GENERALIZED CONFIDENCE INTERVALS FOR THE SCALE PARAMETER OF THE INVERTED EXPONENTIAL DISTRIBUTION

Transcription

1 Internation Journ of Latest Research in Science and Technology ISSN (Online):7- Volume, Issue : Page No.-, November-December 0 (speci Issue Paper ) Speci Issue on Internation conference on Methematics Sciences-0(MSC-0) Confernece Held at Sadguru Gadage Maharaj College, Karad,Maharastra,(India) GENERALIZED CONFIDENCE INTERVALS FOR THE SCALE PARAMETER OF THE INVERTED EXPONENTIAL DISTRIBUTION S.S.Godase a, D. T. Shirke b and D.N.Kashid b a Department of Statistics SGM College, Karad,,INDIA (suwarna_godase@rediffmail.com ) b Department of Statistics Shivaji University, Kolhapur 00, INDIA (dtshirke@gmail.com) b Department of Statistics Shivaji University, Kolhapur 00, INDIA (dnk_stats@unishivaji.ac.in ) Abstract- We consider a k-unit series system with life time of each unit following inverted exponenti distribution with an unknown sce parameter. We provide two generized confidence intervs for the sce parameter based on maximum likelihood estimator and modified maximum likelihood estimator respectively. The performance of proposed generized confidence intervs is evuated using extensive simulation work. The proposed confidence intervs found to perform well for sml to moderate sample sizes. Further the proposed generized confidence intervs perform better than asymptotic confidence interv for sml sample sizes. Keywords: Series system, maximum likelihood estimator, modified maximum likelihood estimator, generized confidence interv. I. INTRODUCTION There is a large amount of literature about the estimation of sce parameter of inverted exponenti distribution using different approaches. Inverted exponenti distribution is life time distribution which is used in the reliability discipline. The inverted exponenti distribution (IED) has been discussed as a life time model by Lin et () in detail. They have obtained maximum likelihood estimators, confidence limits and uniformly minimum variance unbiased estimators for the parameter and reliability function with complete samples. Stefanski () has discussed some basic properties of the IED. We see from the literature review that there is more work on estimation of parameter of inverted exponenti distribution as compared to interv estimation. The main purpose of this article is to develop a generized pivot variable that is simple to use for interv estimation of the parameter in life time distribution of a series system. The concept of generized p- vue was introduced by Tsui and Weerahandi () for hypothesis testing. Weerahandi (3) extended the idea for constructing confidence interv. Weerahandi () gives a detailed discussion ong with numerous examples. The concept of generized confidence intervs have turned out to be very satisfactory for obtaining confidence interv for many complex problems; see Weerahandi(3,), Krishnamoorthy and Mathew (0), Guo and Krishnamoorthy (0), Ng (0), Ye and Wang (00), Kurian et.(00). In this paper, we consider the problem of setting generized confidence interv () for the sce parameter, when lifetime distribution of a unit in a k-unit series system has inverted exponenti distribution. Recently Potdar and Shirke (0) explained reliability estimation of k-unit series system based on progressively censored data. In section, we provide asymptotic confidence interv () based on maximum likelihood estimator (MLE) and modified maximum likelihood estimator (MMLE) for the sce parameter, when lifetime distribution of a unit in a k- unit series system has inverted exponenti distribution. Tiku and Suresh () obtained a new method of estimation for location and sce parameters by using MMLE. R.P.Suresh (00) provides estimation of location and sce parameters in the two parameter exponenti distribution using MMLE. In section 3, generized confidence interv has been developed. In section, we study performance of both confidence intervs (, ) using MLE as well as MMLE for k=, 3 and for sml sample sizes using simulation technique. The proposed s are simple to compute and perform better in sml sample sizes.. Asymptotic Confidence Interv based on MLE and MMLE Consider a k-unit series system with independent and identicly distributed lifetimes of components. Let Y, Y Y k be the lifetimes, where Y i is the lifetime of i th component namely inverted exponenti distribution. Lifetime of the system is X=min (Y, Y Y k ). The cdf of X is The pdf of X is given by, (.) = 0 ; otherwise Here log likelihood of the sample is given by (.) ISSN:7-

