The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the L2 distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the L1 norm.
Also, a geodesic distance can be defined on the cone of strictly positive vectors in ℝn in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean.
That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.
In recent years, a lot of effort has been put into finding suitable mathematical models that fit historical data set. Such models often include coefficients and the accuracy of data approximation depends on them. So the goal is to choose the unknown coefficients to achieve the best possible approximation of data by the corresponding solution of the model. One of the standard methods for coefficient estimation is the least square method. This can provide us data approximation but it can also serve as a starting method for further minimizations such as Matlab function fminsearch.
In this paper, we study a pair of unified and extended fractional integral operator involving the multivariable I-functions and general class of multivariable polynomials. Here, we use Mellin transforms to obtain our main results. Certain properties of these operators concerning to their Mellin-transforms have been investigated. On account of the general nature of the functions involved herein, a large number of known (may be new also) fractional integral operators involved simpler functions can be obtained. We will also quote the particular case of the multivariable H-function.
We are interested in comparing the performance of various nonlinear estimators of parameters of the standard nonlinear regression model. While the standard nonlinear least squares estimator is vulnerable to the presence of outlying measurements in the data, there exist several robust alternatives. However, it is not clear which estimator should be used for a given dataset and this question remains extremely difficult (or perhaps infeasible) to be answered theoretically. Metalearning represents a computationally intensive methodology for optimal selection of algorithms (or methods) and is used here to predict the most suitable nonlinear estimator for a particular dataset. The classification rule is learned over a training database of 24 publicly available datasets. The results of the primary learning give an interesting argument in favor of the nonlinear least weighted squares estimator, which turns out to be the most suitable one for the majority of datasets. The subsequent metalearning reveals that tests of normality and heteroscedasticity play a crucial role in finding the most suitable nonlinear estimator.
We propose an approach for real-time physically semi-realistic animation of strings which directly manipulates the string positions by position based dynamics. The main advantage of a position based dynamics is its controllability. Instability problems of explicit integration schemes can be avoided. Specifically, we offer the following three contributions. We introduce the non-elongating and non-stretchable mass spring dynamics model based on Position Based Dynamics to simulate 1D string. We introduce a method for propagating the twisting angle along the chain of segments. In addition, we solve collision constraints by regularly distributing the spheres along the chain segments followed by particle projection to validate the positions. Proposed strain limiting constraint can handle the strings fixed in multiple locations contrary to single fixed side as is common for hair models. The use of multiple constraints provides an efficient treatment for stiff twisting and non-stretchable mass spring dynamics model.
Gupta et al (2002) suggested an optional randomized response model under the assumption that the mean of the scrambling variable S is ‘unity’ [i.e. µs = 1]. This assumption limits the use of Gupta et al’s (2002) randomized response model. Keeping this in view we have suggested a modified optional randomized response model which can be used in practice without any supposition and restriction over the mean (µs) of the scrambling variables S. It has been shown that the estimator of the mean of the stigmatized variable based on the proposed optional randomized response sampling is more efficient than the Eicchorn and Hayre (1983) procedure and Gupta et al’s (2002) optional randomized technique when the mean of the scrambling S is larger than unity [i.e. µs > 1]. A numerical illustration is given in support of the present study.
In this paper, an effectual and new modification in Laplace Adomian decomposition method based on Bernstein polynomials is proposed to find the solution of nonlinear Volterra integral and integro-differential equations. The performance and capability of the proposed idea is endorsed by comparing the exact and approximate solutions for three different examples on Volterra integral, integro-differential equations of the first and second kinds. The results shown through tables and figures demonstrate the accuracy of our method. It is concluded here that the non orthogonal polynomials can also be used for Laplace Adomian decomposition method. In addition, convergence analysis of the modified technique is also presented.
Fisher information is of key importance in estimation theory. It is used as a tool for characterizing complex signals or systems, with applications, e.g. in biology, geophysics and signal processing. The problem of minimizing Fisher information in a set of distributions has been studied by many researchers. In this paper, based on some rather simple statistical reasoning, we provide an alternative proof for the fact that Gaussian distribution with finite variance minimizes the Fisher information over all distributions with the same variance.