Abstract
We consider improved estimation strategies for a two-parameter inverse Gaussian distribution and use a shrinkage technique for the estimation of the mean parameter. In this context, two new shrinkage estimators are suggested and demonstrated to dominate the classical estimator under the quadratic risk with realistic conditions. Furthermore, based on our shrinkage strategy, a new estimator is proposed for the common mean of several inverse Gaussian distributions, which uniformly dominates the Graybill–Deal type unbiased estimator. The performance of the suggested estimators is examined by using simulated data and our shrinkage strategies are shown to work well. The estimation methods and results are illustrated by two empirical examples.
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References
Ahmad KE, Jaheen ZF (1995) Approximate Bayes estimators applied to the inverse Gaussian lifetime model. Comput Math Appl 29(12):39–47
Ahmad M, Chaubey YP, Sinha BK (1991) Estimation of a common mean of several univariate inverse Gaussian populations. Ann Inst Stat Math 43(2):357–367
Ahmed SE (1998) Large-sample estimation strategies for eigenvalues of a Wishart matrix. Metrika 47:35–45
Ahmed SE, Liu S (2009) Asymptotic theory of simultaneous estimation of Poisson means. Linear Algebra Its Appl 430:2734–2748
Ahmed SE, Nicol C (2012) An application of shrinkage estimation to the nonlinear regression model. Comput Stat Data Anal 56:3309–3321
Ahmed SE, Hussein AA, Sen PK (2006) Risk comparison of some shrinkage M-estimators in linear models. J Nonparametric Stat 18:401–415
Ahmed SE, Doksum KA, Hossain S, You J (2007) Shrinkage, pretest and absolute penalty estimators in partially linear models. Aust NZ J Stat 49:435–454
Balakrishnan N, Leiva V, Sanhueza A, Cabrera E (2009) Mixture inverse Gaussian distribution and its transformations, moments and applications. Statistics 43:91–104
Brown LD, Cohen A (1974) Point and cofidence estimation of a common mean and recovery of inter-block information. Ann Stat 2:963–976
Chhikara RS, Folks JL (1977) The inverse Gaussian distribution as a lifetime model. Technometrics 19: 461–468
Chhikara RS, Folks JL (1989) The inverse Gaussian distribution: theory, methodology, and applications. Marcel Dekker, New York
Chiou P, Miao WW (2005) Shrinakge estimation for the difference between exponential guarantee time parameters. Comput Stat Data Anal 48:489–507
Fisher TJ, Sun XQ (2011) Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix. Comput Stat Data Anal 55:1909–1918
Gao JX, Hitchcock DB (2010) James-stein shrinkage to improve k-means clusters analysis. Comput Stat Data Anal 54:2113–2127
Graybill FA, Deal RB (1959) Combining unbiased estimators. Biometrics 15:543–550
Gross J (2003) Linear regression. Springer, Berlin
Gruber MHJ (2010) Regression estimators: a comparative study, 2nd edn. Johns Hopkins University Press, Baltimore
Gupta RC, Akman HO (1995) Bayes estimation in a mixture inverse Gaussian model. Ann Inst Stat Math 47(3):493–503
Hartung J, Knapp G, Sinha BK (2008) Statistical meta-analysis with applications. Wiley, New York
Hsieh HK, Korwar RM, Rukhin AL (1990) Inadmissibility of the maximum likelihood estimator of the inverse Gaussian mean. Stat Probab Lett 9:83–90
James W, Stein C (1961) Estimation with quadratic loss. In: Proceedings of the fourth Berleley symposium on mathematical statistics and Probability, Vol I. Univ. California Press, Berkeley, pp 361–379
Jorgensen B (1982) Statistical properties of the generalized inverse Gaussian distribution. Springer, New York
Khatri CG, Shah KR (1974) Estimation of the location parameters from two linear models under normality. Commun Stat Theory Methods 3:647–663
