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Success runs in a sequence of exchangeable binary trials
Journal of Statistical Planning and Inference 137 (2007) 2954 – 2963 www.elsevier.com/locate/jspi
Success runs in a sequence of exchangeable binary trials Serkan Eryilmaza,∗ , Sevcan Demirb a Department of Mathematics, Izmir University of Economics, 35330 Balçova, Izmir, ˙ Turkey b Department of Statistics, Ege University, 35100 Bornova, Izmir, ˙ Turkey
Received 3 January 2006; received in revised form 2 September 2006; accepted 26 October 2006 Available online 11 February 2007
Abstract The random variables 1 , 2 , . . . , n are said to be exchangeable (or symmetric) if for each n, P {1 x1 , . . . , n xn } = P {(1) x1 , . . . , (n) xn } for any permutation = ((1), . . . , (n)) of {1, 2, . . . , n} and any xi ∈ R, i = 1, . . . , n, i.e. the joint distribution of 1 , 2 , . . . , n is invariant under permutation of its arguments. In this study, run statistics are considered in the situation for which the elements of an exchangeable sequence 1 , 2 , . . . , n are binary with possible values “1” (success) or “0” (failure). The exact distributions of various run statistics are derived using the fact that the conditional distribution of any run statistic given the number of successes is identical to the corresponding distribution in the independent and identically distributed case. © 2007 Elsevier B.V. All rights reserved. Keywords: Consecutive k-out-of-n system; Exchangeable trials; Longest run; Multicomponent stress–strength model; Polya’s urn model; Run statistics
1. Introduction Various applications of runs and run-based statistics appear in many disciplines such as reliability, statistical quality control, hydrology, psychology, and biology. They have a potential use especially in a sequence of binary outcomes. Many authors have contributed to runs and related problems. For an extensive review of runs and their applications the reader is referred to Balakrishnan and Koutras (2002) as well as Fu and Lou (2003). The derivation of exact distributions of run statistics can be very difficult via traditional combinatorial analysis even in the simplest case of independent and identically distributed (i.i.d.) Bernoulli trials. Fu (1986) and Fu and Koutras (1994) presented the Markov chain imbedding (MCI) technique which is a useful contribution to the runs literature. This technique has been used by many other researchers in finding the distribution of various run statistics. The usage of MCI technique needs to determine a proper finite state space based on the structure of the specified run, a proper partition {Cx } on for each x, a finite Markov chain {Yt (x, i)} and its transition probability matrix. These determinations enable us to obtain the exact distribution of many run statistics. The first component x of the finite Markov chain Yt stands for the number of occurrences of specific patterns and the second component i provides information about the stage of formation of the next pattern. In most cases, the second component is denoted by the number of trailing successes. The sequence Yt has the Markovian property when the initial binary sequence consists of independent or Markov-dependent elements. ∗ Corresponding author. Tel.: +90 232 488 8305; fax: +90 232 279 2626.
E-mail addresses: [email protected] (S. Eryilmaz), [email protected] (S. Demir). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.10.015
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In this study, we consider the runs based on an exchangeable sequence of binary trials 1 , 2 , . . . , n . Since the second component of {Yt (x, i)} does not have the Markovian property in the case of exchangeable trials, MCI technique has not been found an appropriate technique for our investigations related to the derivation of distribution of run statistics in a sequence of exchangeable trials. This prevention is illustrated in the second section. The longest run statistic is one of the most popular run statistics in reliability of consecutive k-out-of -n systems. Exact expression, bounds and approximations for the distribution of the longest run statistic or reliability of consecutive k-out-of -n systems are well studied in the literature whenever the elements of a binary sequence are i.i.d or dependent in a Markovian fashion. See, e.g. Fu (1985), Muselli (2000). In this paper we consider the runs in the case of exchangeable binary trials. We prove in this case that the conditional distribution of any run statistic given the number of successes is identical to the corresponding distribution in the i.i.d. case. This result along with the existing conditional distributions in i.i.d case enable one to obtain the marginal distributions of run statistics for exchangeable trials. In Section 3, we obtain an exact expression and an upper bound for the longest run distribution for the exchangeable case and illustrate the findings for the reliability of a consecutive k-out-of -n system in a multicomponent stress–strength model. The results obtained in this study also enable us to get the distributions of success runs in Polya’s urn model. 2. Distributions of run statistics in exchangeable trials Different enumeration schemes in a sequence of binary trials with possible outcomes “1” (success) or “0” (failure) generate various run statistics such as the number of success runs of length exactly k, the number of success runs of length at least k, the number of nonoverlapping (overlapping) success runs of length k, and the longest success run in n trials denoted, respectively, by En,k , Gn,k , Nn,k , Mn,k , Ln . For the following sequence of 12 outcomes: 111101011000 E12,2 = 1,
E12,3 = 0,
G12,2 = 2,
G12,3 = 1,
N12,2 = 3,
N12,3 = 1,
M12,2 = 4,
M12,3 = 2,
L12 = 4.
The conditional distributions of the statistics defined above, given the number of successes in a sequence of i.i.d. trials have been investigated by some researchers. For example, the distribution of Gn,k given the number of successes Sn = l is given by n−l+1 [(l−kx)/k] x n−l−x+1 n − k(x + j ) j P {Gn,k = x | Sn = l} = n , (2.1) (−1) j n−l l j =0
see, e.g. Balakrishnan and Koutras (2002). Exchangeability is defined as a kind of symmetry between random variables. This type of dependence model has widespread usage in many areas such as reliability, quality control, and actuarial science. See, e.g. Lau (1992), Frostig (2001). In this paper we consider the {0, 1} valued exchangeable random variables. Let 1 , 2 , . . . be a sequence of binary exchangeable trials with two possible outcomes “1” (success) or “0” (failure). (e) (e) The distribution of the random variable Sn = Nn,1 , i.e. the number of successes in n trials is of special importance in our study and it is given by n P {Sn(e) = s} = g(n, s). (2.2) s By inclusion–exclusion principle, g(n, s) can be expressed as g(n, s) =
n−s i=0
(−1)i
n−s i
s+i ,
where y = P {1 = · · · = y = 1} for y = 1, 2, . . . and 0 ≡ 1 (see George and Bowman, 1995). Actually g(n, s) denotes the probability of the event that s of i s are equal to “1” and n − s of i s are equal to “0”. Consider the random variable i which denotes the length of the success run at the ith stage, i.e. {i = k} is equivalent to {i =1, . . . , i−k+1 =1, i−k =0}. {i }i 1 has the Markovian property when 1 , 2 , . . . are independent or dependent
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in a Markovian fashion. But, unfortunately it does not in general have this property when 1 , 2 , . . . are exchangeable. This can be easily verified with the following example: consider the conditional probability P {4 = 2 | 3 = 1, 2 = 0, 1 = 1} =
P {4 = 1, 3 = 1, 2 = 0, 1 = 1} 3 − 4 , = P {3 = 1, 2 = 0, 1 = 1} 2 − 3
however, P {4 = 2 | 3 = 1} =
P {4 = 1, 3 = 1, 2 = 0} 2 − 3 = . P {3 = 1, 2 = 0} 1 − 2
We observe that this type of random variables defined depending on successive events, do not have the Markov property in the case of exchangeable binary trials. This prevents the use of MCI technique. Hence we derive the exact distributions of runs by conditioning on the total number of successes. Consider the following generalization of run statistics which includes many run statistics for the special cases of parameters involved. Definition 2.1. Let 1 a, b, c n, (a, b, c ∈ Z + , a b, bc n) and d ∈ {1, 2} be parameters. If the statistics Xa,b,c,d possess a construction of the form Xa,b,c,d =
b n
Yjmc,d
m=a j =mc
then it is called generalized run statistics (GRS) based on a sequence 1 , 2 , . . . , n , where 1 if j = mc and (d − 1)j +1 = 0, Yjmc,d = 0 otherwise. For various selection of parameters a, b, c, d different enumeration schemes are obtained as a special case. Some well-known schemes are listed below d
a
b
c
Xa,b,c,d
1 1 2 2
k 1 k k
n [ nk ] k n
1 k 1 1
Mn,k Nn,k En,k Gn,k
where [x] denotes the integer part of x. Moreover, for a < b, c = 1, d = 2 Xa,b,c,d gives the number of success runs of length at least a and at most b. Choosing the appropriate parameters, the following representations are obtained for the statistics Mn,k and Nn,k : Xk,n,1,1 ≡ Mn,k =
n n
Yjm,1 ,
X1,[n/k],k,1 ≡ Nn,k =
m=k j =m
[n/k]
n
Yjmk,1 .
