Finding Gaussian Curvature of Lifespan Distribution

Author(s)
William W. S. Chen

ABSTRACT

The objective of this paper is to review the lifespan model. This paper will also suggest four additional general alternative computational methods not mentioned in Kass, R.E. and Vos, P.W. [1] [2]. It is not intended to compare the four formulas to be used in computing the Gaussian curvature. Four different formulas adopted from Struik, D.J. [3] are used and labeled here as (A), (B), (C), and (D). It has been found that all four of these formulas can compute the Gaussian curvature effectively and successfully. To avoid repetition, we only presented results from formulas (B) and (D). One can more easily check other results from formulas (A) and (C).

The objective of this paper is to review the lifespan model. This paper will also suggest four additional general alternative computational methods not mentioned in Kass, R.E. and Vos, P.W. [1] [2]. It is not intended to compare the four formulas to be used in computing the Gaussian curvature. Four different formulas adopted from Struik, D.J. [3] are used and labeled here as (A), (B), (C), and (D). It has been found that all four of these formulas can compute the Gaussian curvature effectively and successfully. To avoid repetition, we only presented results from formulas (B) and (D). One can more easily check other results from formulas (A) and (C).

KEYWORDS

Christoffel Symbols, Gamma, Gaussian Curvature, Inverse Gaussian, Metric Tensor, Mixed Riemann Curvature Tensor, Weibull

Christoffel Symbols, Gamma, Gaussian Curvature, Inverse Gaussian, Metric Tensor, Mixed Riemann Curvature Tensor, Weibull

Cite this paper

Chen, W. (2014) Finding Gaussian Curvature of Lifespan Distribution.*Applied Mathematics*, **5**, 3392-3400. doi: 10.4236/am.2014.521316.

Chen, W. (2014) Finding Gaussian Curvature of Lifespan Distribution.

References

[1] Kass, R.E. and Vos, P.W. (1997) Geometrical Foundations of Asymptotic Inference. John Wiley & Sons, Inc., New York.

http://dx.doi.org/10.1002/9781118165980

[2] Kass, R.E. (1989) The Geometry of Asymptotic Inference (with Discussion). Statistical Science, 4, 188-234.

[3] Struik, D.J. (1961) Lectures on Classical Differential Geometry. 2nd Edition, Dover Publications, Inc., New York.

[4] Lawless, J.F. (1982) Statistical Models and Methods for Lifetime Data. John Wiley & Sons, Hoboken.

[5] Chen, W.W.S. (1980) On the Tests of Separate Families of Hypotheses with Small Sample Size. Journal of Statistical Computation and Simulation, 2, 183-187.

http://dx.doi.org/10.1080/00949658008810406

[6] Chen, W.W.S. (1982) Simulation on Probability Points for Testing of Lognormal or Weibull Distribution with a Small Sample. Journal of Statistical Computation and Simulation, 15, 201-210.

http://dx.doi.org/10.1080/00949658208810583

[7] Chen, W.W.S. (1987) Testing Gamma and Weibull Distribution: A Comparative Study. Estadistica, 39, 1-26.

[8] Chen, W.W.S. (1983) Testing Lognormal and Exponential Distributions: Estimation of Percentile Points. American Journal of Mathematical and Management Sciences, 3, 165-196.

http://dx.doi.org/10.1080/01966324.1983.10737123

[9] Gupta, A.K. (1952) Estimation of the Mean and Standard Deviation of a Normal Population from a Censored Sample. Biometrika, 39, 260-273.

http://dx.doi.org/10.1093/biomet/39.3-4.260

[10] Balakrishnan, N. and Chen, W.W.S. (1999) Handbook of Tables for Order Statistics from Lognormal Distributions with Applications. Kluwer Academic Publishers, Norwell.

http://dx.doi.org/10.1007/978-1-4615-5309-0

[11] Balakrishnan, N. and Chen, W.W.S. (1997) CRC Handbook of Tables for Order Statistics from Inverse Gaussian Distributions with Applications. CRC Press, Boca Raton.

[12] Gray, A. (1993) Modern Differential Geometry of Curves and Surfaces. CRC Press, Inc., Boca Raton.

[1] Kass, R.E. and Vos, P.W. (1997) Geometrical Foundations of Asymptotic Inference. John Wiley & Sons, Inc., New York.

http://dx.doi.org/10.1002/9781118165980

[2] Kass, R.E. (1989) The Geometry of Asymptotic Inference (with Discussion). Statistical Science, 4, 188-234.

[3] Struik, D.J. (1961) Lectures on Classical Differential Geometry. 2nd Edition, Dover Publications, Inc., New York.

[4] Lawless, J.F. (1982) Statistical Models and Methods for Lifetime Data. John Wiley & Sons, Hoboken.

[5] Chen, W.W.S. (1980) On the Tests of Separate Families of Hypotheses with Small Sample Size. Journal of Statistical Computation and Simulation, 2, 183-187.

http://dx.doi.org/10.1080/00949658008810406

[6] Chen, W.W.S. (1982) Simulation on Probability Points for Testing of Lognormal or Weibull Distribution with a Small Sample. Journal of Statistical Computation and Simulation, 15, 201-210.

http://dx.doi.org/10.1080/00949658208810583

[7] Chen, W.W.S. (1987) Testing Gamma and Weibull Distribution: A Comparative Study. Estadistica, 39, 1-26.

[8] Chen, W.W.S. (1983) Testing Lognormal and Exponential Distributions: Estimation of Percentile Points. American Journal of Mathematical and Management Sciences, 3, 165-196.

http://dx.doi.org/10.1080/01966324.1983.10737123

[9] Gupta, A.K. (1952) Estimation of the Mean and Standard Deviation of a Normal Population from a Censored Sample. Biometrika, 39, 260-273.

http://dx.doi.org/10.1093/biomet/39.3-4.260

[10] Balakrishnan, N. and Chen, W.W.S. (1999) Handbook of Tables for Order Statistics from Lognormal Distributions with Applications. Kluwer Academic Publishers, Norwell.

http://dx.doi.org/10.1007/978-1-4615-5309-0

[11] Balakrishnan, N. and Chen, W.W.S. (1997) CRC Handbook of Tables for Order Statistics from Inverse Gaussian Distributions with Applications. CRC Press, Boca Raton.

[12] Gray, A. (1993) Modern Differential Geometry of Curves and Surfaces. CRC Press, Inc., Boca Raton.