Available online at www.sciencedirect.com
Systems Engineering Procedia 5 (2012) 349 – 354
E-Bayesian Estimation and Hierarchical Bayesian Estimation for Estate Probability in Engineering Dan Li, Jianhua Wang* , Difang Chen Department of mathematics,College of Science, Wuhan University of Technology, Wuhan, 430070
Abstract
Dr.Han Ming gives the definition of E-Bayesian estimation of estate probability and the formulas of E-Bayesian estimation, engineering forecast model and its applications in security investment at 2005. He gives hierarchical Bayesian estimation of estate probability and guess that the E-Bayesian estimation asymptotic equal to hierarchical Bayesian estimation of estate probability at 2006. In this paper we give the proof of that the limits of E-Bayesian estimation and hierarchical Bayesian estimation for estate probability are both zero and the hierarchical Bayesian estimation of estate probability which has no observation data is less than that of E-Bayesian estimation, and guess that the hierarchical Bayesian estimation of estate probability which has observation data is also less than that of EBayesian estimation. © © 2012 2011 Published Published by by Elsevier Elsevier Ltd. Ltd. Selection and peer-review under responsibility of Desheng Dash Wu. Open access under CC BY-NC-ND license. Keyworsds: Parameter estimation; estate probability; E-Bayesian estimation; hierarchical Bayesian estimation, engineering
1. Introduction In 2005, Dr. Han Ming gives the definition of E-Bayesian estimation of estate probability ǃ E-Bayesian estimation formula, forecast model and its application in securities investments[1].In 2006, he gives hierarchical Bayesian estimation of state probability asymptotic equal to E-Bayesian estimation[2].the paper is only given the
* Corresponding author. E-mail address:
[email protected].
2211-3819 © 2012 Published by Elsevier Ltd. Selection and peer-review under responsibility of Desheng Dash Wu. Open access under CC BY-NC-ND license. doi:10.1016/j.sepro.2012.04.055
350
Dan Li et al. / Systems Engineering Procedia 5 (2012) 349 – 354
engineering numerical calculation examples, but isn’t given the mathematical proof. In this paper we will give the proof of that the limits of E-Bayesian estimation and hierarchical Bayesian estimation for state probability are both zero .As in the table 2 and table 3 of the paper [2], the numerical calculation of hierarchical Bayesian estimation is error, so Han Ming fail to draw on the size comparative between estate probability’s hierarchical Bayesian estimation value and E-Bayesian estimation value. We can see that hierarchical Bayesian estimation value of state probability is less than that of E-Bayesian estimation value from correct simulation of hierarchical Bayesian estimation .For no observation data, the hierarchical Bayesian estimation of estate probability is less than that of EBayesian estimation. And for observation data, we can guess that the engineering hierarchical Bayesian estimation of estate probability is also less than that of E-Bayesian estimation by engineering numerical examples. 2. E-Bayesian estimation and hierarchical Bayesian estimation of state probability Suppose that x1 , x 2 , " , x n are sample observed values from the same population. Then we divide the nobservations into k groups (1 k n) at a certain distance. Accordingly, the forecast objects are divided into k states, let Ei (i pi (i
1,2, " , k ) be the ith state, there are ri (0 d ri n, i
1,2, " , k ) observed values in state E i .Let
1,2, " , k ) be the probability of forecast objects falling into state Ei (i
1,2, " , k ) , we shall call it estate
probability. Suppose the event whether the forecast objects fall into the state Ei is independent. Then the forecast of securities price can be approximated as n-Bernoulli trials problem, so it can be described as binomial distribution. Let X be the times of forecast objects falling into the state Ei , and its distribution law is P( X
r
ri ) C nri p i i (1 p i ) n ri
, ri
0,1,2, " , n
Where 0 pi 1 , pi is the estate probability. If the prior distribution of estate probability pi is Bata distribution, and its probability density function is
S ( p i | a, b)
pi
a 1
(1 p i ) b 1 B ( a, b)
Where 0 pi 1 , a ! 0, b ! 0 , a and b are hyper-parameters. B(a, b)
1
³t 0
a 1
(1 t ) b 1 dt is Bata distribution.
