On the uniqueness of the M. L. estimates in curved exponential families. (English) Zbl 0657.62031

Pázman, Andrej

Kybernetika 22, 124-132 (1986).

Curved families [cf. B. Efron, Ann. Stat. 6, 362-376 (1978; Zbl 0436.62027)] imbedded in exponential families having full rank differentiable sufficient statistics are considered. It is proved that if the rank is less or equal to the dimension of the sample space, then the maximum likelihood estimate is unique. Examples: the Gaussian nonlinear regression, the Gaussian family with unknown mean and variance, the beta- distribution. Generalized curved exponential families are considered as well.

MSC:

62F10	Point estimation
62J02	General nonlinear regression

Keywords:

Curved families; exponential families; full rank differentiable sufficient statistics; maximum likelihood estimate; Gaussian nonlinear regression; beta-distribution; Generalized curved exponential families

Citations:

Zbl 0436.62027

References:

[1]	S. Amari: Differential geometry of curved exponential families - curvatures and information loss. Ann. Statist. 10 (1982), 357-387. · Zbl 0507.62026
[2]	O. Barndorff-Nielsen: Information and Exponential Families in Statistical Theory. Wiley, Chichester 1979. · Zbl 0387.62011
[3]	B. Efron: The geometry of exponential families. Ann. Statist. 6 (1978), 362-376. · Zbl 0436.62027
[4]	V. Jarník: Diferenciální počet II. (Differential Calculus). NČSAV, Praha 1956.
[5]	A. Pázman: Nonlinear least squares - uniqueness versus ambiguity. Math. Operationsforsch. Statist. Ser. Statist. 15 (1984), 323-336. · Zbl 0562.62053
[6]	A. Pázman: Probability distribution of the multivariate nonlinear least squares estimates. Kybernetika 20 (1984), 209-230. · Zbl 0548.62043
[7]	S. Sternberg: Lectures on Differential Geometry. Second printing. Prentice-Hall, Englewood Cliffs 1965.