Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in
Estimation of Parameters of an Extension of Exponential Distribution Arun Kaushik, Rajwant Kumar Singh & Sandeep Kumar Maurya Department of Statistics and DST-CIMS Banaras Hindu University, Varanasi-221005 Abstract: This paper aims to the estimation of the parameters for an extension of exponential distribution (EED). We have used different estimation methods namely, method of maximum likelihood estimation (MLE), method of Maximum product spacing (MPS) and method of least square (LSE) to obtain the estimates of the parameters. The performances of the estimators have been studied on the basis of Monte Carlo simulation, and finally, a real data set has been used for the illustrative purpose of the study. Keywords: Maximum likelihood estimator, Maximum product spacing (MPS), method of least square (LSE) Mathematics Subject Classification:62F15, 62C10
1. Introduction In life testing phenomenon, it is well known and well-established result that exponential distribution is a most popular distribution for lifetime data and most frequently used. But the utility of this distribution is restricted due to constant hazard rate because in many practical situations it is difficult to have a constant hazard. Therefore, several generalisations based on this distribution have been proposed to cover up the monotone failure rate behaviour see [3], [5], [6], [7], [10], etc. Further, the EED is also another generalisations of exponential distribution, introduced by [1]. The probability density function (pdf) and cumulative distribution function (cdf) of this distribution are given as; f ( x, α , λ ) = αλexp(1 + λx)α −1 {1 − (1 + λx)α } ; x, α , λ > 0 (1)
{
F ( x, α , λ ) = 1 − exp 1 − (1 + λx)α
} ; x, α , λ > 0
(2) respectively. For the family reduce to the exponential distribution. EED is very useful in life testing problem, and it may be used as an alternative to the Weibull and other Exponentiated family of distributions see [1]. The basic properties related to this distribution have been discussed in [1]. Another beauty of this model is that the density function (1) has assumed increasing, decreasing and constant failure rate for different choices of parameters.
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Reliability estimation for lifetime models have greater importance in survival analysis, but here we have not attempted it. The term reliability is the ability of a system or component or its any part to perform its required functions under specified conditions for a stipulated period. The reliability function and hazard function of EED for specified time are given by;
{
R(t , α , λ ) = exp 1 − (1 + λt )α
} ; t,α , λ > 0
(3)
H (t ) = αλexp(1 + λt )
α −1
;t ≥ 0
(4) for a variation of α and λ. It assumes the different shape of the reliability and hazard function. In this paper, our main focused on the consideration of different estimation procedure to obtain the estimates of parameters. For the point estimation, there is various method discussed in statistical literature such as the method of maximum likelihood (MLE), Least Squares Estimation (LSE), Weighted Least Squares estimation (WLSE) and method of moment (MOM). The most popular and widely accepted method is MLE. Despite having very nice properties, easy computation and popularity, various authors have pointed out its limitations in different situations. Pitman [4] discussed that it could not work for ’heavy-tailed’ continuous distribution. [14] and [16] discussed its limitations when density assumes J-shape. For better applicability in such types of situations [11] introduced MPS method which also possesses similar properties to that of MLE. LSE is another method of estimation of parameters which is widely discussed in the literature. This method provides regressionbased estimators of the unknown parameters, which was originally suggested by [18]. The LSE method does not show optimum properties even asymptotically. However, in linear estimation, this method provides good estimators in a small sample. The main objective of this paper is to investigate which method suits most for the considered distribution. It has been observed that the estimators of the parameters cannot be expressed in closed form. Therefore, we have used nonlinear maximisation method to compute them using R software. The rest of the paper is organized as follows: In section 2, we describe the different method of estimation of parameters. Section 3, Page 1222
Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in provides the simulation and numerical result and one real data set has been analysed in Section 4. Finally, the conclusion of the paper is provided in Section 5.
