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Example text

3. 13) over the class of functions with . If you don’t get an explicit solution, can you show that there is a unique solution and can you describe a method to approximate it? 3, show that for , Hint: Interpret the sum of the s as the increase of the cumulative sum diagram between location and and the sum of s as increase of the convex minorant between location and in the diagram. 15) reduces to For a positive random variable, the Laplace transform of its distribution is well defined as . 17). 18).

The method of sampling from a certain population can lead in a natural way to a monotonicity property of the sampling density. 3. 4 deconvolution problems are introduced. In those models, one wants to estimate an unknown distribution function based on a sample from another distribution, which is the convolution of the distribution of interest and another (known) distribution. 6, comes from reliability theory, where one is interested in estimating an increasing hazard rate based on a sample of survival data.

If is differentiable at , any value between the left- and right derivative of function on , taking finite values on an interval at ; otherwise it can take . 23) Note that following theorem. 24) is nonnegative. This yields the following inequality: for all isotonic vectors such that for all , In other words: the isotonic regression of minimize the quadratic function over all isotonic vectors with weight function does not only , but at the same time it minimizes over the set of isotonic vectors with for all .

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