When 0 b 1, a ! 1 , S ( pi | a, b) is increasing in pi , and meets the requirements of hierarchical prior distribution decreasing function’s construction. Prior distribution with thinner tail would make worse robustness of Bayesian distribution. So when 0 b 1 , a should not be too large. Let the upper bound of b be c ( c ! 1 , and c is a constant). The value range of Hyper-parameters a and b is D ^(a, b) 1 a c,0 b 1` . Suppose the prior distribution of a is uniform distribution in (1, c) , and the prior distribution of b is uniform distribution in (0,1) , when a and b are independent, hierarchical prior density distribution of T is given by
S ( pi )
c 1
³³ 1
S ( p i | a,b)S (a,b)dbda
0
1 (c 1)
c 1
³³ 1
0
p i a 1 (1 p i ) b 1 dbda , 0 pi 1 . B ( a, b)
For simplicity, the following only consider the case when a 1 .now the prior distribution of estate probability pi is
S ( pi | b) b(1 pi ) b1
(1)
351
Dan Li et al. / Systems Engineering Procedia 5 (2012) 349 – 354
The hierarchical prior density function of pi is
S ( pi )
c 1 a 1 bpi (1 pi ) b 1db ³ (c 1) 1
(2)
Expected Bayesian estimation of pi is given by theorem 1[1]: Theorem 1 Suppose that x1 , x 2 , " , x n are sample observed values from the same population. Then we divide the n-observations into k groups (1 k n) at a certain distance. Accordingly, the forecast objects are divided into k states, there are ri (0 d ri n, i 1,2, " , k ) observed values in state Ei . If the prior density function S ( pi | b) of pi is given by (1), then we have ri 1 (i) under squared error loss, the Bayesian estimation of pi is pˆ i B (b) n b 1 (ii) if the prior distribution of b is uniform distribution in (1,c),then Expected Bayesian estimation of pi is given by r 1 n c 1 1 c ri 1 ln db i pˆ iEB c 1 n2 c 1 ³1 n b 1 The proof is given by paper [1]. The hierarchical Bayesian estimation of pi is given by theorem 2[2]: Theorem 2 Suppose that x1 , x 2 , " , x n are sample observed values from the same population. Then we divide the n-observations into k groups (1 k n) at a certain distance. Accordingly, the forecast objects are divided into k states, there are ri (0 d ri n, i 1,2, " , k ) observed values in state Ei , if hierarchical prior density function S ( pi ) of p i is given by (2),then under squared error loss, The hierarchical Bayesian estimation of pi is c
pˆ i HB
³1 bB(ri 2, n ri b)db c ³1 bB(ri 1, n ri b)db
The proof is given by paper [2]. 3. The Property of E-Bayesian Estimation and Hierarchical Bayesian Estimation for Estate Probability Han Ming (2006)[2] shows the correlation between E Bayesian estimation of estate probability pi and hierarchical Bayesian estimation of that. But the paper doesn’t give mathematical proof, only give some numerical examples. In this paper we will give the property and the mathematical proof of this property. Theorem 3 The pˆ iHB and pˆ iEB are given by Theorem 1and Theorem 2, then ˆ iEB = lim pˆ iHB ; (i) lim p n of
(ii) when ri
n of
0 , pˆ iHB < pˆ iEB .
Proof (i) By the knowledge of the relation between Bata function with Gamma function[3] ī (a) ī (b) , B ( a, b) ī ( a b)
We have ri 1 B (ri 1, n ri b) , n b 1 ri 1 r 1 r 1 r 1 is decreasing in b, as 1 b c, so i , thus we have i i n b 1 n c 1 n b 1 n 2 c ri 1 r 1 c ri 1 c bB(ri 1, n ri b)db bB(ri 1, n ri b)db i bB(ri 1, n ri b)db ³ 1 n b 1 n c 1 ³1 n b 1 ³1 That is B(ri 2, n ri b)
352
Dan Li et al. / Systems Engineering Procedia 5 (2012) 349 – 354 c
ri 1 n c 1
³1 bB(ri 2, n ri b)db ri 1 , c ³1 bB(ri 1, n ri b)db n 2
Thus c
³ bB(ri 2, n ri b)db lim pˆ iHB = lim 1c n of nof ³1 bB(ri 1, n ri b)db
0,
And because
lim pˆ iEB
nof
lim
nof
ri 1 n c 1 ln c 1 n2
ri 1 ln1 0 , c 1
So
lim pˆ iEB = lim pˆ iHB .
nof
0 , then B(ri 1, n ri b)
(ii) Let ri
B(1, n b)
nof
1 , nb 1 1 , (n b) (n b 1)
1 (n b 1)(n b) c c b nc ³1 bB(1, n b)db ³1 n b db c 1 nln n 1 , c c b c b n 1 c nc ³1 bB(2, n b)db ³1 n b db ³1 n b 1 db (n 1)ln n 2 nln n 1 , B(ri 2, n ri b)
B(2, n b)
So c
pˆ iHB
³1 bB(2, n b)db c ³1 bB(1, n b)db
n 1 c nc nln n 1 . n2 nc c 1 nln n 1
(n 1)ln
For pˆ iHB < pˆ iEB set up, as long as the following established: n 1 c nc nln n2 n 1 1 ln n c 1 nc c 1 n2 c 1 nln n 1
(n 1)ln
(3)
That is ln
1 nc n c 1 n 1 c nc ln ln ln n2 n2 n 1 c 1 n 1
n c 1 , then Let f (n) ln n 1 c ln n c 1 ln n c ln n2
n 1
c 1
n 1
n2
(c 1)(2n 2 c) 1 n c 1 1 nc ln ln n2 (n 2)(n 1 c) n 1 (n 1)(n 2)(n c)(n 1 c) (n 1)(n c)
f c(n)
!
