MODIFIED MAXIMUM LIKELIHOOD ESTIMATION IN LINEAR FAILURE RATE DISTRIBUTION

Transcription

1 MODIFIED MAXIMUM LIKELIHOOD ESTIMATION IN LINEAR FAILURE RATE DISTRIBUTION 1 R.R.L.Kantam, 2 M.Ch.Priya and 3 M.S.Ravikumar 1,2,3 Department of Statistics, Acharya Nagarjuna University, Nagarjunanagar , Guntur-Andhra Pradesh, INDIA. 1 kantam.rrl@gmail.com ; 2 chaitanya.metlapalli@gmail.com ; 3 msrk.raama@gmail.com Abstract: Some estimation procedures for the linear failure rate distribution (LFRD) are considered. The well known classical method maximum likelihood (ML) estimation is attempted resulting in iterative solutions as the MLEs of the parameters. Accordingly, attempts may also be necessary to develop other methods of estimation for LFRD parameters that are reasonably efficient and computationally simple/analytical. We study modified ML estimation to estimate LFRD parameters and compare with exact MLEs in small as well as large samples. Key Words: Linear Failure Rate Distribution, MLE, Modified MLE, Simulation Study. 1. Introduction In reliability studies series systems are one of many popular system configurations. If a series system has two components having independently distributed life time random variables with failure rate functions ଵሺݔሻ and ଶሺݔሻ then it is well known that the reliability of the series system is ௫ ሺݔሻ ݔ ሼ ଵሺݐሻ ൧Ǥݐሻሽ ݐଶሺ (1.1) The corresponding cumulative distribution function, failure density function and failure rate function are respectively given by ௫ (1.2) ൧ǡݐሻሽ ݐଶሺ ሼ ଵሺݐሻ ݔ ሻݔሺܨ ͳ ሺݔሻ ሻǡݔሺܨ (1.3) ௫ ሺݔሻ ሺ௫ሻ ሺ௫ሻ Ǥ (1.4) Taking ଵሺݔሻǡ ଶሺݔሻ, as the failure rates of the well known exponential and Rayleigh distributions in (1.1) we get the most commonly used Linear Failure Rate Distribution (LFRD). More specifically, if ଵሺݔሻ and ଶሺݔሻ ݔ we get, the failure density function, cumulative distribution function, hazard or failure rate function of LFRD as:

2 ሺݔሻ ሺ ሻ ൬௫ మݔ మ ൰ Ǣ ݔ Ͳǡ Ͳǡ Ͳǡ (1.5) ሻݔሺܨ ͳ ൬௫ మ మ ൰ Ǣ ݔ Ͳǡ Ͳǡ Ͳǡ (1.6) ሺݔሻ Ǥݔ (1.7) This distribution has non-zero density at the origin, so that it may be of important use in connection with those types of responses which take place even before observation begins. Listings of similar response time densities are given in Barlow and Proschan (1965). In that sense h(x) is also called the conditional mortality rate if response time is survival time. In the context of competing risks LFRD is the distribution of the minimum of two independent random variables of which one follows exponential and the other follows Rayleigh distribution. Bain (1974) seems to be one of the earliest works that has touched upon LFRD as a model useful for analysis in life testing. Ananda Sen (2005) gave a detailed review along with the distributional characteristics and inferential aspects of LFRD. Some basic features of LFRD are as follows: Mean: (1.8) ሺ Τξ ሻ൯ǡ ට ଶగ మ Τଶ൫ͳ ߤ where ሺǤ ሻ denotes the cumulative distribution function of a standard normal variate. Variance: (1.9) ǡ ଶߤ ሻߤ ଶ ଶ ሺͳ ߪ Mode: (1.10) ሻǡ ሺ ଶܫ ቆට ଵ ቇ ܯ where I(.) denotes indicator function. 100 p th Percentile: ሺሻ ටቀ ቁଶ ଶ୪୭ሺଵ ሻ ଵ ܨ and hence median is ǡ (1.11)

