Sobel, Milton

Uppuluri, V. R. R.

University of Minnesota Digital Conservancy

-ON BONFERRONI-TYPE INEQUALITIES OF THE SAME DEG~£: 
FOR THE PROBABILITY OF A UNION 
by 
and V. R.R. Uppuluri §§ Milton Sobel§ 
University of Minnesota Oak Ridge National Lab. 
Technical Report No. 156 
April 1971 
University of Minnesota 
Minneapolis, Minnesota 
§Research supported by NSF Grant GP-11021. 
§§Visiting Professor, University of Minnesota, partially supported by the 
U.S. Atomic Energy coDD11ission under contract with the Union Carbide 
Corporation. 
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1. Introduction and Sumnary • 
Let Ai (i = 1,2, ••• , k) denote a finite set of events associated 
with the probability space (0, 3, P) and let x.(w) denote the indicator 
1 
random variable of A.. Then max Xt(w) is the indicator random 
1. ·k i=l ,2, ••• , k k 
variable of the set U A .• In [4], Kounias found bounds for P{ U A.) 
i::::l 1 i=l 1. 
in terms of P(Ai) and P(AiAj). Kounias stren~thens the Bc1.1fc_·.,.oni 
inequalities (cf. Feller [2]) by giving lower and upper bounds both of 
the second degree (v = 2). In this paper we excend his result by considering 
k 
both lower and upper bounds on P ( U A.) for any fixed degree v{ \J • 1,2,3, ... ,k) 
i=l 1 
and indicate some applications to Dirichlet integrals. 
In the case of exchangeable random variables xi (w) we let 
p = P {A. A. • • • A. ) • We obtain for any odd degree v < k 
a J1 J2 Ja 
(l.i) 
V~ k V 
'E (-lt-l(k)P + (k-l)P < P{ U A ) < 'E {-l)a-l(k)P 
1 a a v-1 v - 1_1 i - _ 1 a a ex= - a-
and for any even degree v < k 
(1.2) 
V k ~1 ~ (-l)a-l(k)P < P{ U A ) < ~ (-l)a-l(k)P - (k-l)P • 
1 a a - i 1 i - 1 a a v-1 v a= = a= 
One can easily construct examples where these bounds are attained. In 
fact, for v = k the left and right sides are equal in both (1.1) and 
(1.2) so that equality is then attained. It should be noted that the left 
sides of (1.1) and (1.2) can be negative (in which case we use zero for 
the lower bound) and that the right sides can exceed 1 (in which case we 
use 1 for the upper bound). 
Some applications of these bounds to Dirichlet integrals are considered 
in sections 4 and 5. These bounds are applicable for any problem involving a 
union of events and, in particular to problems involving Dirichlet integrals. 
For some further references where these integrals arise the reader may 
refer to the references in [7]. 
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-..J 
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2. Derivation of the Bounds. 
In order to express the results in a compact notation we will define 
a Bonferroni indicator random variable and an operation denoted by *. 
Definition: 
Let B • 
r,J with j ~ k and 1 < r < k denote the Bonfer~o~i function 
of degree r on the j sets A. , Ai , ••• , A. defined by 
1 1 2 1 j 
(2.1) B • = 
r,J + ••• + 
where ~ = ~(w) and 1 <a.< j (i = 1,2, ••• , r). We define B0 . to 
- l. - ,J 
be identically zero. 
Definition: 
(2.2) 
Let B .* B . denote the function defined by 
r,1. s,J 
B . * B . = B . + B • - B .B . • 
r,1. s,J r,1. s,J r,1. s,J 
It is implicitly assumed in this definition that the i sets in B • do 
r' l. 
not include any of the j sets in B . and vice versa i.e., the index 
s,J 
sets are disjoint, although the sets need not be. It is easily verified 
that *2 and we sometimes denote the latter by B1 , 1 • 
It is easily verified that the * product is both associative and 
·co1IDIIUtative. It should be noted that in general B i*B .+B ..• r, s,J r+s,1.+J 
We now state and prove a lemma on certain monotonicities among these 
Bonferroni functions which will help us to deduce (1.1) and (1.2) when 
the indicator random variables are exchangeable. 
Leunna: 
For any fixed k ~ 2 with O ~ v - 1 ~ k and for any partition of 
the index set (1, 2, ••• , k) into 2 parts of sizes 1 and k - 1, we have 
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-(2.3) B <B *B <B <B 
v-1,k - 1,1 v-1,k-1 - k,k - v,k for V odd, 
(2.4) B,._l,k 2: B1· ,l* B > B > B v- v-1,k-1 - k,k - v,k for v even. 