m=1 j =mk
In the following lemma, the range of the random variable Xa,b,c,d is exposed. Lemma 2.1. The range of the random variable Xa,b,c,d is i: i = 0, 1, . . . , b − a + 1, if d = 1, range(Xa,b,c,d ) ≡ n+1 ], if d = 2. i: i = 1, . . . , [ ac+1 Proof. By rewriting the statistics Xa,b,c,d Xa,b,c,d =
n j =ac
Yjac,d +
n j =(a+1)c
(a+1)c,d
Yj
+ ··· +
n j =bc
Yjbc,d ,
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(a+1)c,1
bc,1 ac,1 = 1 since all the outcomes must be = Y(a+1)c = · · · = Ybc for d = 1, Xa,b,c,d takes its maximum value when Yac equal to “1”. So the range of Xa,b,c,d when d = 1 is the set {0, 1, . . . , b − a + 1}.
For d = 2, bm=a nj=mc Yjmc,2 takes its maximum value for small values of mc. Since the minimum value of m is
equal to a, the maximum value of En,ac = nj=ac Yjac,2 is the same with the maximum value of Xa,b,c,2 . We already know that the maximum value of En,ac is [(n + 1)/(ac + 1)]. Hence the proof is completed. (e)
Denote by Xa,b,c,d and Xa,b,c,d generalized run statistics which depend on exchangeable and i.i.d. trials, respectively. (e)
The following lemma shows that the conditional distributions of Xa,b,c,d and Xa,b,c,d given the number of successes are identical. The proof is straightforward since exchangeability implies that all strings with equally many 1’s are equally likely. Lemma 2.2. It is true that (e)
P {Xa,b,c,d = x | Sn(e) = l} = P {Xa,b,c,d = x | Sn = l}. Lemma 2.2 along with the existing results on the conditional distributions of runs enable us to obtain the marginal distributions of run statistics in the case of exchangeable trials. Upon using the law of total probability one obtains (e) (e) P {Xa,b,c,d = x} = P {Xa,b,c,d = x | Sn(e) = l}P {Sn(e) = l}
=
P {Xa,b,c,d = x | Sn = l}P {Sn(e) = l}, (e)
where consists of l n subject to the event {Xa,b,c,d = x, Sn = l} and P {Sn = l} is given by (2.2). For an illustration let us expose the distribution of the number of success runs of length at least k in exchangeable (e) (e) trials. Since P {Gn,k = x | Sn = l} = P {Gn,k = x | Sn = l}, using (2.1) and (2.2) one obtains (e)
P {Gn,k = x} = =
1
P {Gn,k = x | Sn = l}P {Sn(e) = l}
n−l−x+1 j l∈1 j ∈2 i∈3 n − k(x + j ) n−l × l+i , n−l i (−1)j (−1)i
n−l+1 x
where 1 ≡ {l: 0 l n, l − kx 0, n − l + 1 x}, l − kx 2 ≡ j : j n − l − x + 1, j , k 3 ≡ {i: 0 i n − l}. Intersections of the restrictions in 1 , 2 , 3 yields (e) P {Gn,k
n−l−x+1 n−l+1 = x} = (−1) (−1) j x l=kx j =0 i=0 n − k(x + j ) n−l n+1 × l+i , x = 0, 1, . . . , , n−l i k+1 a1 a2 n−l
j
i
where a1 = min(n, n − x + 1), a2 = min([(l − kx)/k], n − l − x + 1) and [b] shows the integer part of b.
(2.3)
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The generalization given in Definition 2.1 provides us with an easy way for the computation of the expected values of run statistics. The result is given in the following lemma. (e)
Lemma 2.3. The expected value of the random variable Xa,b,c,d is ⎧ b−1 n−1
⎪ mc,d ⎪ E(Ymc )+ E(Yjmc,d ) + E(Ynmc,d ) + E(Ynn,d ) if bc = n, ⎪ ⎪ ⎨ m=a j =mc+1 (e) E(Xa,b,c,d ) = ⎪ n−1 b ⎪
⎪ mc,d mc,d mc,d ⎪ E(Ymc ) + E(Yj ) + E(Yn ) if bc < n. ⎩ j =mc+1
m=a
Proof. By the definition of GRS (e)
E(Xa,b,c,d ) =
b n
E(Yjmc,d ) =
m=a j =mc
=
b
b
⎛ ⎝
E(Yjmc,d ) + E(Ynmc,d )⎠ ⎞
n−1
mc,d ⎝E(Ymc )+
⎞
j =mc
m=a
⎛
n−1
E(Yjmc,d ) + E(Ynmc,d )⎠
j =mc+1
m=a
considering the cases bc = n and bc < n the proof is completed.
It can be easily seen that the probability mass function of the run length random variable in the case of exchangeable trials is k − k+1 , k = 0, 1, . . . , i − 1, P {i = k} = i , k = i. The success probabilities associated with the random variables Yjmc,d in the case of exchangeable trials, can be calculated as if d = 1, mc − mc+1 mc,d mc,d E(Yj ) = P {Yj = 1} = mc − 2mc+1 + mc+2 if d = 2 ≡ mc − dmc+1 + (d − 1)mc+2 ,
j > mc
(2.4)
and similarly E(Yjmc,d ) =
mc − (d − 1)mc+1
if j = mc,
mc − mc+1
if j = n.
