Record Values from the Inverse Weibull Lifetime Model: Different Methods of Estimation

Author(s)
Khalaf S. Sultan

ABSTRACT

In this paper, we use the lower record values from the inverse Weibull distribution (IWD) to develop and discuss different methods of estimation in two different cases, 1) when the shape parameter is known and 2) when both of the shape and scale parameters are unknown. First, we derive the best linear unbiased estimate (BLUE) of the scale parameter of the IWD. To compare the different methods of estimation, we present the results of Sultan (2007) for calculating the best linear unbiased estimates (BLUEs) of the location and scale parameters of IWD. Second, we derive the maximum likelihood estimates (MLEs) of the location and scale parameters. Further, we discuss some properties of the MLEs of the location and scale parameters. To compare the different estimates we calculate the relative efficiency between the obtained estimates. Finally, we propose some numerical illustrations by using Monte Carlo simulations and apply the findings of the paper to some simulated data.

In this paper, we use the lower record values from the inverse Weibull distribution (IWD) to develop and discuss different methods of estimation in two different cases, 1) when the shape parameter is known and 2) when both of the shape and scale parameters are unknown. First, we derive the best linear unbiased estimate (BLUE) of the scale parameter of the IWD. To compare the different methods of estimation, we present the results of Sultan (2007) for calculating the best linear unbiased estimates (BLUEs) of the location and scale parameters of IWD. Second, we derive the maximum likelihood estimates (MLEs) of the location and scale parameters. Further, we discuss some properties of the MLEs of the location and scale parameters. To compare the different estimates we calculate the relative efficiency between the obtained estimates. Finally, we propose some numerical illustrations by using Monte Carlo simulations and apply the findings of the paper to some simulated data.

KEYWORDS

Scale Parameter, Location Parameter, Best Linear Unbiased Estimates (BLUEs), Maximum Likelihood Estimates, Relative Efficiency and Monte Carlo Simulations

Scale Parameter, Location Parameter, Best Linear Unbiased Estimates (BLUEs), Maximum Likelihood Estimates, Relative Efficiency and Monte Carlo Simulations

Cite this paper

nullK. Sultan, "Record Values from the Inverse Weibull Lifetime Model: Different Methods of Estimation,"*Intelligent Information Management*, Vol. 2 No. 11, 2010, pp. 631-636. doi: 10.4236/iim.2010.211072.

nullK. Sultan, "Record Values from the Inverse Weibull Lifetime Model: Different Methods of Estimation,"

References

[1] K. N. Chandler, “The Distribution and Frequency of Record Values,” Journal of the Royal Statistical Society Series B, Vol. 14, 1952, pp. 220-228.

[2] V. B. Nevzorov, “Records,” Theory of Probability and Its Applications, Vol. 32, 1988, pp. 201-228.

[3] H. N. Nagaraja, “Record Values and Related Statistics-A Review,” Communications in Statistics-Theory and Methods, Vol. 17, 1988, pp. 2223-2238.

[4] M. Ahsanullah, “Introduction to Record Values,” Ginn Press, Needham Heights, Massachusetts, 1988.

[5] M. Ahsanullah, “Record values, The Exponential Distribution: Theory, Methods and Applications,” In: N. Balakrishnan and A.P. Basu Eds., Gordon and Breach Publishers, Newark, New Jersey, 1995.

[6] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, “A First Course in Order Statistics,” John Wiley Sons, New York, 1992.

[7] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, “Records,” John Wiley Sons, New York, 1998.

[8] N. Balakrishnan, M. Ahsanullah and P. S. Chan, “Relations for Single and Product Moments of Record Values from Gumbel Distribution,” Statistics & Probability Letters, Vol. 15, No. 3, 1992, pp. 223-227.

[9] N. Balakrishnan, P. S. Chan and M. Ahsanullah, “Recurrence Relations for Moments of Record Values from Generalized Extreme Value Distribution,” Communications in Statistics-Theory and Methods, Vol. 22, No. 5, 1993, pp. 1471-1482.

[10] N. Balakrishnan and M. Ahsanullah, “Relations for Single and Product Moments of Record Values from Exponential Distribution,” Journal of Applied Statistical Science, Vol. 2, 1995, pp. 73-87.

[11] N. Balakrishnan, M. Ahsanullah, and P. S. Chan, “On the Logistic Record Values and Associated Inference,” Journal of Applied Statistics, Vol. 2, pp. 233-248.

[12] K. S. Sultan and N. Balakrishnan, “Higher Order Moments of Record Values from Rayleigh and Weibull Distributions and Edgeworth Approximate Inference,” Journal of Applied Statistical Science, Vol. 9, 1999, pp. 193-209.

[13] K. S. Sultan and M. E. Moshref, “Higher Order Moments of Record Values from Generalized Pareto Distribution and Associated Inference,” Metrika, Vol. 51, 2000, pp. 105- 116.

[14] A. A. Soliman, A. H. Abd Ellah and K. S. Sultan, “Comparison of Estimates Using Record Statistics from Weibull Model: Bayesian and Non-Bayesian Approaches,” Computational Statistics and Data Analysis, Vol. 51, No. 3, 2006, pp. 2065-2077.

[15] K. S. Sultan, “Higher Order Moments of Record Values from the Inverse Weibull Lifetime Model and Edgeworth Approximate Inference,” International Journal of Reliability and Applications, Vol. 8, No. 1, 2007, pp. 1-16.

[16] A. Drapella, “Complementary Weibull Distribution: Unknown or Just Forgotten,” Quality and Reliability Engineering International, Vol. 9, 1993, pp. 383-385.

[17] G. S. Mudholkar and G. D. Kollia, “Generalized Weibull family: A Structural Analysis,” Communications in Statistics-Theory and Methods, Vol. 23, 1994, pp. 1149- 1171.

