Computatioal Statistics & Data Aalysis 43 (23) 283 298 www.elsevier.com/locate/csda Modied momet estimatio for the two-parameter Birbaum Sauders distributio H.K.T. Ng a, D. Kudu b, N. Balakrisha a; a Departmet of Mathematics ad Statistics, McMaster Uiversity, Hamilto, Ot., Caada L8S 4K1 b Departmet of Mathematics, Idia Istitute of Techology Kapur, Kapur 2816, Idia Received 1 March 22; received i revised form 1 July 22 Abstract The maximum likelihood estimators ad a modicatio of the momet estimators of a twoparameter Birbaum Sauders distributio are discussed. A simple bias-reductio method is proposed to reduce the bias of the maximum likelihood estimators ad the modied momet estimators. The jackkife techique is also used to reduce the bias of these estimators. Mote Carlo simulatio is used to compare the performace of all these estimators. The probability coverages of codece itervals based o iferetial quatities associated with all these estimators are evaluated usig Mote Carlo simulatios for small, moderate ad large sample sizes. Two illustrative examples ad some cocludig remarks are ally preseted. c 22 Elsevier Sciece B.V. All rights reserved. Keywords: Birbaum Sauders distributio; Jackkife estimate; Bias-reduced estimator; Mote Carlo simulatios; Probability coverage; Codece iterval 1. Itroductio The two-parameter Birbaum Sauders distributio was origially proposed (Birbaum ad Sauders, 1969a) as a failure time distributio for fatigue failure caused uder cyclic loadig. It was also assumed that the failure is due to the developmet ad growth of a domiat crack. A more geeral derivatio was provided by Desmod (1985) based o a biological model. Desmod (1985) also stregtheed the physical justicatio for the use of this distributio by relaxig the assumptios made by Birbaum ad Sauders (1969a). Desmod (1986) ivestigated the relatioship betwee the Birbaum Sauders distributio ad the iverse Gaussia distributio. Some Correspodig author. E-mail address: bala@mcmail.cis.mcmaster.ca (N. Balakrisha). 167-9473/3/$ - see frot matter c 22 Elsevier Sciece B.V. All rights reserved. PII: S167-9473(2)254-2
284 H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 recet works o Birbaum Sauders distributio ca be foud i Chag ad Tag (1993, 1994), Dupuis ad Mills (1998) ad Rieck (1995, 1999), ad a review of these developmets ca be foud i Johso et al. (1995). The cumulative distributio fuctio (CDF) of a two-parameter Birbaum Sauders radom variable T ca be writte as F T (t; ; )= [ 1 { ( t ) 1=2 ( ) }] 1=2 ; t ; ; ; (1) t where ( ) is the stadard ormal CDF. The parameters ad are the shape ad the scale parameters, respectively. It is kow that the desity fuctio of the Birbaum Sauders distributio is uimodal ad although the hazard rate is ot a icreasig fuctio of t but the average hazard rate is early a o-decreasig fuctio of t (Ma et al., 1974, p. 155). The maximum likelihood estimators (MLEs) were discussed origially by Birbaum ad Sauders (1969b) ad their asymptotic distributios were obtaied by Egelhardt et al. (1981). Although the MLEs have several optimal properties, oe still eeds to solve a o-liear equatio i to obtai the solutio; for this purpose, Birbaum ad Sauders (1969b) suggested some iterative schemes to solve the required o-liear equatio. Also, the exact distributio of the MLEs are ot available. Therefore, for costructig codece itervals of the ukow parameters ad, the asymptotic distributios of the MLEs eed to be used. However, it is ot kow how these asymptotic codece