Comparative Study between Generalized Maximum Entropy and Bayes Methods to Estimate the Four Parameter Weibull Growth ModelRead the full article
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Parametric Methodologies for Detecting Changes in Maximum Temperature of Tlaxco, Tlaxcala, México
In this paper, comparison results of parametric methodologies of change points, applied to maximum temperature records from the municipality of Tlaxco, Tlaxcala, México, are presented. Methodologies considered are likelihood ratio test, score test, and binary segmentation (BS), pruned exact linear time (PELT), and segment neighborhood (SN). In order to compare such methodologies, a quality analysis of the data was performed; in addition, lost data were estimated with linear regression, and finally, SARIMA models were adjusted.
The L-Curve Criterion as a Model Selection Tool in PLS Regression
Partial least squares (PLS) regression is an alternative to the ordinary least squares (OLS) regression, used in the presence of multicollinearity. As with any other modelling method, PLS regression requires a reliable model selection tool. Cross validation (CV) is the most commonly used tool with many advantages in both preciseness and accuracy, but it also has some drawbacks; therefore, we will use L-curve criterion as an alternative, given that it takes into consideration the shrinking nature of PLS. A theoretical justification for the use of L-curve criterion is presented as well as an application on both simulated and real data. The application shows how this criterion generally outperforms cross validation and generalized cross validation (GCV) in mean squared prediction error and computational efficiency.
Analytically Simple and Computationally Efficient Results for the GIX/Geo/c Queues
A simple solution to determine the distributions of queue-lengths at different observation epochs for the model GIX/Geo/c is presented. In the past, various discrete-time queueing models, particularly the multiserver bulk-arrival queues, have been solved using complicated methods that lead to incomplete results. The purpose of this paper is to use the roots method to solve the model GIX/Geo/c that leads to a result that is analytically elegant and computationally efficient. This method works well even for the case when the inter-batch-arrival times follow heavy-tailed distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue-lengths at different time epochs.
Analytically Explicit Results for the Distribution of the Number of Customers Served during a Busy Period for Special Cases of the M/G/1 Queue
This paper presents analytically explicit results for the distribution of the number of customers served during a busy period for special cases of the queues when initiated with m customers. The functional equation for the Laplace transform of the number of customers served during a busy period is widely known, but several researchers state that, in general, it is not easy to invert it except for some simple cases such as and queues. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We obtain the distribution of the number of customers served during a busy period for various service-time distributions such as exponential, deterministic, Erlang-k, gamma, chi-square, inverse Gaussian, generalized Erlang, matrix exponential, hyperexponential, uniform, Coxian, phase-type, Markov-modulated Poisson process, and interrupted Poisson process. Further, we also provide computational results using our method. The derivations are very fast and robust due to the lucidity of the expressions.
Hierarchical Models and Tuning of Random Walk Metropolis Algorithms
We obtain weak convergence and optimal scaling results for the random walk Metropolis algorithm with a Gaussian proposal distribution. The sampler is applied to hierarchical target distributions, which form the building block of many Bayesian analyses. The global asymptotically optimal proposal variance derived may be computed as a function of the specific target distribution considered. We also introduce the concept of locally optimal tunings, i.e., tunings that depend on the current position of the Markov chain. The theorems are proved by studying the generator of the first and second components of the algorithm and verifying their convergence to the generator of a modified RWM algorithm and a diffusion process, respectively. The rate at which the algorithm explores its state space is optimized by studying the speed measure of the limiting diffusion process. We illustrate the theory with two examples. Applications of these results on simulated and real data are also presented.
On the Probabilistic Proof of the Convergence of the Collatz Conjecture
A new approach towards probabilistic proof of the convergence of the Collatz conjecture is described via identifying a sequential correlation of even natural numbers by divisions by that follows a recurrent pattern of the form , where represents divisions by 2 more than once. The sequence presents a probability of 50:50 of division by 2 more than once as opposed to division by 2 once over the even natural numbers. The sequence also gives the same 50:50 probability of consecutive Collatz even elements when counted for division by 2 more than once as opposed to division by 2 once and a ratio of 3:1. Considering Collatz function producing random numbers and over sufficient number of iterations, this probability distribution produces numbers in descending order that lead to the convergence of the Collatz function to 1, assuming that the only cycle of the function is 1-4-2-1.