Abstract
In the current paper, we considered the Fisher information matrix from generalized Rayleigh distribution (GR) distribution in moving extremes ranked set sampling (MERSS). The numerical results show that the ranked set sample carry more information about \(\lambda\) and \(\alpha\) than a simple random sample of equivalent size. In order to give more insight into the performance of MERSS with respect to (w.r.t.) simple random sampling (SRS), a modified unbiased estimator and a modified best linear unbiased estimator (BLUE) of scale and shape \(\lambda\) and \(\alpha\) from GR distribution in SRS and MERSS are studied. The numerical results show that the modified unbiased estimator and the modified BLUE of \(\lambda\) and \(\alpha\) in MERSS are significantly more efficient than the ones in SRS.
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References
Abu-Dayyeh W, Assrhani A, Ibrahim K (2013) Estimation of the shape and scale parameters of Pareto distribution using ranked set sampling. Stat Pap 54(1):207–225
Al-Khedhairi A, Sarhan A, Tadj L (2007) Estimation of the generalized Rayleigh distribution parameters. J Inequal Appl 8(2):199–210
Al-Odat MT, Al-Saleh MF (2001) A variation of ranked set sampling. J Appl Stat Sci 10:137–146
Al-Saleh MF, Al-Shrafat K (2001) Estimation of average milk yield using ranked set sampling. Environmetrics 12:395–399
Alshunnar FS, Raqab MZ, Kundu D (2010) On the comparison of the Fisher information of the lognormal and generalized Rayleigh distributions. J Appl Stat 37(3):391–404
Azzalini A (1996) Statistical inference. Chapman and Hall, London
Bader MG, Priest AM (1982) Statistical aspects of fibre and bundle strength in hybrid composites. In: Hayashi T, Kawata K, Umekawa S (eds) Progress in science and engineering of composites. Japanese Society for Composites Materials, Tokyo, pp 1129–1136
Casella G, Berger R (1990) Statistical inference. California, Wads Worth and Brooks
Chen WX, Tian Y, Xie MY (2017) Maximum likelihood estimator of the parameter for a continuous one parameter exponential family under the optimal ranked set sampling. J Syst Sci Complexity 30(6):1350–1363
Chen WX, Yang R, Yao DS, Long CX (2021) Pareto parameters estimation using moving extremes ranked set sampling. Stat Pap 62(3):1195–1211
Chen ZH, Bai ZD, Sinha BK (2003) Ranked set sampling: theorem and applications. Springer, New York
Dong XF, Zhang LY (2020) Estimation of system reliability for exponential distributions based on L ranked set sampling. Commun Stat Theory Methods 49(15):3650–3662
Esemen M, Grler S (2018) Parameter estimation of generalized Rayleigh distribution based on ranked set sample. J Stat Comput Simul 88(4):615–628
He XF, Chen WX, Yang R (2021) Modified best linear unbiased estimator of the shape parameter of log-logistic distribution. J Stat Comput Simul 91(2):383–395
Lesitha G, Thomas PY (2013) Estimation of the scale parameter of a log-logistic distribution. Metrika 76(3):427–448
Kundu D, Raqab MZ (2005) Generalized rayleigh distribution: different methods of estimations. Comput Stat Data Anal 49(1):187–200
Lehmann EL (1983) Theory of point estimation. Wiley, New York
Li NY, Li Y, Li YM et al (2015) Empirical bayes for Rayleigh distribution with unbalanced ranked set samples. Pacific J Appl Math 7(4):279–286
Mahdizadeh M (2015) On entropy-based test of exponentiality in ranked set sampling. Commun Stat Simul Comput 44(4):979–995
McIntyre GA (1952) A method of unbiased selective sampling, using ranked sets. Crop Pasture Sci 3(4):385–390
McIntyre GA (2005) A method of unbiased selective sampling, using ranked sets. The Am Stat 59(3):230–232
Pathak A, Chaturvedi A (2014) Estimation of the reliability function for two-parameter exponentiated Rayleigh or Burr type X distribution. Stat Optim Inf Comput 2(4):305–322
Qian WS, Chen WX, He XF (2021) Parameter estimation for the Pareto distribution based on ranked set sampling. Stat Pap 62(1):395–417