itervals behave i the case of small sample sizes. Moreover, the covetioal momet estimators also have a diculty i that they may ot always exist ad eve if they do, they may ot be uique. For this reaso, modied momet estimators (MMEs) for ad are rst proposed. The MMEs are very easy to compute ad they have explicit expressios i terms of the sample observatios. Ulike the momet estimators, MMEs always exist uiquely. The asymptotic distributios of the MMEs are derived which are the used to costruct co- dece itervals for the ukow parameters. The performace of all these estimators is evaluated through simulatios. Eve though the MLEs ad MMEs are asymptotically ubiased, these simulatio results reveal that they are highly biased i case of small sample sizes. A simple bias correctio techique is proposed which performs quite well eve for small sample sizes. Jackkife procedure is also used to reduce the bias of the MLEs ad MMEs ad is show to work very well i this case; but, this procedure becomes computatioally quite ivolved i case of large sample sizes. 2. The Birbaum--Sauders distributio The CDF of a two-parameter Birbaum Sauders radom variable T is give by (1) ad the correspodig probability desity fuctio (PDF) is [ ( ) 1=2 ( ) ] 3=2 [ 1 f T (t; ; )= 2 + exp 1 ( t 2 t t 2 2 + )] t 2 ; t ;; : (2)
or H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 285 Cosider the followig mootoe trasformatio: [ (T X = 1 ) 1=2 ( ) ] 1=2 T 2 T = (1+2X 2 +2X (1 + X 2 ) 1=2 ); the, from (1), we have X to be a ormally distributed with mea zero ad variace 1 4 2. Usig the above trasformatio, the expected value, variace, ad coeciets of skewess ad kurtosis ca be easily obtaied as E(T)=(1 + 1 2 2 ); (3) Var(T )=() 2 (1 + 5 4 2 ); (4) 1 (T )= 162 (11 2 +6) (5 2 +4) 3 ; (5) 2 (T )=3+ 62 (93 2 + 41) (5 2 +4) 2 : (6) Moreover, if T has a Birbaum Sauders distributio with parameters ad, the T 1 also has a Birbaum Sauders distributio with the correspodig parameters ad 1, respectively (Birbaum ad Sauders, 1969a). Therefore, we also readily have ad E(T 1 )= 1 (1 + 1 2 2 ) (7) Var(T 1 )= 2 2 (1 + 5 4 2 ): (8) 3. Maximum likelihood estimators Let {t 1 ;t 2 ;:::;t } be a radom sample of size from the Birbaum Sauders distributio with the PDF as give i (2). The sample arithmetic ad harmoic meas are deed by s = 1 t i ; r = [ 1 t 1 i ] 1 : Let us further dee the harmoic mea fuctio K by [ ] 1 1 K(x)= (x + t i ) 1 for x ; so that r K().
286 H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 The MLE of (deoted by ˆ) ca be obtaied as the uique positive root of the equatio 2 [2r + K()] + r[s + K()]=: (9) Oce ˆ is obtaied as a solutio of (9), the MLE of (deoted by ˆ) ca be obtaied explicitly as [ s ˆ = ˆ + ˆ ] 1=2 r 2 : Sice (9) is a o-liear equatio i, oe eeds to use a iterative procedure to solve for ˆ. Birbaum ad Sauders (1969b) proposed two iterative procedures (oe simple ad oe complicated) to compute ˆ, but oted that the simple oe works very well for small ( 1 2 ) but may ot work at all for large ( 2). The complicated oe also does ot work i certai rage of the sample space. I this paper, Newto Raphso method is used for the computatio of the MLEs. Egelhardt et al. (1981) showed that the asymptotic joit distributio of ˆ ad ˆ is bivariate ormal ad is give by ( ) ( ) ˆ N ; ˆ where I()=2 2 2 2 [:25+ 2 +I()] {[1 + g(x)] 1 :5} 2 d(x); ; (1) ) 1=2 g(y)=1+ (1+ y2 2 + y y2 : 4 It is iterestig to observe that ˆ ad ˆ are asymptotically idepedet of each other. The asymptotic codece iterval of ca be easily obtaied from (1). Moreover, the asymptotic codece iterval of, for a give, ca also be obtaied from (1). 