Proceedings of the International Seminar on Mathematics and Its Usage in Other Areas November II-12.2010
. ISBN.978-979-1222-9g-2
RARE EVENT MODEL SIMULATION FOR HEAVY TAILED DISTRIBUTION
D o d i Devianto
Department of Mathematics, Andalas University
Limau Manis Campus, Padang-city 25163, West Sumatera, INDONESIA
Email: ddevianto(S).fmipa.unand.ac.id
Abstract.
It is shown that the sum o f rare event random variable X„j generated from generalization o f negative binomial distribution in the scheme o f infinitesimal system o f ^ = { { ^ „ y } y=i2 n) converges to a kind o f stable distribution with skewed property and heavy tailed. The sum of random variables X„j show a good s h ^ to figure out a rare event phenomenon where most of fail events have high probability concentrated around random variables zero, and the remainders (success events) have veiy low probability to appear.
Keywords: rare event, skewed distribution, heavy tailed distribution, stable distribution, generalization of negative binomial distribution, infinitesimal system schemes.
1. Introduction
It is knovm that strongest statistical argument based on the central limit theorem, which states that the sum o f a large number o f independent identically distributed random variables from a finite variance distribution w i l l tend to be normally distributed. However, from empirical reasearch, infinitesimal system o f random variables or triangular array problem has usually heavier tails and for special cases, the distribution o f the row sums from this system does not fit a normal distribution with well because o f its heavy tailed and skewness. This problem only can be solved by generalization o f central limit theorem on the sense o f stable distribution.
Recently, the study o f subclasses from skewed distribution especially for heavy tailed distribution, are booming because its ability to cover empirical data which heavy tailed distribution and it becomes the most popular alternative to Guassian distribution which has been rejected by numerous emperical studies. The strong empirical evidence for these features combined with the generalization o f central limit theorem is used by many papers to justify the use o f stable distribution models, i n economics and finance are given by Mandelbrot (1963), Fama (1965), Fama and R o l l (1970), Embrechts et al. (1997), Rachev and Mittnik (2000), M c C u l l o c h (1996).
The facts above give strong evidence about the importance o f stable distribution to face heavy tailed and skewed distribution performed from empirical data especially for rare event phenomenon that is poorly described by Gaussian distribution, and it is worth to explain i n mathematical theorem and also simulation to confirm the results. Therefore, this paper is devoted to simulate a rare event model for heavy tailed distribution in the scheme o f infinitesimal system.
2. L o w P r o b a b i l i t y and R a r e Event M o d e l
The phenomenon o f low probability (rare) events rely on description o f possible outcomes from something very rare, but they usually give huge impact and nearly impossible to predict from past history o f data set. The rare events sometimes attributed as outlier as it lies outide the realm o f regular expctatiton because nothing in the past can convincingly point to its possibility. For instance, the distribution o f sale o f a product break to market with outstanding high popularity can be recognized as rare event, since this phenomenon occurs with very low probability among many products. The formal mathematical set up for this phenomenon is by setting X(jt) as a random
Mashadi, Syamsudhuha, MDH Gamal dan M. Imran, (Eds)
Proceedings of the International Seminar on Mathematics and Its Usage in Other Areas November 11-12,2010
ISRN 07«.07Q.n??.9S.7
variable for rare event phenomenon such as outstanding high popularity a product above and let event {Rit)} defined on a probability space (il,A,P) and rare in the sense that
x(0 = P r ( i ? ( r ) ) - > 0 as /-^oo . A n estimator for x(t) is a random variable X(t) such that xit) = EXit).
The difficulty in rare event simulation is to produce estimators which not only small variance in term Var{X(t)) but also a small relative error ^Var(X(t))/xit). Assymptotically, the best performance which has been observed in realistic situation is a bounded relative error in the limit / -> 0 0 . The famous simulation to generate rare event simulation performed by the Monte Carlo method, that is to produce « independent and identically distributed replications
X,,X^ X . o f X(t), estimate x-x(t) by its empirical average and form a confidence interval based upon the emeprical variance o f X^ for J = I, 2 ,n .
