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Improved confidence intervals for the scale parameter of Burr XII model based on record values

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Abstract

In this paper some different sorts of confidence intervals are considered for the scale parameter of the Burr type XII distribution based on the upper record values. In this regard, the coverage probability is adopted as a measure of improvement when the endpoints are the same for all types of confidence intervals. Proposed confidence intervals are based on the preliminary test estimator, Thompson shrinkage estimator and Bayes estimator with conjugate prior information. It is nicely demonstrated that the confidence intervals based on the above methodologies are superior to the equal tail confidence interval on specific intervals. Subsequently, to construct a uniformly dominant confidence interval, the result of Kubokawa (Ann Stat 22(1):290–299, 1994) is extended for dependent observations by making use of the information that exists in a covariate record value.

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References

  • Abdel-Hamid AH (2009) Constant-partialy accelereted life tests for Burr XII distribution with progressive type II censoring. J Comput Statist Data Anal 53:2511–2523

    Article  MathSciNet  MATH  Google Scholar 

  • Ahmadi J, Balakrishnan N (2005) Preservation of some reliability properties by certain record statistics. Statistics 39(4):347–354

    Article  MathSciNet  MATH  Google Scholar 

  • Ahmadi J, Razmkhah M (2010) Nonparametric confidence intervals for quantiles and quantile intervals from inversely sampled record breaking data in a multi-sample plan. J Stat Comput Simul 80(9):993–1006

    Article  MathSciNet  MATH  Google Scholar 

  • Akhlaghi MRA, Parsian A (1986) A note on shortest confidence intervals. Commun Stat Simul Comput 15(2):425–433

    Article  MathSciNet  Google Scholar 

  • AL-Hussaini EK, Jaheen ZF (1996) Baysian prediction bounds for the Burr type XII distribution in the presence of outliers. J Stat Plann Inf 55:23–37

    Article  MathSciNet  MATH  Google Scholar 

  • AL-Hussaini EK, Jaheen ZF (1995) Baysian prediction bounds for the Burr type XII failure model. Commun Stat Theory Meth 24:1829–1842

    Article  MATH  Google Scholar 

  • AL-Hussaini EK, Jaheen ZF (1994) Approximate Bayes estimators applied to the Burr model. Commun Stat Theory Meth 23:99–121

    Article  MathSciNet  MATH  Google Scholar 

  • AL-Hussaini EK, Jaheen ZF (1992) Bayes estimate of the parameters, reliability and failure rate function of the Burr type XII failure model. J Stat Comput Simul 41:31–40

    Article  MathSciNet  MATH  Google Scholar 

  • Arabi Belaghi R, Arashi M, Tabatabaey SMM (2012) On the construction of preliminary test estimator based on record values for Burr XII model. Comm Statist Theo Meth, Accepted

  • Arashi M (2009) Preliminary test estimation of the mean vector under balanced loss function. J Stat Res 43(2):55–65

    MathSciNet  Google Scholar 

  • Arashi M (2012) Preliminary test and Stein estimators in simultaneous linear equations. Lin Alg Appl 436(5):1195–1211

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold BC, Balakrishnan N, Nagarja HN (1998) Records. Wiley, New York

    Book  MATH  Google Scholar 

  • Balakrishnan N, Ahsanullah M, Chen PS (1995) On the logistic record values and associate inference. J Appl Stat Sci 2(3):233–248

    MATH  Google Scholar 

  • Bancroft TA (1944) On biases in estimation due to use of preliminary tests of significance. Ann Math 15:190–204

    Article  MathSciNet  MATH  Google Scholar 

  • Brewster JF, Zidek JV (1974) Improving on equvariant estimators? Ann Stat 2:21–38

    Article  MathSciNet  MATH  Google Scholar 

  • Brown LD (1966) On the admisibility of invariant estimators of one or more location parameters. Ann Math Stat 37:1087–1136

    Article  MATH  Google Scholar 

  • Burr IW (1942) Cumulative frequency distribution. Ann Stat 12:69–90

    Google Scholar 

  • Burr IW, Cislak PJ (1968) On general system of distribution: 1. It’s curve-shaped characteristics; 2. the sample median. J Am Stat Assoc 63:627–635

    Article  MathSciNet  Google Scholar 

  • Chandler KN (1952) The distribution and frequency of record values. J R Stat Soc 14:220–228

    MathSciNet  MATH  Google Scholar 

  • Glick N (1978) Breaking records and breaking boards. Am Math Month 85:2–26

    Article  MathSciNet  MATH  Google Scholar 

  • Habibi Rad A, Arghami NR, Ahmadi J (2006) Statistical evidence in experiments and in record values. Commun Stat Theory Meth 35(11):1971–1983

    Article  MathSciNet  Google Scholar 

  • Hwang JT, Casella G (1982) Minimax confidence sets for the mean of a multivariate normal distribution. Ann Stat 10(3):868–881

    Article  MathSciNet  MATH  Google Scholar 

  • Hwang JT, Chen J (1986) Improved confidence sets for the coefficients of a linear model with spherically symmetric errors. Ann Stat 14(2):444–460

    Article  MathSciNet  MATH  Google Scholar 

  • Joshi VM (1967) Inadmisibility of the usual confidence sets for the mean of multivariate normal population. Ann Math Stat 38:1868–1875

    Article  MATH  Google Scholar 

  • Jiang L, Wong ACM (2012) Interval estimations of the two-parameter eponential distribution. J Prob Stat doi:10.1155/2012/734575