4. Modied momet estimators For the usual momet estimators i a two-parameter case, the rst ad secod populatio momets are equated with the correspodig sample momets. I this case, the sample mea ad the sample variace ca be equated to the right-had sides of (3) ad (4), respectively, ad the correspodig momet estimators of ad ca the be obtaied as solutios of ad to these equatios. It ca be easily see from these equatios that if the sample coeciet of variatio is greater tha 5, the the momet estimators do ot exist. If the sample coeciet of variatio is less tha 5, the momet estimators exist; however, the momet estimator of may ot be uique. Istead of usig (3) ad (4), we propose to use (3) ad (7) ad equate them with the correspodig sample estimates to obtai the MMEs. I this case, we have the
H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 287 followig two momet equatios: s = ( 1+ 1 2 2) ; (11) r 1 = 1 (1 + 1 2 2 ): (12) Solvig Eqs. (11) ad (12) for ad, we obtai the MMEs for ad deoted by ad as { [ ( s ) ]} 1=2 1=2 = 2 1 ; r =(sr) 1=2 : The asymptotic joit distributio of ad is bivariate ormal ad is give by ( ) ( ) 2 2 N ; ( 1+ 3 ) : (13) 4 2 () 2 (1+ 1 2 2 ) 2 The proof of this result is preseted i the appedix. Note that the MMEs ad are also asymptotically idepedet of each other, just as i the case of the MLEs. 5. Bias-reduced estimators Based o the results of a extesive Mote Carlo simulatio study, we observed that the MLEs ad the MMEs performed very similarly i terms of both bias ad mea square error, especially for small values of. Upo ispectig the patter of the bias of the MLEs ad MMEs, we observed that Bias(ˆ) Bias( ) ; Bias( ˆ) Bias( ) 2 4 : The, by employig a stadard bias reductio method, we ca simply costruct almost ubiased maximum likelihood estimators (UMLEs, deoted by ˆ ad ˆ ) ad almost ubiased modied momet estimators (UMMEs, deoted by ad )of ad. These bias-reduced estimators are give by ˆ = ( ) 1 ( ) = ; 1 = ) 1 ) 1 ˆ; ˆ = (1+ 2 ˆ (1+ ˆ 2 ˆ; (14) 4 4 (1+ 2 4 ) 1 ) 1 (1+ 2 : (15) 4
288 H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 From the distributioal results preseted i (1), we readily have the asymptotic joit distributio of ˆ ad ˆ to be bivariate ormal ad is give by ( ˆ ) ( ) 2 2( 1) N ; 2 ; (16) ˆ 16 2 (4+ 2 ) 2 [:25+ 2 +I()] similarly, from (13), the asymptotic joit distributio of ad is bivariate ormal ad is give by ( ) ( ) 2 2( 1) 2 N ; 1+ 3 ) : (17) 4 2 ( 16() 2 (4+ 2 ) 2 (1+ 1 2 2 ) 2 6. Jackkife estimators Jackkig is based o sequetially deletig oe sample poit t i ad recomputig the MLEs ad MMEs from the reduced sample of size 1. We remove the poit t j from the data set, ad the recompute r ad s ad also the fuctio K as s (j) = 1 1 i j 1 r (j) = 1 t i = s t j 1 ; i j 1 K (j) (x)= 1 ti 1 1 = (x + t i ) 1 i j r t 1 j 1 ; 1 = K(x) (x + t j) 1 : 1 The, we obtai ˆ (j) as the uique positive root of the equatio 2 [2r (j) + K (j) ()] + r (j) [s (j) + K (j) ()]= ad [ s (j) ˆ (j) = + ˆ ] 1=2 (j) 2 ; ˆ (j) r (j) Similarly, we d { [ (s(j) ) ]} 1=2 1=2 (j) = 2 1 ; r (j) (j) =(s (j) r (j) ) 1=2 :
