ad 1 Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techology, Thammasat Uiversity, Thailad; wararit@mathstat.sci.tu.ac.th
2 118 Wararit Paichkitkosolkul Nadarajah (013) preseted the theorem to costruct the poit estimators for the iverse powers of a ormal mea. Wogkhao et al. (013) proposed two cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio. Their cofidece itervals ca be applied whe the coefficiet of variatio of a cotrol group is kow. Oe of their cofidece itervals is developed based o a exact method i which this cofidece iterval is costructed from the pivotal statistics Z, where Z follows the stadard ormal distributio. The other cofidece iterval is costructed based o the geeralized cofidece iterval (Weerahadi, 1993). Simulatio results show that the coverage probabilities of the two cofidece itervals are ot sigificatly differet. However, the cofidece iterval based o the exact method is shorter tha the geeralized cofidece iterval. The exact method uses Taylor series expasio to fid the expectatio ad variace of the estimator of θ ad uses these results for costructig the cofidece iterval for θ. The lower ad upper limits of the cofidece iterval based o the exact method are difficult to compute sice they deped o a ifiite summatio. Therefore, our mai aim i this paper is to propose a approximate cofidece iterval for the reciprocal of a ormal mea with a kow coefficiet of variatio. The computatio of the ew proposed cofidece iterval is easier tha the exact cofidece iterval proposed by Wogkhao et al. (013). I additio, we also compare the estimated coverage probabilities of the ew proposed cofidece iterval ad existig cofidece iterval usig a Mote Carlo simulatio. The paper is orgaized as follows. I Sectio, the theoretical backgroud of the existig cofidece iterval for θ is discussed. We provide the theorem for costructig the approximate cofidece iterval for θ i Sectio 3. I Sectio 4, the performace of the cofidece itervals for θ is ivestigated through a Mote Carlo simulatio study. Coclusios are provided i the fial sectio. Existig Cofidece Iterval I this sectio, we review the theorem ad corollary proposed by Wogkhao et al. (013) ad use these to costruct the exact cofidece iterval for θ.
3 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 119 Theorem 1. (Wogkhao et al., 013) Let Y,..., 1 Y be a radom sample of size from a ormal distributio with mea µ ad variace σ. The estimator of θ is 1 1 where θˆ Y Y Y. The expectatio of ˆ θ ad j 1 j σ variatio, τ is kow, are respectively µ k ˆ ( k)! τ E( θ) θ 1+ k k 1 k! ˆ θ whe a coefficiet of (1) ad k ˆ (k + 1)! τ ( ). k k 1 k! E θ θ Proof of Theorem 1 See Wogkhao et al. (013) ˆ k From (1), lim E( ˆ θ ( k)! τ θ) θ ad E θ, w where w 1 +. k k 1 k! Thus, the ubiased estimator of θ is ˆ θ 1. w wy Corollary 1. From Theorem 1, θ ˆ var( θ) τ. Proof of Corollary 1 See Wogkhao et al. (013) Now we will use the fact that, from the cetral limit theorem, ˆ θ θ Z N(0,1). var( ˆ θ ) Based o Theorem 1 ad Corollary 1, we get Z ˆ θ θ w N(0,1). () θ τ Therefore, the 100(1 α)% exact cofidece iterval for θ based o Equatio () is ˆ θ ˆ θ CIexact ± z1 α / τ, w k ( k)! τ where w 1+ k k 1 k! ad z 1 α is the 100(1 α / ) percetile of the stadard / ormal distributio.
