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Testing Hypotheses about the Shape Parameter of a Gamma Distribution Author(s): Jerome P. Keating, Ronald E. Glaser, Norma S. Ketchum Source: Technometrics, Vol. 32, No. 1 (Feb., 1990), pp. 67-82 Published by: American Statistical Association and American Society for Quality Stable URL: http://www.jstor.org/stable/1269846 Accessed: 16/11/2009 16:07 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=astata. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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? 1990 American Statistical Association and the American Society for Quality Control
FEBRUARY1990, VOL. 32, NO. 1 TECHNOMETRICS,
HypothesesAbout
Testing
Parameter
of
a
the
Gamma
Shape
Distribution
Jerome P. Keating
Ronald E. Glaser
Norma S. Ketchum
Division of Mathematics University of Texas San Antonio, TX 78285-0664
Lawrence Livermore National Laboratory P.O. Box 808 Livermore,CA 94550
Clinical Sciences Division U.S. Air Force School of Aerospace Medicine Brooks Air Force Base, TX 78235
The purposeof this expositoryarticleis to providegreateraccessto criticalvaluesfor testing hypothesesabout the shape parameterof a gammadistribution.Among the possibletests, we considertests of exponentialityagainstgammaincreasingfailurerate and/or decreasing failurerate alternatives.Extensivetables of criticalvalues are given for individualsample sizes from 2 to 30, and values of the shape parameterin half-integerincrementsfrom A to 10. Medianunbiasedestimatorsare also givenfor the shapeparameterto avoidsmall-sample complicationswith maximumlikelihoodmethods.Powercurvesare given for testingexponentialityagainstgammaalternatives.These curvesillustratecertainadvantagesof this parametrictest over nonparametric alternatives. KEY WORDS: Arithmeticmean;Bartlett'sdistribution;Exponentiality;Geometricmean; Hazardrate.
1.
INTRODUCTION
then knowledge of the coefficient of variation implies knowledge of k. The shape parameter is also known in many applications in renewal theory in which the gamma distribution is used to model times to occurrence of events. Unfortunately, the shape parameter is generally unknown when distributions are fitted to real data. The objective of this expository article is to provide greater accessibility to point and interval estimation of the shape parameter k for the gamma distribution. We also discuss associated power calculations and sample-size determinations. Let TI,. . . , T, be a random sample of size n from a density function in if. Then a minimal sufficient statistic for the parametric pair (k, fl) is the ordered pair (t, 1), where i = (nIi= T)1l" is the sample geometric mean and t = :=1 Tl/n is the sample arithmetic mean. Define the random variable W = l/7, the ratio of the sample geometric mean to the sample arithmetic mean. Note that 0 - W - 1, and (W, t) is also minimallysufficientfor (k, f,). Moreover, Glaser (1976a) showed that W and t are stochastically independent (see Pitman 1937). Greenwood and Durand (1960) gave tables for the MLE of k as a function of W. Wilk, Gnanadesikan, and Huyett (1962) showed that the MLE of k is well approximated by a linear function of 1/(1 - W) for a wide range of values of k. The most powerful invariant test of H(,: k = k( against H1 : k = k, (kl > k,) is expressed in terms
The gamma family provides a widely applicable class of distributions for reliability and survival analysis because it can have a decreasing failure rate (DFR), a constant failure rate, or an increasing failure rate (IFR). But many applied statisticians use the Weibull distribution (see Lawless 1982) because the gamma hazard function and survival function cannot be expressed in closed form. Moreover, maximum likelihood estimators (MLE's) of the parameters must be found as solutions to a system of nonlinear equations. The family of density functions defined by the gamma distribution is given by = {f(t;k, f) : k>Oandf
>0},
where f(t; k, fi) = t-le -'-[r(k)#k],
t > 0,
where k is the shape parameter and /f is the scale parameter. The shape parameter k is especially interesting to reliability analysts because the gamma distribution is DFR, constant, or IFR according to whether (k - 1) is negative, zero, or positive. Scientists have observed that, under repeated observation or sampling, the sample coefficient of variation approaches a deterministic constant (see Linhart 1965). If the underlying model behaves like a gamma distribution, 67
JEROMEP. KEATING,RONALDE. GLASER,AND NORMA S. KETCHUM
68
of the statistic W alone (Lawless 1982). This result is importantin reliability,since k,( = 1 provides a test of exponentialityagainst an IFR memberof '. Moran(1951) firstsuggestedtests for exponentiality basedon W. EngelhardtandBain (1977)showedthat the statistic W gives the uniformlymost powerful unbiased test of k with /f unknown in testing the simplenull hypothesisagainstthe simplealternative. For testing H,,: k = k( againstthe compositealternative HI : k > k0,Lawless(1982) arguedthat large values of W will cause us to reject the null in favor of the alternative. Here we adopt the notation of Lawlessand define the criticalregion C of size a as C = {w : w.k,,,, < w - 1}, where Pr{W > wa.k,), I k
= ko} = a. In this article, we tabulate w,k,l,, for various values
of k,, n, and a. When k, = 1, we will be able to test for exponentialityagainstgammaalternatives,so we can test the practicalhypothesisthat the failurerate is constantagainstthe alternativethatthe failurerate is increasingwithinthe gammafamily. We also construct tests of Ho : k = k, against H, : k < k,. The
subsequentcritical region consists of the values in the left tail of the null distributionof W (i.e., C = {w : 0
w < Wl a,:k ,,}).
is given by g(w; k, n) = Awnk-l(-ln
critical region is the set C = {w : 0 -
w)(n-3)/2E(w),
(1)
where w E [exp(-2nr/n), 1], n-l
This second test allows us
to test the hypothesisthat the failurerate is constant against the alternative that the failure rate is decreasing(DFR) within the gammafamily. Since both upperand lower percentilesof the null distributionof W are given, we can also test the null hypothesisH( : k = k( againstthe two-sidedalternative Hi : k $= k(. The symmetricor equal-tails
A =
n r(k + rln)
n(n-)/2/[r(k)]"-1,
_r=l
and E(w) =
'L,, Vr(-ln w)', with
u, = {r[(n - 1)/2 + r]}-' x
?,
(Z,1z, Z'
'..)/(SI!s3!S5! ...),
w c
Wa/2;k.,n < W C 1}. Thus we can test Wl-a/2:kln} U {w the simplenull hypothesisthatthe underlyinghazard functionis constantagainstthe alternativethat it is not, within the gammafamily. As we have alreadynoted, the estimationof k is complicatedbecause the maximumlikelihood estimates must be obtained throughiterative methods that are nontrivialfor most readers.The asymptotic propertiesof these estimatorsare well known, but the small-samplepropertiesare not. Lilliefors(1971) developed a correctionfactor to reduce the bias of the MLE of k in small samples. Bowmanand Shenton (1988) gave a comprehensivereview of the conventionalestimatorsof k and their associatedproperties. In Section 3, we propose an alternative estimator for the shape parameterk whose smallsample propertiesare well known and whose values depend on the medianof the distributionof W.
2.
the cdf of W. Denote the probabilitydensityfunction (pdf) of W as g(w; k, n) underthe assumptionthat T,, . . ., T, constitutean iid sample of size n from a gamma distribution.The exact distributionof W was obtainedby Glaser(1973, 1976a).Glaser(1976b) showed how the distributionof W was related to Bartlett's (1937a) test statistic for homogeneity of variancesamongindependentrandomsamplesfrom normalpopulations.Becauseof this relationship,we shall call the distributionof W "Bartlett'sdistribution." Glaser's series expression for g(w; k, n) is given in powersof -ln(w). Unfortunately,Glaser's expression yields a conservativeradius of convergence for the series, whichis knownto convergefor all w in the interval[exp(-2n/n), 1]. When we are interestedonly in upper percentilesof the null distributionof W, Glaser'sexpressionwill suffice. Glaser'sexpressionfor the densityfunctionof W
THE CUMULATIVE DISTRIBUTION FUNCTION OF W
To evaluate Wa;k,,.^n the percentiles of the null dis-
tributionof W, we need to give an expressionfor FEBRUARY1990, VOL. 32, NO. 1 TECHNOMETRICS,
where E: indicates summationover all distinct sequences (s,, s3, s5,. . .) of nonnegative integers satisfying is, + 3s3 + 5s5 + .. = r, z, = (-1)'r+[r(r
+ 1)]-'[nr+1
-
]B,,
(r - 1), and B,, B, . . are the Bernoulli numbers.