3 ටቀ ቁଶ ଶ୪୭ሺǤହሻ ܯ ǡ (1.12) In biological sciences this is called 50% survival time denoted by t 50. Recurrence relation for raw moments is ଵ ߤ ƍ ߤ ଵ ଵ ƍ ߤ ଶ ଶǢ Ͳǡͳǡʹ ǥǥ (1.13) The second, third and fourth non-central moments are ƍ (1.14) ሻǡߤ ଶ ଶ ሺͳ ߤ ƍ (1.15) ሻቁǡߤ ሺͳ ߤቀ ଷ ଷ ߤ ସ ߤ ƍ ସమ ߤ మ య ቀଵଶ మ ସయ ቁǡ (1.16) య where µ is the mean of the distribution given by (1.8). It can be seen from (1.10) that LFRD has a non-zero mode only if its parameters a and b satisfy the relation ଶ with a > 0, b > 0. The graphs of LFRD density function for various combinations of the parameters a, b are shown in the following figures. Figure-1.1 Figure-1.2 f(x) a=2.5, b=0.5 f(x) a=3, b= f(x) f(x) Figure-1.3 Figure-1.4 f(x) a=3.5, b=1 f(x) a=5, b= f(x) f(x)

4 Figure 1. 5 Figure f(x) a=5, b=1 1 f(x) a=0.1, b= f(x) f(x) Figure 1.7 Figure f(x) a=0.1, b=4 2 f(x) a=0.1, b= f(x) f(x) Figure 1.9 Figure f(x) a=0.2, b=2 f(x) f(x) a=0.2, b=4 f(x) Figure 1.11 f(x) a=0.2, b= f(x)

5 Maximum Likelihood estimation in LFRD is studied by many authors that include Bain (1974), Salvia (1980), Al-Khedhairi (2008), Sarhan and Kundu (2009), Sarhan and Zaindin (2009), Mahmoud and Alam (2010), and the relevant references therein. All these studies yield iterative solutions as the MLEs of its parameters. It is therefore desirable to develop methods of estimation that are efficient, computationally simpler and more analytical. With this back drop we attempt to suggest and study modifications to likelihood method of estimation for LFRD. The rest of the paper is organized as follows. In Section 2 we present the well known maximum likelihood estimation of parameters. In Section 3 we introduce the MMLE and derive the resulting estimates. In Section 4 we make a comparative study of exact MLE and MMLE of the parameters of Linear Failure Rate Distribution with respect to some sampling measures of dispersion. 2. Maximum Likelihood Estimation Let ݔ ଵ ǡ ݔ ଶ ǡ Ǥ Ǥ ݔ be a random sample of size n drawn from LFRD (a,b) with pdf given by ሺݔሻ ሺ ሻǤݔ ሺ௫ మ మ ሻ Ǣ ݔ Ͳǡ Ͳǡ ͲǤ (2.1) The likelihood function of this sample is Ǣ ǡ ሻǤ ݔሺ ς ଵ ܮ (2.2) Substituting (2.1) in (2.2), we get ሻǤ ሺ௫ మ ݔ ς ቊሺ ܮ మ ሻ ଵ ቋǤ (2.3) The log-likelihood function is ሻ ǡ ݔ ଵ ଶ σ ଵ ሺ ܮ (2.4) where ଵ σ ଵ ݔ ǡ ͳǡʹǥ Therefore, the estimating equations are డ డ డ డ ଵ ଵ σ ଵ Ͳǡ (2.5) ௫ ௫ ଶ σ ଵ ͲǤ (2.6) ௫