Proof: 
By direct substitution of (2.1) into the second expressio--. _,- (2.3) 
we have for any v 
(2.5) ~ 
v-1 
which proves the first inequality in each of (2.3) and (2.4). To prove 
the second inequalities in each of (2.3) and (2.4) we iterate the inequality 
implied by (2.5) for v odd and even obtaining, respectively, 
(2.6) B < B * B < B * B ·< < B < 1 v-1,k - 1,1 v-1,k-l - 2,2 v-1,k-2 - ••• - k,k - ' 
(2.7) 
The remaining inequality in each of (2.3) and (2.4) is the well-known 
Bonferroni bound; this completes the proo~ of the lemma. The first 
inequality of (2.7) with v = 2, namely Bl,k 2: B1, 1* Bl,k-l' is the 
inequality that appears in Kounias [4]. · Of course, this can be further 
improved, possibly by using B3,k, but for the fixed degree 2 neither 
the Bonferroni bound B3,k nor the improved bound in (2.7), B2,2* B1,k_2, 
are allowed,since both are of degree 3. 
For exchangeable indicator random variables we take expectations in 
(2.5) and obtain for any v 
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-· 
(2.8) 
Hence from the last two inequ~lities in (2.3) and (2.4), we obtain 
(2.9) 
~1 · k ~ (-l)a-l(k)P + (k-l1)P < P{ U A.) < E (B k) (for v odd), 1 a a v- v - . 1 1. - v, a= l.= 
(2.10) 
which are equivalent to (1.1) and (1.2). 
It is clear from (2.3) and (2.4) that our bounds are improvements on 
E{Bv-l,k). This has been achieved by increasing the degree from v - 1 
to v for the new lower bound in (2.3) for v odd and for the new upper 
bound in (2.4) for v even. 
3. Further Improvement of the Bounds. 
From the inequalities (2.6) and (2.7) we can actually obtain bounds 
which are as good or better than those in (1.1) and (1.2). Since these are 
quite useful for small v we write them explicitly for v = 2, 3 and 4 
and then give a general expression at the end of this section. 
To derive these improved bounds of connnon second degree we set v = 3 
in (2.6) and v = 2 in (2.7) and obtain after taking expectations 
k 
(3.1) Max(O, E(B2 k)) :5 P{ U A.} :5 Min{E{B1 i* B1 k 1), 1). , i=l]. , , -
For the bounds of coDDI10n third degree we set v = 3 in (2.6), v = 4 
and v = 2 in (2.7) obtaining 
k 
(3.2) Max{O, E{Bl,l* B2 ,k_1)) :5 P{i~iA1) ~ Min{E{B3,k), E{B2,2* Bl,k-2), 1). 
For the bounds of conunon ~ourth degree we set v = 5 and v = 3 in 
(2.6) and v = 4 and 2 in (2.7) obtaining 
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~ 
(3.3) 
k 
Max(O, E{B4,k), E{B2,2* B2,k-2)) ~ P(i~l Ai) 
~ Min(E{B1,1* B3,k-l), E{B3,3* Bl,k-3}, 1). 
In (3.1) no new expressions arise and the result agrees with (1.2) 
for degree 2. In (3.2) two of the expressions already appear in (1.i) for 
degree 3; the new expression for the exchangeabl~ case with any k 2: 3 
is 
(3.4) E{B2 ,2* Bl,k-2) = kP1- (2k-3)P2 + (k-2)P 3 • 
In (3.3) two of the expressions already appear in (1.2) for degree 4; the 
two new expressions for the exchangeable case with any k > 4 are 
(3.5) E{B2,2* B2,k-2) = kPl - (!)P2 + (k-2)2p3 - (k;2)P4, 
(3.6) E{B3, 3* Bl,k-3) = kP 1 - 3(k-2)P2 + (3k-8)P3 - (k-3)P4 • 
It is interesting to note that for any Bonferroni product of two B's as 
in (3.4), (3.5) and (3.6) the sum of the coefficients is identically one. 
In fact, if Q denotes the operation of setting all x. equal ·to 1 
1. 