H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 289 Let us ow dee ˆ ( ) = 1 ˆ (j) ; j=1 ˆ( ) = 1 ˆ (j) ; j=1 ( ) = 1 (j) ; j=1 ( ) = 1 (j) : j=1 The, the bias-corrected jackkife maximum likelihood estimates (JMLEs) of ad (see, for example, Efro, 1982) are give by ˆ J = ˆ ( 1) ˆ ( ) ; ˆ J = ˆ ( 1) ˆ ( ) ; similarly, the bias-corrected jackkife modied momet estimates (JMMEs) of ad are give by J = ( 1) ( ) ; J = ( 1) ( ) : 7. Mote Carlo simulatio results I order to compare the performace of all the above estimators, we performed a simulatio study for dieret sample sizes ad for dieret parameter values. We took the sample size as =5; 1; 2; 5; 1, ad the shape parameter as =:1; :25; :5; 1:; 2:. Sice is the scale parameter, was kept xed at 1., without loss of ay geerality. Of course, the values of bias ad stadard deviatio of estimates of simply eed to be multiplied by (if it is dieret from 1) while the bias ad stadard deviatio of estimates of will ot be aected. All the results were based o 1, Mote Carlo rus. We computed the MLE, MME, UMLE, UMME, JMLE ad JMME for each ru, ad the computed the average estimates ad the stadard deviatios over the 1, rus for all these estimators. The results so obtaied are reported i Tables 1 ad 2. We also computed the 9% ad 95% probability coverages of codece itervals based o iferetial quatities associated with all these estimators usig the asymptotic distributios give earlier. Specically, the 1(1 )% codece itervals for ad based o the MLEs ad UMLEs are give by [ ( 1 ( ) ] 1 z=2 z1 =2 ˆ +1) ; ˆ +1 ; 2 2 ( ) 1 ( ) 1 z =2 ˆ h1 (ˆ) +1 ; ˆ z 1 =2 h1 (ˆ) +1 ;
29 H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 Table 1 Meas of estimates based o Mote Carlo simulatio ( = 1:) Estimate of Estimate of MLE MME UMLE UMME JMLE JMME MLE MME UMLE UMME JMLE JMME 5.1.842.842.153.153.115.115 1.7 1.7 1.4 1.4.9997.9997.25.212.212.2629.2629.2536.2536 1.54 1.54 1.22 1.22.9992.9992.5.4186.4186.5236.5236.568.568 1.225 1.225 1.81 1.81.9984.9984 1..8273.827 1.351 1.348 1.115 1.112 1.835 1.832 1.241 1.242.9927.9922 2. 1.6199 1.6142 2.279 2.21 2.251 2.182 1.2524 1.2524 1.422 1.444.919.9193 1.1.924.924.123.123.15.15 1.2 1.2 1.5 1.5.9997.9997.25.231.231.2557.2557.2513.2513 1.24 1.24 1.2 1.2.9993.9993.5.461.461.513.513.524.524 1.14 1.14 1.61 1.61.9985.9985 1..9168.9166 1.15 1.148 1.46 1.46 1.38 1.379 1.148 1.15.9955.9953 2. 1.8181 1.814 2.134 2.89 2.124 2.11 1.117 1.129 1.25 1.56.973.9738 2.1.964.964.111.111.12.12 1.1 1.1 1. 1..9999.9999.25.248.248.2525.2525.256.256 1.12 1.12 1.5 1.5.9997.9997.5.4811.4811.546.546.512.512 1.53 1.53 1.21 1.21.9994.9994 1..9596.9595 1.64 1.63 1.23 1.23 1.19 1.19 1.59 1.59.9986.9986 2. 1.9122 1.999 2.57 2.32 2.56 2.53 1.466 1.479.9952.9966.9937.9941 5.1.984.984.15.15.999.999.9999.9999 1.1 1.1.9998.9998.25.2459.2459.2513.2513.2497.2497 1. 1. 1.4 1.4.9994.9994.5.4915.4915.523.523.4994.4994 1.13 1.13 1.12 1.12.9989.9989 1..982.982 1.35 1.35.9987.9987 1.59 1.59 1.3 1.3.998.998 2. 1.9615 1.965 2.44 2.34 1.9976 1.9976 1.151 1.157.9981.9987.9966.9966 1.1.992.992.12.12.999.999.9999.9999 1.1 1.1.9998.9998.25.2479.2479.254.254.2498.2498.9999.9999 1.4 1.4.9996.9996.5.4957.4957.57.57.4996.4996 1.4 1.4 1.1 1.1.9992.9992 1..999.999 1.8 1.8.9992.9992 1.25 1.25 1.21 1.21.9986.9986 2. 1.985 1.98 2.3 1.9998 1.9984 1.9984 1.67 1.7.9996 1.1.9978.9978