§ c 1 (c 1)(2n 2 c) 1 (c 1) 2 (c 1) 3 · ¨ ¸ 2 (n 1)(n 2)(n c)(n 1 c) (n 1)(n c) ¨© n 2 2(n 2) 3(n 2) 3 ¸¹
§ c 1 (c 1) 2 · 1 ¨ ¸ (n 2)(n 1 c) ¨© n 1 2(n 1) 2 ¸¹
(c 1) 3 (4n(n 4) (2c 1)(n 1) 15) ! 0 (c ! 1, n t 4) 6(n 1) 2 (n 2) 3 (n c)(n 1 c)
(4)
353
Dan Li et al. / Systems Engineering Procedia 5 (2012) 349 – 354
For f (n) is increasing function, lim f (n) 0 , so f (n) 0 ,thus we have (4), that is to say (3)is right .thus we have nof
pˆ iHB < pˆ iEB , the proof is over. When ri 1, 2 , for different n and c, we can calculate the value of pˆ iHB and pˆ iEB by Mathmetica, then the results can be seen from table 1 and table 2 (the calculation values of pˆ iHB in table 2 and table 3 from paper [2] are wrong). ˆ iHB and pˆ iEB Table1. when ri 1 , the comparison between p n
c
2
3
4
5
6
10
pˆ iEB
0.160085
0.154150
0.148762
0.143841
0.139322
10
pˆ iHB pˆ iEB
0.159561
0.152806
0.146670
0.141143
0.136159
0.038096
0.037740
0.037392
0.037053
0.036723
pˆ iHB pˆ iEB
0.038058
0.037630
0.037201
0.036779
0.036366
0.019510
0.019418
0.019325
0.019233
0.019142
pˆ iHB pˆ iEB
0.019502
0.019388
0.019271
0.019155
0.019039
0.003980
0.003976
0.003972
0.003968
0.003964
pˆ iHB pˆ iEB
0.003980
0.003975
0.003970
0.003965
0.003960
1000
0.001995
0.001994
0.001993
0.001992
0.001991
1000
pˆ iHB
0.001995
0.001994
0.001992
0.001991
0.001990
50 50 100 100 500 500
Table2
2 , the comparison between pˆ iHB and pˆ iEB
when ri
n
c
2
3
4
5
6
10
pˆ iEB pˆ iHB
0.240128
0.231226
0.223143
0.215761
0.208984
0.239506
0.229768
0.221098
0.213423
0.206615
pˆ iEB pˆ iHB pˆ iEB
0.057114
0.056610
0.056089
0.055580
0.055084
0.057089
0.056452
0.055815
0.055191
0.054583
0.029268
0.029127
0.028987
0.028849
0.028713
pˆ iHB pˆ iEB
0.029253
0.029083
0.028909
0.028736
0.028564
0.005970
0.005964
0.005958
0.005952
0.005947
0.005969
0.005962
0.005955
0.005947
0.005940
1000
pˆ iHB pˆ iEB
0.002993
0.002991
0.002990
0.002988
0.002987
1000
pˆ iHB
0.002992
0.002991
0.002989
0.002987
0.002985
10 50 50 100 100 500 500
From table 1 and table 2, we can see when ri 1, 2 , pˆ iHB < pˆ iEB is right. So we can further speculate that for ri t 1 , we may also have pˆ iHB < pˆ iEB . This needs us to study and prove in future. The property of the hierarchical Bayesian estimation and E Bayesian estimation for estate probability is similar to that for engineering reliability parameters under no-failure date[4,5,6]. So it can be proved by the method that has proved the property of estimation of reliability parameters under no-failure date.
Acknowledgements The project was supported by the National Natural Science Foundation of China (30570611, 60773210) and the WHUT Fund (2010-Ia-040), FRFCU, China.
References 1. Han Ming. Expected Bayesian Method for Forecast of Security Investment [J].Operations Research and Management Scienceˈ(2005) 14(5): 98-102
354
Dan Li et al. / Systems Engineering Procedia 5 (2012) 349 – 354
2. Han Ming. E-Bayesian Estimation and Hierarchical Bayesian Estimation of Estate Probability [J]. Operations Research and Management Scienceˈ(2006) 15(5): 70-74 3. Abramowitz M,Stegun I A, Handbook of Mathematical Functions:with Formulas,Graphs, and Mathematical Tables[M].Dover Publications, (1965) 4. Wang Jianhua, Xia Xiaoyan. The property of Hierarchical Bayesian and E-Bayesian Estimation of Exponent Distribution’s Parameter [J]. Mathematica Applicata. (2008) 21S: 33-36 5. Wang Jianhua, Mao Juan. The Property of Hierarchical Bayesian and E-Bayesian Estimation of Binomial Distribution’s Parameter [J] Pure and Applied Mathematics.( 2009) 25(2):223-230 6.Wang Jianhua, Yuan Li. Properties of Hierarchical Bayesian and E-Bayesian Estimations of the Failure Probability in Zero-failure Date[J].Chinese Journal of Engineering Mathematics, (2010) 27(1):78-84