Numericalintegrationof Glaser's expressionfor the density is facilitatedby the observationof Dyer and Keating(1980) that
J
u'"(-n
u)"du
= (p + 1)-'-1 " If
)?)y"e-' -Inz)(p+
dy
I)
= (p + 1)-(q"+')[(q + 1)] x {1 - Go((-ln z)(p + 1); q + 1)}, (2) where Go(x; k) =
f(i [yk-'e-Y/F(k)]
dy is the incom-
plete gamma-functionratio. Thusnumericalintegration can be simplifiedthroughthis observation,and
69
INFERENCESON THE GAMMASHAPE PARAMETER
it yields an infiniteseries expressionfor the cdf: G(w; k, n) , v,[r((n + 2r - 1)/2)1(nk)("+'-''2]
= A
x {1 - G,((-ln
w)nk; (n + 2r - 1)/2)}.
(3)
We use the expressionfor G(w; k, n) in a NewtonRaphson routine to calculate the critical values W,:kn,
the 100(1 - a) percentiles of the distribution
of W. Complicationsarise when we try to determinethe lower critical values w1_, k,, and the median of the
distributionof W for small values of k and/or large valuesof n. The lowercriticalvalueswouldbe useful in testing H(,: k = ko versus H1 : k < k(. The true
criticalvalue may fall outside the radiusof convergence specifiedby Glaser(1976a).Nandi (1980) and Glaser (1980) found series expressionsfor the cdf of Win termsof incompletebetafunctions.Theyshowed that their series convergesuniformlyfor all w in [0, 1]. From Glaser (1976a, cor. 3.1), we note that the
distribution of Z = W" - ni,-1 V, where the V's
are independentlydistributedbeta randomvariables with respectiveparametersk and iln (i = 1, . . . n - 1). Hence it follows that G(w; k, n)
three million values of Z to tabulate w --,k.n. Thus,
with high probability,all of the digits shown in our tables are correct. Anotherviableapproachto the evaluationof G(w; k, n) is the methodsuggestedby Lawless(1982) and employed by Sinclair, Mosimann, and Meeter (1985), who definedthe randomvariablesY2 = T2/ Y,, = T,,/(nt) and expressed W = n[Y, (nt) ... x .. x Y,(1 - 1=,2 Y)]ll1. The random variables Y, . . . Y,, have a joint Dirichlet distribution, and
the most powerfulinvarianttests on k are based on the likelihood ratio of the maximal invariant(Y2, ..., Y,,)underH,0and Hi (Cox and Hinkley 1974). To determineW'slowerpercentagepointsthatfall outside of Glaser'sradiusof convergence,we may employ the resultof Provost(1988). He definedthe randomvariableZ = Wn and found the expression for the jth moment of Z. He also determinedthe probabilitydensity function of Z by invertingthe Mellin transformas h(z; k, n) = 6z-'GC-:,,1(
= A S p,B(k, 6, + j)l(w"; k, 6, + j), /==o
(4)
where B(a, f) is the well-knownbeta function,l(x; a, /f) is the incompletebeta-functionratio, 6,, = (n - 1)/2,
A* =
the rate of convergenceof this series is quite slow, for mostlower however,andthe methodis impractical criticalvalues. To actuallygeneratethe lower criticalvalues that fall outside the radiusof convergencefor the cdf in Equation (3), we used simulationmethods on the randomvariableZ = W".For each pair of k and n (outside the radius of convergence), we generated
n r(k + i/n) - I=1
/
{r((n - 1)/2)[r(k)]"-'},
and ,,-
p/ =
I
, In
i= 1
x
_
((i + 1)n)j(i)
iln + I (lln + j(l))) l= l
i(i)-
x(Z
j(l) _I= 1
,i(i)_
where the summationis taken over all sequences
{j(l), . ., j(n - 1)} such that j(i) is a nonnegative integer and j(l) + *-. + j(n - 1) = j. Note that
.. (a + n - 1), forn = 1,2,... and (a),, = 1. The expressionin (4) converges absolutelyfor all w E [0, 1]. Glaser(1980)showedthat (a), = a(a + 1)
k,
....k (5)
where 6
=
(2n)("-' )/2(nk)/{[r(k)]"n"k- /2} and
G(-l ) is Meijer's G function (see Luke 1969). The connectionbetweenA in Equation(1) and6 in Equation (5) is achieved through the Gauss-Legendre multiplicationformula.Provost'sresultis equivalent to the result of Davis (1979), who developed the resultsin connectionwith the distributionof Wilk's A criterion.Davis gave a numericalsolutionfor the criticalvaluesof Wilk'slikelihoodratiocriterion.His tables, however, are not compatiblewith ours becauseof intrinsicdifferencesin the parametersof the G function. 3.
j(i)!(6( +
k + l/n, k+2/n, . . ., k + (n - 1)1n)
CALCULATIONOF CRITICALVALUES
The expressionsfor the criticalvalues of the onesided hypothesisHI : k > k,)can be found numerically by solvingthe followingnonlinearequationfor G(x; k,, n) = 1 - a.
(6)
FORTRANsubroutineswere writtenfor the pdf (1) and cdf's (3) and (4), respectively,of Bartlett'sdisFEBRUARY1990, VOL. 32, NO. 1 TECHNOMETRICS,
JEROMEP. KEATING,RONALDE. GLASER,AND NORMA S. KETCHUM
70
tribution. These subroutines involve intricate summation techniques used in the calculation of v,. and pj, respectively. The FORTRAN subroutines were used in a Newton-Raphson procedure to solve for x = w,;k.ll,,.An excellent initial approximation to the solution can be found through an adaptation of Bartlett's approximation. An alternative approximation was that of Bain and Engelhardt (1975), who used a two-moment chi-squared approximation to the distribution of - ln(W). The exact critical values were found for values of a = .01, .025, .05, and .10; k( = .5(.5)10; and n = 2(1)30. The values for a of .025 and .05 can also be used to test the null hypothesis H, : k = k, against the two-sided alternative HI : k = k(. In such cases, we reject whenever W > wai2:k,l,or W
