scholarly journals Approximations to Risk Theory's F(x, t) by Means of the Gamma Distribution

1977 ◽  
Vol 9 (1-2) ◽  
pp. 213-218 ◽  
Author(s):  
Hilary L. Seal
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Gamma Distribution ◽  
Gamma Function ◽  
Function Approximation ◽  
Good Accuracy ◽  
Electronic Computer ◽  
Pocket Computer ◽  
Power Approximation ◽  
Aggregate Claims

It seems that there are people who are prepared to accept what the numerical analyst would regard as a shockingly poor approximation to F (x, t), the distribution function of aggregate claims in the interval of time (o, t), provided it can be quickly produced on a desk or pocket computer with the use of standard statistical tables. The so-called NP (Normal Power) approximation has acquired an undeserved reputation for accuracy among the various possibilities and we propose to show why it should be abandoned in favour of a simple gamma function approximation.Discounting encomiums on the NP method such as Bühlmann's (1974): “Everybody known to me who has worked with it has been surprised by its unexpectedly good accuracy”, we believe there are only three sources of original published material on the approximation, namely Kauppi et al (1969), Pesonen (1969) and Berger (1972). Only the last two authors calculated values of F(x, t) by the NP method and compared them with “true” four or five decimal values obtained by inverting the characteristic function of F(x, t) on an electronic computer.

scholarly journals From Aggregate Claims Distribution to Probability of Ruin

1978 ◽  
Vol 10 (1) ◽  
pp. 47-53 ◽  
Author(s):  
Hilary L. Seal
Keyword(s):  
Distribution Function ◽  
Probability Of Ruin ◽  
Type Iii ◽  
Power Approximation ◽  
Pearson Type ◽  
Aggregate Claims ◽  
Image Position ◽  
Independent Distribution

When the distribution of the number of claims in an interval of time of length t is mixed Poisson and the moments of the independent distribution of individual claim amounts are known, the moments of the distribution of aggregate claims through epoch t can be calculated (O. Lundberg, 1940, ch. VI). Several approximations to the corresponding distribution function, F(·, t), are available (see, e.g., Seal, 1969, ch. 2) and, in particular, a simple gamma (Pearson Type III) based on the first three moments has proved definitely superior to the widely accepted “Normal Power” approximation (Seal, 1976). Briefly,where the P-notation for the incomplete gamma ratio is now standard and α, a function of t, is to be found fromthe kappas being the cumulants of F(·, t). An excellent table of the incomplete gamma ratio is that of Khamis (1965).The problem that is solved in this paper is the production of an approximation to U(w, t), the probability of non-ruin in an interval of time of length t, by using the above mentioned gamma approximation to F(·, t).

scholarly journals A Model for Slew Evaluation for On-Chip RC Interconnects using Gamma Distribution Function

2010 ◽  
Vol 1 (10) ◽  
pp. 88-93
Author(s):  
R. Kar ◽  
V. Maheshwari ◽  
Ashis K. Mal ◽  
A.K. Bhattacharjee
Keyword(s):  
Distribution Function ◽  
Gamma Distribution ◽  
Gamma Distribution Function ◽  
On Chip

A model for a system subject to random shocks

1993 ◽  
Vol 30 (4) ◽  
pp. 979-984 ◽  
Author(s):  
Eui Yong Lee ◽  
Jiyeon Lee
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Stochastic Model ◽  
Poisson Process ◽  
The State ◽  
Stationary Case ◽  
Random Shocks ◽  
Random Amount ◽  
Explicit Expressions

A Markovian stochastic model for a system subject to random shocks is introduced. It is assumed that the shock arriving according to a Poisson process decreases the state of the system by a random amount. It is further assumed that the system is repaired by a repairman arriving according to another Poisson process if the state when he arrives is below a threshold α. Explicit expressions are deduced for the characteristic function of the distribution function of X(t), the state of the system at time t, and for the distribution function of X(t), if . The stationary case is also discussed.

scholarly journals From Characteristic Function to Distribution Function: A Simple Framework for the Theory

1991 ◽  
Vol 7 (4) ◽  
pp. 519-529 ◽  
Author(s):  
N.G. Shephard
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Unified Framework ◽  
New Approach ◽  
Inversion Theorem

A unified framework is established for the study of the computation of the distribution function from the characteristic function. A new approach to the proof of Gurland's and Gil-Pelaez's univariate inversion theorem is suggested. A multivariate inversion theorem is then derived using this technique.

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