Abstract
In maximizing a non-linear function G(θ), it is well known that the steepest descent method has a slow convergence rate. Here we propose a systematic procedure to obtain a 1–1 transformation on the variables θ, so that in the space of the transformed variables, the steepest descent method produces the solution faster. The final solution in the original space is obtained by taking the inverse transformation. We apply the procedure in maximizing the likelihood functions of some generalized distributions which are widely used in modeling count data. It was shown that for these distributions, the steepest descent method via transformations produced the solutions very fast. It is also observed that the proposed procedure can be used to expedite the convergence rate of the first derivative based algorithms, such as Polak-Ribiere, Fletcher and Reeves conjugate gradient methods as well.
Similar content being viewed by others
References
Borah, M. (1983). A study of certain discrete probability distributions, Doctoral Dissertation, Gauhati University, Gauhati, India.
Burden, R. (1985). Numerical Analysis, Prindle, Weber and Schmidt, Boston.
Fletcher, R. and Reeves, C. M. (1964). Function minimization by conjugate gradients, Comput. J., 7, 149–154.
Huque, F. (1974). Adaptive estimators for the log-zero-Poisson truncated distribution, Doctoral Dissertation, University of Missouri-Columbia, U.S.A.
Johnson, N. L. and Kotz, S. (1969). Discrete Distributions, Haughton Mifflin, Boston.
Katti, S. K. and Rao, A. V. (1970). The log-zero-Poisson distribution, Biometrics, 26, 801–803.
Martin, D. C. and Katti, S. K. (1965). Fitting of certain contagious distributions to some available data by maximum likelihood method, Biometrics, 21, 34–48.
McGuire, J. V., Brinkley, T. A. and Bancroft, T. A. (1957). The distribution of European corn borer larvae Pyraustra Nubilalis (HBN) in field corn, Biometrics, 13, 65–78.
Plunkett, I. G. and Jain, G. C. (1975). Three generalized negative binomial distributions, Biom. Zeit., 17, 286–302.
Polak, E. and Ribiere, G. (1969). Note sur la convergence de methods de directions conjugees, Rev. Franc Informat Recherche Operationelle, 16, 35–43.
Swell, G. (1988a). Plotting contour surfaces of a function of three variables, ACM Trans. on Math. Software, 14, 33–41.
Swell, G. (1988b). Software for plotting contour surfaces of a function of three variables, ACM Trans. on Math. Software. 14, 42–44.
Author information
Authors and Affiliations
About this article
Cite this article
Habibullah, M., Katti, S.K. A modified steepest descent method with applications to maximizing likelihood functions. Ann Inst Stat Math 43, 391–404 (1991). https://doi.org/10.1007/BF00118644
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00118644