10 or n > 30, we approximate the distribution of W with a beta distribution by employing the technique of Patnaik (1949) (i.e., equate the first two moments of the beta distribution to the first two moments of W). Consequently, we have that a'/(a' + /f') = [F(k + 1n)/F(k)]"-' / - I
x E [r(k + iln)/r(k + l/n + i/n)] = i I
(7)
and + + 1)] a'(a' + 1)/[(a' +' )(' ')(a = [r(k + 2/n)/r(k)]"-' nI-1
x
F + i/n)Ir(k + 2/n + i/n)]. [F(k
(8)
The moments of W can be found in Bartlett (1937b). By solving (7) and (8) for a' and f', we obtain a highly accurate approximation to the sampling distribution of W for k - 10 or n > 30. For example, when k = 10, the absolute difference between the approximate critical value and the exact critical value is less than 2 x 10-4 for n = 2, ... , 30. When n = 30 and k - 2, the absolute difference between the
m I
2 0 m
-n
I
c
w 2
CO) Co
to 0
Table 1. Lower Tenth Percentiles for Bartlett's Distribution
n .1564 .1318 .1311 .1346 .1388 .1431 .1472 .1510 .1547 .1578 .1609 .1639 .1665 .1689 .1713 .1734 .1755 .1774 .1793 .1810 .1824 .1840 .1854 .1869 .1882 .1894 .1908 .1918 .1931
.5 .4359 .3991 .3966 .4006 .4061 .4115 .4165 .4216 .4261 .4302 .4339 .4373 .4405 .4435 .4463 .4487 .4513 .4535 .4556 .4574 .4595 .4613 .4629 .4644 .4662 .4674 .4688 .4700 .4713
1.0 .5928 .5583 .5551 .5582 .5626 .5673 .5717 .5759 .5797 .5831 .5865 .5893 .5922 .5945 .5968 .5989 .6010 .6030 .6048 .6064 .6080 .6094 .6109 .6123 .6137 .6147 .6158 .6169 .6181
1.5 .6842 .6539 .6507 .6530 .6566 .6605 .6642 .6676 .6708 .6737 .6764 .6788 .6811 .6832 .6851 .6871 .6887 .6903 .6918 .6932 .6944 .6957 .6969 .6980 .6990 .7001 .7011 .7019 .7029
2.0 .7429 .7163 .7133 .7151 .7182 .7214 .7245 .7274 .7301 .7326 .7349 .7369 .7389 .7406 .7423 .7438 .7453 .7466 .7479 .7490 .7503 .7513 .7522 .7532 .7541 .7550 .7558 .7566 .7573
2.5 .7834 .7600 .7572 .7587 .7612 .7640 .7667 .7692 .7716 .7737 .7757 .7775 .7791 .7807 .7821 .7834 .7847 .7858 .7869 .7879 .7889 .7898 .7907 .7915 .7923 .7930 .7938 .7944 .7950
3.0 .8130 .7921 .7895 .7908 .7930 .7955 .7978 .8000 .8021 .8040 .8057 .8073 .8087 .8101 .8113 .8125 .8136 .8146 .8156 .8165 .8173 .8181 .8189 .8196 .8203 .8209 .8216 .8221 .8227
3.5 .8356 .8168 .8143 .8154 .8174 .8196 .8217 .8236 .8254 .8271 .8287 .8301 .8314 .8326 .8337 .8347 .8357 .8366 .8375 .8383 .8390 .8398 .8404 .8411 .8417 .8423 .8428 .8433 .8438
4.0 .8533 .8362 .8339 .8349 .8367 .8386 .8405 .8423 .8439 .8454 .8468 .8481 .8430 .8503 .8513 .8523 .8532 .8540 .8548 .8555 .8562 .8568 .8574 .8580 .8584 .8591 .8596 .8600 .8605
4.5 .8676 .8519 .8498 .8507 .8523 .8540 .8557 .8574 .8589 .8602 .8615 .8626 .8637 .8647 .8656 .8665 .8673 .8680 .8687 .8694 .8700 .8706 .8711 .8717 .8722 .8726 .8731 .8735 .8739
5.0 .8794 .8649 .8629 .8637 .8652 .8668 .8683 .8698 .8712 .8724 .8736 .8747 .8757 .8766 .8774 .8782 .8789 .8796 .8802 .8809 .8814 .8820 .8825 .8829 .8834 .8838 .8843 .8847 .8850
5.5 .8892 .8758 .8740 .8746 .8760 .8775 .8789 .8803 .8816 .8827 .8838 .8848 .8857 .8865 .8873 .8880 .8887 .8893 .8899 .8905 .8910 .8915 .8920 .8924 .8928 .8932 .8936 .8940 .8943
6.0 .8976 .8851 .8833 .8840 .8852 .8866 .8879 .8892 .8904 .8915 .8924 .8934 .8942 .8950 .8957 .8964 .8970 .8976 .8981 .8987 .8991 .8996 .9000 .9005 .9009 .9012 .9016 .9019 .9022
6.5 .9048 .8931 .8914 .8920 .8932 .8944 .8957 .8969 .8980 .8990 .8999 .9008 .9015 .9023 .9030 .9036 .9042 .9047 .9052 .9057 .9062 .9066 .9070 .9074 .9078 .9081 .9084 .9088 .9091
7.0 .9110 .9000 .8985 .8990 .9001 .9013 .9025 .9036 .9046 .9055 .9064 .9072 .9079 .9086 .9093 .9098 .9104 .9109 .9114 .9118 .9123 .9127 .9131 .9134 .9138 .9141 .9144 .9147 .9150
7.5
k
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
--I
m I
z
0
Z m --I m -n m
o
CD
CD
0
O
z
0
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
n .0785 .0758 .0824 .0897 .0967 .1028 .1086 .1138 .1187 .1228 .1269 .1306 .1341 .1373 .1402 .1430 .1457 .1481 .1506 .1527 .1547 .1567 .1588 .1603 .1620 .1635 .1653 .1666 .1681
.5
Table 2. Lower Fifth Percentiles for Bartlett's Distribution
1.0 .4780 .4699 .4803 .4921 .5028 .5122 .5204 .5276 .5341 .5398 .5451 .5497 .5541 .5578 .5612 .5644 .5677 .5707 .5732 .5755 .5780 .5801 .5823 .5843 .5862 .5878 .5893 .5909 .5926
1.5 .5845 .5762 .5850 .5952 .6045 .6126 .6197 .6260 .6315 .6365 .6409 .6449 .6485 .6518 .6549 .6579 .6603 .6627 .6649 .6670 .6691 .6709 .6726 .6742 .6758 .6772 .6788 .6801 .6814
2.0 .6563 .6483 .6559 .6646 .6727 .6798 .6860 .6914 .6961 .7004 .7042 .7076 .7108 .7136 .7162 .7186 .7208 .7229 .7249 .7267 .7285 .7300 .7314 .7329 .7343 .7356 .7367 .7378 .7390
2.5 .7075 .7000 .7065 .7142 .7213 .7275 .7329 .7376 .7418 .7455 .7488 .7519 .7546 .7571 .7593 .7614 .7634 .7652 .7669 .7684 .7699 .7713 .7726 .7739 .7749 .7761 .7772 .7782 .7791
3.0 .7456 .7387 .7444 .7512 .7574 .7629 .7677 .7719 .7757 .7790 .7819 .7846 .7870 .7892 .7912 .7930 .7948 .7964 .7978 .7992 .8005 .8017 .8029 .8040 .8050 .8060 .8069 .8078 .8087
3.5 .7751 .7686 .7737 .7798 .7854 .7903 .7946 .7984 .8017 .8047 .8073 .8097 .8119 .8139 .8157 .8173 .8189 .8203 .8216 .8228 .8240 .8251 .8261 .8271 .8280 .8289 .8297 .8305 .8312