(2.5)
Using (2.4) and (2.5) in Lemma 2.3, the expected value of GRS based on a sequence of exchangeable trials is: (e)
E(Xa,b,c,d ) =
b−1
[mc − (d − 1)mc+1 + (n − mc − 1)(mc − dmc+1 + (d − 1)mc+2 )
m=a
+ mc − mc+1 ] + n ,
if bc = n,
and for bc < n (e) E(Xa,b,c,d ) =
b
[mc − (d − 1)mc+1 + (n − mc − 1)(mc − dmc+1 + (d − 1)mc+2 ) + mc − mc+1 ].
m=a
Choosing the appropriate parameters, we obtain the following corollary.
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Corollary 2.1. For k = 1, 2, . . . , n − 1 (e)
n−1
(e)
m=k [n/k]
E(Mn,k ) = E(Nn,k ) =
[m + (n − m)(m − m+1 )] + n , [mk + (n − mk)(mk − mk+1 )],
m=1 (e)
E(En,k ) = (n − k + 1)(k − k+1 ) − (n − k − 1)(k+1 − k+2 ), (e)
E(Gn,k ) =
n−1
[2(m − m+1 ) + (n − m − 1)(m − 2m+1 + m+2 )] + n ,
m=k (e)
(e)
(e)
(e)
and for k = n , E(Mn,k ) = E(Nn,k ) = E(En,k ) = E(Gn,k ) = n . 3. Longest run statistic with reliability application One of the most useful statistics for the reliability of consecutive k-out-of -n systems is the longest run statistic. A
consecutive k-out-of -n system may be defined as follows: consider a coherent binary system =(Z, ) consisting of a component set Z and the structure function : Z → {0, 1}. Suppose that at any instant of time, each component of a system assumes one of two states, operating (1) or failing (0). A consecutive k-out-of -n system is a coherent binary system with the structure function (Z) = 1 if and only if k or more consecutive components operate. Hence, the reliability of a consecutive k-out-of -n system is given by E(Z) = P {Ln k}, where Ln shows the longest success run statistic in n trials. Bounds for the reliability of a consecutive k-out-of -n system whenever the components operate or fail independently of one another have been studied in many papers. Now, suppose that a system consists of n components and Yi (i=1, 2, . . . , n) denotes the strength of the ith component subject to a stress X. The component fails if the applied stress exceeds its strength at any moment, i.e. if Yi > X then the ith component operates otherwise fails. P {Yi > X} gives the reliability of the ith component. Define 1, Yi > X, i = i = 1, 2, . . . , n, 0, Yi X, where Y1 , Y2 , . . . , Yn are i.i.d. random variables and independent of X. It is clear that the random variables 1 , 2 , . . . , n are exchangeable. If in such a system the survival of a system depends on the observation of k or more consecutive operated components, the reliability of the whole system is given by P {L(e) n k}, (e)
where Ln indicates the longest success run statistic based on a sequence of exchangeable binary trials. If in the foregoing system, the survival of a system depends on the observation of k or more (not necessarily consecutive) operated components then the system is called k-out-of -n system and its reliability is given by P {Sn(e) k} =
n− n
=k j =0
(−1)j
n n −
j
+j ,
where m = P {Y1 > X, . . . , Ym > X}. Although the reliability of k-out-of -n system in a multicomponent stress–strength model is considered in some papers (see Bhattacharya and Johnson, 1974; Pandey et al., 1992; Uddin et al., 1993), the reliability of consecutive (e) k-out-of -n system has not been studied in this context yet. In the following we provide some results for P {Ln k}.
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3.1. Exact expression for the system reliability (e)
(e)
(e)
The exact expression for P {Ln k} may instantly come out since P {Ln < k} = P {Gn,k = 0}. By using (2.3), for k = 1, 2, . . . , n P {L(e) n k} = 1 −
n
min([l/k],n−l+1) n−l j =0
l=0
(−1)j (−1)i
i=0
n−l+1 j
n − kj n−l
n−l i
l+i .
(e)
It is possible to get an easier expression for P {Ln k} when 2k n. This result is given in the following lemma. Lemma 3.1. It is true that for 2k n, P {L(e) n k} = (n − k + 1)k − (n − k)k+1 . Proof. One can write for 2k n, P {L(e) n k} =
n−1
(e)
(e)
P {i k, i+1 = 0} + P {(e) n k}
i=k
since (e)
(e)
(e)
(e)
(e)
P {i k, i+1 = 0} = P {i+1 = 0} − P {i < k, i+1 = 0} (e)
= P {i+1 = 0} −
k−1
(e)
(e)
P {i = j, i+1 = 0}
j =0
= (1 − 1 ) −
k−1
(j − 2j +1 + j +2 ) = k − k+1 ,
j =0 (e)
and P {n k} = k , P {L(e) n k} = (n − k + 1)k − (n − k)k+1 .
As seen above, the determination of the probability m = P {1 = · · · = m = 1} is sufficient for the computation of the system reliability. Suppose that P {X x} = FX (x) and P {Yi x} = FY (x), i = 1, 2, . . . , n, where FX , FY are both absolutely continuous. In this case ∞ m = P {Y1 > X, . . . , Ym > X} = (F¯Y (x))m dFX (x). (3.1) −∞
For an illustration let FX (x) = 1 − e− 1 x , FY (x) = 1 − e− 2 x , x > 0. This yields m =
1 .
1 + m 2
Therefore, for k = 1, 2, . . . , n P {L(e) n k} = 1 −
n min([l/k],n−l+1) n−l j =0
l=0
i=0
(−1)j (−1)i
n−l+1 j
and alternative formula for 2k n P {L(e) n k} = (n − k + 1)
1
1 − (n − k) .