H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 291 Table 2 Stadard deviatios of estimates based o Mote Carlo simulatio ( = 1:) Estimate of Estimate of MLE MME UMLE UMME JMLE JMME MLE MME UMLE UMME JMLE JMME 5.1.36.36.383.383.373.373.449.449.445.445.449.449.25.764.764.957.957.935.935.1126.1126.1112.1112.112.112.5.1524.1524.198.198.1883.1883.2263.2263.2216.2216.2225.2223 1..338.334.385.38.3858.3851.4583.4583.4367.4368.4464.4382 2..622.6131.7764.7678.8319.8164.9511.9553.8657.8684 1.513.9397 1.1.219.219.246.246.239.239.314.314.315.315.313.313.25.547.547.616.616.597.597.782.782.784.784.78.78.5.192.192.123.123.12.12.155.155.1544.1544.1534.1533 1..2185.2183.2459.2457.2435.2434.2979.2979.2912.2915.294.2882 2..4445.441.51.4961.56.536.5213.5261.4887.4939.5177.4874 2.1.155.155.165.165.162.162.225.225.223.223.225.225.25.388.388.414.414.45.45.56.56.554.554.559.559.5.776.776.827.827.812.812.111.111.188.188.195.195 1..1554.1553.1655.1654.1635.1635.253.256.22.222.219.216 2..3139.3125.3344.3329.3323.332.328.334.3153.326.3182.3156 5.1.99.99.12.12.11.11.142.142.14.14.142.142.25.249.249.255.255.253.253.354.354.349.349.353.353.5.497.497.59.59.55.55.693.693.684.684.691.691 1..994.994.118.118.114.114.1273.1275.1258.1258.1263.1264 2..1997.1994.244.24.237.237.1935.1979.194.1934.191.193 1.1.7.7.72.72.71.71.1.1.99.99.1.1.25.176.176.18.18.177.177.249.249.246.246.249.249.5.351.351.36.36.354.354.487.487.48.48.486.486 1..72.72.72.72.79.79.89.891.88.88.887.887 2..147.146.1443.1441.1421.1421.1341.1366.1323.1344.133.1349
292 H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 ad [ ( ) 1 ( ) 1 ] ˆ z =2 2 ( 1) +1 ; ˆ z 1 =2 2 ( 1) +1 ; [ ( ) 1 ( ) ] 1 ˆ 4z =2 h 1 (ˆ) (4 +ˆ 2 ) +1 ; ˆ 4z 1 =2 h 1 (ˆ) (4 +ˆ 2 ) +1 ; where h 1 (x)=:25 + x 2 + I(x) ad z p is the 1pth percetile of the stadard ormal distributio. Similarly, the 1(1 )% codece itervals for ad based o the MMEs ad UMMEs are give by [ ( z=2 2 +1) 1 ; ( z1 =2 2 +1 ) ] 1 ; ( ) 1 ( ) 1 z =2 h2 ( ) +1 ; z 1 =2 h2 ( ) +1 ; ad [ ( ) 1 ( ) 1 ] z =2 2 ( 1) +1 ; z 1 =2 2 ( 1) +1 ; [ ( ) 1 ( ) ] 1 4z =2 h 2 ( ) (4 + 2 ) +1 ; 4z 1 =2 h 2 ( ) (4 + 2 ) +1 ; where h 2 (x)= 1+ 3 4 x2 (1 + 1 2 x2 ) : 2 These results are reported i Tables 3 ad 4, respectively. From the simulatio results, it is clear that the performace of the MLEs ad MMEs are almost idetical for dieret sample sizes ad if the shape parameter is ot too large. The average estimates of the MLEs ad MMEs ad their stadard deviatios coicide upto four decimal places if is less tha.5. It is also evidet from these results that the MLEs ad the MMEs are both highly biased if is small ad is large. The bias reductio method works very well i this case for both the parameters eve for small samples; but as expected, it icreases the correspodig stadard deviatio of the estimators. The performace of the JMLE ad JMME are almost idetical at least for small values of, ad they perform better tha UMLE ad UMME i terms of bias. However, JMLE ad JMME possess larger stadard deviatios tha UMLE ad UMME. I additio, UMLE ad UMME also are computatioally simpler while the jackkifed estimators will demad cosiderably high computatioal time i case of large sample sizes. The asymptotic codece itervals do ot work very well whe the sample size is very small as the coverage probabilities are much lower tha the correspodig omial
H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 293 Table 3 Probability coverages of 9% codece itervals based o Mote Carlo simulatio ( = 1:) Probability coverages for Probability coverages for MLE MME UMLE UMME MLE MME UMLE UMME 5.1 86.25 86.25 91.12 91.12 78.57 78.57 86.11 86.11.25 86.25 86.25 91.14 91.14 78.45 78.45 86.3 86.3.5 86.15 86.15 91.32 91.32 78.33 78.36 85.64 85.64 1. 85.57 85.58 91.45 91.49 78.2 78.16 84.25 84.32 2. 83.53 83.72 91.2 91.41 77.4 77.57 8.52 81.74 1.1 88.69 88.69 9.43 9.43 84.91 84.91 88.14 88.14.25 88.73 88.73 9.45 9.45 85. 85. 88.13 88.13.5 88.72 88.72 9.56 9.56 84.78 84.76 88.5 88.3 1. 88.35 88.36 9.73 9.75 84.72 84.87 87.3 87.38 2. 