4 10 Wararit Paichkitkosolkul 3 Proposed Cofidece Iterval To fid a simple approximate expressio for the expectatio of ˆ, θ we use a Taylor series expasio of 1 y aroud µ : 1 y ( ) ( ) ( ) y µ + y µ + O y µ y y y y y y y y µ µ µ (3) Theorem. Let with mea µ ad variace Y,..., 1 Y be a radom sample of size from a ormal distributio σ. The estimator of θ is θˆ Y 1 where 1 Y Yi i 1 The approximate expectatio ad variace of ˆ θ whe a coefficiet of variatio, σ τ is kow, are respectively µ ˆ 1 τ E( θ ) 1+ (3) µ ad var( ˆ θ θ) τ. (4). Proof of Theorem. Cosider radom variable Y where Y has support (0, ). Let ˆ 1. θ Y Fid approximatios for E( ˆ θ ) ad var( ˆ θ ) usig Taylor series expasio of ˆ θ aroud µ as i Equatio (3). The mea of ˆ θ ca be foud by applyig the expectatio operator to the idividual terms (igorig all terms higher tha two), E( ˆ θ ) E 1 Y E E ( Y EY ( )) E ( Y EY ( )) O ( ) Y Y Y Y Y 3 µ µ µ var( Y ) µ ( EY ( )) 1 var( Y + ) µ µ 1 σ 1 + µ µ 1 τ 1 +. µ 3 (4)
5 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 11 A approximatio of the variace of ˆ θ is obtaied by usig the first-order terms of the Taylor series expasio: var( ˆ θ ) var 1 Y 1 1 E E Y Y 1 1 E Y µ E + ( Y EY ( )) µ Y Y µ 1 Y Y var( Y ) 4 µ σ 4 µ var( Y ) µ µ θ τ. (5) It is clear from Equatio (4) that ˆ θ is asymptotically ubiased ( lim E( ˆ θ) θ) ˆ θ τ ad E θ, v where v 1 +. Therefore, the ubiased estimator of θ is From Equatio (5), ˆ θ is cosistet ˆ ( θ ) lim var( ) 0. ˆ θ 1. v vy We the apply the cetral limit theorem ad Theorem, Z ˆ θ θ v N(0,1). θ τ Therefore, it is easily see that the (1 α)100% approximate cofidece iterval for θ is CIapprox ˆ θ ± z v 1 α / ˆ θ τ,
6 1 Wararit Paichkitkosolkul τ where v 1+ ad z1 α / is the 100(1 α / ) percetile of the stadard ormal distributio. 4 Simulatio Study A Mote Carlo simulatio was coducted usig the R statistical software [16] versio 3..1 to compare the estimated coverage probabilities of the ew proposed cofidece iterval ad the exact cofidece iterval. Source code is available i Appedix. The estimated coverage probability (based o M replicates) are give by 1 α #( L θ U) / M, where #( L θ U) deotes the umber of simulatio rus for which the true reciprocal of a ormal mea θ lies withi the cofidece iterval. From two previous sectios, we foud that the legths of both cofidece itervals are equal to z ˆ 1 α / θτ / which the expected legths are ot cosidered i simulatio study. The sets of ormal data were geerated with θ 0.1, 0., 0.5, 1, 5 ad 10, ad the coefficiet of variatio τ 0.05, 0.1, 0., 0.33, 0.5 ad The sample sizes were set at 10, 0, 30, 50, 100 ad 500. The umber of simulatio rus was 10,000 ad the omial cofidece level 1 α was fixed at The results are demostrated i Figure 1 ad Tables 1-4. Both cofidece itervals have estimated coverage probabilities close to the omial cofidece level for almost situatios. However, the estimated coverage probabilities of the exact cofidece iterval are very poor whe the coefficiet of variatio τ is close to 1 ad small sample sizes. Additioally, the estimated coverage probabilities of the cofidece itervals do ot icrease or decrease accordig to the values of τ ad. The estimated coverage probabilities of the proposed cofidece iterval are ot sigificatly differet from these of the exact cofidece iterval i ay situatio. However, the approximate cofidece iterval is very easy to calculate compared with the exact cofidece iterval because the exact cofidece iterval is based o a ifiite summatio.
7 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 13 Coverage Probabilities Exact Approx 0.1 Coverage Probabilities Exact Approx Coverage Probabilities Exact Approx 0.5 Coverage Probabilities Exact Approx Coverage Probabilities Exact Approx 5 Coverage Probabilities Exact Approx Figure 1: Estimated coverage probabilities of cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio whe 30 (solid lie) ad 100 (dash lie)
8 14 Wararit Paichkitkosolkul Table 1: Estimated coverage probabilities of cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio whe θ 0.1 ad 0.. τ θ 0.1 θ 0. Exact Approx. Exact Approx
9 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 15 Table : Estimated coverage probabilities of cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio whe θ 0.5 ad 1. τ θ 0.5 θ 1 Exact Approx. Exact Approx
10 16 Wararit Paichkitkosolkul Table 3: Estimated coverage probabilities of cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio whe θ 5 ad 10. τ θ 5 θ 10 Exact Approx. Exact Approx A Illustrative Example To illustrate a example of two cofidece iterval for the reciprocal of a ormal mea proposed i the previous sectio, we used the weights (i kilograms) of 61 oe-moth old ifats listed as follows:
11 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea The data were take from the study by Ziegler et al. (007) (cited i Ledolter ad Hogg, 010, p.87). From past experiece, we assume that the coefficiet of variatio of the weights of 61 oe-moth old ifats is about The histogram, desity plot, Box-ad-Whisker plot ad ormal quatile-quatile plot are displayed i Figure. Algorithm 1 shows the result of the Shapiro-Wilk ormality test. (a) (b) Frequecy Desity weight weight (c) Sample Quatiles (d) Theoretical Quatiles Figure : (a) Histogram, (b) desity plot, (c) Box-ad-Whisker plot ad (d) ormal quatile-quatile plot of the weight of a oe-moth old ifat
12 18 Wararit Paichkitkosolkul Shapiro-Wilk ormality test data: weight W 0.978, p-value Algorithm 1: Shapiro-Wilk test for ormality of the weight of a oe-moth old ifat The 95% exact ad approximate cofidece itervals for the reciprocal of a ormal mea are calculated ad reported i Table 4. The lower ad upper limits of the both cofidece itervals are ot differet. Table 4: The 95% cofidece itervals for the reciprocal of a ormal mea of the weight of a oe-moth old ifat. Methods Cofidece Itervals Lower Limit Upper Limit Legths Exact Approximate Coclusios I this paper, we proposed a approximate cofidece iterval for the reciprocal of a ormal populatio mea with a kow coefficiet of variatio. Normally, this arises whe the coefficiet of variatio of the cotrol group is kow. The approximate cofidece iterval proposed uses the approximatio of the expectatio ad variace of the estimator. The proposed ew cofidece iterval is compared with the exact cofidece iterval costructed by Wogkhao et al. (013) through a Mote Carlo simulatio study. The approximate cofidece iterval performs as efficietly as the exact cofidece iterval i terms of coverage probability. Moreover, approximate cofidece iterval also is easy to compute compared with the exact cofidece iterval. Appedix: Source R code for all cofidece itervals ci.exact <- fuctio(y,tao,alpha) { <- legth(y) ybar <- mea(y) zeta.hat <- 1/ybar w <- cal.w(tao,)
13 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 19 } z <- qorm(1-alpha/) T1 <- (tao^)/(*(ybar^)) lower <- (zeta.hat/w)-z*sqrt(t1) upper <- (zeta.hat/w)+z*sqrt(t1) out <- cbid(lower,upper) retur(out) ci.approx <- fuctio(y,tao,alpha) { <- legth(y) ybar <- mea(y) zeta.hat <- 1/ybar v <- 1+(tao^)/ z <- qorm(1-alpha/) T1 <- ((zeta.hat^)*(tao^))/ lower <- (zeta.hat/v)-z*sqrt(t1) upper <- (zeta.hat/v)+z*sqrt(t1) out <- cbid(lower,upper) retur(out) } cal.w <- fuctio(tao,) { temp <- rep(0,50) for (k i 1:50) { temp[k] <- (factorial(*k)/((^k)*factorial(k)))*(((tao^)/)^k) } w <- 1+sum(temp) retur(w) } Ackowledgemets The author is grateful to two aoymous referees for their valuable commets ad commets, which have sigificatly ehaced the quality ad presetatio of this paper. The author is also thakful for the support i the form of the research fuds awarded by Thammasat Uiversity. Refereces [1] Ihaka, R. ad Getlema, R. (1996): R: A laguage for data aalysis ad graphics. Joural of Computatioal ad Graphical Statistics, 5,
14 130 Wararit Paichkitkosolkul [] Lamaa, E., Romao, G. ad Sgrbi, C. (1981): Curvature measuremets i uclear emulsios. Nuclear Istrumets ad Methods, 187, [3] Ledolter, L., Hogg, R.V. (010): Applied Statistics for Egieers ad Physical Scietists, Pearso, New Jersey. [4] Weerahadi, S. (1993): Geeralized cofidece itervals, Joural of the America Statistical Associatio, 88, [5] Withers, C.S. ad Nadarajah, S. (013): Estimators for the iverse powers of a ormal mea, Joural of Statistical Plaig ad Iferece, 143, [6] Wogkhao, A., Niwitpog, S. ad Niwitpog, S. (013): Cofidece iterval for the iverse of a ormal mea with a kow coefficiet of variatio. Iteratioal Joural of Mathematical, Computatioal, Statistical, Natural ad Physical Egieer, 7, [7] Zama, A. (1981): Estimators without momets: the case of the reciprocal of a ormal mea. Joural of Ecoometrics, 15, [8] Zama, A. (1985): Admissibility of the maximum likelihood estimate of the reciprocal of a ormal mea with a class of zero-oe loss fuctios. Sakhyā: The Idia Joural of Statistics, Series A, 47, [9] Ziegler, E., Nelso, S.E., Jeter, J.M. (007): Early iro supplemetatio of breastfed ifats, Departmet of Pediatrics, Uiversity of Iowa, Iowa City, USA.
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