Krishnamoorthy K, Tian LL (2008) Inferences on the difference and ratio of the means of two inverse Gaussian distributions. J Stat Plan Inference 138:2082–2089
Kumar S, Tripathi YM, Misra N (2005) James-Stein type estimators for ordered normal means. J Stat Comput Simul 75(7):501–511
Kuriki S, Takemura A (2000) Shrinkage estimation towards a closed convex set with a smooth boundary. J Multivar Anal 75(1):79–111
Lin SH, Wu IM (2011) On the common mean of several inverse Gaussian distributions based on a higher order likelihood method. Appl Math Comput 217:5480–5490
Ma TF, Jia LJ, Su YS (2012) A new estimator of covariance matrix. J Stat Plan Inference 142:529–536
Ma TF, Liu S (2013) Estimation of order-restricted means of two normal populations under the LINEX loss function. Metrika 76:409–425
Ma TF, Ye RD, Jia LJ (2011) Finite-sample properties of the Graybill–Deal estimator. J Stat Plan Inference 141:3675–3680
MacGibbon B, Shorrock G (1997) Shrinkage estimators for the dispersion parameter of the inverse Gaussian distribution. Stat Probab Lett 32:207–214
Maruyama Y, Straederman WE (2005) Necessary conditions for dominating the James–Stein estimator. Ann Inst Stat Math 57:157–165
Masuda H (2009) Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling. Ann Inst Stat Math 61:181–195
Nair KA (1986) Distribution of an estimator of the common mean of two normal populations. Ann Stat 8:212–216
Norwood TE, Hinkelmann K (1977) Estimating the common mean of several normal populations. Ann Stat 5:1047–1050
Prakash G, Singh DC (2006) Shrinkage testimators for the inverse dispersion of the inverse Gaussian distribution under the Linex loss function. Austrian J Stat 35:463–470
Raheem SME, Ahmed SE, Doksum KA (2012) Absolute penalty and shrinkage estimation in partially linear models. Comput Stat Data Anal 56:874–891
Sanhueza A, Leiva V, Balakrishnan N (2008) A new class of inverse Gaussian type distributions. Metrika 68:31–49
Schrodinger E (1915) Zur theorie der Fall-und Steigversuche an Teilchen mit Brownscher Bewegung. Physikalische Zeitschrift 16:289–295
Shapiro CM, Beckmann E, Christiansen N, Bitran JD, Kozloff M, Billings AA, Telfer MC (1987) Immunologic status of patients in remission from Hodgkin’s disease and disseminated malignancies. Amer J Medical Sci 293:366–370
Tutz G, Leitenstorfer F (2006) Response shrinkage estimators in binary regression. Comput Stat Data Anal 50:2878–2901
Tweedie MCK (1957) Statistical properties of inverse Gaussian distributions. Ann Math Stat 28:362–377
Ye RD, Ma TF, Wang SG (2010) Inferences on the common mean of several inverse Gaussian populations. Comput Stat Data Anal 54:906–915
Acknowledgments
The authors would like to thank two anonymous referees and the Editor for many helpful suggestions that have significantly improved the presentation of the manuscript. The research of Tiefeng Ma was supported by Zhejiang Provincial Natural Science Foundation (No. Y6100053) of China. The research of S. E. Ahmed was supported by the Natural Sciences and the Engineering Research Council of Canada.
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Appendices
Appendix A: Proofs
1.1 A.1 Proof for Theorem 1
Denote \(\Delta = R(\mu ,\tilde{\mu }_{\lambda }) - R(\mu ,\bar{X})\). It is easy to see that
Next, we rewrite \(\Delta \) and provide a simple condition for \(\Delta <0\). Using \(\bar{X}\sim IG(\mu ,n\lambda )\) and Lemma 2, we get
where
Therefore, it follows from \(\text { exp }\left\{ \frac{n\lambda -\sqrt{n\lambda (n\lambda +4a)}}{\mu }\right\} <\text { exp }\left\{ \frac{n\lambda -\sqrt{n\lambda (n\lambda +2a)}}{\mu }\right\} \) that a sufficient condition for \(\Delta <0\) is
This is a relaxed condition, and it obviously holds when \(2a \ge c > 0\).