1 + k 2
1 + (k + 1) 2
n − kj n−l
n−l i
1
1 + (l + i) 2
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Table 1 Numerics for the system reliability when n = 10 k
1 = 2, 2 = 4
1 = 3, 2 = 4
1 = 4, 2 = 4
1 = 4, 2 = 2
1 2 3 4 5 6 7 8 9 10
0.7297 0.4602 0.3096 0.2188 0.1608 0.1179 0.0902 0.0712 0.0576 0.0476
0.8465 0.5914 0.4177 0.3034 0.2271 0.1685 0.1300 0.1033 0.0841 0.0698
0.9091 0.6857 0.5048 0.3758 0.2857 0.2143 0.1667 0.1333 0.1091 0.0909
0.9848 0.8751 0.7222 0.5797 0.4643 0.3611 0.2889 0.2364 0.1970 0.1667
(e)
In Table 1 some numerics for Rn,k = P {Ln k} are provided for various values of k, 1 , 2 when n = 10. For large (e) values of n, k factorial terms included in the exact expression of P {Ln k} may prevent its use. Hence, we investigate the bounds. 3.2. An upper bound for the system reliability In Eryilmaz (2005, 2006) the longest run is expressed as the maximum of sample whose members are subject to a Markov chain condition. According to this approach P {Ln < k} = P { max i < k} k i n
= P {k < k, k+1 < k, . . . , n < k}. If we consider the longest run based on a sequence of exchangeable trials (e)
(e)
(e) P {L(e) n < k} = P {k < k, k+1 < k, . . . , n < k}
(3.2)
and (e)
P {(e) n = i + 1 | n−1 = i} = (e)
P {(e) n = 0 | n−1 = i} =
i+1 − i+2 , i − i+1
i − 2i+1 + i+2 , i − i+1
i = 0, 1, . . . , n − 2,
and P {(e) n
= k} =
k − k+1 , k = 0, 1, . . . , n − 1, n , k = n,
where 0 ≡ 1. See Eryilmaz (2003). In this section we use the representation (3.2) for the construction of an upper bound for the system reliability (e) (e) P {Ln k}. Denote by Ai the event {i < k}. In the construction of the bound we use the marginal probabilities P {Ai } and the probabilities of the pairwise intersections P {Aj Aj +1 }. Hence, we first establish the following result. Lemma 3.2. For 1j n − 1 and 1 k n P {Aj Aj +1 } = 1 − 2k + k+1 .
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Table 2 Comparison of the upper bounds to exact or simulated reliabilities
1
2
n
k
Rn,k
u(n, k)
3
4
30
10 12 15
0.1654 0.1275 0.0903
0.1885 0.1358 0.0903
4
4
50
10 15 20 30
0.3558 0.1821 0.0956 0.0471
0.3939 0.1912 0.1126 0.0524
3
4
100
10 15 20 50
0.3157 0.1789 0.1606 0.0258
0.6042 0.2893 0.1691 0.0291
4
4
100
10 15 20 50
0.3898 0.2525 0.1960 0.0257
0.7727 0.3750 0.2208 0.0385
The values of Rn,k for n = 30 are exact and simulated for n = 50, 100.
Proof. (e)
(e)
P {Aj Aj +1 } = P {j < k, j +1 < k} =
k
(e)
(e)
P {j = i − 1, j +1 = 0} +
=
(e)
(e)
P {j = i − 1, j +1 = i}
i=1
i=1 k
k−1
(i−1 − 2i + i+1 ) +
i=1
k−1
(i − i+1 )
i=1
= 1 − 2k + k+1 . We are now ready to construct an upper bound for the system reliability. We use the Worsley’s variant of a Bonferronitype inequality which is given by P
n
Ci
i=1
n
P {Ci } −
i=1
n−1
P {Ci Ci+1 },
(3.3)
i=1
see Worsley (1982). Since P {L(e) n k} = P
n
Aci
.
i=k
Taking Ci = Aci in (3.3) using Lemma 3.2 we obtain P {L(e) n k} (n − k + 1)k − (n − k)k+1 = u(n, k). (e)
In Table 2, bounds for Rn,k = P {Ln k} along with the exact or simulated values—exact values when possible and simulated values for large n and k—are provided. One can conclude from the table that the closer k to n the better approximated reliability.
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4. Success runs in Polya’s urn model Polya’s urn model and its generalizations are of special importance in applied probability. See for example Johnson and Kotz (1977). Suppose that an urn initially contains “r” objects of type I and “s” objects of type II. At each stage an object is randomly chosen, its type is noted, and it is replaced along with another object of the same type. Let 1 if the ith object selected is of type I, i = i 1. 0 if the ith object selected is of type II, It is clear that m = P {1 = 2 = · · · = m = 1} r r +1 r +m−1 = ··· r +s r +s+1 r +s+m−1 (r + m − 1)!(r + s − 1)! = (r − 1)!(r + s + m − 1)! and for any value of k, the random variables 1 , 2 , . . . , k are exchangeable. Now, the usage of m in the formulas derived in the previous sections exposes the distributions and expectations of any success run statistic based on Polya’s urn model. Note that if s = r = 1 then m = 1/(m + 1) which coincides with the probability (3.1) under the assumption of FX = FY . Therefore, Polya’s urn model with s = r = 1 is a special case of the model considered in the previous section. Acknowledgment We are grateful for the comments and suggestions of the referees and an associated editor, whose meticulous reading allowed us to improve the quality of the paper. References Balakrishnan, N., Koutras, M.V., 2002. Runs and Scans with Applications. Wiley Series in Probability and Statistics, Wiley, New york. Bhattacharya, G.K., Johnson, R.A., 1974. Estimation of reliability in a multicomponent stress–strength model. J. Amer. Statist. Assoc. 69, 966–970. Eryilmaz, S., 2003. On the number of success runs of length k in exchangeable binary trials. Internat. Math. J. 3, 989–1000. Eryilmaz, S., 2005. On the distribution and expectation of success runs in nonhomogeneous Markov-dependent trials. Statist. Papers 46, 117–128. Eryilmaz, S., 2006. Some results associated with the longest run statistic in a sequence of Markov-dependent trials. Appl. Math. Comput. 175, 119–130. Frostig, E., 2001. Comparison of portfolios which depend on multivariate Bernoulli random variables with fixed marginals. Insurance: Math. Econ. 29, 319–331. Fu, J.C., 1985. Reliability of large consecutive k–out–of–n : F system. IEEE Trans. Reliability 34, 127–130. Fu, J.C., 1986. Reliability of consecutive k–out–of–n : F system with k − 1 step Markov dependence. IEEE Trans. Reliability R35, 602–606. Fu, J.C., Koutras, M.V., 1994. Distribution theory of runs: Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058. Fu, J.C., Lou, W.Y.W., 2003. Distribution Theory of Runs and Patterns and its Applications, A Finite Markov Chain Imbedding Technique. World Scientific, Singapore. George, E.O., Bowman, D., 1995. A full likelihood procedure for analyzing exchangeable binary data. Biometrics. 51, 512–523. Johnson, N., Kotz, S., 1977. Urn Models and Their Application: An Approach to Modern Discrete Probability Theory. Wiley, New-York. Lau, T.S., 1992. The reliability of exchangeable binary systems. Statist. Probab. Lett. 13, 153–158. Muselli, M., 2000. Useful inequalities for the longest run distribution. Statist. Probab. Lett. 46, 239–249. Pandey, M., Uddin, B.Md., Ferdous, J., 1992. Reliability estimation of an s–out–of–k system with nonidentical component strengths: the Weibull case. Reliability Eng. System Safety 36, 109–116. Uddin, B.Md., Pandey, M., Ferdous, J., Bhviyan, M.R., 1993. Estimation of reliability in a multicomponent stress–strength model. Microelectronics and Reliability 33, 2043–2046. Worsley, K.J., 1982. An improved Bonferroni inequality ad applications. Biometrika 69 (2), 297–302.
