Some special inversion formulas for the laplace transform and their application to the problem of unbiased estimation of the parameters of probability distributions

1990 ◽  
Vol 52 (2) ◽  
pp. 2872-2878
Author(s):  
V. G. Voinov
Keyword(s):  
Laplace Transform ◽  
Probability Distributions ◽  
Unbiased Estimation ◽  
Inversion Formulas ◽  
The Laplace Transform

CERTAIN ISOMETRIES RELATED TO THE BILATERAL LAPLACE TRANSFORM

2006 ◽  
Vol 11 (3) ◽  
pp. 331-346 ◽  
Author(s):  
S. B. Yakubovich
Keyword(s):  
Hilbert Space ◽  
Entire Function ◽  
Laplace Transform ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Inversion Formulas ◽  
Mean Convergence ◽  
The Laplace Transform ◽  
The Mean

We study certain isometries between Hilbert spaces, which are generated by the bilateral Laplace transform In particular, we construct an analog of the Bargmann‐Fock type reproducing kernel Hilbert space related to this transformation. It is shown that under some integra‐bility conditions on $ the Laplace transform FF(z), z = x + iy is an entire function belonging to this space. The corresponding isometrical identities, representations of norms, analogs of the Paley‐Wiener and Plancherel's theorems are established. As an application this approach drives us to a different type of real inversion formulas for the bilateral Laplace transform in the mean convergence sense.

scholarly journals Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

2008 ◽  
Vol 45 (02) ◽  
pp. 531-541 ◽  
Author(s):  
A. G. Rossberg
Keyword(s):  
Laplace Transform ◽  
High Precision ◽  
Computational Efficiency ◽  
Generating Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Random Variables ◽  
Numerical Examples ◽  
Pareto Distributions ◽  
The Laplace Transform

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

Some probability distributions connected with recording apparatus

1948 ◽  
Vol 44 (3) ◽  
pp. 335-341 ◽  
Author(s):  
C. Domb
Keyword(s):  
Probability Distribution ◽  
Laplace Transform ◽  
Asymptotic Behaviour ◽  
Explicit Expression ◽  
Probability Distributions ◽  
Practical Interest ◽  
Observation Time ◽  
Constant Length ◽  
Recording Apparatus ◽  
The Laplace Transform

1. The problem to be discussed in the present paper arises when the finite resolving time of a recording apparatus is taken into account. Events are divided into two classes, recorded and unrecorded. Any recorded event is followed by a dead interval of a certain length of time, during which any other event which occurs will be unrecorded. A typical example is an α-particle counter; a recorded particle causes the chamber to ionize, and no other particle can be recorded until the chamber has deionized. In the case when the dead interval is of constant length τ, if the observation time t lies between (n − 1)τ and nτ, the number of recorded events must be 0, 1, 2, …, n − 1 or n. We shall assume that the probability of an event occurring in the interval [t, t + dt] is λdt, λ being constant; this is the case of most practical interest. The probability distribution of recorded events for dead intervals of constant length has been determined by Ruark and Devol (2). The explicit expression of this distribution is fairly complicated, and it is therefore difficult to manipulate. The method employed in the present paper leads naturally to the Laplace transform of the distribution with respect to the time t, and this is relatively simple. The method can easily be generalized to deal with the case of dead intervals which are not all equal, but follow a probability distribution u(τ) dτ. Finally the Laplace transform is very convenient for determining the asymptotic behaviour of the distribution of recorded events for large times of observation.

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