6 It can be seen that (2.5), (2.6) are to be solved by some numerical iterative methods to get the MLEs of a, b. The asymptotic variances and covariance of the MLEs of a, b are obtained by inverting the information matrix. ଵଶ ܫ ଵଵ ܫ ܫ ൨ (2.7) ଶଶ ܫ ଵଶ ܫ where ቀ డమ ܧ ଵଵ ܫ ቁ ܧ ቀ ଵ డ మ ሺ ௫ሻమቁǡ (2.8) ቀ డమ ܧ ଵଶ ܫ ቁ ܧ ቀ ௫ డడ ሺ ௫ሻమቁǡ (2.9) ቀ డమ ܧ ଶଶ ܫ ቁ ܧ ቀ ௫ మ డ మ ሺ ௫ሻమቁǡ (2.10) The mathematical expectations in (2.8) through (2.10) are to be obtained only through numerical integration. We have used the ten point quadrature formula given in Rao et al. (1966) for this purpose and obtained I ij for n=5, 10, 15, 20; (a=3, b=0.5); (a=0.1, b=6). These are presented in Table (2.1) which in turn would give the asymptotic variances/covariance of MLEs of a, b through inversion of the information matrix. These are given in Table (2.2). Table 2.1: Elements of Information Matrix n a=3, b=0.5 a=0.1, b=6 I 11 I 12 I 22 I 11 I 12 I

7 Table 2.2: Asymptotic Variances & Covariance of MLE s of a and b n a=3, b=0.5 a=0.1, b=6 ሻሺ ( ǡሺ ሻሺ ሻሺ ሻ ǡሺ ሻሺ Modified ML Estimation When the log likelihood equations do not admit analytical expressions as MLEs of the parameters of a density function from complete or censored sample, replacement of certain portions of log likelihood equations by suitable admissible approximations sometimes would lead to simpler and efficient estimates of the parameters. Such estimates in literature are named as approximate or modified MLEs. Tiku (1967); Mehrotra and Nanda (1974); Pearson and Rootzen (1977); Tiku and Suresh (1992); Rosaiah et al. (1993a); Rosaiah et al. (1993b); Kantam and Srinivasa Rao (1993); Kantam and Srinivasa Rao (2002); Kantam and Sriram (2003) and the references therein are some of the works in this direction. We adopt this concept of MML estimation for LFRD by considering its reparameterised version as given in Ananda Sen (2005). The pdf of LFRD after reparameterisation is ሺݔሻ ቀͳ ௫ ఏ మቁ ௫ ቀଵ where ߠ Τ ξ assumed to be known. The corresponding cdf is ሻݔሺܨ ͳ మഇ మቁ Ǣ ݔ Ͳǡ Ͳǡ (3.1) ௫ሺଵ మഇ మሻ Ǥ (3.2) The likelihood function of a random sample of size n is given by ς ܮ ሺͳ ௫ ሻ ଵ ς ௫ ఏ మ ଵ ς ሺଵ మഇ మሻ ଵ Ǥ (3.3) The log likelihood equation to estimate a is (since è is assumed to be known) is

8 డ Ͳ σ ቀ ௫ డ ଵ ቁ ቂσ ఏ మ ௫ ଵ ݔ σ ଵ ቃ ଶఏ మ (3.4) This equation needs to be solved iteratively to get the MLEs of a for a known è. To overcome the iterative techniques that may sometimes lead to convergence problems, we propose an alternative procedure as described below. ௫ In equation (3.4) the expression σ ቀ ௫ ଵ can be identified to be the cause for ఏ మ ௫ ቁ iterative solution for a. It is scale invariant also. It can be rewritten as ሺ ሻܩ ఏ మ ǡ (3.5) where ݔ. We suggest to approximate G(Z i ) as a linear function in Z i say (3.6) Ǥ ߚ ߙ ሺ ሻܩ Substituting (3.6) in (3.5) solving it for a after simplification we get σసభ ቂσసభ ௫ σ సభమഇ మ σసభ ఉ ௫ ቃ Ǥ (3.7) We call in (3.7) as MMLE of a which can be obtained if ߙ ǡ ߚ are known. Here we propose two methods to get ߙ ǡ ߚ. In both of these methods, we need to order the sample and then proceed. Therefore without loss of generality we assume that ݔ ଵ ݔ ଶ ڮ ݔ is the given ordered complete sample. Method-I: Let ଵ ǡ ͳǡʹǡ Ǥ Ǥ Ǥ Let כ ǡ ככ be the solutions of the following equations ǡ ככ ሻ ככሺ ܨ ǡ כ ሻ כሺ ܨ where כ ට ǡ ככ ට ǡ F(.) is the cdf of reparameterised LFRD, and ݍ ͳ Ǥ The expressions for כ ǡ ככ are