Success runs in a sequence of exchangeable binary trials Serkan Eryilmaza,∗ , Sevcan Demirb a Department of Mathematics, Izmir University of Economics, 35330 Balçova, Izmir, ˙ Turkey b Department of Statistics, Ege University, 35100 Bornova, Izmir, ˙ Turkey
Received 3 January 2006; received in revised form 2 September 2006; accepted 26 October 2006 Available online 11 February 2007
Abstract The random variables 1 , 2 , . . . , n are said to be exchangeable (or symmetric) if for each n, P {1 x1 , . . . , n xn } = P {(1) x1 , . . . , (n) xn } for any permutation = ((1), . . . , (n)) of {1, 2, . . . , n} and any xi ∈ R, i = 1, . . . , n, i.e. the joint distribution of 1 , 2 , . . . , n is invariant under permutation of its arguments. In this study, run statistics are considered in the situation for which the elements of an exchangeable sequence 1 , 2 , . . . , n are binary with possible values “1” (success) or “0” (failure). The exact distributions of various run statistics are derived using the fact that the conditional distribution of any run statistic given the number of successes is identical to the corresponding distribution in the independent and identically distributed case. © 2007 Elsevier B.V. All rights reserved. Keywords: Consecutive k-out-of-n system; Exchangeable trials; Longest run; Multicomponent stress–strength model; Polya’s urn model; Run statistics
1. Introduction Various applications of runs and run-based statistics appear in many disciplines such as reliability, statistical quality control, hydrology, psychology, and biology. They have a potential use especially in a sequence of binary outcomes. Many authors have contributed to runs and related problems. For an extensive review of runs and their applications the reader is referred to Balakrishnan and Koutras (2002) as well as Fu and Lou (2003). The derivation of exact distributions of run statistics can be very difficult via traditional combinatorial analysis even in the simplest case of independent and identically distributed (i.i.d.) Bernoulli trials. Fu (1986) and Fu and Koutras (1994) presented the Markov chain imbedding (MCI) technique which is a useful contribution to the runs literature. This technique has been used by many other researchers in finding the distribution of various run statistics. The usage of MCI technique needs to determine a proper finite state space based on the structure of the specified run, a proper partition {Cx } on for each x, a finite Markov chain {Yt (x, i)} and its transition probability matrix. These determinations enable us to obtain the exact distribution of many run statistics. The first component x of the finite Markov chain Yt stands for the number of occurrences of specific patterns and the second component i provides information about the stage of formation of the next pattern. In most cases, the second component is denoted by the number of trailing successes. The sequence Yt has the Markovian property when the initial binary sequence consists of independent or Markov-dependent elements. ∗ Corresponding author. Tel.: +90 232 488 8305; fax: +90 232 279 2626.
E-mail addresses: [email protected] (S. Eryilmaz), [email protected] (S. Demir). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2006.10.015
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In this study, we consider the runs based on an exchangeable sequence of binary trials 1 , 2 , . . . , n . Since the second component of {Yt (x, i)} does not have the Markovian property in the case of exchangeable trials, MCI technique has not been found an appropriate technique for our investigations related to the derivation of distribution of run statistics in a sequence of exchangeable trials. This prevention is illustrated in the second section. The longest run statistic is one of the most popular run statistics in reliability of consecutive k-out-of -n systems. Exact expression, bounds and approximations for the distribution of the longest run statistic or reliability of consecutive k-out-of -n systems are well studied in the literature whenever the elements of a binary sequence are i.i.d or dependent in a Markovian fashion. See, e.g. Fu (1985), Muselli (2000). In this paper we consider the runs in the case of exchangeable binary trials. We prove in this case that the conditional distribution of any run statistic given the number of successes is identical to the corresponding distribution in the i.i.d. case. This result along with the existing conditional distributions in i.i.d case enable one to obtain the marginal distributions of run statistics for exchangeable trials. In Section 3, we obtain an exact expression and an upper bound for the longest run distribution for the exchangeable case and illustrate the findings for the reliability of a consecutive k-out-of -n system in a multicomponent stress–strength model. The results obtained in this study also enable us to get the distributions of success runs in Polya’s urn model. 2. Distributions of run statistics in exchangeable trials Different enumeration schemes in a sequence of binary trials with possible outcomes “1” (success) or “0” (failure) generate various run statistics such as the number of success runs of length exactly k, the number of success runs of length at least k, the number of nonoverlapping (overlapping) success runs of length k, and the longest success run in n trials denoted, respectively, by En,k , Gn,k , Nn,k , Mn,k , Ln . For the following sequence of 12 outcomes: 111101011000 E12,2 = 1,
E12,3 = 0,
G12,2 = 2,
G12,3 = 1,
N12,2 = 3,
N12,3 = 1,
M12,2 = 4,
M12,3 = 2,
L12 = 4.
The conditional distributions of the statistics defined above, given the number of successes in a sequence of i.i.d. trials have been investigated by some researchers. For example, the distribution of Gn,k given the number of successes Sn = l is given by n−l+1 [(l−kx)/k] x n−l−x+1 n − k(x + j ) j P {Gn,k = x | Sn = l} = n , (2.1) (−1) j n−l l j =0
see, e.g. Balakrishnan and Koutras (2002). Exchangeability is defined as a kind of symmetry between random variables. This type of dependence model has widespread usage in many areas such as reliability, quality control, and actuarial science. See, e.g. Lau (1992), Frostig (2001). In this paper we consider the {0, 1} valued exchangeable random variables. Let 1 , 2 , . . . be a sequence of binary exchangeable trials with two possible outcomes “1” (success) or “0” (failure). (e) (e) The distribution of the random variable Sn = Nn,1 , i.e. the number of successes in n trials is of special importance in our study and it is given by n P {Sn(e) = s} = g(n, s). (2.2) s By inclusion–exclusion principle, g(n, s) can be expressed as g(n, s) =
n−s i=0
(−1)i
n−s i
s+i ,
where y = P {1 = · · · = y = 1} for y = 1, 2, . . . and 0 ≡ 1 (see George and Bowman, 1995). Actually g(n, s) denotes the probability of the event that s of i s are equal to “1” and n − s of i s are equal to “0”. Consider the random variable i which denotes the length of the success run at the ith stage, i.e. {i = k} is equivalent to {i =1, . . . , i−k+1 =1, i−k =0}. {i }i 1 has the Markovian property when 1 , 2 , . . . are independent or dependent
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in a Markovian fashion. But, unfortunately it does not in general have this property when 1 , 2 , . . . are exchangeable. This can be easily verified with the following example: consider the conditional probability P {4 = 2 | 3 = 1, 2 = 0, 1 = 1} =
P {4 = 1, 3 = 1, 2 = 0, 1 = 1} 3 − 4 , = P {3 = 1, 2 = 0, 1 = 1} 2 − 3
however, P {4 = 2 | 3 = 1} =
P {4 = 1, 3 = 1, 2 = 0} 2 − 3 = . P {3 = 1, 2 = 0} 1 − 2
We observe that this type of random variables defined depending on successive events, do not have the Markov property in the case of exchangeable binary trials. This prevents the use of MCI technique. Hence we derive the exact distributions of runs by conditioning on the total number of successes. Consider the following generalization of run statistics which includes many run statistics for the special cases of parameters involved. Definition 2.1. Let 1 a, b, c n, (a, b, c ∈ Z + , a b, bc n) and d ∈ {1, 2} be parameters. If the statistics Xa,b,c,d possess a construction of the form Xa,b,c,d =
b n
Yjmc,d
m=a j =mc
then it is called generalized run statistics (GRS) based on a sequence 1 , 2 , . . . , n , where 1 if j = mc and (d − 1)j +1 = 0, Yjmc,d = 0 otherwise. For various selection of parameters a, b, c, d different enumeration schemes are obtained as a special case. Some well-known schemes are listed below d
a
b
c
Xa,b,c,d
1 1 2 2
k 1 k k
n [ nk ] k n
1 k 1 1
Mn,k Nn,k En,k Gn,k
where [x] denotes the integer part of x. Moreover, for a < b, c = 1, d = 2 Xa,b,c,d gives the number of success runs of length at least a and at most b. Choosing the appropriate parameters, the following representations are obtained for the statistics Mn,k and Nn,k : Xk,n,1,1 ≡ Mn,k =
n n
Yjm,1 ,
X1,[n/k],k,1 ≡ Nn,k =
m=k j =m
[n/k]
n
Yjmk,1 .