4.0 .7984 .7924 .7970 .8025 .8076 .8121 .8160 .8194 .8224 .8251 .8275 .8296 .8316 .8334 .8350 .8365 .8379 .8392 .8404 .8415 .8426 .8436 .8445 .8454 .8462 .8470 .8477 .8484 .8491
4.5 .8175 .8118 .8160 .8210 .8257 .8298 .8333 .8365 .8392 .8417 .8438 .8458 .8476 .8492 .8507 .8521 .8534 .8545 .8556 .8566 .8576 .8585 .8594 .8602 .8609 .8616 .8623 .8629 .8636
5.0 .8332 .8280 .8317 .8364 .8407 .8444 .8477 .8506 .8531 .8554 .8574 .8592 .8608 .8623 .8637 .8650 .8661 .8672 .8682 .8691 .8700 .8709 .8716 .8724 .8731 .8737 .8743 .8749 .8755
5.5 .8465 .8415 .8450 .8493 .8533 .8568 .8598 .8625 .8648 .8669 .8688 .8704 .8719 .8733 .8746 .8758 .8768 .8778 .8788 .8796 .8805 .8812 .8819 .8826 .8833 .8839 .8845 .8850 .8855
6.0 .8578 .8532 .8564 .8604 .8641 .8673 .8701 .8726 .8748 .8767 .8785 .8800 .8814 .8827 .8839 .8850 .8860 .8869 .8878 .8886 .8893 .8901 .8907 .8914 .8920 .8925 .8931 .8936 .8941
6.5 .8676 .8632 .8662 .8699 .8734 .8764 .8790 .8814 .8834 .8852 .8868 .8883 .8896 .8908 .8919 .8929 .8939 .8947 .8955 .8963 .8970 .8977 .8983 .8989 .8994 .9000 .9005 .9010 .9014
7.0 .8761 .8719 .8747 .8782 .8815 .8843 .8868 .8890 .8909 .8926 .8941 .8955 .8967 .8979 .8989 .8998 .9007 .9015 .9023 .9030 .9037 .9043 .9049 .9054 .9060 .9065 .9069 .9074 .9078
7.5
k
.3122 .3058 .3173 .3299 .3413 .3515 .3601 .3684 .3758 .3818 .3877 .3929 .3977 .4019 .4058 .4094 .4130 .4162 .4194 .4219 .4249 .4273 .4296 .4317 .4341 .4360 .4380 .4398 .4416
--1
m I z 0
m H
Cn
-n m co
C
(0 0
0
rCOA)
N) z 0
Table 3. Lower 2.5 Percentiles for Bartlett's Distribution
n .0393 .0442 .0527 .0609 .0685 .0753 .0815 .0873 .0928 .0973 .1017 .1060 .1097 .1134 .1167 .1197 .1229 .1255 .1284 .1307 .1330 .1353 .1372 .1396 .1414 .1433 .1452 .1466 .1483
.5 .2222 .2352 .2555 .2737 .2894 .3027 .3141 .3244 .3336 .3416 .3488 .3555 .3615 .3667 .3717 .3763 .3808 .3847 .3889 .3919 .3953 .3983 .4012 .4039 .4067 .4092 .4116 .4140 .4162
1.0 .3831 .3961 .4271 .4357 .4515 .4648 .4761 .4860 .4950 .5023 .5090 .5151 .5210 .5260 .5304 .5345 .5387 .5422 .5454 .5488 .5516 .5546 .5572 .5597 .5623 .5644 .5661 .5683 .5704
1.5 .4966 .5080 .5272 .5442 .5584 .5703 .5804 .5891 .5967 .6034 .6094 .6145 .6198 .6240 .6282 .6321 .6351 .6384 .6414 .6442 .6466 .6489 .6511 .6532 .6552 .6572 .6591 .6607 .6624
2.0 .5772 .5870 .6042 .6192 .6318 .6424 .6513 .6590 .6656 .6715 .6767 .6814 .6856 .6894 .6929 .6961 .6990 .7019 .7044 .7068 .7090 .7111 .7130 .7149 .7167 .7183 .7198 .7212 .7225
2.5 .6364 .6449 .6602 .6736 .6849 .6942 .7022 .7090 .7149 .7200 .7246 .7287 .7324 .7358 .7388 .7416 .7442 .7466 .7488 .7509 .7528 .7548 .7564 .7581 .7594 .7610 .7624 .7636 .7647
3.0 .6815 .6889 .7026 .7147 .7248 .7332 .7403 .7463 .7516 .7562 .7603 .7640 .7673 .7702 .7730 .7755 .7778 .7799 .7818 .7837 .7854 .7870 .7885 .7899 .7913 .7925 .7938 .7950 .7962
3.5 .7168 .7233 .7358 .7467 .7558 .7634 .7698 .7753 .7801 .7842 .7879 .7912 .7942 .7968 .7993 .8015 .8036 .8055 .8073 .8089 .8104 .8119 .8132 .8145 .8157 .8168 .8179 .8189 .8199
4.0 .7451 .7510 .7623 .7723 .7806 .7876 .7934 .7984 .8027 .8065 .8098 .8128 .8155 .8179 .8202 .8222 .8241 .8258 .8274 .8289 .8303 .8316 .8328 .8339 .8350 .8361 .8370 .8380 .8388
4.5 .7684 .7737 .7841 .7933 .8009 .8073 .8126 .8172 .8211 .8246 .8277 .8304 .8328 .8351 .8371 .8390 .8407 .8423 .8437 .8451 .8464 .8476 .8487 .8497 .8507 .8517 .8526 .8534 .8542
5.0 .7878 .7926 .8023 .8107 .8178 .8236 .8286 .8328 .8364 .8396 .8425 .8450 .8473 .8493 .8512 .8529 .8545 .8559 .8573 .8585 .8597 .8608 .8618 .8628 .8637 .8646 .8654 .8662 .8669
5.5 .8042 .8087 .8176 .8255 .8320 .8375 .8420 .8460 .8493 .8523 .8549 .8573 .8594 .8613 .8630 .8646 .8660 .8674 .8686 .8698 .8709 .8719 .8729 .8738 .8746 .8754 .8762 .8769 .8776
6.0 .8183 .8224 .8307 .8381 .8442 .8493 .8536 .8572 .8604 .8631 .8656 .8677 .8697 .8715 .8731 .8746 .8759 .8772 .8783 .8794 .8804 .8814 .8823 .8831 .8839 .8846 .8853 .8860 .8866
6.5 .8305 .8343 .8421 .8490 .8548 .8595 .8635 .8669 .8699 .8725 .8748 .8768 .8786 .8803 .8818 .8832 .8845 .8856 .8867 .8877 .8887 .8896 .8904 .8912 .8919 .8926 .8933 .8939 .8945
7.0 .8412 .8447 .8521 .8586 .8640 .8684 .8722 .8754 .8782 .8806 .8828 .8847 .8864 .8880 .8894 .8907 .8919 .8930 .8940 .8950 .8958 .8967 .8975 .8982 .8989 .8995 .9002 .9007 .9013
7.5
k
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
rrl m I z2
0
K m 0) I -n
cn m
C; 31) w CD 00
r-
0 CA)
z
INo
p
Table 4. Lower First Percentiles for Bartlett's Distribution
n .0157 .0219 .0296 .0370 .0443 .0510 .0571 .0626 .0684 .0729 .0775 .0819 .0860 .0897 .0931 .0964 .0996 .1026 .1054 .1082 .1106 .1131 .1152 .1180 .1197 .1219 .1238 .1256 .1276
.5 .1411 .1670 .1927 .2157 .2344 .2505 .2644 .2763 .2876 .2973 .3059 .3139 .3211 .3275 .3334 .3393 .3446 .3491 .3542 .3580 .3619 .3655 .3690 .3724 .3761 .3786 .3817 .3843 .3868
1.0 .2843 .3165 .3475 .3729 .3937 .4110 .4263 .4389 .4498 .4597 .4679 .4755 .4829 .4886 .4946 .4999 .5049 .5094 .5135 .5174 .5211 .5248 .5279 .5311 .5343 .5371 .5391 .5417 .5444
1.5 .3984 .4304 .4607 .4850 .5046 .5208 .5343 .5458 .5558 .5648 .5727 .5793 .5860 .5913 .5966 .6015 .6056 .6097 .6133 .6170 .6200 .6233 .6260 .6285 .6312 .6334 .6356 .6378 .6402