2 Internation Journ of Latest Research in Science and Technology. The MLE of è can be obtained by solving = 0, where where is the parameter of interest and is a vector of nuisance parameters. A generized pivot where (.3) The solution can be obtained by Newton-Raphson Method by taking initi solution.then Fisher Information is given by = E = (.) By using asymptotic norm distribution of MLE, we construct confidence interv for è. Let is the MLE of è. Therefore by Cramer () where be the asymptotic variance. Therefore, 00(-á) % asymptotic confidence interv for è is given by (.) where is the upper 00(á/) th percentile of standard norm distribution. In the following we discuss based on MMLE on the lines of Tiku and Suresh (). The likelihood equation is given by where (.) The maximum likelihood equation (.) does not have explicit solution for è. This is due to the fact that the term is intractable. In this paper, we use the MML approach to derive approximate MLE for è by linearizing the term using Taylor series expansion around the quantile point of F with reference to Tiku et. Al. (), Tiku and Suresh (), R.P.Suresh (00). The linearization is done in such a way that the derived MML estimators retain l the desirable asymptotic properties of the maximum likelihood estimators. Here MMLE is (.7) where a =, b =. For more details one may refer to Tiku and Suresh () and Suresh (00). Lemma.: Distribution of, both are free from è. Proof: The proof is similar to the one given by Gulati and Mi (00). While constructing generized pivot this lemma can be used. 3. Generized Confidence Intervs x is a observed vue of X, for interv estimation defined by Weerahandi (), has the following properties: has a distribution free of unknown parameters. ii) The vue of The percentiles of can then be used to obtain confidence intervs for. Such confidence intervs are referred to as generized confidence intervs. For example, if denotes the th percentile of, then is a generized upper confidence limit for. A lower confidence limit or two-sided confidence limits can be similarly defined. Thus is obtained by using a generized pivot. The generized pivot quantity based on is Obviously, the observed vue of Q i is è. Moreover, the distribution of Q i does not depend on unknown parameter. Therefore, Q i is a generized pivot for è. Computing gorithm For a given data set X, X X n, the generized confidence interv can be computed by the following steps. Here N is the number of simulations. Algorithm to obtain :. Input N, n, k, è.. Generate independently and identicly distributed observations (U, U,,Un) from U(0,). 3. For the given vue of the parameter è, set. for i=,,,n.. Then (x, x,, xn) is the required sample from the distribution of a k-unit series system with inverted exponenti distribution as the component life distribution.. Compute MLE of è ( ). 7. Generate N samples from F (.) (as given in (.)) by setting è= and for each of the sample compute MLE (say ).. Using and, i=,,,n. compute 0. Arrange Qi in ascending order as Q [], Q[],, Q[N].. Compute for è as [Q([Ná/]), Q([ N(-á/])].. Extending above gorithm one can estimate ge probability of the proposed. Here [Q([Ná/]), Q([ N(-á/]) ] is a two-sided 00(-á) percent based on MLE.. In the above gorithm, we can replace MLE by MMLE and obtain, based on MMLE. Suppose that X=(X, X,, X n ) form a random sample from a distribution which depends on the parameters ISSN:7-

3 . Simulation study Internation Journ of Latest Research in Science and Technology. Table. Mean ge of and when è=, k=. We conduct extensive simulation experiments to evuate performance of s based on MLE and MMLE. We choose different vues of è, k, n and á. Results are tabulated in Table (-). Figures in the st row are based on MLE, while figures in the nd row are based on MMLE. From tables -, we observe that simulated ge of does not differ significantly whether it can be computed from MLE as well as MMLE. However, large sample approach underestimates the ge probabilities for most of the scenarios, especily when the sample size is sml and (or) the parameter è is large. Also the performance of the proposed does not depend on è. As the sample size is large, the two estimators (MLE, MMLE) are equly efficient.the results reported in this paper can be extended to other members of inverted sce family of distributions given by Potdar and Shirke (0). able. Mean ge of and by using MLE and MMLE when è=, k= Table 3. Mean ge of and when è=, k=3. ge ISSN:7-3