m=1 j =mk
In the following lemma, the range of the random variable Xa,b,c,d is exposed. Lemma 2.1. The range of the random variable Xa,b,c,d is i: i = 0, 1, . . . , b − a + 1, if d = 1, range(Xa,b,c,d ) ≡ n+1 ], if d = 2. i: i = 1, . . . , [ ac+1 Proof. By rewriting the statistics Xa,b,c,d Xa,b,c,d =
n j =ac
Yjac,d +
n j =(a+1)c
(a+1)c,d
Yj
+ ··· +
n j =bc
Yjbc,d ,
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(a+1)c,1
bc,1 ac,1 = 1 since all the outcomes must be = Y(a+1)c = · · · = Ybc for d = 1, Xa,b,c,d takes its maximum value when Yac equal to “1”. So the range of Xa,b,c,d when d = 1 is the set {0, 1, . . . , b − a + 1}.
For d = 2, bm=a nj=mc Yjmc,2 takes its maximum value for small values of mc. Since the minimum value of m is
equal to a, the maximum value of En,ac = nj=ac Yjac,2 is the same with the maximum value of Xa,b,c,2 . We already know that the maximum value of En,ac is [(n + 1)/(ac + 1)]. Hence the proof is completed. (e)
Denote by Xa,b,c,d and Xa,b,c,d generalized run statistics which depend on exchangeable and i.i.d. trials, respectively. (e)
The following lemma shows that the conditional distributions of Xa,b,c,d and Xa,b,c,d given the number of successes are identical. The proof is straightforward since exchangeability implies that all strings with equally many 1’s are equally likely. Lemma 2.2. It is true that (e)
P {Xa,b,c,d = x | Sn(e) = l} = P {Xa,b,c,d = x | Sn = l}. Lemma 2.2 along with the existing results on the conditional distributions of runs enable us to obtain the marginal distributions of run statistics in the case of exchangeable trials. Upon using the law of total probability one obtains (e) (e) P {Xa,b,c,d = x} = P {Xa,b,c,d = x | Sn(e) = l}P {Sn(e) = l}
=
P {Xa,b,c,d = x | Sn = l}P {Sn(e) = l}, (e)
where consists of l n subject to the event {Xa,b,c,d = x, Sn = l} and P {Sn = l} is given by (2.2). For an illustration let us expose the distribution of the number of success runs of length at least k in exchangeable (e) (e) trials. Since P {Gn,k = x | Sn = l} = P {Gn,k = x | Sn = l}, using (2.1) and (2.2) one obtains (e)
P {Gn,k = x} = =
1
P {Gn,k = x | Sn = l}P {Sn(e) = l}
n−l−x+1 j l∈1 j ∈2 i∈3 n − k(x + j ) n−l × l+i , n−l i (−1)j (−1)i
n−l+1 x
where 1 ≡ {l: 0 l n, l − kx 0, n − l + 1 x}, l − kx 2 ≡ j : j n − l − x + 1, j , k 3 ≡ {i: 0 i n − l}. Intersections of the restrictions in 1 , 2 , 3 yields (e) P {Gn,k
n−l−x+1 n−l+1 = x} = (−1) (−1) j x l=kx j =0 i=0 n − k(x + j ) n−l n+1 × l+i , x = 0, 1, . . . , , n−l i k+1 a1 a2 n−l
j
i
where a1 = min(n, n − x + 1), a2 = min([(l − kx)/k], n − l − x + 1) and [b] shows the integer part of b.
(2.3)
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The generalization given in Definition 2.1 provides us with an easy way for the computation of the expected values of run statistics. The result is given in the following lemma. (e)
Lemma 2.3. The expected value of the random variable Xa,b,c,d is ⎧ b−1 n−1
⎪ mc,d ⎪ E(Ymc )+ E(Yjmc,d ) + E(Ynmc,d ) + E(Ynn,d ) if bc = n, ⎪ ⎪ ⎨ m=a j =mc+1 (e) E(Xa,b,c,d ) = ⎪ n−1 b ⎪
⎪ mc,d mc,d mc,d ⎪ E(Ymc ) + E(Yj ) + E(Yn ) if bc < n. ⎩ j =mc+1
m=a
Proof. By the definition of GRS (e)
E(Xa,b,c,d ) =
b n
E(Yjmc,d ) =
m=a j =mc
=
b
b
⎛ ⎝
E(Yjmc,d ) + E(Ynmc,d )⎠ ⎞
n−1
mc,d ⎝E(Ymc )+
⎞
j =mc
m=a
⎛
n−1
E(Yjmc,d ) + E(Ynmc,d )⎠
j =mc+1
m=a
considering the cases bc = n and bc < n the proof is completed.
It can be easily seen that the probability mass function of the run length random variable in the case of exchangeable trials is k − k+1 , k = 0, 1, . . . , i − 1, P {i = k} = i , k = i. The success probabilities associated with the random variables Yjmc,d in the case of exchangeable trials, can be calculated as if d = 1, mc − mc+1 mc,d mc,d E(Yj ) = P {Yj = 1} = mc − 2mc+1 + mc+2 if d = 2 ≡ mc − dmc+1 + (d − 1)mc+2 ,
j > mc
(2.4)
and similarly E(Yjmc,d ) =
mc − (d − 1)mc+1
if j = mc,
mc − mc+1
if j = n.
