Model Assisted Statistics and Applications 11 (2016) 179–184 DOI 10.3233/MAS-150362 IOS Press
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Parameters estimators of irregular right-angled triangular distribution Kartlos J. Kachiashvilia,b and Alexander L. Topchishvilic,∗ a
Georgian Technical University, Tbilisi, Georgia I. Vekua Institute of Applied Mathematics of the Tbilisi State University, Tbilisi, Georgia c DolMa, District Committee of Administrative District Marburg-Biedenkopf, Marburg, Germany b
Abstract. We obtained and investigated consistent, unbiased and efficient estimators of the parameters of irregular right-angled triangular distribution on the basis of maximum likelihood estimators. Some computation results realized on the basis of simulation of the appropriate random samples demonstrate theoretical outcomes. Keywords: Irregular right-angled triangular distribution, maximum likelihood estimator, consistency, unbiasedness, efficiency
1. Introduction When solving many problems, necessity of identification of probability distribution laws of random variables, which characterize the phenomenon under study, arises. In particular, metrological certification of measurement devices means the establishment just as of probability distribution function (density) of random components of the measurement errors, so of their separate numerical characteristics for determination of quantitative indicators of the measurement accuracy [7]. Electronic identification systems of persons have a very wide application in many spheres of human activity [1,9]. In particular, such systems are used as electronic voting systems in many countries, as security systems of banks and other secret objects for various purposes. Their application stably increases annually. When identifying persons with the help of these systems, a decision is made by comparing of measurement results of some characteristics of persons with the patterns stored in the system. The deviations of measurement results from the proper patterns in such systems are quite well approximated by normal distribution (see, for example, Maltoni et al. [6]; Boll et al. [1]) so that such deviations from the improper patterns are distributed (can be approximated) by right-angled triangular distribution. The right angled triangle distribution is also used, for example, for approximation of stochastic components of measurement errors of some measurement devices [7]. This distribution is irregular as its domain of definition depends on the distribution parameters. Obtaining of high-quality estimations of the parameters of irregular distributions is an actual problem. Standard triangular distribution and some of its characteristics are considered in Jonson et al. [5]. Consistent, unbiased and efficient estimations of the parameters of symmetric triangular distribution are given in Primak et al. [7]. For irregular right-angled triangle distribution the situation is more complicated and the results obtained in Primak et al. [7] are not applicable. Therefore, the current investigation is devoted to the solution of the mentioned problem for irregular right-angled triangular distribution. The results of investigation, in particular, the consistent, unbiased and efficient estimators of the parameters of irregular right-angled triangular distribution and the results of simulation are given in Section 2. Made conclusions are given in Sections 3. ∗ Corresponding
author: Alexander Topchishvili, Friedrich-Ebert-Str. 88, 35039 Marburg, Germany. E-mail:
[email protected].
c 2016 – IOS Press and the authors. All rights reserved ISSN 1574-1699/16/$35.00
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2. Estimators of parameters of right-angled triangular distribution Let us consider the right-angled triangular distribution with the following distribution density function and cumulative density function, respectively [5], pT (x) =
2(x − a) , at a x b, (b − a)2
(1)
and ⎧ ⎪ ⎨ 0 at x2 < a, FT (x) = (x−a) (b−a)2 , at a x b, ⎪ ⎩ 1 at x > b.
(2)
It is not difficult to see that the right-angled triangular distribution is the special case of Beta distribution [4,5] pB (x) =
(x − a)p−1 (b − x)q−1 1 · , at a x b, B(p, q) (b − a)p+q−1
B(p, q) =
1
0
tp (1 − t)q−1 dt,
when the parameters p = 2 and q = 1. Now introduce the following designations: x1 , x2 , . . . , xn is the sample of the right-angled triangular distributed random variable X; x(1) , x(2) , . . . , x(n) are the order statistics, where x(1) = xmin = min xi and {i}
x(n) = xmax = max xi . Let us consider the maximum likelihood estimations of the parameters a and b of the {i}
irregular right-angled triangular distribution and investigate such properties of these estimations as consistency, unbiasedness and efficiency. The likelihood function is n 2n (xi − a). L(x|a, b) = (b − a)2n i=1
The estimations of maximum likelihood of the parameters a and b are solutions of the following problem: L(x|a, b) =
n 2n (xi − a) ⇒ max, (b − a)2n i=1 {a,b}
(3a)
subject to a min xi = xmin and b max xi = xmax . {i}
{i}
(3b)
It is easy to see that by solving the problem Eq. (3) we obtain aML = xmin and bML = xmax .
