Advances and Applications in Statistics
Volume 65, Issue 1, Pages 1 - 17
(November 2020) http://dx.doi.org/10.17654/AS065010001 |
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ESTIMATING AN EXPONENTIALLY DECAYING FUNCTION OF RATE PARAMETER OF A POISSON PROCESS
I. C. Kipchirchir
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Abstract: In this paper, we consider point estimation of an exponentially decaying function of rate parameter λ of Poisson process using discrete (increments) and continuous (interarrival times) variables under maximum likelihood and minimum variance paradigms. It is found that for increments, moments of estimators are in terms of elementary function - the exponential function whereas for interarrival times, moments of estimators are in terms of special functions -modified Bessel function of the third kind for maximum likelihood estimators and confluent hypergeometric function for the uniformly minimum variance unbiased estimators. Behaviourally, the moments mirror the exponentially decaying function of λ. The maximum likelihood estimators are biased, however, it is found that asymptotic unbiasedness for fixed nλ where n is the sample size corresponds to a limiting Poisson process which is degenerate at zero. Simulations reveal that maximum likelihood estimates and uniformly minimum variance unbiased estimates exhibit similar behaviour, moreover, they converge to the exponentially decaying function of λ as sample size increases. |
Keywords and phrases: Poisson process, increments, interarrival times, exponentially decaying function, estimation, maximum likelihood, minimum variance, moments, simulation, behaviour.
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