The probability distribution this kind o f event, as an example, can be approuched by taking probability p o f "failure" close to one from generalization o f negative binomial distribution having distribution as follows
P r ( X = r ) = v(v + l) ...(v + r - l ) ^ ^ "
r!
where v = \/n for large enough n, r takes integer values starting from zero and p is probability of failure with 0 < p <\ , q = \ - p . This probability distribution has mean
fi = E{X) = y p / ( l - p) and variance
-p +p
n
(7^=E(iX-tif) = -
- + p
V
relatively small when v is getting small for large enough ti.
Probability Distribution 1.0
0.8
0.6
0.4
0.2
Random Variable X 10 20 30 40 50
Figure 1. Generalization negative binomial distribution (skewed graph, red plot) and classical negative binomial distribution (waved graph, blue plot) with probability o f fail p = 0.75 and n = 5. Generalization negative binomial distribution has low probabilities except at x = o , that is to figure out the rare event phenomenon with very heavy tailed.
Beside the characterization o f that kind o f distribution with low probabilities, it is very fascinating to see the properties o f their limit distribution o f sums o f independent random variables.
There are many establish theorem, one o f them is central limit theorem that it is necessary and sufficient conditions for sums o f independent random variables with fixed mean and finite variance converges to normal distribution. In the very special case, e.g. random variables from rare event phenomenon with skewed distribution, limit distribution the sums o f this independent random variables converges to some special distribution. In many cases with large number o f random variables are systemized going to be small or most o f them close to zero, then it is
Proceedings of the hitemational Seminar on Mathematics and Its Usage in Other Areas November 11-12,2010
. ISBN.978.979.1777.0S.9 interesting to draw a distribution tendency from this s u m s o f independent random variables, that
we call the system as infinitesimal system o f random v a r i a b l e s . T o cover problem above for a special case, Devianto and Takano (2007) have derived necessary and sufficient conditions for convergence o f row sums o f an infinitesimal triangular a r r a y o f random variables to the geometric distribution where the proof o f its theorem based on the L e v y representation o f infinitely divisible characteristic function o f geometric distribution.
Next, let us treat random variables fi-om the g e n e r a l i z a t i o n o f negative binomial distribution in infinitesimal system scheme, which is by setting {{X„j };=i.2 n) ^ a sequence o f row wise independent identically distributed random variables w i t h X„j has generalization o f negative binomial distribution as the following probabilities,
Pr iX„j = r ) = V ( V + 1 ) ... ( V + r - 1) ^ (1 - / ' ) "
where 0<p<\, r = 0, 1, 2, ... and v = 1/n for p o s i t i v e integer n. The system o f random variables {{v^„y}y=i,2 „} is triangular array or infinitesimal system because it is seen that for every J] >0
max P r{|^„,|>7;} = l - ( 1 - / ' ) ' ' - > 0
y=l, 2 ,n '
as « -> 0 0 . The parameter v in this term often to be c a l l e d as over (under) dispersion parameter, that occurred when observed variance is higher (lower) t h a n variance i n theoretical model. In this scheme o f infinitesimal system random variables we fixed over (under) dispersion parameter
v = l / « .
Base on the infinitesimal system {{X„^.}y=,_2 „ } „ = i , 2.... where random variable X„j has generalization o f negative binomial distribution, then w e have important example on convergence to the geometric distribution i n the following theorem.
Theorem (Devianto and T a k a n o , 2007). The system o f independent identically distributed random variables {{A'„y}y=i.2 „}„=i,2,... is the infinitesimal system o f random variables and the sequence o f distribution functions of sums o f independent random variables
Z „ = A ' „ , + Z „ 2 +... + X„„ converges completely to the geometric distribution.
We have known that random sample from generalization o f negative binomial distribution can be recognized as a rare event case. If we set an infinitesimal system o f random variables
{ { ^ n ; } > i . 2 «}n=i, 2 . . . ^hcre X„j has generalization o f negative binomial distribution, then by Theorem above random variable Z „ = X „ , + X „ 2+ . . . c o n v e r g e s completely to the geometric distribution. This fact gives us new evidence that sums o f independent identicuil>
distributed random variables from rare event phenomenon has tendency not only its heavy tailed distribution but also on convergence to kind of memory less distribution, that is convergence to the geometric distribution.