Qiu GX, Eftekharian A (2021) Extropy information of maximum and minimum ranked set sampling with unequal samples. Commun Stat-Theory Methods 50(13):2979–2995
Shen BL, Chen WX, Zhou YW, Deng CH (2022) Fisher information for generalized Rayleigh distribution in ranked set sampling design with application to parameter estimation. Appl Math-A J Chin Univ Ser B 37(4):615–630
Surles JG, Padgett WJ (1998) Inference for P(Y\(<\)X) in the Burr type X model. J Appl Stat Sci 7(4):225–238
Surles JG, Padgett WJ (2005) Some properties of a scaled Burr type X distribution. J Stat Plan Inference 128:271–280
Yang R, Chen WX, Dong YF (2023) Log-extended exponential-geometric parameters estimation using simple random sampling and moving extremes ranked set sampling. Commun Stat-Simul Comput 52(1):267–277
Yang R, Chen WX, Yao DS et al (2020) The efficiency of ranked set sampling design for parameter estimation for the log-extended exponential-geometric distribution. Iranian J Sci Technol Trans Sci 44(2):497–507
Zamanzade E (2019) EDF-based tests of exponentiality in pair ranked set sampling. Stat Pap 60(6):2141–2159
Acknowledgements
On behalf of all authors, the corresponding author states that there is no conflict of interest. The authors are thankful to the editor in chief, an associate editor and reviewers for their valuable comments and suggestions to improve the paper.
Funding
This research was supported by National Science Foundation of China (Grant Nos. 12261036 and 11901236), Scientific Research Fund of Hunan Provincial Education Department (Grant No. 21A0328), Provincial Natural Science Foundation of Hunan (Grant No. 2022JJ30469) and Young Core Teacher Foundation of Hunan Province (Grant No. [2020]43).
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Appendix
Appendix
The log-likelihood function for a
Proof of Theorem 1
single observation
$$\begin{aligned}&\displaystyle L_{MERSS}^* = d_0 + 2\sum \limits _{i = 1}^m {\ln i} + 2m\ln \alpha + 4m\ln \lambda \\&\quad + 2\sum \limits _{i = 1}^m {\ln X_{ii} } - \sum \limits _{i = 1}^m {\lambda ^2 X_{ii}^2 } + \sum \limits _{i = 1}^m {\left( {\alpha i - 1} \right) } \ln \left( {1 - e^{ - \left( {\lambda X_{ii} } \right) ^2 } } \right) \\&\quad + \sum \limits _{i = 1}^m {\ln Y_{1i} } - \sum \limits _{i = 1}^m {\lambda ^2 Y_{1i}^2 } + \left( {\alpha - 1} \right) \sum \limits _{i = 1}^m {\ln \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) } \\&\quad + \sum \limits _{i = 1}^m {\left( {i - 1} \right) \ln \left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) }, \end{aligned}$$
where \(d_0\) is a value which is free of \(\alpha\) and \(\lambda\). In order to compute the Fisher information matrix in MERSS, the first derivative and the second derivative of \(L_{MERSS}^*\) are respectively computed as
$$\begin{aligned}&\displaystyle \frac{{\partial L_{MERSS}^* }}{{\partial \lambda }} = \frac{{4m}}{\lambda } - 2\lambda \sum \limits _{i = 1}^m {X_{ii}^2 } + \sum \limits _{i = 1}^m {\left( {\alpha i - 1} \right) \frac{{2\lambda X_{ii}^2 e^{ - \left( {\lambda X_{ii} } \right) ^2 } }}{{1 - e^{ - \left( {\lambda X_{ii} } \right) ^2 } }}} \\&\quad - 2\lambda \sum \limits _{i = 1}^m {Y_{1i}^2 } + \left( {\alpha - 1} \right) \sum \limits _{i = 1}^m {\frac{{2\lambda Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } }}{{1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } }}} \\&\quad - \sum \limits _{i = 1}^m {\left( {i - 1} \right) \frac{{2\alpha \lambda Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} }}{{1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha }}},\\&\quad \displaystyle \frac{{\partial L_{MERSS}^* }}{{\partial \alpha }} = \frac{{2m}}{\alpha } \\&\quad + \sum \limits _{i = 1}^m {i\ln \left( {1 - e^{ - \left( {\lambda X_{ii} } \right) ^2 } } \right) } + \sum \limits _{i = 1}^m {\ln \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) } \\&\quad - \sum \limits _{i = 1}^m {\left( {i - 