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Confidence Intervals for a Two-Parameter Exponential Distribution: One- and Two-Sample Problems

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Confidence Intervals for a Two-Parameter Exponential Distribution: One- and Two-Sample Problems Communications in Statistics - Theory and Methods ISSN: 0361-0926 (Print) 1532-415X (Online) Journal homepage: http://www.tandfonline.com/loi/lsta20 Confidence intervals for a two-parameter exponential distribution: One- and two-sample problems K. Krishnamoorthy & Yanping Xia To cite this article: K. Krishnamoorthy & Yanping Xia (2017): Confidence intervals for a two- parameter exponential distribution: One- and two-sample problems, Communications in Statistics - Theory and Methods, DOI: 10.1080/03610926.2017.1313983 To link to this article: http://dx.doi.org/10.1080/03610926.2017.1313983 Accepted author version posted online: 07 Apr 2017. Published online: 07 Apr 2017. Submit your article to this journal Article views: 22 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lsta20 Download by: [K. Krishnamoorthy] Date: 13 September 2017, At: 09:00 COMMUNICATIONS IN STATISTICS—THEORY AND METHODS , VOL. , NO. , – https://doi.org/./.. Confidence intervals for a two-parameter exponential distribution: One- and two-sample problems K. Krishnamoorthya and Yanping Xiab aDepartment of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, USA; bDepartment of Mathematics, Southeast Missouri State University, Cape Girardeau, MO, USA ABSTRACT ARTICLE HISTORY The problems of interval estimating the mean, quantiles, and survival Received February probability in a two-parameter exponential distribution are addressed. Accepted March Distribution function of a pivotal quantity whose percentiles can be KEYWORDS used to construct confidence limits for the mean and quantiles is Coverage probability; derived. A simple approximate method of finding confidence intervals Generalized pivotal quantity; for the difference between two means and for the difference between MNA approximation; two location parameters is also proposed. Monte Carlo evaluation stud- Tolerance limits. ies indicate that the approximate confidence intervals are accurate even for small samples. The methods are illustrated using two examples. 1. Introduction The two-parameter exponential distribution is a commonly used model in lifetime data anal- ysis and reliability studies. Inferential methods for the exponential distribution, and applica- tions in the context of life-testing and reliability, have been addressed in several articles and books; for example, see Bain and Engelhardt (1991), Balakrishnan and Basu (1995), Meeker and Escobar (1998), Lawless (2003), and the references therein. As the two-parameter expo- nential distribution is a member of the location–scale family, pivotal quantities based on the maximum likelihood estimates (MLEs) are easy to find, and pivotal-based approach to find exact test or confidence intervals (CIs) for the parameters and for the quantiles are readily available; for example see Lawless (2003, Chapter 4). Toreviewthepivotal-basedapproach,letX1,...,Xn be a sample from a two-parameter Downloaded by [K. Krishnamoorthy] at 09:00 13 September 2017 exponential distribution with the probability density function 1 f (x|a, b) = exp(−(x − a)/b), x > a, b > 0. (1) b The MLEs of a and b are given by n 1 ¯ a = X( ) and b = (X − X( ) ) = X − X( ), (2) 1 n i 1 1 i=1 CONTACT K. Krishnamoorthy [email protected] University of Louisiana at Lafayette, Lafayette, LA, USA. Supplemental data for this article can be accessed on the publisher’s website. © Taylor & Francis Group, LLC 2 K. KRISHNAMOORTHY AND Y. XIA where X(1) is the smallest of the Xi’s.ItiswellknownthattheMLEsa and b are independent with (− ) 2n a a 2 2nb 2 ∼ χ and ∼ χ − . (3) b 2 b 2n 2 Closed-form CIs for a and b canbeeasilyobtainedonthebasisoftheabovedistributional results. A pivotal quantity for estimating the mean or a quantile can be obtained as follows. Since the mean of the two-parameter exponential distribution is μ = a + b,andthepth quantile is a − ln(1 − p)b,0< p < 1,letusdevelopapivotalquantityforageneralcasea + cb,where c is a known positive constant. Using the distributional results in (3), we find + − − (− )/ − χ 2 a cb a c a a b 2nc 2 = ∼ = f , , say, (4) χ 2 n c b b/b 2n−2 where the Chi-square random variables are independent. For 0 <α<0.5, let fn,c;α denote the α quantile of f , .Then n c a + fn,c;αb, a + fn,c;1−αb (5) is an exact 1 − 2α CI for a + cb. Guenther, Patil, and Uppuluri (1976)haveevaluatedthecumulativedistributionfunction (cdf) of fn,c for calculating one-sided tolerance limits. Specifically, these authors have evalu- ated the cdf of fn,c when c =−log(p) and c =−ln(1 − p), for calculating upper tolerance limitsandlowertolerancelimits,respectively.However,wefoundthatthecalculationof percentiles using their expression of the cdf poses some problems. Specifically, to find the per- centiles using an iterative scheme, the initial value should be very close to the true percentile; otherwise, the iterative scheme could