9 כ ߠ ଶ ߠඥߠ ଶ ʹ ǡ ሺͳ כ ሻǡ (3.8) ככ ߠ ଶ ߠඥߠ ଶ ʹ ǡ ሺͳ ככ ሻǤ (3.9) The slope ߚ and intercept ߙ of the linear approximation in the equation (3.6) are given by ൯ ሺ ככ ൫ ߚ ሻ כ ככ (3.10) ǡ כ (3.11) Ǥ ߚ ሺ ሻܩ ߙ where â i is given by (3.10). The values of á i and â i in this method for n = 5, 10, 15, 20 for è=0.1(0.1) 3.0 are computed by us and those for n=5 and 10 are only given in Table 3.1. Table 3.1: Method I for n=5 è â i á i è â i á i è â i á i

10

11 Table 3.1: Method I (Continued) for n=10 è â i á i è â i á i è â i á i

12

13

14 Method-II: Considering Taylor s expansion of G(Zi) upto its first derivative w.r.t Z i in a neighborhood of population quantile corresponding to p i, we get (3.12) ሺ ሻǡ ሖܩ ߚ ప (3.13) ǡ ߚ ప ሺ ሻܩ ߙ

15 where Zi is the quantile of LFRD given as the solution of the equation ሺ ሻܨ Ǥ Here ߠߠ ߠඥ ଶ ʹሺͳ ሻǤ The values of ߙ and ߚ in this method for n=5,10,15,20 for è=0.1(0.1)3.0 are computed by us and those for n=5,10 are only given in Table 3.2. Table 3.2: Method-II for n=5 è â i á i è â i á i è â i á i

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17 Table 3.2: Method-II (Continued) for n=10 è â i á i è â i á i è â i á i

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19

20 Comparative Study In the two modified methods, of Section-3, the basic principle is that certain expressions in the log likelihood equation are linearised in a neighborhood of the population quantile which depends on the size of the sample also. The larger the size, the narrower the neighborhood and hence is the closer the approximation. That is, the exactness of the approximation becomes finer and finer for large values of n. Hence the approximate log likelihood equation and the exact log likelihood equation tend to each other as n. Hence the exact and modified MLEs are asymptotically identical (Tiku et al. 1986). The asymptotic identical nature of exact and modified MLEs may not be true in small samples and these are to be assessed with the help of small sample characteristics of the MMLEs. Because of non-availability of analytical sampling variances, we compared the modified ML method with exact ML method through Monte Carlo simulation. 10,000 random samples of size n= 5 (5) 20 each are generated from LFRD with (a=0.2, b=6); (a=0.2, b=4); (a=0.2, b=2); (a=0.1, b=6); (a=0.1, b=4); (a=0.1, b=2); i.e., è = , 0.1, , , 0.05, in succession. For each sample at the corresponding è the á i and â i of Method-I (Method-II) given in Table 3.1 (3.2) are borrowed and used in equation 3.7 to get the

21 modified MLE of a by Method-I (Method-II). The MMLE of a with known value of è is used in ߠ Τ ξ to get the corresponding MMLE of b by Method-I (Method-II). The empirical mean, variance, mean square error of MMLE s of a, b by Method-I and Method-II are respectively given in Tables 4.1 and 4.2 for n=5,10 only. Table 4.1: Means and Variances of and based on MMLE: Method-I n a b ሻሺ ሻሺ ሻሺ ሻሺ ሻሺ ሻሺ Generalised Variance ሺ ǡ ሻ Generalised MSE ሺ ǡ ሻ E E E E E E Table 4.2: Means and Variances of and based on MMLE: Method-II n a b ሻሺ ሻሺ ሻሺ ሻሺ ሻሺ ሻሺ Generalised Variance ሺ ǡ ሻ Generalised MSE ሺ ǡ ሻ E E E