3. The M o d e l and Simulation Results
We have explained that generalization o f negative binomial distribution has a good shape to figt .c out a rare event phenomenon where most o f "fail" events have high probability concentrated on random variables with value zero, and the remainders (success events) have very low probability to appear. N o w , by using this fact we generate random samples X„^ from this rare event phenomenon by using acceptance-rejection algorithm then setting them into new random variables
Y„ defined at time t as follows
Mashadi, Syamsudhuha, MDH Gamal dan M. Imran, (Eds)
Proceedings of the International Seminar on Mathematics and Its Usage in Other Areas November II-12,2010
TSRN<?78.g7g-n7%<)s.? : ,
\+X„j w i t h p r o b a b i l i t y 1/2 w i t h p r o b a b i l i t y 1 / 2 .
We are interested i n probability distribution o f annual growth o f the sums o f independent random variables Y„j ,that is Z „ = 7„, + Y„2 + ... +Y„„. We expect that the statistical properties of the growth o f distribution Z„ still depend on properties o f random samples X„j, since it is natural that magnitude o f fluctuations Z„ w i l l increase while the sharpness o f probability distribution is going more skewed than probability half-geometric distribution, where half geometric distribution is defined as follows
PriX = n) = ( l / 2 ) ( l - / 7 ) p " for « = 0 , 1 , 2 , . . il/2)(l-p)p-" for « = - l , - 2 , where p is a fixed probability o f "failure" on any single attempt.
Figure 2. Plot P D F o f z, (red plot) and half-geometric distribution (blue plot) with iV^ = 10'' random samples and p = 0.9. The difference shape between distribution o f Z„ and half-geometric distribution occurred because o f effect summation random samples by adding plus or minus while generating random variable Y„j.
J
20Figure 3. Plot C D F o f Z„ (red plot) and half-geometric distribution (blue plot) with A'^ = 10'' random samples and p = 0.9. The difference shape between distribution o f Z„ and half-geometric distribution occurred because o f effect summation random samples by adding plus or minus while generating random variable Y„j.
Proceedings of the International Seminar on Mathematics and Its Usage in Other Areas November II-12,2010 ISBN. 978-979-1222-95-2
Figure 4. The Log-plot P D F o f Z„ (red plot) performs very close to straight line, while half-geometric distribution (blue plot) collapse onto two single straight line.
Figure 5. The Log-plot P D F o f 1 - C D F Z„ (blue plot), the tail o f Log-log-plot is expected to be a straight line.
The model o f rare event above is to show growth dynamics by random samples Z „ , that strongly performs a kind o f stable distribution, this satble distribution has implication o f heavy tailed distribution, where the tail show a kind o f power lawa behavior. The log-log plot o f the distributions (Figure 4) collapses onto straight line. This result suggest the universality, the equivalence o f power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power law relation i n the tail behavior. This is a new alternative point o f view in the study o f skewed distribution related to firm size distribution for rare event phenomenon, and it is the worth result to study the size o f business firms distribution for growth dynamics.
4. Conclusion
The row sum o f independent random variables X„j is generated from generalization o f negative binomial distribution in the scheme o f infinitesimal system o f X = { { ^ „ y } y= i 2 «} converges a kind o f stable distribution with skewed property and heavy tailed, and the tails show a powe.
behavior as implication of stable distribution properties. The behavior of sum of random variables X„j show a good shape to figure out a rare event phenomenon where most o f feil events have high jMobability concentrated around random variables zero, and the remainders (success events) have veiy low probability to appear. This is a new alternative point o f view in the study o f skewed distribution related to firm size distribution for growth dynamics.
R E F E R E N C E S
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Mashadi, Syamsudhuha, MDH Gamal dan M. Imran, (Eds)
Proceedings of the International Seminar on Mathematics and Its Usage in Other Areas November II-12,2010
TSRN.97S.97q.1777.0S-?
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