[18] R. Jiag, D. N. P. Murthy and P. Ji, “Models Involving two Inverse Weibull Distributions,” Reliability Engineering and System Safety, Vol. 73, No. 1, 2001, pp. 73-81.

[19] R. Calabria and G. Pulcini, “On the Maximum Likelihood and Least-Squares Estimation in the Inverse Weibull Distributions,” Statistica Applicata, Vol. 2, No. 1, 1990, pp. 53-66.

[20] M. Maswadah, “Conditional Confidence Interval Estimation for the Inverse Weibull Distribution Based on Censored Generalized Order Statistics,” Journal of Statistical Computation Simulation, Vol. 73, No. 12, 2003, pp. 887- 898.

[21] D. N. P. Murthy, M. Xie and R. Jiang, “Weibull Models,” John Wiley & Sons, New York, 2004.

[22] E. M. Nigm and R. F. Khalil, “Record Values from Inverse Weibull Distribution and Associated Inference,” Bulletin of the Faculty of Science, 2006.

[23] N. Balakrishnan and A. C. Cohen, “Order Statistics and Inference: Estimation Methods,” Academic Press, San Diego, 1991.

[24] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” Dover, New York, 1972.

[1] K. N. Chandler, “The Distribution and Frequency of Record Values,” Journal of the Royal Statistical Society Series B, Vol. 14, 1952, pp. 220-228.

[2] V. B. Nevzorov, “Records,” Theory of Probability and Its Applications, Vol. 32, 1988, pp. 201-228.

[3] H. N. Nagaraja, “Record Values and Related Statistics-A Review,” Communications in Statistics-Theory and Methods, Vol. 17, 1988, pp. 2223-2238.

[4] M. Ahsanullah, “Introduction to Record Values,” Ginn Press, Needham Heights, Massachusetts, 1988.

[5] M. Ahsanullah, “Record values, The Exponential Distribution: Theory, Methods and Applications,” In: N. Balakrishnan and A.P. Basu Eds., Gordon and Breach Publishers, Newark, New Jersey, 1995.

[6] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, “A First Course in Order Statistics,” John Wiley Sons, New York, 1992.

[7] B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, “Records,” John Wiley Sons, New York, 1998.

[8] N. Balakrishnan, M. Ahsanullah and P. S. Chan, “Relations for Single and Product Moments of Record Values from Gumbel Distribution,” Statistics & Probability Letters, Vol. 15, No. 3, 1992, pp. 223-227.

[9] N. Balakrishnan, P. S. Chan and M. Ahsanullah, “Recurrence Relations for Moments of Record Values from Generalized Extreme Value Distribution,” Communications in Statistics-Theory and Methods, Vol. 22, No. 5, 1993, pp. 1471-1482.

[10] N. Balakrishnan and M. Ahsanullah, “Relations for Single and Product Moments of Record Values from Exponential Distribution,” Journal of Applied Statistical Science, Vol. 2, 1995, pp. 73-87.

[11] N. Balakrishnan, M. Ahsanullah, and P. S. Chan, “On the Logistic Record Values and Associated Inference,” Journal of Applied Statistics, Vol. 2, pp. 233-248.

[12] K. S. Sultan and N. Balakrishnan, “Higher Order Moments of Record Values from Rayleigh and Weibull Distributions and Edgeworth Approximate Inference,” Journal of Applied Statistical Science, Vol. 9, 1999, pp. 193-209.

[13] K. S. Sultan and M. E. Moshref, “Higher Order Moments of Record Values from Generalized Pareto Distribution and Associated Inference,” Metrika, Vol. 51, 2000, pp. 105- 116.

[14] A. A. Soliman, A. H. Abd Ellah and K. S. Sultan, “Comparison of Estimates Using Record Statistics from Weibull Model: Bayesian and Non-Bayesian Approaches,” Computational Statistics and Data Analysis, Vol. 51, No. 3, 2006, pp. 2065-2077.

[15] K. S. Sultan, “Higher Order Moments of Record Values from the Inverse Weibull Lifetime Model and Edgeworth Approximate Inference,” International Journal of Reliability and Applications, Vol. 8, No. 1, 2007, pp. 1-16.

[16] A. Drapella, “Complementary Weibull Distribution: Unknown or Just Forgotten,” Quality and Reliability Engineering International, Vol. 9, 1993, pp. 383-385.

[17] G. S. Mudholkar and G. D. Kollia, “Generalized Weibull family: A Structural Analysis,” Communications in Statistics-Theory and Methods, Vol. 23, 1994, pp. 1149- 1171.

[18] R. Jiag, D. N. P. Murthy and P. Ji, “Models Involving two Inverse Weibull Distributions,” Reliability Engineering and System Safety, Vol. 73, No. 1, 2001, pp. 73-81.

[19] R. Calabria and G. Pulcini, “On the Maximum Likelihood and Least-Squares Estimation in the Inverse Weibull Distributions,” Statistica Applicata, Vol. 2, No. 1, 1990, pp. 53-66.

[20] M. Maswadah, “Conditional Confidence Interval Estimation for the Inverse Weibull Distribution Based on Censored Generalized Order Statistics,” Journal of Statistical Computation Simulation, Vol. 73, No. 12, 2003, pp. 887- 898.

[21] D. N. P. Murthy, M. Xie and R. Jiang, “Weibull Models,” John Wiley & Sons, New York, 2004.

[22] E. M. Nigm and R. F. Khalil, “Record Values from Inverse Weibull Distribution and Associated Inference,” Bulletin of the Faculty of Science, 2006.

[23] N. Balakrishnan and A. C. Cohen, “Order Statistics and Inference: Estimation Methods,” Academic Press, San Diego, 1991.

[24] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” Dover, New York, 1972.