Soit un bâton de longueur 1 cassé en fragments uniformément au hasard. Quelle est la distribution de la longueur du plus long fragment?
Plus formellement, soit soit IID , et soit les statistiques d'ordre associées, c'est-à - dire que nous commandons simplement l'échantillon dans d'une manière que . Laisser .
Je m'intéresse à la distribution de . Les moments, les résultats asymptotiques ou les approximations de sont également intéressants.
Réponses:
Avec les informations fournies par @Glen_b, j'ai pu trouver la réponse. En utilisant les mêmes notations que la question
où si a > 0 et 0 sinon. Je donne également l'attente et la convergence asymptotique à la distribution de Gumbel ( NB : pas Beta)a+=a a>0 0
The material of the proofs is taken from several publications linked in the references. They are somewhat lengthy, but straightforward.
1. Proof of the exact distribution
Soit des variables aléatoires uniformes IID dans l'intervalle ( 0 , 1 ) . En les commandant, nous obtenons les statistiques d'ordre k notées ( U ( 1 ) , … , U ( k ) ) . Les espacements uniformes sont définis comme Δ i = U ( i ) - U ( i - 1 ) , avec U ((U1,…,Uk) (0,1) k (U(1),…,U(k)) Δi=U(i)−U(i−1) U(0)=0 and U(k+1)=1 . The ordered spacings are the corresponding ordered statistics Δ(1)≤…≤Δ(k+1) . The variable of interest is Δ(k+1) .
For fixedx∈(0,1) , we define the indicator variable 1i=1{Δi>x} . By symmetry, the random vector (11,…,1k+1) is exchangeable, so the joint distribution of a subset of size j is the same as the joint distribution of the first j . By expanding the product, we thus obtain
We will now prove thatE(∏ji=11i)=(1−jx)k+ , which will establish the distribution given above. We prove this for j=2 , as the general case is proved similarly.
IfΔ1>x , the k breakpoints are in the interval (x,1) . Conditionally on this event, the breakpoints are still exchangeable, so the probability that the distance between the second and the first breakpoint is greater than x is the same as the probability that the distance between the first breakpoint and the left barrier (at position x ) is greater than x . So
2. Expectation
For distributions with finite support, we have
Integrating the distribution ofΔ(k+1) , we obtain
The last equality is a classic representation of harmonic numbersHi=1+12+…+1i , which we demonstrate below.
With the change of variableu=1−x and expanding the product, we obtain
3. Alternative construction of uniform spacings
In order to obtain the asymptotic distribution of the largest fragment, we will need to exhibit a classical construction of uniform spacings as exponential variables divided by their sum. The probability density of the associated order statistics(U(1),…,U(k)) is
If we denote the uniform spacingsΔi=U(i)−U(i−1) , with U(0)=0 , we obtain
By definingU(k+1)=1 , we thus obtain
Now, let(X1,…,Xk+1) be IID exponential random variables with mean 1, and let S=X1+…+Xk+1 . With a simple change of variable, we can see that
DefineYi=Xi/S , such that by a change of variable we obtain
Integrating this density with respect tos , we thus obtain
So the joint distribution ofk+1 uniform spacings on the interval (0,1) is the same as the joint distribution of k+1 exponential random variables divided by their sum. We come to the following equivalence of distribution
4. Asymptotic distribution
Using the equivalence above, we obtain
whereTk+1=X1+…+Xk+1k+1−1 . This variable vanishes in probability because E(Tk+1)=0 and Var(log(k+1)Tk+1)=(log(k+1))2k+1↓0 . Asymptotically, the distribution is the same as that of X(k+1)−log(k+1) . Because the Xi are IID, we have
5. Graphical overview
The plot below shows the distribution of the largest fragment for different values ofk . For k=10,20,50 , I have also overlaid the asymptotic Gumbel distribution (thin line). The Gumbel is a very bad approximation for small values of k so I omit them to not overload the picture. The Gumbel approximation is good from k≈50 .
6. References
The proofs above are taken from references 2 and 3. The cited literature contains many more results, such as the distribution of the ordered spacings of any rank, their limit distribution and some alternative constructions of the ordered uniform spacings. The key references are not easily accessible, so I also provide links to the full text.
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Ce n'est pas une réponse complète, mais j'ai fait quelques simulations rapides, et voici ce que j'ai obtenu:
Cela semble remarquablement beta-ish, et cela a un peu de sens, car les statistiques d'ordre des distributions uniformes iid sont beta wiki .
Cela pourrait donner un point de départ pour dériver le pdf résultant.
Je mettrai à jour si j'arrive à une solution finale fermée.
À votre santé!
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I produced the answer for a conference in Siena (Italy) in 2005. The paper (2006) is presented on my web-site here (pdf). The exact distributions of all the spacings (smallest to largest) are found on pages 75 & 76.
I'm hoping to give a presentation on this topic at the RSS Conference in Manchester (England) in September 2016.
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