produce inaccurate percentiles. Since the calculation of exact tolerance limits by Guenther, Patil, and Uppuluri (1976) is numerically involved, Engehardt and Bain (1978)haveproposedalargesampleapproximationforthelowerand upper tolerance factors. However, our simulation studies (not reported here) indicated that such approximations are not satisfactory even for large samples of sizes around 50 or more. For estimating a survival probability, Roy and Mathew (2005) have developed a pivotal quan- tity using the generalized variable approach and proposed a simulation approach to obtain confidence limits. In general, a confidence limit for a survival probability can be deduced from appropriate one-sided tolerance limits. In fact, for the present setup, we can obtain an Downloaded by [K. Krishnamoorthy] at 09:00 13 September 2017 exact solution for the problem of estimating a survival probability from the derivation of the distribution function of fn,c. Toward this, we give an alternate expression for the cdf of fn,c whichiseasytocompute,anditcanbeusedtofindthepercentilesforcalculatingCIsfor a + cb for any given c > 0. In particular, the cdf can be used to find CIs for the mean and confidence bounds for quantiles. Regarding two-sample problems, a test for the ratio of two scale parameters can be read- ily obtained from the distributional results in (3). Kumar and Patel (1971)haveproposedan exact test for equality of two threshold (location) parameters assuming that the scale parame- ters are equal. Kharrati-Kopaei (2015), Kharrati-Kopaei, Malekzadeh, and Sadooghi-Alvandi (2013), and Maurya, Goyal, and Gill (2011) have developed simultaneous CIs for successive differences of location parameters, and a CI for the difference between two location param- eters can be obtained as a special case. To the best of our knowledge, no interval estimation method or test was proposed for the difference between two means when the scale parameters are unknown and arbitrary. COMMUNICATIONS IN STATISTICS—THEORY AND METHODS 3 Therestofthearticleisorganizedasfollows.Inthefollowingsection,wedescribethe recent modified normal-based approximation (MNA) by Krishnamoorthy (2016)tofind the percentiles of a linear combination of independent random variables. In Section 3,we provide an exact expression for finding the cdf of fn,c in (4). In Section 4,wepresentCIsfor the mean, one-sided confidence limits for quantiles and confidence limits for a survival prob- ability. The necessary table values for constructing CIs for the mean and quantiles are also provided. In Section 5,wedevelopCIsforthedifferencebetweentwolocationparameters when the scale parameters are unknown and arbitrary, and for the difference between two means using the generalized variable approach (Weerahandi 1993), and the MNA. The pro- posedCIsareevaluatedintermsofcoverageprobabilities.Themethodsareillustratedusing two examples in Section 6, and some concluding remarks are given in Section 7. 2. Modified normal-based approximations ,..., Let X1 Xk be independent continuous random variables not necessarily from the same α = ,..., = k w family of distributions. Let Xi;α denote the quantile of Xi, i 1 k.LetQ i=1 iXi, where wi’sareknownconstants.Then,anapproximationtotheα quantile is given by 1 k w ( ) − k w2 ( ) − ∗ 2 2 , <α≤ . , i=1 iE Xi i=1 i [E Xi Xi ] for 0 5 Qα 1 (6) k w ( ) + k w2 ( ) − ∗ 2 2 , . <α< , i=1 iE Xi i=1 i [E Xi Xi ] for 0 5 1 ∗ = w > w < where Xi Xi;α if i 0andisXi;1−α if i 0. Furthermore, P(Qα/2 ≤ Q ≤ Q1−α/2 ) 1 − α, for 0 <α<1. Krishnamoorthy (2016) has obtained the above approximation, and it can be readily verified that the above approximate percentile in (6) is exact for normally distributed independent random variables. His numerical studies indicated that these MNAs are very satisfactory for manycommonlyuseddistributionssuchasthebeta,Student’st,andthenoncentralF. An approximation to the percentiles of the ratio of two independent random variables can also be obtained from (6). Let X and Y be independent continuous random variables with means μx and μy, respectively, and assume that Y is a positive random variable. Let Xα (Yα ) denote the 100α percentile of X (Y ),andletRα denote the 100α percentile of X/Y.Byequating the approximate 100α percentile of X − RαY tozero,andsolvingtheequationforRα,wefind Downloaded by [K. Krishnamoorthy] at 09:00 13 September 2017 ⎧ 1 ⎪ Y −α 2 α 2 2 ⎪ r− r2− 1− 1− 1 r2− r− X ⎪ μy μy ⎪ , <α≤ . , ⎨ Y −α 2 0 5 1− 1− 1 μy Rα 1 (7) ⎪ Y −α 2 X −α 2 2 ⎪ r+ r2− 1− 1− 1 r2− r− 1 ⎪ μy μy ⎪ ,.<α< , ⎩ Y −α 2 5 1 1− 1− 1 μy where r = μx/μy. For more details, see Krishnamoorthy (2016). 3. Evaluation of the cdf of fn,c = ( − χ 2 )/χ 2 As noted earlier, the percentiles of fn,c 2nc 2n−2 (see expression (4)) with appro- priate choices of c canbeusedtoconstructconfidenceboundsforthemeanorquantiles. 4 K. KRISHNAMOORTHY AND Y. XIA To find the percentiles of fn,c, let us first derive the cdf of fn,c. As shown in the appendix, ⎧ − ⎪ e nc , ≤ , ⎨ (1−t)n−1 t 0 ( ≤ ) = − ( ), = , P fn,c t ⎪ 1 F2n 2nc t 1 ⎩ − / 2nc e nc 2nc t n−2 −0.5(1−t)y − − + , < < > .
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