In a sample of size *N*, let *n*_{j} for
be the observed frequencies in the *j*-th cell of a
*m*-cell multinomial distribution with corresponding
parameter
.
The maximum likelihood estimator of
is
where
for
.
Asymptotically,
has *m*-variate normal distribution with zero mean vector and
variance-covariance matrix
with elements

Therefore, the corresponding correlation matrix has singular negative product structure with , where , for , and rank

(4) |

and

(5) |

for , respectively, while Bailey (1980) suggested the other two confidence intervals for as

(6) |

and

(7) |

for , where , , and . Based on the Bonferroni inequality, without consideration of the singular correlation structure of , Goodman's (1965) criterion (hereafter called GC) is usually used to determine for the above four techniques in (4)-(7), where is defined as the percentage point of the chi-square distribution with 1 degree of freedom.

After taking into account the singular correlation structure of
,
Kwong and Iglewicz (1996) proposed another criterion (hereafter called
KIC) of setting
in (4)-(7),where
is
defined as the two-sided
percentage point of the standardized
*m*-variate normal distribution with the singular negative equi-correlated
structure. Kwong and Iglewicz (1996) conducted a simulation study to compare
the performances of these two criteria. The study concluded that KIC provides
less conservative and shorter confidence intervals than GC does when the
sample size is sufficiently large.

However, based on the conjecture that the least favorable parameter vector having all the elements equal to one another, KIC uses only part of singular correlation structure. Intuitively, it is possible to improve KIC if the singular negative product correlation structure, instead of singular equi-correlated structure, is incorporated into the evaluation of critical values.

For a given confidence level ,
let

By the central limit theorem,

Since the correlation structure of is dependent on the unknown parameter , the sample proportions are used to replace the parameters in order to estimate the correlation structure. Therefore, based on the result in Section 2.2, we numerically solve

for and then use it to construct the simultaneous confidence intervals for multinomial proportions when the sample size is relatively large. The numerical solution of this equation requires the use of a nonlinear equation solving method. We considered the use of variations on the secant method for this problem because the computation of the derivative of

The Pegasus method requires an initial bounding interval for the
solution point *t*^{*} (
,
with
). A crude initial bounding
interval can be determined from the simple Bonferroni bound.
Using notation similar to that in Section 2.1, we let
,
and then
.
If we let
,
then the simple Bonferroni bound
is
.
A simple lower bound is
.
Combining these bounds, and using
,
a crude bounding
interval for *t*^{*} is given by
(*t*_{CL}, *t*_{GC}), where
(equivalent to GC previously discussed)
and
.
When *m* is large, the repeated evaluation of *h*(*t*) requires expensive
numerical integrations, so in order to reduce the number of evaluations of
*h*(*t*), we looked for better bounds that could be used to reduce the size of
the initial bounding interval, thereby reducing the number of steps required
for the convergence of the Pegasus method. Better bounds can be inexpensively
computed from combinations of univariate and bivariate probabilities. The
best bounds using these probabilities are the Hunter-Worsley
bound for an upper *t*^{*} limit (see Hsu, 1996, p. 229), and a modified
Bonferroni bound for a lower *t*^{*} limit (see Kwerel, 1975). If we let
,
the Hunter-Worsley and modified Bonferroni bounds guarantee that

where , and

Another problem that arises when an iterative numerical method like
the Pegasus method is combined with a numerical integration method, is the
problem of balancing the errors from the two methods. We let
be
an approximation to *t*^{*}, and
be the
numerical integration estimate of *h*(*t*). When carrying out the iterations for
the Pegasus method, if we attempt to compute *h*(*t*)
at a point
close to *t*^{*}, we actually compute

If we want to be able to stop the hybrid numerical procedure when for some small , then we need an estimate of

Selected test results are summarized in Table 2. We used the same sets of
*p*_{j}'s that were used for the tests that were done to produce Table 1. We also
added two other *p*_{j} sets for *m*=5.
The error tolerance
was set at 0.001 for all of the tests.
In the cases where
,
no numerical integration
was necessary, and we set
*t*^{*} = (*t*_{HW}+*t*_{MB})/2.
For most of these problems, the computation time required to compute the
required *t*^{*} values to within the
requested
accuracy was not significant.
In most cases the modified Bonferroni and Hunter-Worseley bonds
could be used to quickly provide *t*^{*} values to within the
,
and the KIC value was very accurate. The two additional
*p*_{j} sets were added for *m*=5 to provide examples where the KIC value
is not as accurate. This occurs when there is significant
variation in the *p*_{j} values.

p_{j}'s |
t_{CL} |
t_{MB} |
t^{*} |
t_{KIC} |
t_{HW} |
t_{GC} |
f-Values | |

(.2, .1, .4, .3) | .100 | 1.645 | 2.189 | 2.190 | 2.193 | 2.208 | 2.241 | 30016 |

.050 | 1.960 | 2.465 | 2.466 | 2.468 | 2.476 | 2.498 | 38864 | |

.010 | 2.576 | 3.011 | 3.013 | 3.013 | 3.014 | 3.023 | 23520 | |

.005 | 2.807 | 3.219 | 3.220 | 3.221 | 3.221 | 3.227 | 0 | |

(.1, .2, .2, .2, .3) |
.100 | 1.645 | 2.288 | 2.288 | 2.289 | 2.308 | 2.326 | 44512 |