  • Kibria BMG (2004) Performance of the shrinkage preliminary tests ridge regression estimators based on the conflicting of W, LR and LM tests. J Stat Comput Simul 74(11):793–810

    Article  MathSciNet  MATH  Google Scholar 

  • Kibria BMG, Saleh AK Md Ehsanes (2010) Preliminary test estimation of the parameters of exponential and Pareto distributions for censored samples. Stat Papers 51:757–773

    Article  MathSciNet  MATH  Google Scholar 

  • Kibria BMG, Salah AK Md Ehsanes (2006) Optimum critical value for pre-test estimators. Commun Stat Theory Meth 35(2):309–320

    Article  MATH  Google Scholar 

  • Kibria BMG, Saleh AK Md Ehsanes (2005) Comparison between Han-Bancroft and Brook method to determine the optimum significance level for pre-test estimator. J Prob Stat Sci 3:293–303

    Google Scholar 

  • Kibria BMG, Saleh AK Md Ehsanes (2004) Preliminary test ridge regression estimators with Student’s t errors and conflicting test-statistics. Metrika 59(2):105–124

    Article  MathSciNet  MATH  Google Scholar 

  • Kibria BMG, Saleh AK Md Ehsanes (1993) Performance of shrinkage preliminary test estimator in regression analysis. Jahangrinagar Rev. A17:133–148

    Google Scholar 

  • Kubokawa T (1994) A unified approach to improving the equivariant estimators. Ann Stat 22(1):290–299

    Article  MathSciNet  MATH  Google Scholar 

  • Nagaraja HN (1988) Record values and related statistics—a review. Commun Stat Theory Meth 17(7):2223–2238

    Article  MathSciNet  Google Scholar 

  • Petropoulos C, Kourouklis K (2012) New classes of improved confidence intervals for the variance of a normal distribution. Metrica 75:491–506

    Article  MathSciNet  MATH  Google Scholar 

  • Petropoulos C (2011) New classes of improved confidence intervals for the scale parameter of a two-parameter exponential distribution. Stat Method 8:401–410

    Article  MathSciNet  MATH  Google Scholar 

  • Razmkhah M, Ahmadi J (2011) Comparing two sampling schemes based on entropy of record statistics, Stat Papers (in press)

  • Rodriguez N (1977) A guide to the Burr type XII distributions. Biometrika 64:129–134

    Article  MathSciNet  MATH  Google Scholar 

  • Saleh AK Md Ehsanes (2006) Theory of preliminary test and stein-type estimations with applications. Wiley, New York

    Book  Google Scholar 

  • Saleh AK Md Ehsanes, Kibria BMG (1993) Performance of some new preliminary test ridge regression estimators and their properties. Commun Stat Theory Meth 22(10):2747–2764

    Article  MathSciNet  MATH  Google Scholar 

  • Saleh AK Md Ehsanes, Sen PK (1978) Nonparametric estimation of location parameter after a preliminary test on regression. Ann Stat 6:154–168

    Article  MathSciNet  MATH  Google Scholar 

  • Shao Q (2004) Notes on maximum likelihood estimation for the three-parameter Burr XII distribution. Comput Stat Data Anal 45:675–687

    Article  MATH  Google Scholar 

  • Soliman AA (2005) Estimation of life from proressively censored data using Burr XII model. IEEE Trans Rel 54(1):34–42

    Article  MathSciNet  Google Scholar 

  • Stein C (1962) Confidence sets for the mean of a multivariate normal distribution. J R Stat Soc Ser B 24:265–296

    MATH  Google Scholar 

  • Tadikamalla PR (1980) A look at the Burr and related distributions. Int Stat Rev 48:337–344

    Article  MathSciNet  MATH  Google Scholar 

  • Thompson JR (1968a) Some shrinkage techniques for estimating the mean. J Am Stat Assoc 63:113–122

    Article  Google Scholar 

  • Wang FK, Keats JB, Zimmer WJ (1996) Maximum likelihood estimation of the Burr XII distribution with censored and uncensored data. Microelectron Reliab 36:359–362

    Article  Google Scholar 

  • Wang L, Shi Y (2010) Empirical Bayes inference for the Burr model based on records. Appl Math Sci 4(34):1663–1670

    MathSciNet  MATH  Google Scholar 

  • Wingo, D. R. (1993). Maximum likelihood methods for fitting the Burr Type XII distribution to multiply (progressively) censored life test data, metrical, 40, 203–210

  • Wu S-J, Chen Y-J, Chang C-T (2007) Statistical inference based on progressively censored samples with random removals from the Burr type XII distribution. J Stat Comput Simul 77(1):19–27

    Article  MathSciNet  MATH  Google Scholar 

  • Wu J-W, Yu H-Y (2005) Statistical inference about the shape parameter of Burr type XII distribution under the failure-censored sampling plan. App Math Comput 163(1):443–482

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank the editor and anonymous referees for their helpful suggestions which greatly improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, P.O. Box 91775-1159, Mashhad, Iran

    R. Arabi Belaghi & S. M. M. Tabatabaey

  2. Department of Statistics, School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran

    M. Arashi

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  1. R. Arabi Belaghi
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  2. M. Arashi
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  3. S. M. M. Tabatabaey
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Correspondence to R. Arabi Belaghi.

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Arabi Belaghi, R., Arashi, M. & Tabatabaey, S.M.M. Improved confidence intervals for the scale parameter of Burr XII model based on record values. Comput Stat 29, 1153–1173 (2014). https://doi.org/10.1007/s00180-014-0484-3

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