87.53 87.7 9.13 9.34 83.96 84.35 85.39 85.99 2.1 89.87 89.87 9.31 9.31 87.9 87.9 89.29 89.29.25 89.89 89.89 9.32 9.32 87.9 87.9 89.27 89.27.5 89.83 89.83 9.39 9.39 87.14 87.18 89.7 89.8 1. 89.62 89.64 9.42 9.45 86.97 87.3 88.75 88.73 2. 89.16 89.21 89.97 9.18 86.66 86.74 87.63 88.22 5.1 89.33 89.33 9.14 9.14 88.58 88.58 89.96 89.96.25 89.33 89.33 9.14 9.14 88.66 88.66 89.84 89.84.5 89.28 89.28 9.19 9.19 88.78 88.79 89.72 89.68 1. 89.25 89.24 9.11 9.11 88.67 88.8 89.41 89.56 2. 89.4 89.1 89.93 89.98 88.18 88.53 88.81 89.24 1.1 9.5 9.5 89.65 89.65 89.43 89.43 9.54 9.54.25 9.6 9.6 89.67 89.67 89.45 89.44 9.51 9.52.5 9.4 9.4 89.64 89.64 89.36 89.38 9.43 9.47 1. 9.3 9.3 89.64 89.64 89.21 89.4 9.11 9.26 2. 89.96 89.98 89.69 89.7 88.95 89.33 89.46 9.1 levels. But for sample sizes 2 or more, the performaces are quite satisfactory for codece itervals for both ad. The bias reductio techique deitely helps to improve the coverage probabilities i both cases to a certai extet. 8. Illustrative examples Practical applicatio of the above estimators is illustrated here with two examples with oe ivolvig a large sample ad the other with a small sample. Example 1. The data set is give by Birbaum ad Sauders (1969b) o the fatigue life of 661-T6 alumium coupos cut parallel to the directio of rollig ad oscillated at 18 cycles per secod (cps). The data set cosists of 11 observatios with maximum stress per cycle 31; psi. The data are preseted i Table 5.
294 H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 Table 4 Probability coverages of 95% codece itervals based o Mote Carlo simulatio ( = 1:) Probability coverages for Probability coverages for MLE MME UMLE UMME MLE MME UMLE UMME 5.1 93.86 93.86 95.6 95.6 84.89 84.89 9.56 9.56.25 93.87 93.87 95.68 95.68 84.75 84.75 9.43 9.43.5 93.79 93.79 95.7 95.7 84.38 84.38 89.81 89.81 1. 93.65 93.67 95.94 96.5 83.37 83.34 88.48 88.51 2. 92.24 92.38 95.91 96.1 81.54 81.89 86.15 86.63 1.1 94.46 94.46 95.43 95.43 9.33 9.33 93.15 93.15.25 94.46 94.46 95.46 95.46 9.36 9.36 93.17 93.17.5 94.46 94.46 95.5 95.5 9.25 9.27 92.79 92.8 1. 94.25 94.25 95.45 95.5 89.6 89.72 92.9 92.12 2. 93.45 93.56 95.17 95.34 88.49 88.68 9.54 9.99 2.1 95.19 95.19 95.32 95.32 92.9 92.9 94.12 94.12.25 95.17 95.17 95.34 95.34 92.78 92.78 94.16 94.15.5 95.16 95.16 95.41 95.41 92.59 92.55 94.23 94.21 1. 95.6 95.7 95.41 95.42 92.32 92.36 93.68 93.71 2. 94.73 94.8 95.17 95.26 91.44 91.73 92.69 93.2 5.1 94.72 94.72 95.8 95.8 93.81 93.81 94.65 94.65.25 94.7 94.7 95.7 95.7 93.82 93.82 94.61 94.61.5 94.72 94.72 95.4 95.4 93.77 93.8 94.57 94.54 1. 94.59 94.59 95.4 95.5 93.56 93.63 94.2 94.26 2. 94.45 94.47 94.98 95.3 93.35 93.33 93.88 93.88 1.1 95. 95. 94.7 94.7 94.44 94.44 95.1 95.1.25 94.99 94.99 94.71 94.71 94.46 94.46 94.93 94.94.5 94.98 94.98 94.71 94.71 94.44 94.42 95.2 95. 1. 94.96 94.96 94.73 94.73 94.46 94.4 94.7 94.85 2. 94.94 94.95 94.71 94.73 94.21 94.39 94.27 94.67 Table 5 Fatigue lifetime data preseted by Birbaum ad Sauders (1969b) 7 9 96 97 99 1 13 14 14 15 17 18 18 18 19 19 112 112 113 114 114 114 116 119 12 12 12 121 121 123 124 124 124 124 124 128 128 129 129 13 13 13 131 131 131 131 131 132 132 132 133 134 134 134 134 134 136 136 137 138 138 138 139 139 141 141 142 142 142 142 142 142 144 144 145 146 148 148 149 151 151 152 155 156 157 157 157 157 158 159 162 163 163 164 166 166 168 17 174 196 212 I summary, we have i this case =11, s=133:73267, ad r =129:93321. For this example, the poit estimates of ad obtaied by all the methods are summarized i Table 6.