(3.7) Q{B * B .} = Q{l - (1-B )(1-B .)) = 1 - Q(l-B )Q(l-B .) = 1 r,r s,J r,r s,J r,r s,J 
r 
since Q{l - B ) = ~ (-l)a(r) = O. 
r,r O ot ot= 
More generally, it can be shown (the proof is omitted) that in symbolic 
notation we can write the ~ degree bounds as 
(3.8) 
• "9-j 
E{Bj,j* Bv .. j,k-j) = 1 - (1-P)J ~ (-l)i(k7j)Pi, 
i::O 1. 
where, after expanding,we replace p°' by p Q' for all ct. For any fixed 
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degree v this gives a lower (upper) bound if v - j is even (odd). 
As j varies we obtain all of our bounds. 
In the next section we shall illustrate these bounds by means of a 
numerical example involving Dirichlet integrals. 
4. Application to Dirichlet Integrals. 
Suppose n balls are dropped independently into k + 1 cells, k 
of which have a common single-trial cell probab!.lity p ~ 1/k {and one 
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with probability 1 - kp). Let Min(k, :n) denote the observed minimum 
of the k cell frequencies in this multinomial problem. Then it is shown 
in [6] that for r > 1 and n > kr 
(4.1) P{Min{k, n) > r) = I(k){r, n), 
- p 
where 
(4.2) 
For n < kr we define I(k){r, n) to be zero. Similarly we let Max(k, n) p 
denote the observed maximum of the k cell frequencies. In [7] the properties 
of this type I-Dirichlet integral (4.2) have been investigated and tables 
have been computed in [3]; for k = 1 
function, I(l)(r, n) = I {r, n-r+l). p p 
it reduces to the incomplete beta 
If we let the set Ai denote the event that the frequency in the 
k 
1th cell is at least r, then U Ai denotes the event that Max(k, n) ~ r 
i=l 
and the event A A ••• A 
Q'l Q'2 a j 
{for any collection of size j) denotes the 
event that Min{k, n) ~ r. Hence by using the inclusion-exclusion principle, 
it is easy to see that 
(4.3) k P{Max(k, n) ~ r) = E{Bk,k) = ~ (-l)a-1c:)P(Min(a, n) ~ r) 
a=l 
If n < kr then some of the terms on the right side of (4.3) will vanish 
so that we need only sum up to [n/r], the integer part of n/r. The 
distribution of Max{k, n) can be obtained from (4.3) using tables for 
I{a)(r, n) {a< k). However for higher values of k some of these tables p -
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are not available and it becomes useful to obtain bounds for the left side 
of (4.3), preferably ef the same degree. Thus we utilize the results of 
sections 2 and 3 as illustrated below. 
Suppose, for example, we wish to find lower and upper bounds of degree 
3 for P{Max(k, n) ~ r} when k = 10, r = 2 and n = 8. Using 
(3.2) and (3.4) for the upper bound we obtain .98715 since 
(4.4) 
For the lower bound we use (3.2) or (1.1) and obtain 
(4.5) 
The exact value, using (4.3) and tabulated values in [3], is .98186. 
The comparable bounds given by Mallows [5] are 
(4.6) 
and for the above example with k = 10, r = 2, n = 8 this gives 
(4.7) Max(.84572, .79311)::: P{Max(lO, 8) ~ 2)::: Min(l.86895, 1), 
so that both of our bounds for the third degree are closer to the exact 
value than the corresponding bounds in [5]. 
For many values of k, r, and n it should be noted that 'degeneracies' 
will enter into our problem in different possible ways. For example, if 
n > kr then 
(4.8) P{Max(k, n) ~ r} = E{Bk,k) = 1, 
since for any distribution of n balls in k cells the maximum frequency 
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is at least [n/k] ~ r. Another type of 'degeneracy' is when E{B k) 
"' 
already gives the exact answer. For example, if [n/r] >" then I(j)(r, n) = 0 
- p 
for j >" and we seen from (4.3) that 
(4.9) P{Max(k, n} > r) = E{Bk k} = E{B k), 
- , "' 
so that the th v-- degree bounds give exact answers. In this ::,~ .. ,e the 
bounds in (1.1) and (1.2) cannot be further sharpened. 
5. Other Applications to Dirichlet Integrals. 
In the previous section we developed bounds of the th v- degree for 
the P(Max(k, n) ~ r) in terms of P{Min(i, n} ~ r) for 1 < i < v. 
However, if k is large, tables of P{Min(k, n) ~ r) itself are not 
available and it is useful to have th v-- order bounds for this quantity 
(for any k) in terms of P{Min(i, n) ~ r) with 1::: i::: v. We first 
develop these bounds in the context of exchangeable random variables and 
then specialize to the case of multinomial probabilities. 
