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By Mary Phipps, Malcolm Quine

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This Optimization algorithm needs to incorporate any constraints on the parameters to be estimated (for example, for the exponential distribution there is the constraint X > 0). Furthermore, it is not guaranteed that the likelihood function is unimodal. It may have several local maxima in addition to a global maximum. Therefore, a global optiniization algorithm such as a genetic algorithm [Gold891or interval arithmetic [LiiLlOO]may be in order. From a practical point of view, however, such approaches may imply very long calculation times.

One just has to add the function which sets the parameter that defines the boundary under investigation to the current value. 5 Real Data All estimation algorithms presented so far have implicitly assumed a so-called full sample which means that all events of interest are included in the sample. Furthermore, in all cases there is the underlying assumption of knowing the exact occurrence time of an event 5 , . Data that are collected in real-world applications are not always of that kind but may be grouped and censored.

BASICS OF PROBABILITY AND STATISTICS 21 The importance of the exponential distribution is based on the fact that it is the only continuous distribution that possesses the memoryless property: 3 P ( X 5 u + t I X > u)= 1 - exp -= = P ( X 5 t ) . 24) As an example for an application of Eq. 24), consider a bus stop with the following schedule: Buses arrive with exponentially distributed interarrival times and identical mean Now if you have already been waiting in vain for u units of time for the bus to come, the probability of a bus arrival within the next t units of time is the same as if you had just shown up at the bus stop, that is, you can forget about the past or about the time already spent waiting.

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