2.0 .4850 .5149 .5430 .5653 .5832 .5978 .6100 .6203 .6293 .6370 .6439 .6500 .6555 .6605 .6653 .6695 .6733 .6767 .6799 .6829 .6858 .6886 .6912 .6933 .6957 .6976 .6996 .7013 .7030
2.5 .5513 .5787 .6045 .6248 .6410 .6542 .6652 .6744 .6824 .6894 .6955 .7009 .7058 .7102 .7142 .7179 .7212 .7243 .7274 .7301 .7327 .7350 .7371 .7393 .7410 .7430 .7448 .7464 .7477
3.0 .6031 .6282 .6518 .6704 .6851 .6970 .7069 .7153 .7224 .7287 .7342 .7391 .7434 .7474 .7510 .7542 .7575 .7600 .7626 .7650 .7672 .7693 .7712 .7732 .7750 .7765 .7780 .7797 .7810
3.5 .6445 .6676 .6892 .7062 .7197 .7305 .7395 .7471 .7536 .7593 .7643 .7687 .7727 .7762 .7794 .7824 .7851 .7876 .7899 .7921 .7941 .7959 .7977 .7993 .8009 .8023 .8037 .8050 .8064
4.0 .6783 .6996 .7195 .7351 .7475 .7575 .7657 .7726 .7786 .7838 .7883 .7923 .7959 .7992 .8021 .8048 .8073 .8095 .8116 .8136 .8154 .8171 .8187 .8202 .8216 .8229 .8242 .8254 .8265
4.5 .7063 .7260 .7445 .7590 .7703 .7795 .7871 .7935 .7990 .8037 .8079 .8116 .8149 .8179 .8206 .8230 .8253 .8274 .8293 .8311 .8328 .8343 .8358 .8371 .8384 .8396 .8408 .8419 .8429
5.0 .7299 .7483 .7654 .7789 .7894 .7979 .8050 .8109 .8160 .8203 .8242 .8276 .8307 .8334 .8359 .8382 .8403 .8422 .8440 .8456 .8471 .8486 .8499 .8512 .8524 .8535 .8545 .8555 .8565
5.5 .7501 .7672 .7832 .7958 .8056 .8135 .8201 .8256 .8303 .8344 .8380 .8411 .8440 .8465 .8488 .8510 .8529 .8547 .8563 .8578 .8593 .8606 .8618 .8630 .8641 .8651 .8661 .8670 .8679
6.0 .7674 .7835 .7985 .8103 .8195 .8269 .8330 .8382 .8426 .8464 .8497 .8527 .8553 .8577 .8599 .8619 .8637 .8653 .8668 .8683 .8696 .8708 .8720 .8731 .8741 .8751 .8760 .8768 .8777
6.5 .7825 .7977 .8118 .8229 .8315 .8385 .8443 .8491 .8532 .8568 .8599 .8627 .8652 .8674 .8694 .8713 .8730 .8745 .8760 .8773 .8785 .8797 .8808 .8818 .8827 .8837 .8845 .8853 .8861
7.0 .7958 .8101 .8235 .8339 .8421 .8486 .8541 .8586 .8625 .8659 .8688 .8714 .8738 .8759 .8778 .8795 .8811 .8825 .8839 .8851 .8863 .8874 .8884 .8894 .8903 .8911 .8919 .8927 .8934
7.5
k
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
m z 0
-I
m 0
C)
C
-< CD co 0
0 r o.1
0
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
n .9999 .9900 .9590 .9183 .8774 .8398 .8063 .7768 .7508 .7277 .7073 .6890 .6726 .6577 .6443 .6325 .6207 .6104 .6011 .5920 .5838 .5762 .5693 .5622 .5558 .5499 .5447 .5392 .5344
.5 1.0 1.5 2.0
.9999 1.0000 1.0000 .9959 .9974 .9981 .9826 .9891 .9921 .9646 .9777 .9837 .9458 .9656 .9749 .9280 .9541 .9664 .9115 .9433 .9584 .8967 .9335 .9511 .8832 .9246 .9445 .8710 .9165 .9384 .8600 .9091 .9329 .8500 .9023 .9278 .8408 .8961 .9231 .8324 .8904 .9188 .8246.8 851 .9148 .81 .8802 .9111 .8109 .8756 .9076 .8047 .8714 .9044 .7989 .8674 .9013 .7936 .8637 .8985 .7885 .8 958 .7838 .8568 .8568 933 .7793 .8537 .8909 .7750 .8507 .8886 .7710 .8479 .8865 .7672 .8453 .8844 .7637 .8427 .8825 .7602 .8403 .8806 .7571 .8380 .8789
Table 5. Upper First Percentiles for Bartlett's Distribution
2.5 1.0000 .9988 .9949 .9895 .9837 .9781 .9729 .9681 .9637 .9597 .9560 .9526 .9495 .9466 .9439 .9414 .9391 .9369 .9349 .9329 .9311 .9294 .9278 .9263 .9248 .9234 .9221 .9208 .9196
3.0 1.0000 .9990 .9957 .9911 .9861 .9814 .9769 .9728 .9691 .9656 .9625 .9596 .9569 .9544 .9521 .9500 .9480 .9461 .9443 .9427 .9411 .9397 .9383 .9369 .9357 .9345 .9333 .9323 .9312
3.5 1.0000 .9991 .9962 .9922 .9880 .9838 .9800 .9763 .9731 .9700 .9673 .9647 .9624 .9602 .9582 .9563 .9546 .9529 .9514 .9500 .9486 .9473 .9461 .9449 .9438 .9428 .9418 .9408 .9399
4.0 1.0000 .9992 .9967 .9931 .9893 .9857 .9822 .9790 .9761 .9735 .9710 .9687 .9667 .9647 .9629 .9613 .9597 .9583 .9569 .9556 .9544 .9532 .9521 .9511 .9501 .9492 .9483 .9475 .9466
4.5 1.0000 .9993 .9970 .9938 .9904 .9871 .9841 .9812 .9786 .9762 .9740 .9719 .9701 .9683 .9667 .9652 .9638 .9625 .9613 .9601 .9590 .9580 .9570 .9561 .9552 .9543 .9535 .9528 .9520
5.0 1.0000 .9994 .9973 .9944 .9913 .9883 .9855 .9829 .9806 .9784 .9764 .9745 .9728 .9713 .9698 .9684 .9671 .9659 .9648 .9638 .9628 .9618 .9609 .9601 .9593 .9585 .9578 .9571 .9564
5.5 1.0000 .9994 .9975 .9949 .9921 .9893 .9868 .9844 .9822 .9802 .9784 .9767 .9751 .9737 .9723 .9711 .9699 .9688 .9678 .9668 .9659 .9650 .9642 .9634 .9627 .9620 .9613 .9607 .9601
6.0 1.0000 .9995 .9977 .9953 .9927 .9902 .9878 .9856 .9836 .9818 .9801 .9785 .9771 .9757 .9745 .9733 .9723 .9712 .9703 .9694 .9686 .9678 .9670 .9663 .9656 .9649 .9643 .9637 .9632
6.5 1.0000 .9995 .9979 .9956 .9932 .9909 .9887 .9867 .9848 .9831 .9815 .9801 .9787 .9775 .9763 .9753 .9743 .9733 .9724 .9716 .9708 .9701 .9694 .9687 .9681 .9675 .9669 .9663 .9658
7.0 1.0000 .9995 .9980 .9959 .9937 .9915 .9895 .9876 .9858 .9842 .9828 .9814 .9802 .9790 .9779 .9769 .9760 .9751 .9743 .9735 .9728 .9721 .9714 .9708 .9702 .9697 .9691 .9686 .9681
7.5
k
1.0000 .9985 .9938 .9872 .9802 .9735 .9672 .9614 .9561 .9513 .9468 .9428 .9390 .9355 .9323 .9293 .9266 .9239 .9215 .9192 .9171 .9150 .9131 .9112 .9095 .9079 .9063 .9048 .9033 1
m --
z
0 m I
m c3 Cr) -n
0 oco c rCO
z
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