Let Ai denote the complement of Ai (i = 1,2, ••• , k) and let B j 
r, 
with j::: k and 1 < r < k denote the same Bonferroni function as in 
(2.1) for the j 
B's replaced by 
sets 
i's 
A. , A1 , ••• ,Ai. We now use (3.2) with all 1 1 2 j 
to obtain, after taking expectations, for the case 
" = 3 
k (5.1) Max(O, E{B1,1* i2,k-l))::: P(i~l Ai)::: Min(E{B3,k), E(B2,2* il,k-2),1). 
For the exchangeable case we now let Qi 
obtain from (5.1) as in (3.3) 
denote P(A A 
Q'l Q'2 ••• 
- ) Aa. 
]. 
and 
(5.2) 
k 
Max(o, k<Ti- <~>~2 + ck;1><r3> ~ PC1~1 A11 
--- k--- k-- ...... --- ...... 
::: Min(kQ1- (2 )Q2 + ( 3)Q3, kQ1 - (2k-3)Q2 + (k-2)Q3, 1). 
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-Using the inclusion-exclusion principle, we find that Qi for 
any i can be written as 
(5.3) - i j i Qi = ~ ( -1) ( . )P j 
• 0 J J= 
where P. = P{A · A ••• A ) as in section 1 and P·_O = 1. Sab~tituting 
J a1 a2 aj 
these in (5.2) gives us for the third degree 
(5.4) k 2 1- 1 k -Max(O, 1 - (; )P1 + {k-l)(k-3)P2 - (; )P 3) :SP( U A.) i=l 1 
For the multinomial case {where k cells have common probability p and 
one cell has probability 1 - kp), Ai denotes the event that the frequency 
k 
of the i th cell is less than r, i.e., Ni < r and hence U Ai 
,n . 1 (k) l.= 
denotes the event that Min(k, n) < r. Since I (r, n) = P{Min(k,n} > r} = Pk, 
p -
we take the complement of all three expressions in (5.4) and obtain the 
final result for k ~ 3 
where we have replaced the upper bound 1 by P 3 :S 1. Fot' k = 3 we obtain 
from (5.5) the "identity check" P3 :S P3 :S P3 and in this sense the bounds 
are sharp. 
For v = 2 and v = 4 we give the corresponding bounds for Pk in 
terms of P 1 , ••• , P v without derivation. For v = 2 and any k > 2 
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(5.6) Max((k-l)P2- (k-2)P1, 0) :S Pk :S Min{P2 , (:)P2- k(k-2)P1+ (k;l)]. 
For v = 4 and any k 2: 4 
(5.7) Max{(k;l)P4- (k-4){k;l)P3+ {k-l){k;
3)P2+ (k;
2)P1, (k-3)P4-(k-4)P3, 0) 
:S Pk :S Min(P4, {k;
2)P4- (k-2){k-4)P3+ (k;3)P2, (t)P.1 - (k-4)(;)P3 
{k>ck-3> ,k-2> ck-1>1 + 2 2. P2- k ·3 Pl+ 4 • 
In a similar manner we can obtain bounds of the th v-- degree for Pk in 
terms of P1, P2 , ••• , P" for any v. In fact, a general expression (given 
without proof) which gives all of our bounds for any degree v is given by 
(5.8) Max{L1, L3, ••• , Lv) ~Pk~ Min(L2 , L4 , ••• , Lv }, 0 e 
where 
(5.9) 
v0 (ve) is the largest odd (even) integer < v and 
f3 
L = ~ {-l)a(k-v-l+a)(k-v+f3)p • 
f3 O 0t a-a v-a a= 
A device to express (5.9) more succinctly is to write the generating 
function of Lf3 
(5.10) " L{x) = ~ L xf3 
e=o a 
in symbolic notation. It can be shown {proof is omitted) that we can 
write L{x) symbolically as 
{5.11) L{x) = (-l)f3pv-f3(1-x)-{k-v){l-Px)k-v+f3 
where, after obtaining the coefficient of xe, we replace Pj by P .• In 
J 
the interesting special case k - v = 1 we have 
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-(5.12) 
and hence from (5.8) we can writ·e all of our bounds in the single 
symbolic form for all a. ( 0 ~ a ~ V) 
(5.13) . 
An i111Dediate corollary of (5.13) for a= 1 is that the sequence P1 , P2 , P3, ••• 
is convex with respect to the subscript since for any v < k 
(5.14) p l+P 1>2P • V- v+ - V 
It should be noted that all of the above inequalities (5.1) - (5.14) 
are valid for any set of exchangeable events, though we are discussing 
them in the context of the t111ltinouq.al distribution. 
