Computational Statistics & Data Analysis 2 (1984) 191-206 North-Holland 191 prZL&d confidence we to 0 intervals for the Bernoulli parameter Olga ALVAREZ, Andrzej MA’T’IJSZEWSKI and David SOTRES Colegio de Postgradmdos, Centro de Estadisticay Calculo, Chapingo, Mexico 56230 Abstrmt: The paper proposes a practical procedure for obtaining a confidence interval (CI) for the parameter v of the Bernoulli distribution. Let x be the observed number of successes of a random sample of size n from this distribution. The procedure is as follows: use Table 1 to determine whether the given pair (n , x) is a sma!l or a large sample pair. If the small sample situation applies then use Table 2 which gives the Steme-Crow CI. Otherwise, use the Anscombe CI for which practical formulas are given. Keywords: Confidence interval, Bernoulli parameter. I. Introduction Recently Matuszewski and Sotres (1983) have made a comparison of the most well known confidence intervals (CI’s) for ?T, the parameter of the Bernoulli distribution. Among others they considered the uniformly most accurate unbiased (UMAU) CI (Eudey (1949)), the minimum total length conservative CI (Steme-Crow (1956)), two Bayesian-type CI’s, four well know approximate CI’s (Normal, arcsine, Anscombe (1948) and Borges (19X))), the Clopper-Pearson CI (1934), a fiducial-type CI (Stevens-Lehmann (1950)), and the Bl.yth and Still CI (1983). The comparison was done taking into account different aspects such as optimality, average length, actual confidence level, computational simplicity, etc. As a result of thi j comparison they recommend the use of any of the following four methods: Eudey (1949), Stevens-Lehmann (1950), Steme-Crow (1956) or Anscombe (1948) (in what follows these will be referenced as the ‘best’ methods). In choosing one of these four CI’s they suggest using your personal preferences with regard to the corresponding properties of these methods. Taking this recommendation as a basis the following procedure is proposed for obtaining. a CI for the parameter n: for small sample situations use the Sterne-Crow CI (see Table 2), otherwise use Anscombe’s CI (see formula (2.1)). Table 1 of Section 2 indicates if a given pair (n, x) is a large or a small sam,?le situation. A precise definition of a large sample pair (n, x) is also given. 0167-9473/84/$3.00 0 1984, Elsevier Science Publishers B.V. ((North-Holland)
Transcript
Computational Statistics & Data Analysis 2 (1984) 191-206 North-Holland
191
prZL&d confidence
we to 0 intervals for the Bernoulli
parameter
Olga ALVAREZ, Andrzej MA’T’IJSZEWSKI and David SOTRES Colegio de Postgradmdos, Centro de Estadistica y Calculo, Chapingo, Mexico 56230
Abstrmt: The paper proposes a practical procedure for obtaining a confidence interval (CI) for the parameter v of the Bernoulli distribution. Let x be the observed number of successes of a random sample of size n from this distribution. The procedure is as follows: use Table 1 to determine whether the given pair (n , x) is a sma!l or a large sample pair. If the small sample situation applies then use Table 2 which gives the Steme-Crow CI. Otherwise, use the Anscombe CI for which practical formulas are given.
Recently Matuszewski and Sotres (1983) have made a comparison of the most well known confidence intervals (CI’s) for ?T, the parameter of the Bernoulli distribution. Among others they considered the uniformly most accurate unbiased (UMAU) CI (Eudey (1949)), the minimum total length conservative CI (Steme-Crow (1956)), two Bayesian-type CI’s, four well know approximate CI’s (Normal, arcsine, Anscombe (1948) and Borges (19X))), the Clopper-Pearson CI (1934), a fiducial-type CI (Stevens-Lehmann (1950)), and the Bl.yth and Still CI (1983). The comparison was done taking into account different aspects such as optimality, average length, actual confidence level, computational simplicity, etc. As a result of thi j comparison they recommend the use of any of the following four methods: Eudey (1949), Stevens-Lehmann (1950), Steme-Crow (1956) or Anscombe (1948) (in what follows these will be referenced as the ‘best’ methods). In choosing one of these four CI’s they suggest using your personal preferences with regard to the corresponding properties of these methods.
Taking this recommendation as a basis the following procedure is proposed for obtaining. a CI for the parameter n: for small sample situations use the Sterne-Crow CI (see Table 2), otherwise use Anscombe’s CI (see formula (2.1)). Table 1 of Section 2 indicates if a given pair (n, x) is a large or a small sam,?le situation. A precise definition of a large sample pair (n, x) is also given.
192 0. Alvarez et al. / Confidence intervals for the Bernoulli parameter
Althoq# the two recommended randomized CI’s, i.e. those derived by Eudey and Stevens-Lehmann, do not appear explicitly in the proposed procedure, they do appear implicitly because, as we shall see in Subsection 2.2., they were used to obtain Table 1.
Table 2 gives the Sterne-Crow CI’s with 95% and 99% confidence levels for small sample pairs, obtained by using a FORTRAN program based on the algo- rithm given by Crow (1956).