(2.5)
Using (2.4) and (2.5) in Lemma 2.3, the expected value of GRS based on a sequence of exchangeable trials is: (e)
E(Xa,b,c,d ) =
b−1
[mc − (d − 1)mc+1 + (n − mc − 1)(mc − dmc+1 + (d − 1)mc+2 )
m=a
+ mc − mc+1 ] + n ,
if bc = n,
and for bc < n (e) E(Xa,b,c,d ) =
b
[mc − (d − 1)mc+1 + (n − mc − 1)(mc − dmc+1 + (d − 1)mc+2 ) + mc − mc+1 ].
m=a
Choosing the appropriate parameters, we obtain the following corollary.
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Corollary 2.1. For k = 1, 2, . . . , n − 1 (e)
n−1
(e)
m=k [n/k]
E(Mn,k ) = E(Nn,k ) =
[m + (n − m)(m − m+1 )] + n , [mk + (n − mk)(mk − mk+1 )],
m=1 (e)
E(En,k ) = (n − k + 1)(k − k+1 ) − (n − k − 1)(k+1 − k+2 ), (e)
E(Gn,k ) =
n−1
[2(m − m+1 ) + (n − m − 1)(m − 2m+1 + m+2 )] + n ,
m=k (e)
(e)
(e)
(e)
and for k = n , E(Mn,k ) = E(Nn,k ) = E(En,k ) = E(Gn,k ) = n . 3. Longest run statistic with reliability application One of the most useful statistics for the reliability of consecutive k-out-of -n systems is the longest run statistic. A
consecutive k-out-of -n system may be defined as follows: consider a coherent binary system =(Z, ) consisting of a component set Z and the structure function : Z → {0, 1}. Suppose that at any instant of time, each component of a system assumes one of two states, operating (1) or failing (0). A consecutive k-out-of -n system is a coherent binary system with the structure function (Z) = 1 if and only if k or more consecutive components operate. Hence, the reliability of a consecutive k-out-of -n system is given by E(Z) = P {Ln k}, where Ln shows the longest success run statistic in n trials. Bounds for the reliability of a consecutive k-out-of -n system whenever the components operate or fail independently of one another have been studied in many papers. Now, suppose that a system consists of n components and Yi (i=1, 2, . . . , n) denotes the strength of the ith component subject to a stress X. The component fails if the applied stress exceeds its strength at any moment, i.e. if Yi > X then the ith component operates otherwise fails. P {Yi > X} gives the reliability of the ith component. Define 1, Yi > X, i = i = 1, 2, . . . , n, 0, Yi X, where Y1 , Y2 , . . . , Yn are i.i.d. random variables and independent of X. It is clear that the random variables 1 , 2 , . . . , n are exchangeable. If in such a system the survival of a system depends on the observation of k or more consecutive operated components, the reliability of the whole system is given by P {L(e) n k}, (e)
where Ln indicates the longest success run statistic based on a sequence of exchangeable binary trials. If in the foregoing system, the survival of a system depends on the observation of k or more (not necessarily consecutive) operated components then the system is called k-out-of -n system and its reliability is given by P {Sn(e) k} =
n− n
=k j =0
(−1)j
n n −
j
+j ,
where m = P {Y1 > X, . . . , Ym > X}. Although the reliability of k-out-of -n system in a multicomponent stress–strength model is considered in some papers (see Bhattacharya and Johnson, 1974; Pandey et al., 1992; Uddin et al., 1993), the reliability of consecutive (e) k-out-of -n system has not been studied in this context yet. In the following we provide some results for P {Ln k}.
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3.1. Exact expression for the system reliability (e)
(e)
(e)
The exact expression for P {Ln k} may instantly come out since P {Ln < k} = P {Gn,k = 0}. By using (2.3), for k = 1, 2, . . . , n P {L(e) n k} = 1 −
n
min([l/k],n−l+1) n−l j =0
l=0
(−1)j (−1)i
i=0
n−l+1 j
n − kj n−l
n−l i
l+i .
(e)
It is possible to get an easier expression for P {Ln k} when 2k n. This result is given in the following lemma. Lemma 3.1. It is true that for 2k n, P {L(e) n k} = (n − k + 1)k − (n − k)k+1 . Proof. One can write for 2k n, P {L(e) n k} =
n−1
(e)
(e)
P {i k, i+1 = 0} + P {(e) n k}
i=k
since (e)
(e)
(e)
(e)
(e)
P {i k, i+1 = 0} = P {i+1 = 0} − P {i < k, i+1 = 0} (e)
= P {i+1 = 0} −
k−1
(e)
(e)
P {i = j, i+1 = 0}
j =0
= (1 − 1 ) −
k−1
(j − 2j +1 + j +2 ) = k − k+1 ,
j =0 (e)
and P {n k} = k , P {L(e) n k} = (n − k + 1)k − (n − k)k+1 .
As seen above, the determination of the probability m = P {1 = · · · = m = 1} is sufficient for the computation of the system reliability. Suppose that P {X x} = FX (x) and P {Yi x} = FY (x), i = 1, 2, . . . , n, where FX , FY are both absolutely continuous. In this case ∞ m = P {Y1 > X, . . . , Ym > X} = (F¯Y (x))m dFX (x). (3.1) −∞
For an illustration let FX (x) = 1 − e− 1 x , FY (x) = 1 − e− 2 x , x > 0. This yields m =
1 .
1 + m 2
Therefore, for k = 1, 2, . . . , n P {L(e) n k} = 1 −
n min([l/k],n−l+1) n−l j =0
l=0
i=0
(−1)j (−1)i
n−l+1 j
and alternative formula for 2k n P {L(e) n k} = (n − k + 1)
1
1 − (n − k) .
1 + k 2
1 + (k + 1) 2
n − kj n−l
n−l i
1
1 + (l + i) 2
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Table 1 Numerics for the system reliability when n = 10 k
1 = 2, 2 = 4
1 = 3, 2 = 4
1 = 4, 2 = 4
1 = 4, 2 = 2
1 2 3 4 5 6 7 8 9 10
0.7297 0.4602 0.3096 0.2188 0.1608 0.1179 0.0902 0.0712 0.0576 0.0476
0.8465 0.5914 0.4177 0.3034 0.2271 0.1685 0.1300 0.1033 0.0841 0.0698
0.9091 0.6857 0.5048 0.3758 0.2857 0.2143 0.1667 0.1333 0.1091 0.0909
0.9848 0.8751 0.7222 0.5797 0.4643 0.3611 0.2889 0.2364 0.1970 0.1667
(e)
In Table 1 some numerics for Rn,k = P {Ln k} are provided for various values of k, 1 , 2 when n = 10. For large (e) values of n, k factorial terms included in the exact expression of P {Ln k} may prevent its use. Hence, we investigate the bounds. 3.2. An upper bound for the system reliability In Eryilmaz (2005, 2006) the longest run is expressed as the maximum of sample whose members are subject to a Markov chain condition. According to this approach P {Ln < k} = P { max i < k} k i n
= P {k < k, k+1 < k, . . . , n < k}. If we consider the longest run based on a sequence of exchangeable trials (e)
(e)
(e) P {L(e) n < k} = P {k < k, k+1 < k, . . . , n < k}
(3.2)
and (e)
P {(e) n = i + 1 | n−1 = i} = (e)
P {(e) n = 0 | n−1 = i} =
i+1 − i+2 , i − i+1
i − 2i+1 + i+2 , i − i+1
i = 0, 1, . . . , n − 2,
and P {(e) n
= k} =
k − k+1 , k = 0, 1, . . . , n − 1, n , k = n,
where 0 ≡ 1. See Eryilmaz (2003). In this section we use the representation (3.2) for the construction of an upper bound for the system reliability (e) (e) P {Ln k}. Denote by Ai the event {i < k}. In the construction of the bound we use the marginal probabilities P {Ai } and the probabilities of the pairwise intersections P {Aj Aj +1 }. Hence, we first establish the following result. Lemma 3.2. For 1j n − 1 and 1 k n P {Aj Aj +1 } = 1 − 2k + k+1 .