2. Estimation of the parameters
be
an
n
i =1
n
n
i =1
i =1
(5)
(α − 1)∑log (1 + λxi ) − ∑(1 + λxi ) α Maximum likelihood estimates of the parameters are obtained by differentiating the log of likelihood function w.r.t. to the parameters and equating them to zero. Thus, we get normal equations as, n n n + ∑log (1 + λxi ) − ∑(1 + λxi )α log (1 + λxi ) = 0 α i =1 i =1
(6) and n n n + (α − 1)∑log (1 + λxi ) −1 − ∑αxi (1 + λxi )α −1 = 0 λ i =1 i =1
Above likelihood, the equation of α and λ form an implicit system and does not exist a unique root for above system of equations, so they can not be solved analytically. Thus, maximum likelihood estimates are obtained by using any iterative procedures. Here, we suggest using nlm() function.
{
}
D1 = F ( x1 ) = 1 − exp 1 − (1 + λx1 )α D( n +1) = 1 − F ( xn ) = exp{1 − (1 + λxn )α }
{
}
Di = F ( xi ) − F ( x(i −1) ) = exp 1 − (1 + λxi −1 )α − = 1.
(11)
the
logarithm
of
G,
we
get, (12)
Or we may write S as n 1 S = ln D1 + ∑ ln Di + ln Dn +1 (n + 1) i=2 ln[1 − exp{1 − (1 + λx1 ) α }] + 1 n = (n + 1) ∑ ln{exp{1 − (1 + λxi −1 ) α }− exp{1 − (1 + λxi ) α }} i=2 1 α {1 − (1 + λxn ) } + (n + 1) (13) After differentiating the above equation with respect to the parameter α and λ respectively and then equating them to zero, we get the normal equation as follows: Let
{
}
U (α , λ ) i = exp 1 − (1 + λxi )α ,
{
}
U (α ) i = −(1 + λxi ) exp 1 − (1 + λxi )α log (1 + λxi ), α
'
U (λ ) i = − xiα (1 + λxi ) '
α −1
Let V (λ ) = (1 + λxi ) , and (7) obtain the likelihood equation
S α'
{
}
exp 1 − (1 + λxi )α .
V ' (λ ) = xi then we
U ' (α ) 1 − + n + 1 1 − U (α , λ ) n U ' (α ) i −1 − U ' (α ) i − 1 ∑ ' ' =0 i = 2 U (α , λ ) i −1 − U (α , λ ) i n +1 α V (λ )log (V (λ ))
=
S λ'
(
(9)
(9)
(10)
MPS method choose that θ which maximises the product of spacings or in other words to maximise
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1 U (λ ) − + n + 1 1 − U (α , λ ) n U ' (λ ) i −1 − U ' (λ ) i − 1 ∑ ' ' i = 2 U (α , λ ) i −1 − U (α , λ ) i =0 n +1 α − 1 ' αV (λ )V (λ ) '
(8)
And the general term of spacings is given by,
i
i.e.,
(14)
This Method was introduced by [11] as an alternative to Method of MLE. The method is briefly described as follows. The CDF of the EED is given by the equation (2), and the uniform spacings are defined as follows:
∑D
spacings,
n +1 G = ∏Di i =1
2.2 Method of Product Spacing
such that
the
S = 1/(n + 1)∑ ln Di
independently
L( x | α , λ ) = ∏ f ( xi , α , λ ) = n(1 + log (αλ )) +
}
of
n +1
identically distributed (iid) random sample of size n observed from EED defined in (1). Thus, the likelihood function of α and λ for the observed samples is given as
{
mean
1/n +1
i =1
x1 , x2 ,..., xn
exp 1 − (1 + λxi )α
geometric
Taking
2.1 Maximum likelihood estimation Suppose
the
=
(15) The above likelihood equations cannot be solved analytically. Therefore, we use the same iterative method to obtained the solution. (10)
2.3 Method of Least Square Let x1 < x2 < , < xn be the n ordered random sample of any distribution with cdf F(x), we get
E[ F ( xi )] =
i n +1
(16)
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Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in The least square minimising
estimates
n i P(α , λ ) = ∑ F ( xi ) − n +1 i =1
are
obtained
by
2
(17)
Putting the cdf of EED in equation (17) we get
{
}
i P(α , λ ) = ∑1 − exp 1 − (1 + λxi )α − n +1 i =1 n
2
(18) In order to minimise equation (18), we have to differentiate it with respect to α and λ , which gives the following equation: n 1 ' U (α ) = 0 − ∑ 1 − U (α , λ ) − n + 1 i =1 n 1 ' U (λ ) = 0 . − ∑ 1 − U (α , λ ) − n + 1 i =1
(19) (20)