(4)
Let us investigate the above mentioned properties of these estimations. Observation results xmin and xmax are order statistics. Therefore, their distribution laws are expressed by laws (1) and (2) and have accordingly the following forms [2,3]: n F1 (x) = 1 − [1 − F (x)] and Fn (x) = F n (x). The appropriate densities are n−1 2(x − a) (x − a)2 f1 (x) = n 1 − , a x b, (5a) (b − a)2 (b − a)2
K.J. Kachiashvili and A.L. Topchishvili / Parameters estimators of irregular right-angled triangular distribution
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Table 1 Computation results of the estimations of the parameters and their accuracy characteristics n 3 5 10 15 20 25 30 50 100 130 200 500
aM L 16.4702 15.6889 14.7808 14.3111 14.0749 13.8746 13.7473 13.4168 13.0915 12.9884 12.8529 12.6432
bM L 20.1700 20.6647 21.0530 21.1927 21.2602 21.3061 21.3386 21.3962 21.4415 21.4525 21.4641 21.4784
aunb 12.2418 12.2829 12.2957 12.2470 12.2853 12.2639 12.2746 12.2702 12.2784 12.2734 12.2768 12.2785
bunb 21.4913 21.5029 21.4908 21.4909 21.4846 21.4869 21.4897 21.4875 21.4873 21.4878 21.4871 21.4876
Δn (a) 4.2105 3.4024 2.4892 2.0566 1.7918 1.6085 1.4720 1.1458 0.8132 0.7138 0.5761 0.3648
Δn (b) 1.3158 0.8337 0.4386 0.2971 0.2246 0.1806 0.1510 0.0912 0.0458 0.0353 0.0230 0.0092
E(aM L − a)2 25.2405 17.6112 10.2915 7.2604 5.8542 4.8346 4.2059 2.7951 1.6527 1.3574 0.9965 0.4011
E(bM L − b)2 4.3454 2.0840 0.7975 0.4640 0.3244 0.2463 0.1946 0.1078 0.0500 0.0378 0.0240 0.0093
E(aunb − a)2 14.1260 7.0253 2.9691 1.7947 1.2931 0.9964 0.8157 0.4580 0.2124 0.1620 0.1012 0.0394
E(bunb − b)2 2.1499 0.7605 0.1956 0.0872 0.0514 0.0344 0.0224 0.0085 0.0021 0.0013 0.0005 0.0001
22.0000
AML
20.0000
BML Aunb Bunb
Value
18.0000
16.0000
14.0000
12.0000 3
5
10
15
20
25
30
50
100
130
200
500
n Fig. 1. Parameter estimations dependences on the sample size n. AML – Maximum likelihood estimation of parameter a; BML – Maximum likelihood estimation of parameter b; Aunb – Unbiased estimation of parameter a; Bunb – Unbiased estimation of parameter b.
n−1 (x − a)2 2(x − a) , a x b. (b − a)2 (b − a)2 It is not difficult to define mathematical expectations of maximum likelihood estimations Eq. (4) fn (x) = n
E(aML ) = E(amin ) = a + (b − a) ·
n j=0
(5b)
j
Cnj
where according to Prudnikov et al. [8] j=n j (−1) 1 n j=1 j An = = j=n Cnj j=0 2 j + 0.5 j=1 (j + 0.5)
(−1) = a + (b − a) · An , 2j + 1
(6)
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5.0000
DelAn DelBn
4.0000
Value
3.0000
2.0000
1.0000
0.0000 3
5
10
15
20
25
30
50
100
130
200
500
n Fig. 2. Dependences of the bias of the maximum likelihood estimations of the sample size n. DelAn – the bias of parameter a; DelBn – the bias of parameter b.