n .9992 .9751 .9245 .8708 .8227 .7816 .7468 .7171 .6916 .6695 .6502 .6332 .6182 .6047 .5925 .5814 .5710 .5618 .5538 .5456 .5384 .5316 .5256 .5194 .5140 .5088 .5041 .4994 .4953
.5
Table 6. Upper 2.5 Percentiles for Bartlett's Distribution
3.5 .9999 .9978 .9929 .9874 .9820 .9771 .9727 .9688 .9653 .9621 .9593 .9567 .9543 .9522 .9502 .9483 .9466 .9450 .9435 .9421 .9408 .9396 .9384 .9373 .9363 .9353 .9344 .9335 .9326
4.0 .9999 .9980 .9937 .9888 .9841 .9797 .9758 .9724 .9692 .9664 .9639 .9616 .9595 .9576 .9558 .9541 .9526 .9512 .9499 .9486 .9475 .9464 .9453 .9444 .9434 .9425 .9417 .9409 .9401
4.5 .9999 .9982 .9944 .9900 .9857 .9818 .9783 .9752 .9724 .9699 .9676 .9655 .9636 .9619 .9603 .9588 .9574 .9561 .9549 .9538 .9528 .9518 .9508 .9500 .9491 .9483 .9476 .9468 .9462
5.0 1.0000 .9984 .9949 .9909 .9870 .9835 .9803 .9775 .9749 .9727 .9706 .9687 .9670 .9654 .9639 .9626 .9613 .9602 .9591 .9581 .9571 .9562 .9553 .9545 .9538 .9531 .9524 .9517 .9511 .9511
5.5 5.5 1.0000 .9985 .9954 .9917 .9881 .9849 .9820 .9794 .9771 .9750 .9731 .9713 .9698 .9683 .9670 .9657 .9646 .9635 .9625 .9616 .9607 .9599 .9591 .9584 .9577 .9570 .9564 .9558 .9552 .9552
6.0 1.0000 .9987 .9957 .9923 .9891 .9861 .9834 .9810 .9789 .9769 .9752 .9736 .9721 .9708 .9695 .9684 .9673 .9664 .9654 .9646 .9638 .9630 .9623 .9616 .9609 .9603 .9597 .9592 .9586 .9586
6.5 1.0000 .9988 .9960 .9929 .9899 .9871 .9846 .9824 .9804 .9786 .9770 .9755 .9741 .9729 .9717 .9707 .9697 .9688 .9679 .9671 .9664 .9656 .9650 .9643 .9637 .9632 .9626 .9621 .9616 .9616
7.0 1.0000 .9988 .9963 .9934 .9906 .9880 .9857 .9836 .9817 .9800 .9785 .9771 .9759 .9747 .9736 .9727 .9717 .9709 .9701 .9693 .9686 .9680 .9673 .9667 .9662 .9656 .9651 .9646 .9642 .9642
7.5 1.0 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9
8
k 3.0 .9999 .9974 .9919 .9855 .9793 .9737 .9687 .9642 .9602 .9566 .9533 .9504 .9477 .9452 .9429 .9408 .9389 .9371 .9354 .9338 .9323 .9309 .9296 .9283 .9271 .9260 .9249 .9239 .9229
8
2.5 .9999 .9970 .9904 .9830 .9757 .9691 .9632 .9580 .9533 .9491 .9453 .9419 .9387 .9359 .9332 .9308 .9285 .9264 .9244 .9226 .9208 .9192 .9177 .9162 .9149 .9135 .9123 .9111 .9100
7.5
2.0 .9999 .9963 .9883 .9792 .9705 .9626 .9555 .9492 .9436 .9386 .9340 .9299 .9261 .9227 .9195 .9166 .9139 .9114 .9090 .9068 .9048 .9028 .9010 .8993 .8977 .8961 .8946 .8932 .8919
7.0
1.5 .9999 .9953 .9852 .9736 .9626 .9526 .9437 .9359 .9288 .9225 .9168 .9117 .9070 .9028 .8989 .8952 .8919 .8888 .8859 .8832 .8806 .8782 .8760 .8739 .8718 .8699 .8681 .8664 .8648
6.5
1.0 .9998 .9935 .9796 .9639 .9489 .9355 .9236 .9130 .9037 .8953 .8878 .8810 .8748 .8692 .8641 .8593 .8550 .8509 .8471 .8436 .8403 .8372 .8342 .8315 .8289 .8264 .8241 .8218 .8197
6.0
.9997 .9896 .9676 .9430 .9200 .8995 .8815 .8657 .8518 .8394 .8284 .8185 .8096 .8015 .7941 .7873 .7810 .7752 .7698 .7648 .7601 .7559 .7515 .7477 .7442 .7406 .7374 .7344 .7315
---I m C) z 0I 0
m
5
cn -n 0
c CD c30
0
z 0 p
n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Table 7. Upper Fifth Percentiles for Bartlett's Distribution
1.5 .9994 .9905 .9759 .9615 .9488 .9379 .9285 .9203 .9131 .9068 .9012 .8961 .8916 .8875 .8837 .8803 .8771 .8742 .8714 .8689 .8665 .8643 .8622 .8602 .8584 .8566 .8550 .8534 .8519
2.0 .9996 .9926 .9811 .9697 .9596 .9509 .9433 .9368 .9310 .9259 .9214 .9174 .9137 .9104 .9073 .9045 .9019 .8996 .8973 .8953 .8934 .8915 .8898 .8882 .8867 .8853 .8839 .8826 .8814
2.5 .9996 .9939 .9844 .9750 .9666 .9594 .9531 .9476 .9428 .9386 .9348 .9314 .9283 .9255 .9230 .9206 .9185 .9165 .9146 .9129 .9112 .9097 .9083 .9069 .9057 .9044 .9033 .9022 .9012
3.0 .9997 .9948 .9868 .9787 .9716 .9654 .9600 .9553 .9512 .9476 .9443 .9414 .9387 .9363 .9341 .9321 .9302 .9285 .9269 .9254 .9240 .9227 .9214 .9203 .9192 .9181 .9171 .9162 .9153
3.5 .9997 .9955 .9885 .9815 .9753 .9698 .9651 .9610 .9574 .9542 .9514 .9488 .9465 .9444 .9425 .9407 .9390 .9375 .9361 .9348 .9336 .9324 .9313 .9303 .9293 .9284 .9275 .9267 .9259
4.0 .9998 .9960 .9898 .9836 .9781 .9733 .9691 .9655 .9623 .9594 .9569 .9546 .9525 .9506 .9489 .9473 .9459 .9445 .9433 .9421 .9410 .9400 .9390 .9381 .9372 .9364 .9356 .9349 .9342
4.5 .9998 .9964 .9909 .9853 .9804 .9760 .9723 .9690 .9661 .9635 .9613 .9592 .9573 .9556 .9541 .9527 .9513 .9501 .9490 .9479 .9469 .9460 .9451 .9443 .9435 .9428 .9421 .9414 .9408
5.0 .9998 .9968 .9917 .9867 .9822 .9783 .9748 .9719 .9692 .9669 .9648 .9630 .9613 .9597 .9583 .9570 .9558 .9547 .9537 .9527 .9518 .9509 .9501 .9494 .9487 .9480 .9473 .9467 .9462
5.5 .9998 .9970 .9924 .9878 .9837 .9801 .9770 .9742 .9718 .9697 .9678 .9661 .9645 .9631 .9618 .9606 .9595 .9585 .9575 .9566 .9558 .9550 .9543 .9536 .9530 .9523 .9517 .9512 .9507
6.0 .9998 .9973 .9930 .9887 .9850 .9817 .9788 .9763 .9740 .9721 .9703 .9687 .9673 .9660 .9648 .9637 .9626 .9617 .9608 .9600 .9592 .9585 .9578 .9572 .9566 .9560 .9555 .9549 .9545
6.5 .9999 .9975 .9935 .9896 .9861 .9830 .9803 .9780 .9759 .9741 .9724 .9710 .9696 .9684 .9673 .9663 .9653 .9645 .9636 .9629 .9622 .9615 .9609 .9603 .9597 .9592 .9587 .9582 .9577
7.0 .9999 .9977 .9940 .9903 .9870 .9842 .9817 .9795 .9775 .9758 .9743 .9729 .9717 .9705 .9695 .9685 .9677 .9668 .9661 .9654 .9647 .9641 .9635 .9629 .9624 .9619 .9614 .9610 9605.