To illustrate numerically the third degree bounds in (5.5), consider 
the multinomial example with k = 10, r = 2, n = 40 and p = .09. In 
(10)( ) this case the exact value of P10 = 1_09 2, 40, available from [3], is 
.24434, Our lower bound is Max(.23809, .02826, 0) = .23809 and the 
upper bound is Min(.29487, ,68801) = .29487. 
This idea of bounding Pk can also be used for the case of a 
multinomial distribution with unequal cell probabilities 
k 
f.= (p1 , P2 , ••• , pk; 1-~pi). Alam [l] has already shown the interesting 1 
property that 
(5.15) rik)(r, n) '.:: 4k)(r, n) 
where P = (p1 + p2 + ••• + pk)/k and I~k)(r, n) is defined by 
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l..i 
~ 
\al 
~ 
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_, 
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-i 
la 
\al 
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ialll 
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(5.16) I(k)(r, n) = 
1?_ 
p . 
r(n+l) J l 
[r(r)]~(n-kr+l) o 
. . . 
pk k k j (1 - ~ x. t-kr TT x:-ldx .• 
0 i=l 1 i=l 1 1 
In getting an upper bound for the left-side of {5.15) it may happen (for 
example, when k is large) that the right side of (5.15) is unavailable. 
Then our bound for {k)(r, n) given for v = 3 in (5.5) can then be 
used as an alternative upper bound for the left side of (5.1~,. 
.For the lower bound Alam [1] gives 
(5.17) 
* 
I(k)(r, n) > r{n+l) rp 
E. - [r(r)]~(n-kr+l) '6 
* ** k 
JP JP n kr k r 1 ••• (1 - ~xi) - Tf xi- dx1 , 0 0 1 i=l 
* * ** k where p = min(p1 , p2 , ••• , pk) and (k-l)p + p = ~ pi. In this i=l 
case our lower bound cannot be directly used for the right side of (5.17) 
since we do not have homogeneity in the k cell probabilities. A lower 
bound can, however, be obtained by writing 
k k-1 . 
(5.18) P{ U Ai)~ P( U Ai)+ P{¾) 
i=l i=l 
where Ai denotes the event that the frequency Ni,n of the 1!!! cell 
is at least r. We apply this to the case of a multinomial distribution 
* th with k - 1 cells having probability p and the k- cell having 
probability p**, and obtain, by taking complements in (5.18), 
(5.19) P{Min{k, n) 2: r) 2: P{Min(k-1, n) ~ r) + P{Nk,n ?:. r) - 1 
(k-1) (1) 
=I* (r, n) +I** {r, n) - 1. 
p p 
Since the right side of (5.17) equals the left side of (5.19) we can use 
the right side of (5.19) as a lower bound for Iik)(r, n) in (5.17) 
(when both members of (5.17) are not available). If k is large, so 
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' . 
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that I(k-l){r, n) is also not available, then we can substitute the 
p* 
lower bounds from (5.5) into (5.19). 
Acknowledgement. 
The authors wish to thank Dr. G. P. Steck of Sandia Corporation for 
making available to us his tables [8]. 
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. :; 
REFERENCES 
[l] Alam, K. (1970). Monotonicity properties of the multinomial distribution. 
Ann. Math. Statist. 41 315-317. 
- --- = 
[2] Feller, W. (1968). An Introduction to Probability Theory and its 
Applications, Volume 1. Wiley, New York. 
[3) Frankowski, K., Sobel, M., and Uppuluri, V. R.R. (1971~. :ables 
of the Type I -Dirichlet integral, and applications. (In preparation.) 
[4] Kounias, E.G. (1968). Bounds for the probability of a union, with 
applications. ~- ~- Statist. ~ 2154-2158. 
[5] Mallows, c. L. (1968). An inequality involving multinomial probabilities. 
Biometrika 22, 422-424. 
[6] Olkin, I. and Sobel, M. (1965). Integral expressions for tail probabilities 
of the multinomial and negative multinomial distributions. 
Biometrika ~ 167-179. 
[7] Sobel, M. and Uppuluri, V. R. R. (1971). Some properties and applications 
of Dirichlet integrals. (In preparation.) 
[8] Steck, G. P. (196o). Tables of the distribution of the maximum of r 
jointly multinomially distributed variables with all cell 
probabilities equal, r = 2(1)50, N = 1(1)50. Unpublished. 
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On Bonferroni-Type Inequalities of the same Degree for the Probability of a Union

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On Bonferroni-Type Inequalities of the same Degree for the Probability of a Union

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