22 E E E E As seen from the elements of the information matrix or the asymptotic dispersion matrix of MLEs as given in Tables 2.1 and 2.2, we can say that the MLEs of a, b do have a generalized variance defined as the sum of the individual variances (the trace of the asymptotic dispersion matrix). Proceeding on similar lines we may think of the notion of generalized variance for MMLEs also. As MMLEs are biased estimators we may consider the MSE in the place of variance thus, motivating us to observe the trend of the sum of the MSEs for a better comprehension of the performance of biased estimators. As an analogy we may call it by name generalized MSE ሺ ǡ ሻ. These are given in the last two columns of Tables 4.1 and 4.2. These columns would indicate that both the methods I & II of modifications are resulting in MMLEs with almost the same variance (same MSE) except at a few cases. However, the general trend in the calculations indicates that MMLE by Method-I is marginally more efficient than MMLE by Method-II w.r.t generalized MSE. References: [1] Al-Khedhairi, A. (2008). Parameters Estimation for a Linear Exponential Distribution Based on grouped Data. International Mathematical Forum, Vol. 3, No. 33, [2] Ananda Sen. (2005). Linear failure rate distribution. Encyclopedia of Statistical Sciences, (eds. Kotz, Balakrishnan, Read, Vidakovic), Vol. 6, [3] Bain Lee J. (1974). Analysis for the Linear Failure-Rate Life-Testing Distribution. Technometrics, Vol.16, No. 4, [4] Barlow, R.E., and Proschan, F. (1965). Mathematical Theory of Reliability, J.Wiley and Sons, New York. [5] Kantam, R.R.L., and Srinivasa Rao, G. (1993): Reliability Estimation in Rayleigh Distribution with Censoring Some Approximations to ML Method. Proceedings of II Annual Conference of Society for Development of Statistics, [6] Kantam, R.R.L., and Srinivasa Rao, G. (2002): Log-logistic Distribution: Modified Maximum Likelihood Estimation. Gujarat Statistical Review, Vol. 29, No. 1 & 2, [7] Kantam, R.R.L., and Sriram, B. (2003): Maximum Likelihood Estimation from Censored Samples Some Modifications in Length Biased Version of Exponential Model. Statistical Methods, Vol. 5, No. 1,

23 [8] Mahmoud, M.A.W., and Farouq Mohammad A. Alam. (2010): The Generalized Linear Exponential Distribution, Statistics and Probability Letters, Vol. 80, [9] Mehrotra, K.G., and Nanda, P. (1974): Unbiased Estimation of Parameters by Order Statistics in the case of Censored Samples, Biometrika, Vol. 61, [10] Pearson, E.S., and Rootzen, H. (1977): Simple and Highly Efficient Estimators for a Type-I Censored Normal Sample. Biometrika, Vol. 64, No. 1, [11] Rao, C.R., Mathai, A., and Mitra, S.K. (1966): Formulae and Tables for Statistical Work. Statistical Publishing Society, Calcutta, India. [12] Rosaiah, K., Kantam, R.R.L., and Narasimham, V.L. (1993a): ML and Modified ML Estimation in Gamma Distribution with a Known Prior Relation Among the Parameters. Pakistan Journal of Statistics, Vol. 9, No. 3(B), [13] Rosaiah, K., Kantam, R.R.L., and Narasimham, V.L. (1993b): On Modified Maximum Likelihood Estimation of Gamma Parameters. Journal of Statistical Research, Bangladesh, Vol. 27, No. 1&2, [14] Salvia Anthony, A. (1980). Maximum Likelihood Estimation of Linear Failure Rate. IEEE Transactions on Reliability, R- Vol. 29, No. 1, [15] Sarhan A.M., and Kundu, D. (2009). Generalized Linear Failure Rate Distribution. Communications in Statistics: Theory and Methods, Vol. 38, No. 5, [16] Sarhan, A.M., and Zaindin, M. (2009). Modified Weibull Distribution. Journal of Applied Sciences, Vol. 11, [17] Tiku, M.L (1967): Estimating the Mean and Standard Deviation from a Censored Normal Sample. Biometrika, Vol. 54, [18] Tiku, M.L., and Suresh, R.P. (1992): A New Method of Estimation for Locatio and Scale Parameters. Journal of Statistical Planning & Inference, Vol. 30,