This completes the proof.
1.2 A.2 Proof for Theorem 2
Similar to the proof of Theorem 1, we get
where
Obviously, \(\Delta <0\) when \(L_{1}<0\) and \(L_{2}<0\). Note that
Then \(\Delta <0\) is true if the following two inequalities hold
It is easy to see that (8.5b) is a sufficient condition for (8.5a). Then we get that \(\Delta <0\) if (8.5b) is true. Next, we discuss (8.5b).
Note that \(\sqrt{n\lambda +2a}-\sqrt{n\lambda }=\frac{2a}{\sqrt{n\lambda +2a}+\sqrt{n\lambda }}>\frac{a}{\sqrt{n\lambda +4a}}\). After some manipulation, a sufficient condition for (8.5b) is found to be
Obviously, (8.6) is equivalent to
Using \(c\le 2a\), from (8.5) to (8.6), we get a sufficient condition for \(\Delta <0\)
It follows from Lemma 1 that
where the equality holds when \(n\lambda S\) is distributed as \(\chi _{n-1}^{2}\).
From (8.8) and (8.9), it is therefore enough to see that \(\Delta <0\) holds for \(0<c\le \frac{2an}{n+1}\).
This completes the proof.
1.3 A.3 Proof for Theorem 3
Note that
Then it is only required to prove \(I_{2}-2I_{1}\le 0\) if we want to get the dominance result of \(\tilde{\mu }_\text {S}\).
As \(\bar{X}_{i}'s\) are mutually independent and \(E(\bar{X}_{i})=\mu \), using Lemma 2 we get
By Cauchy inequality and Lemma 2, we have
So we get \(I_{2}-2I_{1}\le \frac{1}{\mu ^{3}}E\left[ \left( \frac{\mu }{\sum _{j=1}^{k}\frac{n_{j}-1}{S_{j}}}\right) ^{2}\right. \)
Note that \(\text { exp }\left\{ \frac{n_{i}\lambda _{i}-\sqrt{n_{i}\lambda _{i}(n_{i}\lambda _{i}+4a_{i})}}{\mu }\right\} <\text { exp }\left\{ \frac{n_{i}\lambda _{i}-\sqrt{n_{i}\lambda _{i}(n_{i}\lambda _{i}+2a_{i})}}{\mu }\right\} \). Then the following two inequalities is a sufficient condition for \(I_{2}-2I_{1}\le 0\)
\( i=1,\ldots ,k.\)
Note that (8.14) is a sufficient condition for (8.15) and the fact that \(\sqrt{n_{i}\lambda _{i}+2a_{i}}-\sqrt{n_{i}\lambda _{i}}>\frac{a_{i}}{\sqrt{n_{i}\lambda _{i}+4a_{i}}}\). So, a sufficient condition for \(I_{2}-2I_{1}\le 0\) is given for \(i=1,\ldots ,k\) by
After some manipulation, for each \(i=1,\ldots ,k\), (8.16) is equivalent to
As \({\small \left( \frac{\mu }{S_{i}\sum _{j=1}^{k}\frac{n_{j}-1}{S_{j}}}\right) ^{2}\frac{c_{i}(n_{i}-1)^{2}}{(n_{i}+4a_{i}S_{i})^{2}}}\) is a continuous nonnegative decreasing function of \(S_{i}\), using Lemma 1 we get for \(kc_{i}\le 2a_{i}, i=1,\ldots ,k,\)
Then, from (8.18) we get a sufficient condition for (8.17)
This completes the proof.
Appendix B: Derivation of (3.5)
The derivation of (3.5) is as follows:
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Ma, T., Liu, S. & Ahmed, S.E. Shrinkage estimation for the mean of the inverse Gaussian population. Metrika 77, 733–752 (2014). https://doi.org/10.1007/s00184-013-0462-8
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DOI: https://doi.org/10.1007/s00184-013-0462-8