On estimating common mean of several inverse Gaussian distributions. (English) Zbl 1493.62097
Summary: Estimation of the common mean of inverse Gaussian distributions with different scale-like parameters is considered. We study finite sample properties, second-order admissibility and Pitman closeness properties of the Graybill-Deal estimator of the common mean. The best asymptotically normal estimator of the common mean is derived when the coefficients of variations are known. When the scale-like parameters are unknown but ordered, an improved estimator of the common mean is proposed. We also derive estimators of the common mean using the modified profile likelihood method. A simulation study has been performed to compare among the estimators.
MSC:
| 62F10 | Point estimation |
| 62F30 | Parametric inference under constraints |
| 62C15 | Admissibility in statistical decision theory |
Keywords:
Graybill-Deal estimator; second-order admissible; Pitman nearness; coefficient of variation; common mean; profile likelihoodReferences:
| [1] | Ahmad, M.; Chaubey, YP; Sinha, BK, Estimation of a common mean of several univariate inverse Gaussian populations, Ann Inst Stat Math, 43, 2, 357-367 (1991) · Zbl 0850.62240 · doi:10.1007/BF00118641 |
| [2] | Barndorff-Nielsen, O., On a formula for the distribution of the maximum likelihood estimator, Biometrika, 70, 2, 343-365 (1983) · Zbl 0532.62006 · doi:10.1093/biomet/70.2.343 |
| [3] | Bayarri, MJ; Berger, JO, The interplay of Bayesian and frequentist analysis, Stat Sci, 19, 1, 58-80 (2004) · Zbl 1062.62001 · doi:10.1214/088342304000000116 |
| [4] | Chang, YT; Shinozaki, N., Estimation of two ordered normal means under modified Pitman nearness criterion, Ann Inst Stat Math, 67, 5, 863-883 (2015) · Zbl 1440.62084 · doi:10.1007/s10463-014-0479-4 |
| [5] | Chen, MH; Shao, QM, Monte Carlo estimation of Bayesian credible and HPD intervals, J Comput Graph Stat, 8, 1, 69-92 (1999) |
| [6] | Chhikara, RS; Folks, JL, The inverse Gaussian distribution (1989), New York: Marcel Dekker, New York · Zbl 0701.62009 |
| [7] | Dabli, PV, Use of known coefficients of variation in the estimation of the common mean of two normal populations, Biom J, 23, 8, 731-736 (1981) · Zbl 0491.62023 · doi:10.1002/bimj.4710230802 |
| [8] | Gacula, MC Jr; Kubala, JJ, Statistical models for shelf life failures, J Food Sci, 40, 2, 404-409 (1975) · doi:10.1111/j.1365-2621.1975.tb02212.x |
| [9] | Ghosh, JK; Sinha, BK, A necessary and sufficient condition for second order admissibility with applications to Berkson’s bioassay problem, Ann Stat, 9, 6, 1334-1338 (1981) · Zbl 0484.62047 · doi:10.1214/aos/1176345650 |
| [10] | Kazemi, MR; Jafari, AA, Inference about the shape parameters of several inverse Gaussian distributions: testing equality and confidence interval for a common value, Metrika, 82, 5, 529-545 (2019) · Zbl 1421.62030 · doi:10.1007/s00184-018-0693-9 |
| [11] | Khan, RA, A note on estimating the mean of a normal distribution with known coefficient of variation, J Am Stat Assoc, 63, 323, 1039-1041 (1968) · Zbl 0162.49704 |
| [12] | Krishnamoorthy, K.; Tian, L., Inferences on the difference and ratio of the means of two inverse Gaussian distributions, J Stat Plan Inference, 138, 7, 2082-2089 (2008) · Zbl 1134.62019 · doi:10.1016/j.jspi.2007.09.005 |
| [13] | Kubokawa, T., Closer estimators of a common mean in the sense of Pitman, Ann Inst Stat Math, 41, 3, 477-484 (1989) · Zbl 0702.62023 · doi:10.1007/BF00050663 |
| [14] | Lin, SH; Wu, IM, On the common mean of several inverse Gaussian distributions based on a higher order likelihood method, Appl Math Comput, 217, 12, 5480-5490 (2011) · Zbl 1207.62044 |
| [15] | Ma, T.; Ye, R.; Jia, L., Finite-sample properties of the Graybill-Deal estimator, J Stat Plan Inference, 141, 11, 3675-3680 (2011) · Zbl 1219.62043 · doi:10.1016/j.jspi.2011.06.004 |
| [16] | Ma, T.; Liu, S.; Ahmed, SE, Shrinkage estimation for the mean of the inverse Gaussian population, Metrika, 77, 6, 733-752 (2014) · Zbl 1304.62046 · doi:10.1007/s00184-013-0462-8 |
| [17] | Nair, KA, An estimator of the common mean of two normal populations, J Stat Plan Inference, 6, 2, 119-122 (1982) · Zbl 0489.62027 · doi:10.1016/0378-3758(82)90079-9 |
| [18] | Nelson, WB, Applied life data analysis (2003), New York: Wiley, New York · Zbl 0579.62089 |
| [19] | Niu, C.; Guo, X.; Xu, W.; Zhu, L., Testing equality of shape parameters in several inverse Gaussian populations, Metrika, 77, 6, 795-809 (2014) · Zbl 1304.62042 · doi:10.1007/s00184-013-0465-5 |
| [20] | Oono, Y.; Shinozaki, N., Estimation of two order restricted normal means with unknown and possibly unequal variances, J Stat Plan Inference, 131, 2, 349-363 (2005) · Zbl 1061.62033 · doi:10.1016/j.jspi.2004.02.007 |
| [21] | Pal, N.; Lim, WK, A note on second-order admissibility of the Graybill-Deal estimator of a common mean of several normal populations, J Stat Plan Inference, 63, 1, 71-78 (1997) · Zbl 1090.62510 · doi:10.1016/S0378-3758(96)00202-9 |
| [22] | Pitman, EJ, The closest estimates of statistical parameters, Math Proc Camb, 33, 2, 212-222 (1937) · JFM 63.0515.03 · doi:10.1017/S0305004100019563 |
| [23] | Raqab, MZ; Madi, MT, Inference for the generalized Rayleigh distribution based on progressively censored data, J Stat Plan Inference, 141, 10, 3313-3322 (2011) · Zbl 1216.62151 · doi:10.1016/j.jspi.2011.04.016 |
| [24] | Robertson, T.; Wright, FT; Dykstra, RL, Order restricted statistical inference (1988), New York: Wiley, New York · Zbl 0645.62028 |
| [25] | Severini, TA, Likelihood functions for inference in the presence of a nuisance parameter, Biometrika, 85, 3, 507-522 (1998) · Zbl 0926.62014 · doi:10.1093/biomet/85.3.507 |
| [26] | Tian, L., Testing equality of inverse Gaussian means under heterogeneity, based on generalized test variable, Comput Stat Data Anal, 51, 2, 1156-1162 (2006) · Zbl 1157.62336 · doi:10.1016/j.csda.2005.11.012 |
| [27] | Tian, L.; Wu, J., Inferences on the mean response in a log-regression model: the generalized variable approach, Stat Med, 26, 28, 5180-5188 (2007) · doi:10.1002/sim.2911 |
| [28] | Ventura, L.; Cabras, S.; Racugno, W., Prior distributions from pseudo-likelihoods in the presence of nuisance parameters, J Am Stat Assoc, 104, 486, 768-774 (2009) · Zbl 1388.62060 · doi:10.1198/jasa.2009.0133 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