2.4 Estimator for the Reliability function and hazard function The estimation of reliability and hazard function using MPS for specified value of t can be easily developed by utilising the discussions due to [11] and [20], that MPS shows the invariance property just like MLE. The estimation of reliability and hazard using MPS is also discussed by [17]. Thus the MPS estimate of the reliability and hazard function can be given as follows: The MLEs of the reliability and hazard function can be evaluated by using invariance principle. Therefore, Maximum likelihood estimators for reliability and hazard function are given as;
αˆ
(21)
λˆt
e − αˆ ˆ λˆe λt ˆ H M (t ) = λˆt e − αˆ
(22)
respectively. Similarly, the maximum product spacing estimators for reliability and hazard functions are obtained by taking αˆ = αˆ MP and
λˆ = λˆMP
in (21) and (22). Here, we only provide
Simulation Study
In this section, we have observed the performances of the proposed estimators based on the simulation study. For this purpose, we have generated samples from EED for a different of the sample sizes. It may be mentioned here that the exact expression of MSE can not be obtained because estimates are not found in close forms. It may also be noted here that MSE will depend on sample size n , scale parameter λ and shape parameter α respectively. In this study different variation of sample size(n) say n(=20,30,50,70,100,150), shape parameter α say α (=0.5,2,3,4,5) and scale parameter λ say λ (=0.5,2,3,4,5) have been considered. Here, αˆ ML and
The above normal equations cannot be solved analytically. Therefore we use nlm(.) function to obtained the solution.
Rˆ M (t ) =
3
λˆML
are ML estimates of
α
and
λ
respectively and-and λˆMP are MPS estimates of the parameters.On the basis of the extensive study, it is observed that the MSEs of all the estimators are decreased as values of sample size increases which is obvious (see Table 2, 3). From the Table 2 and 3, we observed that MSE of the MPS estimators for scale parameter is smaller as compared to that of the MLEs and LSEs. But for large values of n the MSEs of the MLEs of scale parameter α is least as compared to other methods of estimation. From Table 3, we observe that the MLE estimators are provided with the efficient result as compared to all methods taken under consideration for a different combination of the shape and scale parameters when we sample size n is 25. But for scale parameter λ MPS perform better. From Table 4, it is observed that for fixed value of n MSEs of MPS is smaller for lower settings of parameters but the higher value of the parameter, MLEs are better regarding MSE.
4
Real Illustration
In this section, we analyse one data set to demonstrate how the proposed methods can be utilised in the real-life phenomenon. The data shows the powerful earthquakes in Iran. This data set is also used by [1] to illustrate their studies. It has been verified that the given data set provides a better fit. Further, we have calculated the Maximum likelihood estimates, Maximum product spacings estimates, Least squares estimates of the parameter, reliability
Table 1: Estimates of the considered characteristics for the considered real data set when t=78 hrs. R(t) H(t) R(t) H(t) 8
0.8492
0.0022
0.0002
0.7656
the expression, not dealt its estimation.
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0.8305
0.0023
0.0032
3.6637
0.0005
characteristics which are presented in Table 1.
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Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in
5
Conclusion
In this paper, we have considered the different classical estimation technique for estimating the unknown lifetime characteristics from the extension of the exponential distribution. Among all the considered methods, we have found that maximum likelihood estimation and maximum product spacing estimation procedures both perform efficiently. But, MPS estimation provides more efficient result than the method of MLE for scale parameter, as well as for small sample size and perform approximatively same when the sample size is large. For fixed n = 25 , we recommend a method of MPS for scale parameter and MLE for Shape parameter. Overall MLE is recommended.