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S. Eryilmaz, S. Demir / Journal of Statistical Planning and Inference 137 (2007) 2954 – 2963
Table 2 Comparison of the upper bounds to exact or simulated reliabilities
1
2
n
k
Rn,k
u(n, k)
3
4
30
10 12 15
0.1654 0.1275 0.0903
0.1885 0.1358 0.0903
4
4
50
10 15 20 30
0.3558 0.1821 0.0956 0.0471
0.3939 0.1912 0.1126 0.0524
3
4
100
10 15 20 50
0.3157 0.1789 0.1606 0.0258
0.6042 0.2893 0.1691 0.0291
4
4
100
10 15 20 50
0.3898 0.2525 0.1960 0.0257
0.7727 0.3750 0.2208 0.0385
The values of Rn,k for n = 30 are exact and simulated for n = 50, 100.
Proof. (e)
(e)
P {Aj Aj +1 } = P {j < k, j +1 < k} =
k
(e)
(e)
P {j = i − 1, j +1 = 0} +
=
(e)
(e)
P {j = i − 1, j +1 = i}
i=1
i=1 k
k−1
(i−1 − 2i + i+1 ) +
i=1
k−1
(i − i+1 )
i=1
= 1 − 2k + k+1 . We are now ready to construct an upper bound for the system reliability. We use the Worsley’s variant of a Bonferronitype inequality which is given by P
n
Ci
i=1
n
P {Ci } −
i=1
n−1
P {Ci Ci+1 },
(3.3)
i=1
see Worsley (1982). Since P {L(e) n k} = P
n
Aci
.
i=k
Taking Ci = Aci in (3.3) using Lemma 3.2 we obtain P {L(e) n k} (n − k + 1)k − (n − k)k+1 = u(n, k). (e)
In Table 2, bounds for Rn,k = P {Ln k} along with the exact or simulated values—exact values when possible and simulated values for large n and k—are provided. One can conclude from the table that the closer k to n the better approximated reliability.
S. Eryilmaz, S. Demir / Journal of Statistical Planning and Inference 137 (2007) 2954 – 2963
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4. Success runs in Polya’s urn model Polya’s urn model and its generalizations are of special importance in applied probability. See for example Johnson and Kotz (1977). Suppose that an urn initially contains “r” objects of type I and “s” objects of type II. At each stage an object is randomly chosen, its type is noted, and it is replaced along with another object of the same type. Let 1 if the ith object selected is of type I, i = i 1. 0 if the ith object selected is of type II, It is clear that m = P {1 = 2 = · · · = m = 1} r r +1 r +m−1 = ··· r +s r +s+1 r +s+m−1 (r + m − 1)!(r + s − 1)! = (r − 1)!(r + s + m − 1)! and for any value of k, the random variables 1 , 2 , . . . , k are exchangeable. Now, the usage of m in the formulas derived in the previous sections exposes the distributions and expectations of any success run statistic based on Polya’s urn model. Note that if s = r = 1 then m = 1/(m + 1) which coincides with the probability (3.1) under the assumption of FX = FY . Therefore, Polya’s urn model with s = r = 1 is a special case of the model considered in the previous section. Acknowledgment We are grateful for the comments and suggestions of the referees and an associated editor, whose meticulous reading allowed us to improve the quality of the paper. References Balakrishnan, N., Koutras, M.V., 2002. Runs and Scans with Applications. Wiley Series in Probability and Statistics, Wiley, New york. Bhattacharya, G.K., Johnson, R.A., 1974. Estimation of reliability in a multicomponent stress–strength model. J. Amer. Statist. Assoc. 69, 966–970. Eryilmaz, S., 2003. On the number of success runs of length k in exchangeable binary trials. Internat. Math. J. 3, 989–1000. Eryilmaz, S., 2005. On the distribution and expectation of success runs in nonhomogeneous Markov-dependent trials. Statist. Papers 46, 117–128. Eryilmaz, S., 2006. Some results associated with the longest run statistic in a sequence of Markov-dependent trials. Appl. Math. Comput. 175, 119–130. Frostig, E., 2001. Comparison of portfolios which depend on multivariate Bernoulli random variables with fixed marginals. Insurance: Math. Econ. 29, 319–331. Fu, J.C., 1985. Reliability of large consecutive k–out–of–n : F system. IEEE Trans. Reliability 34, 127–130. Fu, J.C., 1986. Reliability of consecutive k–out–of–n : F system with k − 1 step Markov dependence. IEEE Trans. Reliability R35, 602–606. Fu, J.C., Koutras, M.V., 1994. Distribution theory of runs: Markov chain approach. J. Amer. Statist. Assoc. 89, 1050–1058. Fu, J.C., Lou, W.Y.W., 2003. Distribution Theory of Runs and Patterns and its Applications, A Finite Markov Chain Imbedding Technique. World Scientific, Singapore. George, E.O., Bowman, D., 1995. A full likelihood procedure for analyzing exchangeable binary data. Biometrics. 51, 512–523. Johnson, N., Kotz, S., 1977. Urn Models and Their Application: An Approach to Modern Discrete Probability Theory. Wiley, New-York. Lau, T.S., 1992. The reliability of exchangeable binary systems. Statist. Probab. Lett. 13, 153–158. Muselli, M., 2000. Useful inequalities for the longest run distribution. Statist. Probab. Lett. 46, 239–249. Pandey, M., Uddin, B.Md., Ferdous, J., 1992. Reliability estimation of an s–out–of–k system with nonidentical component strengths: the Weibull case. Reliability Eng. System Safety 36, 109–116. Uddin, B.Md., Pandey, M., Ferdous, J., Bhviyan, M.R., 1993. Estimation of reliability in a multicomponent stress–strength model. Microelectronics and Reliability 33, 2043–2046. Worsley, K.J., 1982. An improved Bonferroni inequality ad applications. Biometrika 69 (2), 297–302.