1} \right) \frac{{\left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha \ln \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) }}{{1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha }}},\\&\quad \displaystyle \frac{{\partial ^2 L_{MERSS}^* }}{{\partial \lambda ^2 }} = - \frac{{4m}}{{\lambda ^2 }} - 2\sum \limits _{i = 1}^m {X_{ii}^2 } + \sum \limits _{i = 1}^m {\left( {\alpha i - 1} \right) \frac{{2X_{ii}^2 e^{ - \left( {\lambda X_{ii} } \right) ^2 } - 2X_{ii}^2 e^{ - 2\left( {\lambda X_{ii} } \right) ^2 } - 4\lambda ^2 X_{ii}^4 e^{ - \left( {\lambda X_{ii} } \right) ^2 } }}{{\left( {1 - e^{ - \left( {\lambda X_{ii} } \right) ^2 } } \right) ^2 }}} - 2\sum \limits _{i = 1}^m {Y_{1i}^2 } \\&\quad + \left( {\alpha - 1} \right) \sum \limits _{i = 1}^m {\frac{{2Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } - 2Y_{1i}^2 e^{ - 2\left( {\lambda Y_{1i} } \right) ^2 } - 4\lambda ^2 Y_{1i}^4 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } }}{{\left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^2 }}} \\&\quad - \sum \limits _{i = 1}^m {\left( {i - 1} \right) } \left( {\frac{{4\alpha ^2 \lambda ^2 Y_{1i}^4 e^{ - 2\left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 2} }}{{\left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^2 }}} \right. \\&\quad + \left. {\frac{{2\alpha Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} - 4\alpha \lambda ^2 Y_{1i}^4 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} - 4\alpha \lambda ^2 Y_{1i}^4 e^{ - 2\left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 2} }}{{1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha }}} \right) ,\\&\quad \displaystyle \frac{{\partial ^2 L_{MERSS}^* }}{{\partial \lambda \partial \alpha }} = \sum \limits _{i = 1}^m {i\frac{{2\lambda X_{ii}^2 e^{ - \left( {\lambda X_{ii} } \right) ^2 } }}{{1 - e^{ - \left( {\lambda X_{ii} } \right) ^2 } }}} + \sum \limits _{i = 1}^m {\frac{{2\lambda Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } }}{{1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } }}}\\&\quad - \sum \limits _{i = 1}^m {\left( {i - 1} \right) \left( {\frac{{2\lambda Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} }}{{1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha }} + \frac{{2\alpha \lambda Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} \ln \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) }}{{\left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^2 }}} \right) } \end{aligned}$$
and
$$\begin{aligned} \displaystyle \frac{{\partial ^2 L_{MERSS}^* }}{{\partial \alpha ^2 }} = - \frac{{2m}}{{\alpha ^2 }} - \sum \limits _{i = 1}^m {\left( {i - 1} \right) \frac{{\left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha \ln ^2 \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) }}{{\left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^2 }}}. \end{aligned}$$
Then under the assumed regularity conditions of Theorem 1
$$\begin{aligned} \displaystyle&I_{11,~MERSS}= - E\left( {\frac{{\partial ^2 L_{MERSS}^* }}{{\partial \lambda ^2 }}} \right) \\&\quad = \frac{{4m}}{{\lambda ^2 }} \\&\quad + 2E\left( {\sum \limits _{i = 1}^m {X_{ii}^2 } } \right) - E\left( {\sum \limits _{i = 1}^m {\left( {\alpha i - 1} \right) \frac{{2X_{ii}^2 e^{ - \left( {\lambda X_{ii} } \right) ^2 } - 2X_{ii}^2 e^{ - 2\left( {\lambda X_{ii} } \right) ^2 } - 4\lambda ^2 X_{ii}^4 e^{ - \left( {\lambda X_{ii} } \right) ^2 } }}{{\left( {1 - e^{ - \left( {\lambda X_{ii} } \right) ^2 } } \right) ^2 }}} } \right) \\&\quad + 2E\left( {\sum \limits _{i = 1}^m {Y_{1i}^2 } } \right) \\&\quad - \left( {\alpha - 1} \right) E\left( {\sum \limits _{i = 1}^m {\frac{{2Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } - 2Y_{1i}^2 e^{ - 2\left( {\lambda Y_{1i} } \right) ^2 } - 4\lambda ^2 Y_{1i}^4 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } }}{{\left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^2 }}} } \right) \\&\quad + E\left[ {\sum \limits _{i = 1}^m {\left( {i - 