7.5
k 1.0 .9992 .9869 .9670 .9475 .9304 .9157 .9031 .8923 .8828 .8745 .8671 .8605 .8545 .8492 .8443 .8398 .8356 .8318 .8283 .8250 .8220 .8191 .8164 .8138 .8115 .8092 .8071 .8052 .8031
.5 .9969 .9987 .9502 .9790 .8803 .9477 .8170 .9176 .7653 .8915 .7233 .8695 .6890 .8508 .6605 .8348 .6366 .8210 .6161 .8089 .5984 .7982 .5826 .7887 .5694 .7802 .5574 .7726 .5464 .7656 .5366 .7593 .5274 .7535 .5196 .7482 .5123 .7432 .5055 .7388 .4988 .7343 .4931 .7305 .4877 .7265 .4827 .7230 .4778 .7199 .4735 .7167 .4695 .7137 .4655 .7110 .4617 .7083
H-
m
z
m
0
Cl) '1
m CC
(0 0
0 CA)
1%)
0
3.0 3.5 4.0
k
.9994
6.5
.9529
.9681 .9665 .9651 .9638 .9627 .9616 .9607 .9598 .9590 .9582 .9575 .9569 .9563 .9557 .9552 .9547 .9542 .9538 .9533
.9994 .9948 .9893 .9845 .9805 .9772 .9744 .9720
7.0
.9561
.9995 .9952 .9900 .9855 .9818 .9787 .9761 .9739 .9719 .9702 .9688 .9674 .9662 .9652 .9642 .9633 .9625 .9617 .9610 .9604 .9598 .9592 .9587 .9582 .9577 .9573 .9568 .9564
7.5
.9699
.9497
.9944 .9885 .9833 .9790 .9754 .9724 .9698 .9676 .9656 .9639 .9624 .9610 .9598 .9587 .9576 .9567 .9558 .9550 .9543 .9536 .9529 .9523 .9517 .9512 .9507 .9502 .9493
Table 8. Upper Tenth Percentiles for Bartlett's Distribution
2.5
.9477
6.0
2.0
.9455 .9435 .9418 .9402 .9387 .9373 .9361 .9350 .9339 .9329 .9320 .9311 .9303 .9295 .9288 .9281 .9274 .9268
5.5
1.5 .9989 .9908 .9810 .9724 .9654 .9596 .9547 .9505 .9469 .9438 .9411 .9386 .9364 .9344 .9326 .9310 .9295 .9281 .9268 .9256 .9245 .9234 .9225 .9215 .9207 .9199 .9191 .9184 .9177
5.0
1.0 .9988 .9894 .9781 .9684 .9603 .9536 .9481 .9433 .9392 .9357 .9325 .9297 .9272 .9250 .9229 .9211 .9193 .9178 .9163 .9149 .9137 .9125 .9114 .9103 .9094 .9084 .9076 .9067 .9059
4.5
.b .9986 .9875 .9743 .9628 .9534 .9457 .9392 .9336 .9289 .9247 .9211 .9179 .9150 .9124 .9100 .9078 .9058 .9040 .9023 .9007 .8993 .8979 .8966 .8954 .8943 .8932 .8922 .8913 .8903
n .9983 .9848 .9688 .9550 .9437 .9344 .9266 .9200 .9143 .9094 .9051 .9012 .8978 .8947 .8918 .8893 .8869 .8848 .8828 .8809 .8792 .8775 .8760 .8746 .8733 .8720 .8708 .8697 .8686 .9991 .9918 .9831 .9756 .9694 .9642 .9598 .9561 .9529 .9501
.9978 .9806 .9603 .9430 .9289 .9172 .9076 .8994 .8923 .8862 .8809 .8762 .8719 .8681 .8646 .8615 .8586 .8559 .8535 .8512 .8491 .8471 .8453 .8435 .8419 .8403 .8389 .8375 .8362 .9993 .9939 .9875 .9818 .9772 .9733 .9700 .9672 .9648 .9627 .9609 .9592 .9578 .9564 .9552 .9541 .9531 .9521 .9513 .9505 .9497 .9490 .9483 .9477 .947 1 .9466 .9460 .9455 .9451
.9969 .9733 .9458 .9225 .9036 .8881 .8754 .8646 .8554 .8475 .8405 .8343 .8288 .8239 .8194 .8154 .8117 .8082 .8051 .8022 .7995 .7969 .7946 .7923 .7903 .7882 .7864 .7848 .7830 .9992 .9934 .9863 .9801 .9750 .9708 .9672 .9642 .9616 .9593 .9573 .9555 .9539 .9524 .9511 .9499 .9488 .9478 .9468 .9459 .9451 .9443 .9436 .9430 .9423 .9417 .9411 .9406 .9401
.9950 .9574 .9147 .8794 .8512 .8285 .8099 .7944 .7812 .7699 .7601 .7514 .7437 .7368 .7306 .7250 .7199 .7151 .7108 .7070 .703 1 .6998 .6964 .6933 .6906 .6878 .6854 .6830 .6808 .9992 .9927 .9849 .9781 .9725 .9678 .9639 .9606 .9577 .9552 .9530 .9510 .9492 .9476 .9462 .9449 .9436 .9425 .9415 .9405 .9396 .9388 .9380 .9372 .9365 .9359 .9353 .9347 .9341
2 .9877 3 .9008 4 .8102 5 .7406 6 .6884 7 .6483 8 .6167 9 .5911 10 .5699 11 .5521 12 .5371 13 .5234 14 .5121 15 .5020 16 .4929 17 .4845 18 .4775 19 .4708 20 .4648 21 .4591 22 .4538 23 .4491 24 .4447 25 .4407 26 .4367 27 .4331 28 .4298 29 .4266 30 .4235
I
z
0 m H -I
O
cn C)
C
m co
cc CD 0 o
0 rco N)
z
-h
0
Table 9. Fiftieth Percentiles for Bartlett's Distribution
n .7071 .5175 .4417 .4022 .3781 .3620 .3502 .3417 .3349 .3293 .3252 .3211 .3182 .3154 .3133 .3110 .3094 .3078 .3064 .3050 .3039 .3028 .3019 .3010 .3003 .2994 .2988 .2981 .2975
.5 .8660 .7533 .6998 .6693 .6498 .6362 .6262 .6186 .6126 .6077 .6037 .6003 .5974 .5949 .5929 .5908 .5892 .5877 .5863 .5850 .5841 .5831 .5821 .5811 .5805 .5797 .5792 .5784 .5778
1.0 .9148 .8375 .7987 .7760 .7612 .7508 .7430 .7371 .7324 .7286 .7254 .7227 .7204 .7184 .7167 .7152 .7139 .7127 .7116 .7106 .7097 .7088 .7082 .7076 .7069 .7063 .7058 .7053 .7049
1.5 .9378 .8793 .8492 .8314 .8196 .8113 .8051 .8003 .7965 .7934 .7908 .7887 .7868 .7852 .7838 .7826 .7815 .7805 .7796 .7788 .7781 .7774 .7768 .7763 .7757 .7753 .7748 .7745 .7740
2.0 .9510 .9041 .8796 .8650 .8552 .8484 .8432 .8392 .8361 .8335 .8313 .8295 .8280 .8266 .8254 .8244 .8235 .8227 .8219 .8213 .8207 .8201 .8196 .8191 .8187 .8183 .8179 .8176 .8173
2.5 .9597 .9205 .8999 .8875 .8792 .8733 .8689 .8655 .8628 .8606 .8588 .8572 .8559 .8547 .8537 .8528 .8521 .8514 .8507 .8501 .8496 .8491 .8487 .8483 .8479 .8476 .8473 .8470 .8467
3.0 .9657 .9321 .9143 .9036 .8964 .8913 .8875 .8845 .8821 .8802 .8786 .8772 .8761 .8750 .8742 .8734 .8727 .8721 .8715 .8710 .8706 .8702 .8698 .8694 .8691 .8688 .8685 .8683 .8680
3.5 .9702 .9408 .9251 .9156 .9093 .9048 .9014 .8988 .8967 .8950 .8935 .8923 .8913 .8904 .8896 .8889 .8883 .8878 .8873 .8868 .8864 .8860 .8857 .8854 .8851 .8848 .8846 .8844 .8841
4.0 .9736 .9475 .9335 .9250 .9194 .9153 .9123 .9099 .9080 .9065 .9052 .9041 .9032 .9024 .9017 .9011 .9005 .9000 .8996 .8992 .8988 .8985 .8982 .8979 .8976 .8974 .8972 .8970 .8968
4.5 .9764 .9529 .9402 .9325 .9274 .9237 .9210 .9189 .9171 .9157 .9146 .9136 .9128 .9120 .9114 .9108 .9103 .9099 .9095 .9091 .9088 .9085 .9082 .9079 .9077 .9075 .9073 .9071 .9069
5.0 .9786 .9572 .9457 .9387 .9340 .9307 .9281 .9262 .9246 .9233 .9222 .9214 .9206 .9199 .9193 .9188 .9184 .9180 .9176 .9173 .9170 .9167 .9164 .9162 .9160 .9158 .9156 .9154 .9153
5.5 .9804 .9608 .9503 .9438 .9395 .9364 .9341 .9323 .9309 .9297 .9287 .9279 .9272 .9265 .9260 .9255 .9251 .9247 .9244 .9241 .9238 .9235 .9233 .9231 .9229 .9227 .9225 .9224 .9222
6.0 .9820 .9639 .9541 .9482 .9442 .9413 .9391 .9375 .9361 .9350 .9341 .9334 .9327 .9321 .9316 .9312 .9308 .9304 .9301 .9298 .9296 .9294 .9291 .9289 .9287 .9286 .9284 .9283 .9281
6.5 .9833 .9665 .9574 .9519 .9481 .9455 .9435 .9419 .9407 .9397 .9388 .9381 .9375 .9369 .9365 .9361 .9357 .9354 .9351 .9348 .9346 .9343 .9341 .9340 .9338 .9336 .9335 .9333 .9332
7.0 .9844 .9688 .9603 .9551 .9516 .9491 .9472 .9458 .9446 .9437 .9429 .9422 .9416 .9411 .9407 .9403 .9399 .9396 .9394 .9391 .9389 .9387 .9385 .9383 .9382 .9380 .9379 .9377 .9376
7.5
k
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
JEROMEP. KEATING,RONALDE. GLASER,AND NORMAS. KETCHUM
80
4.1
An IFR Example
The datain operatinghoursfor planenumber7909 are as follows: Lr
0
0.6-
90 10 14 24 44 59 310 76 130 208
0.4-
0.2 0.0 1.0
1.5
I 2.0
I 2.5
I 3.0
I 3.5
I 4.0
k Figure 1. Power Curves for Testing Exponentiality Against Gamma IFR Alternatives.