.050 | 1.960 | 2.554 | 2.554 | 2.555 | 2.565 | 2.576 | 95200 | |

.010 | 2.576 | 3.084 | 3.084 | 3.084 | 3.086 | 3.090 | 46768 | |

.005 | 2.807 | 3.286 | 3.287 | 3.287 | 3.288 | 3.291 | 0 | |

(.3, .1, .05, .5, .05) |
.100 | 1.645 | 2.288 | 2.288 | 2.289 | 2.308 | 2.326 | 44512 |

.050 | 1.960 | 2.554 | 2.554 | 2.555 | 2.565 | 2.576 | 95200 | |

.010 | 2.576 | 3.084 | 3.084 | 3.084 | 3.086 | 3.090 | 46768 | |

.005 | 2.807 | 3.286 | 3.287 | 3.287 | 3.288 | 3.291 | 0 | |

(.1, .1, .2, .2, .2, .2) | .100 | 1.645 | 2.362 | 2.363 | 2.363 | 2.383 | 2.394 | 38256 |

.050 | 1.960 | 2.621 | 2.621 | 2.621 | 2.632 | 2.638 | 50960 | |

.010 | 2.576 | 3.139 | 3.139 | 3.139 | 3.142 | 3.144 | 70336 | |

.005 | 2.807 | 3.339 | 3.340 | 3.339 | 3.340 | 3.341 | 0 | |

(.10, .30, .05, .50, .05) |
.100 | 1.645 | 2.280 | 2.282 | 2.289 | 2.296 | 2.326 | 15728 |

.050 | 1.960 | 2.547 | 2.549 | 2.555 | 2.555 | 2.576 | 13712 | |

.010 | 2.576 | 3.079 | 3.079 | 3.084 | 3.080 | 3.090 | 0 | |

.005 | 2.807 | 3.282 | 3.283 | 3.287 | 3.283 | 3.291 | 0 | |

(.05, .05, .05, .05, .80) |
.100 | 1.645 | 2.277 | 2.280 | 2.289 | 2.290 | 2.326 | 97088 |

.050 | 1.960 | 2.546 | 2.547 | 2.555 | 2.551 | 2.576 | 137872 | |

.010 | 2.576 | 3.079 | 3.079 | 3.084 | 3.080 | 3.090 | 0 | |

.005 | 2.807 | 3.283 | 3.283 | 3.287 | 3.283 | 3.291 | 0 | |

(.1, .1, .2, .2, .2, .1, |
.100 | 1.645 | 2.421 | 2.422 | 2.422 | 2.441 | 2.450 | 72272 |

.1) | .050 | 1.960 | 2.675 | 2.675 | 2.676 | 2.685 | 2.690 | 112208 |

.010 | 2.576 | 3.185 | 3.185 | 3.185 | 3.187 | 3.189 | 18544 | |

.005 | 2.807 | 3.382 | 3.383 | 3.382 | 3.383 | 3.384 | 0 | |

(.1, .1, .1, .1, .15, .05, |
.100 | 1.645 | 2.471 | 2.472 | 2.473 | 2.490 | 2.498 | 160144 |

.2, .2) | .050 | 1.960 | 2.721 | 2.722 | 2.721 | 2.730 | 2.734 | 154880 |

.010 | 2.576 | 3.224 | 3.225 | 3.224 | 3.226 | 3.227 | 20048 | |

.005 | 2.807 | 3.419 | 3.419 | 3.419 | 3.420 | 3.421 | 0 | |

(.01, .02, .07, .1, .15, .05, |
.100 | 1.645 | 2.514 | 2.515 | 2.516 | 2.531 | 2.539 | 33680 |

.3, .2, .1) | .050 | 1.960 | 2.760 | 2.761 | 2.761 | 2.768 | 2.773 | 50000 |

.010 | 2.576 | 3.258 | 3.259 | 3.258 | 3.259 | 3.261 | 0 | |

.005 | 2.807 | 3.451 | 3.451 | 3.451 | 3.452 | 3.452 | 0 | |

(.1, .05, .05, .04, .06, .1, |
.100 | 1.645 | 2.553 | 2.554 | 2.554 | 2.570 | 2.576 | 55536 |

.15, .15, .1, .2) | .050 | 1.960 | 2.796 | 2.796 | 2.796 | 2.804 | 2.807 | 484416 |

.010 | 2.576 | 3.288 | 3.289 | 3.288 | 3.290 | 3.291 | 0 | |

.005 | 2.807 | 3.479 | 3.480 | 3.479 | 3.480 | 3.481 | 0 | |

(.02, .08, .04, .06, .1, .1, |
.100 | 1.645 | 2.586 | 2.588 | 2.587 | 2.604 | 2.609 | 691008 |

.16, .14, .15, .1, .05) | .050 | 1.960 | 2.827 | 2.827 | 2.827 | 2.835 | 2.838 | 489952 |

.010 | 2.576 | 3.315 | 3.316 | 3.315 | 3.317 | 3.317 | 0 | |

.005 | 2.807 | 3.505 | 3.505 | 3.505 | 3.506 | 3.506 | 0 | |

(.01, .03, .06, .05, .05, .1, |
.100 | 1.645 | 2.617 | 2.618 | 2.618 | 2.634 | 2.638 | 501248 |

.15, .05, .1, .14, .16, .1) | .050 | 1.960 | 2.855 | 2.857 | 2.855 | 2.863 | 2.865 | 74544 |

.010 | 2.576 | 3.339 | 3.340 | 3.339 | 3.341 | 3.341 | 0 | |

.005 | 2.807 | 3.528 | 3.529 | 3.528 | 3.529 | 3.529 | 0 |

2001-02-09