1} \right) } \left( {\frac{{4\alpha ^2 \lambda ^2 Y_{1i}^4 e^{ - 2\left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 2} }}{{\left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^2 }}} \right. } \right. \\&\quad + \left. {\left. {\frac{{2\alpha Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} - 4\alpha \lambda ^2 Y_{1i}^4 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} - 4\alpha \lambda ^2 Y_{1i}^4 e^{ - 2\left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 2} }}{{1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha }}} \right) } \right] \\&\quad = \frac{{4m}}{{\lambda ^2 }} + 4\alpha \lambda ^2 \sum \limits _{i = 1}^m i \int _0^\infty {t^3 e^{ - \left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha i - 1} } dt \\&\quad - 2\alpha \lambda ^2 \sum \limits _{i = 1}^m {\left( {\alpha i^2 - i} \right) } \int _0^\infty {\left( {2t^3 e^{ - 2\left( {\lambda t} \right) ^2 } - 2t^3 e^{ - 3\left( {\lambda t} \right) ^2 } - 4\lambda ^2 t^5 e^{ - 2\left( {\lambda t} \right) ^2 } } \right) } \\&\quad \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha i - 3} dt\\&\quad + 4\alpha \lambda ^2 \sum \limits _{i = 1}^m i \int _0^\infty {t^3 e^{ - \left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha - 1} } \left( {1 - \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^\alpha } \right) ^{i - 1} dt \\&\quad - 2\left( {\alpha ^2 - \alpha } \right) \lambda ^2 \sum \limits _{i = 1}^m {i\int _0^\infty {\left( {2t^3 e^{ - 2\left( {\lambda t} \right) ^2 } - 2t^3 e^{ - 3\left( {\lambda t} \right) ^2 } - 4\lambda ^2 t^5 e^{ - 2\left( {\lambda t} \right) ^2 } } \right) } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha - 3} \left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^{i - 1} dt}\\&\quad + \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) } \int _0^\infty {8\alpha ^3 \lambda ^4 t^5 e^{ - 3\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 3} }\\&\quad \left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^{i - 3} + \left( {4\alpha ^2 \lambda ^2 t^3 e^{ - 2\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 2} } \right. \\&\quad - \left. {8\alpha ^2 \lambda ^4 t^5 e^{ - 2\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 2} - 8\alpha ^2 \lambda ^4 t^5 e^{ - 3\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 3} } \right) \left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^{i - 2} dt \\&\quad = \frac{{4m}}{{\lambda ^2 }} \\&\quad + 4\alpha \lambda ^2 \sum \limits _{i = 1}^m i \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \int _0^\infty {t^3 e^{ - \left( {j + 1} \right) \left( {\lambda t} \right) ^2 } } dt \\&\quad - 2\alpha
\lambda ^2 \sum \limits _{i = 1}^m {\left( {\alpha i^2 - i} \right) } \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 3} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \int _0^\infty {\left( {2t^3 e^{ - \left( {j + 2} \right) \left( {\lambda t} \right) ^2 } - 2t^3 e^{ - \left( {j + 3} \right) \left( {\lambda t} \right) ^2 } - 4\lambda ^2 t^5 e^{ - \left( {j + 2} \right) \left( {\lambda t} \right) ^2 } } \right) } dt \\&\quad + 4\alpha \lambda ^2 \sum \limits _{i = 1}^m i \sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \int _0^\infty {t^3 e^{ - \left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha j + \alpha - 1} } dt \\&\quad - 2\left( {\alpha ^2 - \alpha } \right) \lambda ^2 \sum \limits _{i = 1}^m {i\sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \int _0^\infty {\left( {2t^3 e^{ - 2\left( {\lambda t} \right) ^2 } - 2t^3 e^{ - 3\left( {\lambda t} \right) ^2 } - 4\lambda ^2 t^5 e^{ - 2\left( {\lambda t} \right) ^2 } } \right) } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha j + \alpha - 3} dt} \\&\quad + \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) } \left[ {\sum \limits _{j = 0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \int _0^\infty {8\alpha ^3 \lambda ^4 t^5 e^{ - 3\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha j + 