approximateand the exact criticalvalues is less than 10-3 and usually less than 5 x
10-4.
The power functioncan also be calculated,since we have Expressions(3) and (4) for the distribution of W. Let ;'(k; a, k(, n) denote the power function when H, : k = k(,versusHI : k > k,, a is the level of significanceof the test, and n is the sample size. Therefore, it follows that '(k; a, k(, n) = 1 - G(w,:k,,; k, n).
(9)
The power function for testing the null hypothesis of exponentialityagainstthe statedalternativeat the 5% level of significanceis graphedfor varioussample sizes in Figure 1. In Figure1, one can see the graphical informationthat confirmsthe results of Engelhardtand Bain (1977) that the test is unbiased. FromFigure1, we see the anticipatedresultswhen the sample size is small, say 5. In this case, when k = 4 the power is still less than one-half. When n is as small as 20, however, we observe that the ascent of the power curve is quite steep. In fact, the power exceeds .70 in detectinga true k as small as 2 when the sample size is 20 and the size of the test is
.05. For n = 30, the resultsare even more remarkable. Likewise, one can also calculateminimumsample sizes needed to detect a value of k as largeas k' with a maximumType II errorrate of /. For example, to detect a value of k as large as 2 with at least 75% power, the minimumsamplesize requiredis n = 22. To detect a value of k as large as 3 with at least 95% power, the minimumsample size requiredis
n = 17.
4.
DATA EXAMPLES
We shall illustratethe methodsgiven in this article through data provided by Proschan(1963) on the times between successivefailuresof air conditioning equipmentin 13 Boeing 720 aircraft. FEBRUARY1990, VOL. 32, NO. 1 TECHNOMETRICS,
60 186 61 49 56 20 79 84 29 118 25 156 26 44 23 62 70 101 208
In this sample, the geometricmean i = 60.15 hours, and the arithmeticmean t = 83.52 hours. Thus the test statisticW, the ratio of geometricto arithmetic mean, is W = it/t = .7203.
Lawless (1982), using nonparametricmethods, concludedthat there is a lack of evidenceagainstthe null hypothesisof exponentiality.Let us test H( : k = 1 againstthe alternativeH, : k > 1. FromTables 5 and 7, we find the critical values as w.01 1.29 = .7602 and w.(51.29 = .7110. Consequently, we reject H( at
the 5% level of significancebut not at the 1% level. Unlike Lawless, we use the restrictedalternativeof the gamma family for which k > 1. The p value associatedwith this test is .0384. Since we have rejectedH( : k = 1, it now seems appropriateto provide a point estimate of k. The MU estimatorthat is a functionof w can be found by solving for k*, where w 5^.k29= .7203. From Table 9, we can see that w5:1.5.29< .7203 < w.2.29. When k, < kK < k2 and tabled values are known for wa, .l,, and Wk,;.,,, an excellent interpolation scheme for w,;.k,, can be developed from the results of Wilk,
Gnanadesikan,and Huyett (1962). To approximate the critical value wa:kn,, linearly interpolate between the points (k,, 1/(1 - W:k.n)) and (k2, 1/(1 wa:k_,,)). This interpolation scheme is motivated by
the previouslynoted resultthat the MLE of k is well approximatedby a linearfunctionof 1/(1 - W). We shallreferto thismethodas Wilkinterpolation,which is decidedlysuperiorto ordinarylinearinterpolation. Wilk interpolationin Table 9 provides an approximate MU estimatorof k as 1.5876. Using the secant method, we arrive at the MU estimate of kMU=
1.5881. Since H(, is rejected, it seems worthwhileto calculate an estimatedpower of this test when n = 29 and k = 1.59. Using the methodologygiven in Section 3, we obtain the power as 1 - G(w. 15;,29;1.59,
29) = .5602. If the true value of k is 1.59, the probabilityof makinga Type II error(i.e., acceptingthe false null of exponentiality)is little less than .44 for a randomsample of size 29. From Table 7, we can also place a 95% lower confidenceboundon k. In thiscase, we need to solve for k', where w.(:k.29 = .7203. From Table 7, we can see that W.051.29< .7203 < w.(51.5.29, which places k'
INFERENCESON THE GAMMASHAPE PARAMETER
in the interval (1, 1.5). The Wilk-interpolated approximation to k' is given as 1.0345. The exact 95% lower confidence bound found numerically is 1.0350.
81
erate percentiles of W in all cases of (k, n) for which the percentile fell outside the radius of convergence. [Received May 1988. Revised July 1989.1
4.2
A DFR Example
Pressure vessels constructed of fiber/epoxy composite materials wrapped around metal liners have lifetimes that may be modeled by a gamma distribution. The shape parameter depends on such factors as the applied pressure and the composite wall thickness. For suitably high pressures and/or thin walls, a DFR condition may occur; that is, the mean residual life for a vessel having survived a wear-in period exceeds the expected life of a new vessel. The following data (in hours) report the failure times of 20 similarly constructed vessels subjected to a certain constant pressure: 274 28.5 1.7 20.8 871 363 1,311 1,661 236 828 458 290 54.9 175 1.787 970 .75 1,278 776 126
The arithmetic and geometric means are computed to be - = 575.53 and i = 196.94, so w = t/t = .3422. The test for exponentiality against DFR alternatives (H( : k = 1 vs. H, : k < 1) is rejected at the 1% level, since w < w 01 ,20 = .3542. In fact, the MU estimate k*, which solves W^5k*^20= .3422, is found by repeated simulations to be k* = .544, and analogously, the 90% upper confidence bound on k, which solves w.k',20 = .3422, is seen to be k' = .758. (Wilk interpolation using the percentile tables results in the approximations k: = .540 and k' = .744.) It is rather apparent that these vessels exhibit DFR lifetimes. When k < 1, Wilk interpolation tends to underestimate the true upper confidence bounds, whereas linear interpolation overestimates the true upper confidence bounds. Consequently, we recommend a hybrid interpolation scheme based on these methods. Using this interpolation scheme and the percentile tables 9 and 1, we obtain respective approximations of k* = .546 and k' = .760. Moreover, this hybrid interpolation can be used for values of k in [0, .5) by setting w, _:a,,
= 0.
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