358 W.J.CONOVER,MARK E.JOHNSON,AND MYRLE M.JOHNSON Table 6.For Double Exponential and(Dbl.Exp.)2 Distributions,Proportion of Times the Null Hypothesis of Equal Variances Was Rejected by the Various Tests,Under the Null Hypothesis(test size)and(in parentheses)Under the Alternative Hypothesis(power),at a=.05 Dbl.Exp.Distribution:Symmetric (Db1.Exp.)2:Asymmetric n=(5,5,5,5) (10,10,10,10) (20,20,20,20) (5,5,20,20) (10,10,10,10) (5,5,20,20) TABLE 1 TESTS N-P .316 (.553) ,339(,7131 ,316(,836) .333(,801） .876 （.912) 883(.933 N-P:med ,322 .556 .340 (.713 .317 .835 .333 (.795 .881 .914 .886 .934 Bar .157 .395 .273 （.661 288 .821 ,233 .710 .856 802 .832 897 Bar:med .164 .397 .275 .659 ,292 .822 .240 .697 ,860 .891 .830 .898 Coch 154 .3R6 .232 .592 .214 762 .721 .723 .773 ,798 .856 Coch:med .164 .381 .236 .593 .212 .764 ,712 ,724 .7 .799 .855 Hart .134 .345 .248 .624 .264 806 .460 .815 .845 .888 .908 .950 Hart:med ,139 ,348 .252 .629 267 .806 .457 .807 .849 .888 .906 ,950 Bar3 ,161 .402 .275 ,661 288 .821 .236 .711 .856 .892 .833 (.897 Bar3:med .170 ,401 .275 .659 293 ,822 .699 .862 .892 .834 898 Sam .135 .364 .261 .653 .284 .819 .670 .853 .888 .808 .884 Sam:med 145 .365 265 .650 .820 663 ,853 .887 .805 .885 Lehl .275 .504 .315 .687 .831 :3 .790 .868 .907 .866 .929 Lehl:med .278 .507 313 689 .828 .779 ,874 .910 .872 .930 Leh2 .404 .620 .361 .728 .841 .809 .888 ,921 ,883 .939 Leh2:med ,401 627) 366 .727 ,341 .840j ,356 .803 ,889 （.924 .886 (.939 TABLE 2 TESTS Barl ,450 .553 .273 .641) ,169 （.727 .179 (.605 .696 .766 .439 (.557 8a med 470 8 144 .701 133 .546 .551 .654 .332 .454 .129 .050 .492 .046 ,289 .230 .153 Bar2 med 165 440 .020 .229 .099 .073 Schl .668 .739 .254 .717 .758 .819 .759 Schl med 548 .250 603 .170 .718 .210 .663 .652 .741 8 .693 Sch2 .298 .101 ,322j .079 .511 .116 .486 361 ,453 284 .468 Sch2:med 087 （.176 .069 .243j .058 .460) ,082 .416j .249 (.310j ,188 .355 TABLE 3 TESTS Levl .097 （.268 .077 （.415 .068 （.645) .087 （.396 .473 （.579 .384 (.420 Levl:med .008 .051 .033 ,291 .039 .591 .035 .325 .048 .092 .060 .057 Lev2 .057 .155 .048 ,266 .040 .524 .079 .194 Lev2 .074 149 .077 med .010 .051 .024 .184 .027 115 .473 .048 43 .012 .024 .078 (.027 .098 .229 .121 83 .326 .078 498 .087 .404 ,741 .805 .729 .836 .290 .419 .082 .630 .092 .458 715 803 ,688 .797 Lev4:med .000 .008 .306 .041 562 .047 .413 Mi]l .045 145 .226) .099 .199 .046 （,136 .067 .319) .087 .537) .107 .419 .195 （.240)】 .214 .291) as the power of the three tests mentioned previously these bids(Dougherty and Lohrenz 1976).If it is for those same symmetric distributions.Therefore,the reasonable to assume that the variance of the log of three tests,Levl:med,F-K:med X2,and F-K:med F, the bids on each lease is constant within a sale,then again appear to be the best tests to use on the basis of the scale parameter of the lognormal distribution can robustness and power be estimated using all the bids in the sale. 4.APPLICATION TO THE LPR DATA BASE The bids in 40 sales were examined.These included all the sales held from October 13,1954 to October 27 Since 1954 the United States government has 1977,which is the date of the last sale recorded in the periodically held sales in which offshore leases have data base at the time of this study.We considered only been offered to the highest bidder for the production leases within a sale receiving two or more bids on the of oil and gas.The lease,production,and revenue lease.The 40 sales averaged about 50 leases per sale, (LPR)data base includes detailed information on the with a range from 5 to 133.Although some of the bids submitted,as well as the yearly production and leases have as many as 12 or 13 bids,small numbers of revenue data on each lease.Our interest is in the bids bids are the general rule,with about half of the leases submitted on the various leases within each sale. examined having only two bids submitted on them. Often,the lognormal distribution is used to model For example,the sale held on July 21,1970 was the TECHNOMETRICS©，VOL.23,NO.4,NOVEMBER1981 This content downloaded from 61.190.7.73 on Mon,30 Sep 2013 22:38:50 PM All use subject to JSTOR Terms and ConditionsW. J. CONOVER, MARK E. JOHNSON, AND MYRLE M. JOHNSON Table 6. For Double Exponential and( Dbl. Exp.)2 Distributions, Proportion of Times the Null Hypothesis of Equal Variances Was Rejected by the Various Tests, Under the Null Hypothesis (test size) and (in parentheses ) Under the Alternative Hypothesis (power), at a = .05 Dbl. Exp. Distribution: Symmetric (Dbl. Exp.)2: Asymmetric n=(5,5,5,5) (10,10,10,10) (20,20,20,20) (5,5,20,20) (10,10,10,10) TABLE 1 TESTS N-P N-P:med Bar Bar:med Coch Coch:med Hart Hart:med Bar3 Bar3:med Sam Sam:med Lehl Lehl :med Leh2 Leh2:med TABLE 2 TESTS Barl Barl :med Bar2 Bar2:med Schl Schl :med Sch2 Sch2:med TABLE 3 TESTS Levl Levl:med Lev2 Lev2 :med Lev3 Lev4 Lev4 :med Mill as the power of the three tests mentioned previously for those same symmetric distributions. Therefore, the three tests, Levi: med, F-K: med X2, and F-K: med F, again appear to be the best tests to use on the basis of robustness and power. 