H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 295 Table 6 Poit estimates of ad for Example 1 Estimator MLE.17385 131.818792 MME.17385 131.819255 UMLE.17289 131.8913 UMME.17289 131.89593 JMLE.1726 131.798227 JMME.1726 131.798661 Table 7 Stadard deviatios of estimates ad iterval estimates of ad for Example 1 Estimator SD 9% CI 95% CI SD 9% CI 95% CI MLE.12 (.1527,.1927) (.1497,.1976) 2.2267 (128.2552,135.5861) (127.5944,136.3325) MME.12 (.1527,.1927) (.1497,.1976) 2.2267 (128.2556,135.5866) (127.5948,136.333) UMLE.122 (.1541,.1949) (.1511,.1999) 2.2487 (128.2116,135.6143) (127.5448,136.3685) UMME.122 (.1541,.1949) (.1511,.1999) 2.2487 (128.2121,135.6148) (127.5452,136.369) Table 8 Fatigue lifetime data preseted by McCool (1974) 152.7 172. 172.5 173.3 193. 24.7 216.5 234.9 262.6 422.6 From Eqs. (1), (13), (16) ad (17), the asymptotic variace of the estimators ca be readily obtaied, ad also the codece itervals for ad based o the MLEs, MMEs, UMLEs ad UMMEs ca be readily costructed usig the asymptotic ormality. The results so obtaied are preseted i Table 7. Example 2. This example is from McCool (1974) o the fatigue life i hours of 1 bearigs of a certai type. These data were used as a illustrative example for the three-parameter Weibull distributio by Cohe et al. (1984). The data are preseted i Table 8. I this case, we d = 1; s = 22:48 ad r = 23:8853. The poit estimates ad iterval estimates obtaied from this data are summarized i Tables 9 ad 1, respectively. It is hearteig to observe i both these examples that the MMEs are very early the same as the MLEs ad also the correspodig codece itervals. Also, while the ubiased estimators correct for the bias, they do result i a larger stadard deviatio.
296 H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 Table 9 Poit estimates of ad for Example 2 Estimator MLE.282489 212.4984 MME.282489 212.2378 UMLE.313877 211.52897 UMME.313877 211.499462 JMLE.32368 211.47377 JMME.32368 211.446452 Table 1 Stadard deviatios of estimates ad iterval estimates of ad for Example 2 Estimator SD 9% CI 95% CI SD 9% CI 95% CI MLE.632 (.265,.4468) (.1964,.529) 18.7527 (185.127,248.1452) (18.7241,256.511) MME.632 (.265,.4468) (.1964,.529) 18.754 (185.954,248.1121) (18.6993,256.476) UMLE.78 (.2228,.538) (.2111,.6118) 2.7356 (182.2191,252.726) (177.574,261.6814) UMME.78 (.2228,.538) (.2111,.6118) 2.7332 (182.194,252.394) (177.4828,261.6472) 9. Cocludig remarks Sice the maximum likelihood estimators ad the modied momet estimators behave very similarly i almost all cases cosidered, we recommed the use of the modied momet estimators because they are explicit estimators ad are very easy to compute. Although jackkife estimators also work very well, they caot be recommeded for large sample sizes. If the sample size is very small (say, less tha 1), the bias corrected modied momet estimators or bias corrected jackkife estimators should be used as i this case the origial estimators ca be highly biased. The asymptotic codece itervals behave very well for large sample sizes (at least 2), but ot for small sample sizes. I the latter case, oe may rely o simulated percetage poits rather tha o the asymptotic ormality. Appedix. Derivatio of the asymptotic distributio of the modied momet estimators Let T; T 1 ;:::;T be idepedet ad idetically distributed Birbaum Sauders radom variables with desity fuctio as i (2). Let us dee the radom variables [ ] 1 S = 1 1 T i ad R = Ti 1 : From the strog law of large umbers, it is kow that S ad R 1 coverge almost surely to E(T ) ad E(T 1 ), respectively. Also from the cetral limit theorem, we