The bounds of the intervals were obtaine to four decimal places, then rounded off to two decimal places. Unfortunately the original table for 1 I n s 30 given in Crow (1956) contains many errors.
2. i. ?%e Anscombe CI
As was mentioned in the introduction the proposed procedure uses the kscombe =CI for the iarge sample situations. We now give practical formulas for this Sle?~erv~k Anscombe (1948) suggested the use the of the transformation ?__ - T,( ;fj 1= il: + $ ( X + +)/( .FT + $) ] to get asymptotically constant vari- ance up to The second order of n. For the Anscombe CI we mean the irr.te?a.ls! *which is obtained through the pivotal method along tvith the assumption that T’(X) has exactly normal distribution. Practical formulas for this interval at the 1 - (Y level are the following:
LB(x) -=: sin* [msx(O O, (T,(x) - $71_.,2)/~~)],
UB(r)==sin2[min(900,(T,(x:+~~,_,/2)~~)], (2 1) .
where z1 __ a/2 is the lOO(1 - o/q +percentile of the standard normal distribution. It has been found (see Matuszewski and Sotres (1983)j that this CI has better bchavio~ &an the more traditional normal approximations (one of the approxi-. mations considered was that recommended by Ghosh (1979)). To demonstrate this the criterion ofl fiducial interpretation was used.
2.2. Approximation of the best methods by Anscombe’s CI
For a fixed 1 - (Y aald for a given pair (n, x), the ‘best’ methods result in numerically different intervals. Of course, these differences decrease as the sample size n increases. Also one can observe that the differences are smaller for central values of x when compared with those corresponding to values of x near 0 or n. Owe these differences between the four intervals are considered to be small from a practical point of view, all four methods can be represented by just one. Those combinations (n, x) for which such a situation occurs v.ill be called large sample cases. In what follows the criterion we have used to determine whether the pair (n j X) is a large sa le c?se or not is precisely defined.
0. Ahurez et al. / Confidence intervals for the Bernoulli parameter 193
First note that for a given 1 - a:, ii and X, the Sterne-CrcIw and Anscombe intervals are completely determined, while the Eudey and Stevens-Le intervals are not, because, in addition, they depend on u E [QJ]. However, since the bounds of these last two intervals are increasing in u it is sufficient to consider both intervals for u = 0 and u = 1. The Eudey (u = 01, Eudey (u = l), Stevens-Lehmann ( u = 0), Stevens-Lehmann (u = l), Sterne-Crow and Anscombe intervals will be denoted (LB,, UBi), i = 1,. . . ,6, respectively. We will say that a pair ( n, X) is a large sample case if
D( X, n) = max{ max ILBi - L&I, rrnxs IUBi - UB,() < 0.01 l iSi
Table 1 was obtained using this criterion. Therefore, for any large sample pair (n, x) the Anscombe CI is a good representative of the ‘best’ methods; and so one can say that the Anscombe CI has approximately all the optimal properties that the ‘best’ meihods have. For this reason and for its computational simpkity we recommend the use the Anscombe CI in the large sample situation.
2.3 The SE&~ sample situation
In lhis subsection the definition of the proposed procedure will be completed by choosing one of the 4 ‘best’ methods to be used in the small sample situation,
‘Table 1 Values of k such that Anscombe’s CI is a good representattve of the ‘best’ methods for k s x I ti - k, i.e. those pairs (n, x) for which D( n, x) -C 0.01
(Y = 0.95 a = 0.99
n k n k
52 23 56 26 53 19 57 24 s4 18 58 22 55 17 59 to 63 17 56 16 64 to 67 16 57 to 58 15 68 to 71 15 59 14 72 to 74 14 60 to 61 i3 75 to 77 13 62 to 64 12 78 to Xi 12 65 10 82 to R3 11 66 io 67 9 84 to 85 10 68 to 69 8 86 11 70 to 72 7 87 to 92 9 73 to 74 6 93 to 103 7 75 to 70 5 104 5 77 to 78 6 105 to 110 4 79 to 84 4 111 to 118 3 85 to 91 3 119 tc 139 2 92 to 109 2 140 to 200 1
110 to 200 1 -_-
19:’
Table 2A
0. Alvai.c’,- et al. j Confidence intervals for the Bernoulli parameter
Steme-Crow confider:ce intervals for 95% confidence levei -
i.e. when the actual pair (n, x) is not included in Table 1. Since we are interested in the small sample situation we must prefer an exact
method: this eliminates Anscombe’s CI. Now considering that randomized rules are generally not acceptable to practitioners (they often ask why a CI should depend on a random mechanism independent of the experimental data), one can see that the Sterne-Crow CI is the appropriate method for this situation
[ll F.J. Anseombe, The transformation of Poisson, binomial and negative-binomial data, &.I-
metrika 35 (3/4) (1948) 246-254.
206 0. Alvarez et al. / Confidence intervals for the Bernoulli parameter
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