4 Internation Journ of Latest Research in Science and Technology. Table. Mean ge of and when è=, k=. Table. Mean ge of and by using MLE Table. Mean ge of and when è=, k= CONCLUSION Generized confidence intervs are provided for the sce parameter of life time distribution of k-unit series system, when unit life time distribution is inverted exponenti. The proposed confidence interv performs satisfactory for sml to moderate sample sizes. These intervs are superior to the asymptotic confidence intervs. when è=, k=3. ge REFERENCES. Bhattacharya G.K. () The asymptotics of maximum likelihood and related estimators based on Type II censored data, Journ of American Statistic Association, 0,3-0.. Cramer H. () Mathematic Methods of Statistics, Princeton University Press, Prinnceton, N.J. 3. Guo H. and Krishnamoorthy K. (0) Comparison between two quantiles: Norm and Exponenti cases. Communications in Statistics, Simulation and computation, 3,3-.. Jordan S. M. and Krishnamoorthy, K. () Exact confidence intervs for the common mean of sever norm populations. Biometrics,, Krishnamoorthy K., Mathew T., Ramchandran G. (00) Generized p-vues and confidence intervs: A Novel approach for anyzing log normly distributed exposure data. Journ of Occupation and Environment Hygiene, 3, -0.. Krishnamoorthy K., Mathew T (0) Inferences on the means of lognorm distributions using generized p-vues and generized confidence intervs. Journ of Statistic Planning and Inference,, Krishnamoorthy K. and Mathew T. (00) One-Sided tolerance limits in banced and unbanced one-way random models based on generized confidence limits. Technometrices,, -.. Krishnamoorthy K., Mukherjee S. and Guo H. (0) Inference on reliability intwo-parameter exponenti stress-strength model. Metrika, vol., -.. Kumbhar, R. R. and Shirke, D. T., (00) Tolerance limits for lifetime distribution of k-unit parlel system, Journ of Statistic Computation and Simulation, 7, Kurian K. M., Mathew T. and Sebastian, G. (00) Generized confidence intervs for process capability indices in the one-way random model. Metrika, 7, 3-.. Lin C.T., Duran B.S., Lewis T.O.() Inverted gamma as life distribution, Microelectron Reliability, (), -.. Ng C.K. (0) Performance of the three methods of the interv estimation of coefficient of variation. Interstat. ISSN:7-

5 Internation Journ of Latest Research in Science and Technology.. Potdar K.G., Shirke D.T. (0) Inference for the parameters of generized inverted family of distributions. Probstat Forum, 0, -.. Potdar K.G., Shirke D.T. (0) Reliability estimation of k-unit series system based on progressively censored data, Electronic journ of applied statistic anysis,,-3.. Suresh R.P (00) Estimation of location and sce parameters in a two parameter exponenti distribution from a censored sample, Statistic Methods, (),-.. Suresh R.P (7) On approximate likelihood estimators in censored norm samples. Gujarat Statistic Review,, Tian L.L., Cappelleri J.C. (00) A new approach for interv estimation and hypothesis testing of a certain intraclass correlation coefficient: the generized variable method. Statistics in Medicine, 3,.. Tiku M.L. (7) Estimating the mean and standard deviation from a censored sample. Biometrica,, -.. Tiku M.L. () Estimating the parameters of norm and logistic distribution from censored samples, Austrian Journ of Statistic, 0, Tiku M.L., Suresh R.P. () A new method of estimation for location and sce parameters, Journ of Statistic Planning and Inference, 30, -. Tiku M.L., Tan W.Y., Bkrishnan N. () Robust Inference, Marvel Delker, Inc, New York.. Stefanski L. A. () A note on the arithmetic-geometricharmonic means inequities. The American Statistician, 0(3), Singh S.K., Singh U., Kumar D.() Bayes estimators of the reliability function and parameter of inverted exponenti distribution using informative and non-informative priors, Journ of Statistic Computation and Simulation, DOI 0,00/ Tian L.L., Cappelleri J.C. (00) A new approach for interv estimation and hypothesis testing of a certain intraclass correlation coefficient: the generized variable method. Statistics in Medicine, 3,.. Tsui K., Weerahandi S. () Generized p-vues in significance testing of hypotheses in the presence of nuisance parameters Journ of American Statistic Association,,0.. R.P.Suresh (00) Estimation of location and sce parameters in a two parameter exponenti distribution from a censored sample, (), Verrill S. and Johnson R.A. (0) Confidence bounds and hypothesis tests for norm distribution coefficients of variation. Communications in Statistics, 3, -0.. Weerahandi S. (3) Generized confidence intervs. Journ of American Statistic Association,.. Weerahandi S. () Exact Statistic methods for Data Anysis. Springer,New York. 30. Weerahandi S., Johnson R. A. () Testing reliability in a stress-strength model when X and Y are normly distributed. Technometrics, 3, 3.. Yu P. L.H., Sun Y. and Sinha B. K. () On exact confidence intervs for the common mean of sever norm populations. Journ of Statistic Planning and Inference., Ye R.D., and Wang S.G. (00) Generized inferences on the common mean in the MANOVA models. Communications in statistics theory and methods, 37, -3. ISSN:7-