Two Integrals Arising in Generalized Inverse Gaussian Model and Heat Conduction Problems (M. A. Chaudry AND S. M. Zubair)

SIAM Review ◽  
1993 ◽  
Vol 35 (3) ◽  
pp. 503-504 ◽  
Author(s):  
M. L. Glasser
Keyword(s):  
Heat Conduction ◽  
Generalized Inverse ◽  
Gaussian Model ◽  
Inverse Gaussian ◽  
Generalized Inverse Gaussian

ROLLER-COASTER FAILURE RATES AND MEAN RESIDUAL LIFE FUNCTIONS WITH APPLICATION TO THE EXTENDED GENERALIZED INVERSE GAUSSIAN MODEL

2010 ◽  
Vol 25 (1) ◽  
pp. 103-118 ◽  
Author(s):  
Ramesh C. Gupta ◽  
Weston Viles
Keyword(s):  
Failure Rate ◽  
Generalized Inverse ◽  
Residual Life ◽  
Gaussian Model ◽  
Change Points ◽  
Mean Residual Life ◽  
Additional Parameter ◽  
Inverse Gaussian ◽  
Generalized Inverse Gaussian ◽  
The Relationship

The investigation in this article was motivated by an extended generalized inverse Gaussian (EGIG) distribution, which has more than one turning point of the failure rate for certain values of the parameters. In order to study the turning points of a failure rate, we appeal to Glaser's eta function, which is much simpler to handle. We present some general results for studying the reationship among the change points of Glaser's eta function, the failure rate, and the mean residual life function (MRLF). Additionally we establish an ordering among the number of change points of Glaser's eta function, the failure rate, and the MRLF. These results are used to investigate, in detail, the monotonicity of the three functions in the case of the EGIG. The EGIG model has one additional parameter, δ, than the generalized inverse Gaussian (GIG) model's three parameters; see Jorgensen [7]. It has been observed that the EGIG model fits certain datasets better than the GIG of Jorgensen [7]. Thus, the purpose of this article is to present some general results dealing with the relationship among the change points of the three functions described earlier. The EGIG model is used as an illustration.

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