4. APPLICATION TO THE LPR DATA BASE Since 1954 the United States government has periodically held sales in which offshore leases have been offered to the highest bidder for the production of oil and gas. The lease, production, and revenue (LPR) data base includes detailed information on the bids submitted, as well as the yearly production and revenue data on each lease. Our interest is in the bids submitted on the various leases within each sale. Often, the lognormal distribution is used to model these bids (Dougherty and Lohrenz 1976). If it is reasonable to assume that the variance of the log of the bids on each lease is constant within a sale, then the scale parameter of the lognormal distribution can be estimated using all the bids in the sale. The bids in 40 sales were examined. These included all the sales held from October 13, 1954 to October 27, 1977, which is the date of the last sale recorded in the data base at the time of this study. We considered only leases within a sale receiving two or more bids on the lease. The 40 sales averaged about 50 leases per sale, with a range from 5 to 133. Although some of the leases have as many as 12 or 13 bids, small numbers of bids are the general rule, with about half of the leases examined having only two bids submitted on them. For example, the sale held on July 21, 1970 was the TECHNOMETRICS ?, VOL. 23, NO. 4, NOVEMBER 1981 (5,5,20,20) .316 .322 .157 .164 .154 .164 .134 .139 .161 .170 .135 .145 .275 .278 .404 .401 (.553) (.556) (.395) (.397) (.386) (.381) (.345) (.348) (.402) (.401) (.364) (.365) (.504) (.507) (.620) (.627) .339 .340 .273 .275 .232 .236 .248 .252 .275 .275 .261 .265 .315 .313 .361 .366 (.713) (.713) (.661) (.659) (.592) (.593) (.624) (.629) (.661) (.659) (.653) (.650) (.687) (.689) (.728) (.727) .316 .317 .288 .292 .214 .212 .264 .267 .288 .293 .284 .285 .314 .315 .334 .341 (.836) (.835) (.821) (.822) (.762) (.764) (.806) (.806) (.821) (.822) (.819) (.820) (.831) (.828) (.841) (.840) .333 .333 .233 .240 .324 .340 .460 .457 .236 .243 .213 .231 .317 .314 .357 .356 (.801) (.795) (.710) (.697) (.721) (.712) (.815) (.807) (.711) (.699) (.670) (.663) (.790) (.779) (.809) (.803) .876 .881 .856 .860 .723 .724 .845 .849 .856 .862 .853 .853 .868 .874 .888 .889 (.912) (.914) (.892) (.891) (.773) (.777) (.888) (.888) (.892) (.892) (.888) (.887) (.907) (.910) (.921) (.924) .883 .886 .832 .830 .798 .799 .908 .906 .833 .834 .808 .805 .866 .872 .883 .886 (.933) (.934) (.897) (.898) (.856) (.855) (.950) (.950) (.897) (.898) (.884) (.885) (.929) (.930) (.939) (.939) .450 .238 .047 .010 .470 .325 .167 .087 (.553) (.470) .129) .041) (.671) (.548) (.298) (.176) .273 .199 .054 .016 .313 .250 .101 .069 (.641) (.563) (.232) (.165) (.668) (.603) (.322) (.243) .169 .144 .050 .033 .190 .170 .079 .058 (.727) (.701) (.492) (.440) (.739) (.718) (.511) (.460) .179 .133 .046 .020 .254 .210 .116 .082 (.605) (.546) (.289) (.229) (.717) (.663) (.486) (.416) .696 .551 .172 .071 .758 .652 .361 .249 (.766) (.654) .230) .099) (.819) (.741) (.453) (.310) .439 .332 .100 .024 .598 .514 .284 .188 (.557) (.454) (.153) .073) (.759) (.693) (.468) (.355) .097 .008 .057 .010 .098 .121 .000 .046 (.268) (.051) (.155) (.051) (.229) (.290) (.008) (.136) .077 .033 .048 .024 .077 .093 .045 .067 (.415) (.291) (.266) (.184) (.326) (.419) (.306) (.319) .068 .039 .040 .027 .078 .082 .041 .087 (.645) (.591) (.524) (.473) (.498) (.630) (.562) (.537) .087 .035 .079 .048 .087 .092 .047 .107 (.396) (.325) (.194) (.143) (.404) (.458) (.413) (.419) .473 .048 .074 .012 .741 .715 .145 .195 (.579) (.092) (.115) (.024) (.805) (.803) (.226) (.240) .384 .060 .149 .078 .729 .688 .099 .214 (.420) (.057) (.077) (.027) (.836) (.797) (.199) (.291) 358 This content downloaded from 61.190.7.73 on Mon, 30 Sep 2013 22:38:50 PM All use subject to JSTOR Terms and Conditions