2\alpha - 3} } dt+ \sum \limits _{j = 0}^{i - 2} {\left( {\begin{array}{*{20}c} {i - 2} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j } \right. \\&\quad \left. {\int _0^\infty {\left( {4\alpha ^2 \lambda ^2 t^3 e^{ - 2\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 2} - 8\alpha ^2 \lambda ^4 t^5 e^{ - 2\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 2} - 8\alpha ^2 \lambda ^4 t^5 e^{ - 3\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 3} } \right) } dt} \right] \\&\quad = \frac{{4m}}{{\lambda ^2 }} + \frac{{2\alpha }}{{\lambda ^2 }}\sum \limits _{i = 1}^m i \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \left( {j + 1} \right) ^{ - 2} \int _0^\infty {te^{ - t} } dt \\&\quad - \frac{{2\alpha }}{{\lambda ^2 }}\sum \limits _{i = 1}^m {\left( {\alpha i^2 - i} \right) } \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 3} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \\&\quad \left[ {\left( {j + 2} \right) ^{ - 2} \int _0^\infty {te^{ - t} dt} - \left( {j + 3} \right) ^{ - 2} \int _0^\infty {te^{ - t} dt} - 2\left( {j + 2} \right) ^{ - 3} \int _0^\infty {t^2 e^{ - t} dt} } \right] \\&\quad + 4\alpha \lambda ^2 \sum \limits _{i = 1}^m i \sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 1} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \int _0^\infty {t^3 e^{ - \left( {k + 1} \right) \left( {\lambda t} \right) ^2 } } dt \\&\quad - 2\left( {\alpha ^2 - \alpha } \right) \lambda ^2 \sum \limits _{i = 1}^m {i\sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \int _0^\infty {\left( {2t^3 e^{ - \left( {k + 2} \right) \left( {\lambda t} \right) ^2 } - 2t^3 e^{ - \left( {k + 3} \right) \left( {\lambda t} \right) ^2 } } \right. } } \\&\quad -\left. {4\lambda ^2 t^5 e^{ - \left( {k + 2} \right) \left( {\lambda t} \right) ^2 } } \right) dt \\&\quad + \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) } \left[ {\sum \limits _{j = 0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \int _0^\infty {8\alpha ^3 \lambda ^4 t^5 e^{ - \left( {k + 3} \right) \left( {\lambda t} \right) ^2 } } dt} \right. \\&\quad + \sum \limits _{j = 0}^{i - 2} {\left( {\begin{array}{*{20}c} {i - 2} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \left( {\sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 2} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \int _0^\infty {4\alpha ^2 \lambda ^2 t^3 e^{ - \left( {k + 2} \right) \left( {\lambda t} \right) ^2 } - 8\alpha ^2 \lambda ^4 t^5 e^{ - \left( {k + 2} \right) \left( {\lambda t} \right) ^2 } } dt} \right. \\&\quad - \left. {\left. {\sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \int _0^\infty {8\alpha ^2 \lambda ^4 t^5 e^{ - \left( {k + 3} \right) \left( {\lambda t} \right) ^2 } } dt} \right) } \right] \\&\quad = \frac{{4m}}{{\lambda ^2 }} + \frac{{2\alpha }}{{\lambda ^2 }}\sum \limits _{i = 1}^m i\\&\quad \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \left( {j + 1} \right) ^{ - 2} - \frac{\alpha }{{\lambda ^2 }}\sum \limits _{i = 1}^m {\left( {\alpha i^2 - i} \right) } \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 3} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \left[ {2\left( {j + 2} \right) ^{ - 2} } \right. \\&\quad - \left. {2\left( {j + 3} \right) ^{ - 2} - 8\left( {j + 2} \right) ^{ - 3} } \right] \\&\quad + \frac{{2\alpha }}{{\lambda ^2 }}\sum \limits _{i = 1}^m i \sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 1} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {k + 1} \right) ^{ - 2} \int _0^\infty {te^{ - t} } dt \\&\quad - \frac{2}{{\lambda ^2 }}\left( {\alpha ^2 - \alpha } \right) \\&\quad \sum \limits _{i = 1}^m {i\sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {\left( {k + 2} \right) ^{ - 2} \int _0^\infty {te^{ - t} dt} - \left( {k + 3} \right) ^{ - 2} \int _0^\infty {te^{ - t} dt} } \right. } \\&\quad - \left. {2\left( {k + 2} \right) ^{ - 2} \int _0^\infty {t^2 e^{ - t} dt} } \right) dt \\&\quad + \frac{1}{{\lambda ^2 }}\sum \limits _{i = 1}^m {\left( {i^2 - i} \right) } \left[ {4\alpha ^3 \sum \limits _{j = 0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {k + 3} \right) ^{ - 3} \int _0^\infty {t^2 e^{ - t} } dt} \right. \\&\quad + \alpha ^2 \sum \limits _{j = 0}^{i - 2} {\left( {\begin{array}{*{20}c} {i - 2} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \left( {\sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 2} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {2\left( {k + 2} \right) ^{ - 2} \int _0^\infty {te^{ - t} dt} - 4\left( {k + 2} \right) ^{ - 3} \int _0^\infty {t^2 e^{ - t} dt} } \right) } \right. \\&\quad - 4\left. {\left. {\sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {k + 3} \right) ^{ - 3} \int _0^\infty {t^2 e^{ - t} dt} } \right) } \right] \\&\quad = \frac{{4m}}{{\lambda ^2 }} \\&\quad + \frac{{2\alpha }}{{\lambda ^2 }}\sum \limits _{i = 1}^m i \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \left( {j + 1} \right) ^{ - 2} - \frac{\alpha }{{\lambda ^2 }}\sum \limits _{i = 1}^m {\left( {\alpha i^2 - i} \right) }\\&\quad \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 3} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \left[ {2\left( {j + 2} \right) ^{ - 2} } \right. \\&\quad - \left. {2\left( {j + 3} \right) ^{ - 2} - 8\left( {j + 2} \right) ^{ - 3} } \right] + \frac{{2\alpha }}{{\lambda ^2 }}\sum \limits _{i = 1}^m i \\&\quad \sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 1} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {k + 1} \right) ^{ - 2} \\&\quad - \frac{{\alpha ^2 - \alpha }}{{\lambda ^2 }}\sum \limits _{i = 1}^m {i\sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {2\left( {k + 2} \right) ^{ - 2} - 2\left( {k + 3} \right) ^{ - 2} } \right. } - \left. {8\left( {k + 2} \right) ^{ - 2} } \right) \\&\quad + \frac{1}{{\lambda ^2 }}\sum \limits _{i = 1}^m {\left( {i^2 - i} \right) } \left[ {8\alpha ^3 \sum \limits _{j = 0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {k + 3} \right) ^{ - 3} } \right. \\&\quad + \alpha ^2 \sum \limits _{j = 0}^{i - 2} {\left( {\begin{array}{*{20}c} {i - 2} \\ j \\ \end{array}} \right) } \left( { - 1} \right) ^j \\&\quad \left( {\sum \limits _{k = 0}^{\infty } {\left(
{\begin{array}{*{20}c} {\alpha j + 2\alpha - 2} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {2\left( {k + 2} \right) ^{ - 2} - 8\left( {k + 2} \right) ^{ - 3} } \right) } \right. - 8\left. {\left. {\sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 3} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \left( {k + 3} \right) ^{ - 3} } \right) } \right] , \\&\quad \displaystyle I_{12,~MERSS}=-E\left( {\frac{{\partial ^2 L_{MERSS}^* }}{{\partial \lambda \partial \alpha }}} \right) \\&\quad = - E\left[ {\sum \limits _{i = 1}^m {i\frac{{2\lambda X_{ii}^2 e^{ - \left( {\lambda X_{ii} } \right) ^2 } }}{{1 - e^{ - \left( {\lambda X_{ii} } \right) ^2 } }}} } \right] \\&\quad - E\left[ {\sum \limits _{i = 1}^m {\frac{{2\lambda Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } }}{{1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } }}} } \right] \\&\quad + E\left[ {\sum \limits _{i = 1}^m {\left( {i - 1} \right) \left( {\frac{{2\lambda Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} }}{{1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha }} + \frac{{2\alpha \lambda Y_{1i}^2 e^{ - \left( {\lambda Y_{1i} } \right) ^2 } \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^{\alpha - 1} \ln \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) }}{{\left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^2 }}} \right) } } \right] \\&\quad =-4\alpha \lambda ^3 \sum \limits _{i = 1}^m {i^2 \int _0^\infty {t^3 e^{ - 2\left( {\lambda t} \right) ^2 } } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha i - 2} dt} \\&\quad - 4\alpha \lambda ^3 \sum \limits _{i = 1}^m {i\int _0^\infty {t^3 e^{ - 2\left( {\lambda t} \right) ^2 } } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha - 2} \left( {1 - \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^\alpha } \right) ^{i - 1} dt} \\&\quad + \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \int _0^\infty \left[ {4\alpha \lambda ^3 t^3 e^{ - 2\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 2} \left( {1 - \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^\alpha } \right) ^{i - 2} } \right. } \\&\quad + \left. {4\alpha ^2 \lambda ^3 t^3 e^{ - 2\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 2} \ln \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) \left( {1 - \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^\alpha } \right) ^{i - 3} } \right] \\&\quad =-4\alpha \lambda ^3 \sum \limits _{i = 1}^m {i^2 } \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 2} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \int _0^\infty {t^3 e^{ - \left( {j + 2} \right) \left( {\lambda t} \right) ^2 } } dt} \\&\quad - 4\alpha \lambda ^3 \sum \limits _{i = 1}^m {i\sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } } \\&\quad \int _0^\infty {t^3 e^{ - 2\left( {\lambda t} \right) ^2 } } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha j + \alpha - 2} dt \\&\quad + \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \left[ {\sum \limits _{j = 0}^{i - 2} {\left( {\begin{array}{*{20}c} {i - 2} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } \int _0^\infty {4\alpha \lambda ^3 t^3 e^{ - 2\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha j + 2\alpha - 2} dt} } \right. } \\&\quad + \left. {\sum \limits _{j = 0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \int _0^\infty {4\alpha ^2 \lambda ^3 t^3 e^{ - 2\left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha j + 2\alpha - 2} \ln \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) dt} } } \right] \\&\quad =-\frac{{2\alpha }}{\lambda }\sum \limits _{i = 1}^m {i^2 } \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 2} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \left( {j + 2} \right) ^{ - 2} \int _0^\infty {te^{ - t} } dt} \\&\quad - 4\alpha \lambda ^3 \sum \limits _{i = 1}^m {i\sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } } \\&\quad \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 2} \\ k \\ \end{array}} \right) } \left( { - 1} \right) ^k \int _0^\infty {t^3 e^{ - \left( {k + 2} \right) \left( {\lambda t} \right) ^2 } } dt \\&\quad + \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \left[ {\sum \limits _{j = 0}^{i - 2} {\left( {\begin{array}{*{20}c} {i - 2} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 2} \\ k \\ \end{array}} \right) \left( { - 1} \right) ^k } } \right. } \\&\quad \int _0^\infty {4\alpha \lambda ^3 t^3 e^{ - \left( {k + 2} \right) \left( {\lambda t} \right) ^2 } dt}\\&\quad - \left. {\frac{{2\alpha ^2 }}{\lambda }\sum \limits _{j = 0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \int _0^1 {\ln t \cdot t\left( {1 - t} \right) ^{\alpha j + 2\alpha - 2} \ln \left( {1 - t} \right) dt} } } \right] \\&\quad =-\frac{{2\alpha }}{\lambda }\sum \limits _{i = 1}^m {i^2 } \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 2} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \left( {j + 2} \right) ^{ - 2} }-\frac{{2\alpha }}{\lambda }\sum \limits _{i = 1}^m {i\sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \sum \limits _{k=0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 2} \\ k \\ \end{array}} \right) \left( { - 1} \right) ^k } } } \\ \left( {k + 2} \right) ^{ - 2} \int _0^\infty {te^{ - t} } dt\\&\quad + \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \left[ {\frac{{2\alpha }}{\lambda }\sum \limits _{j = 0}^{i - 2} {\left( {\begin{array}{*{20}c} {i - 2} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 2} \\ k \\ \end{array}} \right) \left( { - 1} \right) ^k } \left( {k + 2} \right) ^{ - 2} \int _0^\infty {te^{ - t} dt} } \right. } \\&\quad - \left. {\frac{{2\alpha ^2 }}{\lambda }\sum \limits _{j = 0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \left( {\int _0^1 {\ln t \cdot t^{\alpha j + 2\alpha - 2} \ln \left( {1 - t} \right) dt - \int _0^1 {\ln t \cdot t^{\alpha j + 2\alpha - 1} \ln \left( {1 - t} \right) dt} } } \right) } } \right] \\&\quad =-\frac{{2\alpha }}{\lambda }\sum \limits _{i = 1}^m {i^2 } \sum \limits _{j = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha i - 2} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \left( {j + 2} \right) ^{ - 2} }-\frac{{2\alpha }}{\lambda }\sum \limits _{i = 1}^m {i\sum \limits _{j = 0}^{i - 1} {\left( {\begin{array}{*{20}c} {i - 1} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + \alpha - 2} \\ k \\ \end{array}} \right) \left( { - 1} \right) ^k } } \left( {k + 2} \right) ^{ - 2} } \\&\quad + \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \left[ {\frac{{2\alpha }}{\lambda }\sum \limits _{j = 0}^{i - 2} {\left( {\begin{array}{*{20}c} {i - 2} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } \sum \limits _{k = 0}^{\infty } {\left( {\begin{array}{*{20}c} {\alpha j + 2\alpha - 2} \\ k \\ \end{array}} \right) \left( { - 1} \right) ^k } \left( {k + 2} \right) ^{ - 2} } \right. } - \left. {\frac{{2\alpha ^2 }}{\lambda }\sum \limits _{j = 0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j \left( {\sum \limits _{n = 1}^\infty {\frac{1}{{n\left( {\alpha j + 2\alpha - 1 + n} \right) ^2 }}-\sum \limits _{n = 1}^\infty {\frac{1}{{n\left( {\alpha j + 2\alpha + n} \right) ^2 }}} } } \right) } } \right] \end{aligned}$$
and
$$\begin{aligned}&\displaystyle I_{22,~MERSS}= - \sum \limits _{i = 1}^m {E\left( {\frac{{\partial ^2 L_{MERSS}^* }}{{\partial \alpha ^2 }}} \right) }\\&\quad = \frac{{2m}}{{\alpha ^2 }} + E\left[ {\sum \limits _{i = 1}^m {\left( {i - 1} \right) \frac{{\left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha \ln ^2 \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) }}{{\left( {1 - \left( {1 - e^{ - \left( {\lambda Y_{1i} } \right) ^2 } } \right) ^\alpha } \right) ^2 }}} } \right] \\&\quad = \frac{{2m}}{{\alpha ^2 }} + \alpha \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \int _0^\infty {2\lambda ^2 } te^{ - \left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{2\alpha - 1} \ln ^2 \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) \left( {1 - \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^\alpha } \right) ^{i - 3} } dt \\&\quad = \frac{{2m}}{{\alpha ^2 }} + \alpha \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \sum \limits _{j=0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } \int _0^\infty {2\lambda ^2 } te^{ - \left( {\lambda t} \right) ^2 } \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) ^{\alpha j + 2\alpha - 1} \ln ^2 \left( {1 - e^{ - \left( {\lambda t} \right) ^2 } } \right) } dt \\&\quad = \frac{{2m}}{{\alpha ^2 }} + \alpha \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \sum \limits _{j=0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } \int _0^1 {t^{\alpha j + 2\alpha - 1} \ln ^2 tdt} } \\&\quad = \frac{{2m}}{{\alpha ^2 }} + \alpha \sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \sum \limits _{j=0}^{i - 3} {\left(
{\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } \frac{2}{{\left( {\alpha j + 2\alpha } \right) ^3 }}} \\&\quad = \frac{{2m}}{{\alpha ^2 }} + \frac{2}{{\alpha ^2 }}\sum \limits _{i = 1}^m {\left( {i^2 - i} \right) \sum \limits _{j=0}^{i - 3} {\left( {\begin{array}{*{20}c} {i - 3} \\ j \\ \end{array}} \right) \left( { - 1} \right) ^j } \left( {j + 2} \right) ^{ - 3} }. \end{aligned}$$
The Theorem is proved by combining \(I_{11,~MERSS}\), \(I_{12,~MERSS}\) with \(I_{22,~MERSS}\).
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Shen, B., Chen, W., Zhou, Y. et al. Sampling Information for Generalized Rayleigh Distribution with Application to Parameter Estimation. Iran J Sci 47, 515–529 (2023). https://doi.org/10.1007/s40995-023-01428-4
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DOI: https://doi.org/10.1007/s40995-023-01428-4