Abstract
In a recent paper [JUKIĆ, D.: A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set, J. Comput. Appl. Math. 375 (2020)], a new existence level was introduced and then was used to obtain a necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set. In this paper, we determined that existence level for the residual sum of squares of the power-law regression with an unknown location parameter, and so we obtained a necessary and sufficient condition which guarantee the existence of the least squares estimate.
This work was supported by the Croatian Science Foundation through research grant IP-2016-06-6545.
(Communicated by Tomasz Natkaniec)
References
[1] Bates, D. M.—Watts, D. G.: Nonlinear Regression Analysis and Its Applications, Wiley, New York, 1988.10.1002/9780470316757Search in Google Scholar
[2] Demidenko, E.: Optimization and Regression, Nauka, Moscow, 1989 (in Russian).Search in Google Scholar
[3] Demidenko, E.: Criteria for unconstrained global optimization, J. Optim. Theory Appl. 136 (2008), 375–395.10.1007/s10957-007-9298-6Search in Google Scholar
[4] Demidenko, E.: Criteria for global minimum of sum of squares in nonlinear regression, Comput. Statist. Data Anal. 51 (2006), 1739–1753.10.1016/j.csda.2006.06.015Search in Google Scholar
[5] Demidenko, E.: On the existence of the least squares estimate in nonlinear growth curve models of exponential type, Comm. Statist. Theory Methods 25 (1996), 159–182.10.1080/03610929608831686Search in Google Scholar
[6] Jukić, D.: A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set, J. Comput. Appl. Math. 375 (2020), Art. ID 112791., https://doi.org/10.1016/j.cam.2020.112791.Search in Google Scholar
[7] Jukić, D.: A simple proof of the existence of the best estimator in a quasilinear regression model, J. Optim. Theory Appl. 162 (2014), 293–302. https://doi.org/10.1007/s10957-013-0434-1.Search in Google Scholar
[8] Jukić, D.: On the existence of the best discrete approximation in lp norm by reciprocals of real polynomials, J. Approx. Theory 156 (2009), 212–222. https://doi.org/10.1016/j.jat.2008.05.001.Search in Google Scholar
[9] Jukić, D.—Kralik, G.—Scitovski, R.: Least squares fitting Gompertz curve, J. Comput. Appl. Math. 169 (2004), 359–375. https://doi.org/10.1016/j.cam.2003.12.030.Search in Google Scholar
[10] Jukić, D.: A necessary and sufficient criteria for the existence of the least squares estimate for a 3-parametric exponential function, Appl. Math. Comput. 147 (2004), 1–17., https://doi.org/10.1016/S0096-3003(02)00482-4.Search in Google Scholar
[11] Nakamura, T.: Existence theorems of a maximum likelihood estimate from a generalized censored data sample, Ann. Inst. Statist. Math. 36 (1984), 375–393.10.1007/BF02481977Search in Google Scholar
[12] Nievergelt, Y.: On the existence of best Mitscherlich, Verhulst, and West growth curves for generalized least-squares regression, J. Comput. Appl. Math. 248 (2013), 31–46. https://doi.org/10.1016/j.cam.2013.01.005.Search in Google Scholar
[13] Ratkowsky, D. A.: Handbook of Nonlinear Regression Models, Marcel Dekker, New York, 1989.Search in Google Scholar
[14] Reitan, T.—Petersen-Øverleir, A.: Bayesian power-law regression with a location parameter, with applications for construction of discharge rating curves, Stochast. Environ. Res. Risk Assess. 22(3) (2008), 351–365.10.1007/s00477-007-0119-0Search in Google Scholar
[15] Ross, G. J. S.: Nonlinear Estimation, Springer, New York, 1990.10.1007/978-1-4612-3412-8Search in Google Scholar
[16] Seber, G. A. F.—Wild, C. J.: Nonlinear Regression, John Wiley, New York, 1989.10.1002/0471725315Search in Google Scholar
[17] Todinov, M.: Reliability and Risk Models: Setting Reliability Requirements, Wiley, New Jersey, 2016.10.1002/9781118873199Search in Google Scholar
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