6
[9] A. Wood, Predicting software reliability, IEEE Transactions on Software Engineering, 22(1996),6977. [10] R. E. Barlow and F. Proschan (1975), Statistical theory of reliability and life testing probability models. Holt, Rinehart and Winston, New York. [11] Cheng, R. C. H. and Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society B 45, 394-403. [12] Ranneby, B., (1984). The Maximum Spacings Method. An Estimation Method Related to the Maximum Likelihood Method. Scand. J. Stat., 11, 93-112.
References
[1] Firoozeh Haghighi, Samira Sadeghi. An Exponential Extension. 41‘emes Journ´ees de Statistique, SFdS, Bordeaux, 2009, Bordeaux, France, France. 2009. [2] G. S. Rao, M. E. Ghitany and R. R. L. Kantam, Reliability test plans for Marshall-Olkin extended exponential distribution, Applied Mathematical Science, Vol. 3, 2009, no. 55, 2745-2755 [3] R. D Gupta and D. Kundu, Generalised exponential distribution: Existing result and some recent development, J.Stat. Plan. Inf. 137 (2007), pp. 3537-3547. [4] S. K. Singh, U. Singh and Abhimanyu S. Yadav, Bayesian Estimation of Marshall-Olkin Extended Exponential Parameters Under Various Approximation Techniques, accepted in Hacettepe Journal of Mathematics and Statistics, an article in press July 2013. [5] Gupta, R.D., Kundu, D., Exponentiated exponential distribution: an alternative to gamma and Weibull distributions. Biometrical J. 43 (1), 117-130, 2001. [6] Gupta, R.D., Kundu, D., Generalised exponential distributions: different methods of estimations. J.Statist. Comput. Simulations 69 (4), 315-338, 2001. [7] Gupta, R.D. and Kundu, D. Generalized exponential distributions; Statistical Inferences. Journal of Statistical Theory and Applications, 1, 101-118, 2002. [8] Birnbaum, Z. W. and Saunders, S. C. (1969). Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability, 6, 328-347.
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[13] Shah, A. and Gokhale, D. V., On Maximum Product of Spacings Estimation for Burr XII Distributions. Commun. Stat. Simulator. Computat., 22(3), 615-641, 1993. [14] Harter, H. L. and Moore, A. H., Maximum likelihood estimation of the parameters of Gamma and Weibull populations from complete and from censored samples,(1965). [15] Anatolyev, Stanislav and Grigory Kosenok, An alternative to maximum likelihood based on spacings, Econometric Theory, Vol. 21, No. 2, pp. 472-476, 2005. [16] Huzurbazar, V. S., The likelihood equation, consistency and the maxima of the likelihood function. Ann. Eugenics, 14, 185-200, (1948). [17] U. Singh, S. K. Singh and R. K. Singh, Comparative study of traditional estimation method and maximum product spacing method in Generalised inverted exponential Distribution, J. Stat. Appl. Pro. 3, No. 2, 1-17,(2014). [18] Swain, J., Venkatraman, S. and Wilson, J. (1988) “Least squares estimation of distribution function in Johnson’s translation system”, Journal of Statistical Computation and Simulation, 29, 271-297. [19] Ghosh, S.R. Jammalamadaka (2001) A general estimation method using spacings. Journal of Statistical Planning and Inference 93. [20] F.P.A Coolen and M.J Newby, “A note on the use of the product of spacings in Bayesian inference”.
