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Sampling
Published in A. C. Faul, A Concise Introduction to Machine Learning, 2019
This approach can also be taken, if the probability density function f is log concavelog concave, i.e. log (f) is a concave function. Many probability density functions are log concave. Assume that the function depicted in Figure 3.5 is log (f(x)). The piecewise linear envelope can be transformed back into the space of f, by applying the exponential function. The result is shown in Figure 3.6. The envelope is a piecewise exponential function. Samples from this can be obtained by using the exponential distributionexponential distributiondistribution!exponential and inverse transform sampling.
Setting Fulfillment-Time Guarantees for Accepting Customer Orders in a Periodic-Review Base-Stock Inventory System
Published in IISE Transactions, 2023
Yanyi Xu, Doğan A. Serel, Arnab Bisi, Maqbool Dada
Given that (5a, b) involve convolutions of distributions, establishing unimodality in R appears quite challenging. However, as shown in Theorem 4 below, if the demand distribution is from the monotone convolution ratios (MCR) family, which is defined in Appendix A, the cost functions in (3a)-(3c) are unimodal in R. This result is quite general because it is shown in Proposition 1–2(ii) of Rosling (2002) that if F(x) is log-concave, F(x) is MCR, so that MCR is a generalization of log-concavity. Therefore, most commonly used discrete distributions, for example, Poisson, uniform, binomial, negative binomial, hypergeometric, as well as continuous distributions such as normal, exponential, gamma, Weibull, beta distribution with parameters (r, q) such that , lognormal, uniform, and their translations, convolutions, and truncations that have log-concave F(x) belong to the MCR family. After analyzing the partial derivative of with respect to R as in (5a, b) and invoking the MCR property, we are able to show the following:
Quasiconcavity of a separable product of utility functions
Published in Optimization, 2018
Mohammed Berdi, Abdelhak Hassouni
One can also consider the function . The two problems just above are equivalent to the quasiconcavity of the product function Transposing convexity indices to the multiplicative case, Crouzeix-Kebbour [6] have introduced multiplicative indices of concavity for positive functions. If f is quasiconcave, then all are log-concave, except perhaps one which is then log-concave transformable. These results are easily derived of the results on the additive case. Recall that log-concave functions and distributions are of first importance in stochastic optimization. The multiplicative aspect of separability seems to be easier in application, and has a great interest in many models of mathematical programming with probabilistic constraints. See Précopa[7] and Norkin-Roenko [8].
Nonlinear Tikhonov regularization in Hilbert scales with balancing principle tuning parameter in statistical inverse problems
Published in Inverse Problems in Science and Engineering, 2019
The log-concave probability distributions include normal, exponential, logistic, chi-square, chi and Laplace. A survey of these can be found in [42]. The stochastic error is assumed to be a zero mean stochastic process with bounded covariance operator. Therefore, considering a general class of distributions larger than the Gaussian fit into the general setting of our problem. Due to Lemma 4.3, the overall rate of convergence of the two-step method will depend of the class of probability distributions to which the discretized noise term belongs to.