and E(bML ) = E(xmax ) = b − (b − a) ·
1 = b − (b − a) · Bn , (2n + 1)
(7)
where Bn =
1 . 2n + 1
The maximum likelihood estimation of the range of observation results is WML = xmax − xmin , and its mathematical expectation is E(WML ) = (b − a)(1 − Bn − An ). The unbiased estimation of the range is Wunbiased = (xmax − xmin )/(1 − Bn − An ). Finally, for unbiased estimations of the parameters a and b, we have aunbiased = xmin − (xmax − xmin ) · An /(1 − Bn − An ), bunbiased = xmax + (xmax − xmin ) · Bn /(1 − Bn − An ).
(8)
K.J. Kachiashvili and A.L. Topchishvili / Parameters estimators of irregular right-angled triangular distribution
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EffAML
30.0000
EffBML VAunb VBunb
25.0000
Value
20.0000
15.0000
10.0000
5.0000
0.0000 3
5
10
15
20
25
30
50
100
130
200
500
n Fig. 3. Dependences of mean square deviations (m.s.d.) of parameter estimations of the sample size n. EffAML – m.s.e. of maximum likelihood estimation of parameter a; EffBML – m.s.e. of maximum likelihood estimation of parameter b; VAunb – m.s.e. of unbiased estimation of parameter a; VBunb – m.s.e. of unbiased estimation of parameter b.
From formulae (6) and (7) we can see that the estimations (4) are biased with the displacement Δan = (b − a) · An and Δbn = (b − a) · Bn , respectively. It is clear that An → 0 and Bn → 0 when n → ∞. Therefore Δan → 0 and Δbn → 0 when n → ∞. As An /(1 − Bn − An ) → 0 and Bn /(1 − Bn − An ) → 0 when n → ∞, unbiased estimations (8) have the properties of consistency as well. For comparison of estimations (4) and (8) by efficiency, let us use the mean square deviations of these estimations from their true values. The mean square deviations are related with variance and bias of the estimation as follows: E(aML − a)2 = V (aML ) + Δan ,
E(bML − b)2 = V (bML ) + Δbn ,
E(aanbiased − a)2 = V (aanbiased ), E(banbiased − b)2 = V (banbiased ).
(9)
For comparison of mean square deviations of estimations (4) and (8) and computation of the above-considered characteristics we generated the sequence of random numbers distributed by right-angled triangular distribution with the parameters a = 12.2772 and b = 21.4876. The number of samples used for computation of characteristics (9) for each sample with the size n was equal to ten thousand. The appropriate computation results are given in Table 1. From this table we can see that the unbiased estimations are more accurate and more efficient than maximum likelihood estimations. Parameter estimations dependences on the sample size n are shown in Fig. 1; dependences of the displacements of the maximum likelihood estimations on the sample size n are given in Fig. 2 and dependences of mean square deviations (m.s.d.) of the parameter estimations on the sample size n for maximum likelihood and unbiased estimations are shown in Fig. 3.
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3. Conclusion The consistent, unbiased and efficient estimators of the parameters of irregular right-angled triangular distribution have been obtained and investigated. These estimations are consistent like maximum likelihood estimations but they are more accurate and more efficient than the last ones. Computation results realized on the simulation basis of the appropriate random samples with ten thousand observations clearly demonstrate theoretical outcomes.
Acknowledgments The authors are very grateful to Shota Rustaveli National Science Foundation of Georgia as the current research was supported by this Foundation, grant AR/183/4-100/13. The authors would like to express sincere gratitude to Dr. Stan Lipovetsky, co-editor-in-chief of the journal “Model Assisted Statistics and Applications” and the anonymous referees of the paper for their thorough and in-depth study of the paper, constructive comments and remarks, and valuable suggestions, which gave us possibility to improve significantly the presentation of the manuscript and directed us for the future thoughts and research.
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