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Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in Table 2: Average estimates of the parameters using MPS, MLE and LSE respectively for fixed value of α = 0.2 and λ = 0.2 for different choices of sample size.
n
MSE
MSE
MSE
MSE
MSE
MSE
20
0.2225
0.0011
0.1911
0.0113
0.2112
0.0008
0.1693
0.0105
0.1879
0.0007
0.2435
0.0145
30
0.2188
0.0008
0.1838
0.0077
0.2088
0.0006
0.1742
0.0073
0.1948
0.0005
0.2152
0.0092
50
0.2123
0.0004
0.1924
0.005
0.2052
0.0003
0.1889
0.0047
0.1966
0.0003
0.2117
0.0061
70
0.209
0.0003
0.1928
0.0035
0.2038
0.0002
0.1917
0.0034
0.197
0.0002
0.2094
0.0042
100
0.2069
0.0002
0.1925
0.0023
0.2023
0.0002
0.1928
0.0023
0.1986
0.0002
0.2051
0.0031
0.0001
0.2054
0.0025
0.0001
0.2028
0.0021
120
0.2056
0.0001
0.1943
0.0021
0.2017
0.0001
0.1953
0.002
0.1983
150
0.2049
0.0001
0.1944
0.0017
0.2015
0.0001
0.1957
0.0016
0.1991
Table 3: Average estimates of the parameters using MPS, MLE and LSE respectively for fixed value of α = 0.5 and λ = 0.5 for different choices of sample size. n
MSE
MSE
MSE
MSE
MSE
MSE
20
0.6104
0.0218
1.5432
1.0852
0.5588
0.0146
1.6085
0.984
0.4525
0.0543
2.3641
1.3152
30
0.5827
0.0126
1.6452
0.7428
0.539
0.0091
1.7251
0.6558
0.4747
0.0122
2.1586
0.9228
50
0.5547
0.0068
1.7323
0.4577
0.5208
0.0049
1.831
0.4079
0.4814
0.0015
2.086
0.5496
70
0.5397
0.0046
1.8001
0.3078
0.5155
0.0036
1.8867
0.2869
0.4882
0.0005
2.0848
0.4004
100
0.5326
0.003
1.8394
0.2192
0.5117
0.0025
1.9176
0.2087
0.4935
0.0002
2.0423
0.2866
120
0.5259
0.0025
1.859
0.1819
0.5087
0.0021
1.9315
0.1675
0.4941
0.0004
2.0323
0.243
150
0.5217
0.0019
1.8766
0.1419
0.5063
0.0017
1.9487
0.1343
0.4964
0.0001
2.0213
0.1951
Table 4: Average estimates of the parameters using MPS, MLE and LSE respectively for fixed value of sample size n = 25 for different choices of parameter α , λ . α,λ
MSE
0.5,0.5
0.5824
0.5,2 .5,2.5
MSE
MSE
MSE
MSE
MSE
0.0139
0.4189
0.0521
0.5337
0.0107
0.435
0.0488
0.4485
0.0099
0.6063
0.0757
0.5934
0.016
1.6276
0.8718
0.5448
0.0111
1.6917
0.7886
0.4636
0.0315
2.2972
1.1372
0.5951
0.0165
1.9842
1.4415
0.5423
0.0113
2.1271
1.2298
0.469
0.0316
2.7459
1.1923
2,0.5
1.9461
0.6353
0.578
0.0806
1.8138
0.6174
0.5734
0.0801
1.0484
0.4528
1.1918
0.4786
2,1
2.0033
0.7206
1.1004
0.3541
1.8152
0.6626
1.1436
0.3517
1.0758
0.1336
2.2471
1.5553
2,2
2.0832
0.7523
2.1011
1.4983
1.9366
0.7026
2.1571
1.4985
1.1438
2.1967
4.2387
5.0118
2,3
2.12
0.7896
3.0396
3.1696
1.99
0.7561
3.0634
3.3796
1.1421
3.9541
6.3079
10.9423
3,3
2.5076
2.0676
4.1632
4.1419
2.459
2.0846
3.9617
4.2024
1.2449
1.5014
9.1821
38.2181
4,4
2.7708
4.1793
6.8929
9.2574
2.6808
4.5685
6.5322
9.8147
1.2318
1.6807
16.5563
157.6596
5,5
2.9481
7.4873
10.1758
26.7888
2.9578
8.0842
9.5146
20.3815
1.2602
30.8192
24.6905
387.7141
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