H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 297 readily observe that S ad R 1 are asymptotically ormally distributed; furthermore, ay liear combiatio of the form as + br 1 = 1 [at i + bti 1 ] is also asymptotically ormally distributed for all a ad b. Therefore, (S R 1 ) is asymptotically distributed as bivariate ormal. Specially, we have ( ) [( ) ] S E(T ) N ; ; R 1 E(T 1 ) where ( ) 11 12 = ; 21 22 11 = Var(T )=() 2 (1 + 5 4 2 ); 12 = 21 = Cov(T; T 1 )=E(1) E(T)E(T 1 )=1 (1 + 1 2 2 ) 2 ; 22 = Var(T 1 )= 2 2 (1 + 5 4 2 ): We ow eed to d the asymptotic joit distributio of ( ) [ ] f1 (S; R) = ; f 2 (S; R) where { [ ( ) 1=2 1=2 x f 1 (x; y)= 2 1]} y ad f 2 (x; y)=(xy) 1=2 : By usig the Taylor series expasio, we obtai ( ) [( ) ] N ; ; where @f 1 @f 1 @f 1 @f 1 @x @y @x @y = @f 2 @x @f 2 @y @f 2 @x 2 2 = ( () 2 1+ 3 4 2 (1+ 1 2 2 ) 2 @f 2 @y ) : T x=e(t );y=e(t 1 )
298 H.K.T. Ng et al. / Computatioal Statistics & Data Aalysis 43 (23) 283 298 Of course, we are usig here the followig expressios: @f 1 = 1 @x 2 ; @f 1 @y @f 2 @x @f 2 @y x=e(t );y=e(t 1 ) = x=e(t );y=e(t 1 ) 2 ; x=e(t );y=e(t 1 ) x=e(t );y=e(t 1 ) = 1 2+ 2 ; = 2 2+ 2 : Refereces Birbaum, Z.W., Sauders, S.C., 1969a. A ew family of life distributio. J. Appl. Probab. 6, 319 327. Birbaum, Z.W., Sauders, S.C., 1969b. Estimatio for a family of life distributios with applicatios to fatigue. J. Appl. Probab. 6, 328 347. Chag, D.S., Tag, L.C., 1993. Reliability bouds ad critical time for the Birbaum Sauders distributio. IEEE Tras. Reliab. 42, 464 469. Chag, D.S., Tag, L.C., 1994. Percetile bouds ad tolerace limits for the Birbaum Sauders distributio. Comm. Statist. Theory Methods 23, 2853 2863. Cohe, A.C., Whitte, B.J., Dig, Y., 1984. Modied momet estimatio for the three-parameter Weibull distributio. J. Qual. Techol. 16, 159 167. Desmod, A.F., 1985. Stochastic models of failure i radom eviromets. Caad. J. Statist. 13, 171 183. Desmod, A.F., 1986. O the relatioship betwee two fatigue-life models. IEEE Tras. Reliab. 35, 167 169. Dupuis, D.J., Mills, J.E., 1998. Robust estimatio of the Birbaum Sauders distributio. IEEE Tras. Reliab. 47, 88 95. Efro, B., 1982. The Jackkife, the Bootstrap ad Other Resamplig Plas. Society for Idustrial ad Applied Mathematics, Philadelphia. Egelhardt, M., Bai, L.J., Wright, F.T., 1981. Ifereces o the parameters of the Birbaum Sauders fatigue life distributio based o maximum likelihood estimatio. Techometrics 23, 251 255. Johso, N.L., Kotz, S., Balakrisha, N., 1995. Cotiuous Uivariate Distributios Vol. 2, 2d Editio. Wiley, New York. Ma, N.R., Schafer, R.E., Sigpurwalla, N.D., 1974. Methods for Statistical Aalysis of Reliability ad Life Data. Wiley, New York. McCool, J.I., 1974. Iferetial techiques for Weibull populatios. Aerospace Research Laboratories Report ARL TR74-18. Wright Patterso Air Force Base, Dayto, OH. Rieck, J.R., 1995. Parametric estimatio for the Birbaum Sauders distributio based o symmetrically cesored samples. Comm. Statist. Theory Methods 24, 1721 1736. Rieck, J.R., 1999. A momet-geeratig fuctio with applicatio to the Birbaum Sauders distributio. Comm. Statist. Theory Methods 28, 2213 2222.