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Random Variables, Distributions & Linear Regression
Published in Nailong Zhang, A Tour of Data Science, 2020
The quantile function is the inverse of CDF, i.e., QX(p)=inf{x∈ℝ:p≤FX(x)}.
Bayesian Modeling and Inference for One-Shot Experiments
Published in Technometrics, 2023
Jonathan Rougier, Andrew Duncan
For the link function we reason as follows. Current practice is satisfied with a symmetric sigmoidal link function (Probit). Therefore, we will also use a symmetric sigmoidal link function, although the resulting sensitivity function will not be symmetric, because is not an affine function of x. We use the quantile function of a distribution, which satisfies our requirement that for . This choice implies that for ; that is, sufficiently high stimulus values are certain to cause a “go,” although these values may be well above , the top of the range of interest. This particular Beta was chosen because of its width, but the result will not be sensitive to the choice because the spline coefficients can adjust.
Value at risk approach to producer's best response in an electricity market with uncertain demand
Published in Optimization, 2022
Martin Branda, René Henrion, Miroslav Pištěk
For a one-dimensional random variable X on a probability space , we denote its distribution by μ which is defined as for all Borel measurable subsets . This distribution induces the distribution function the inverse of which is the quantile function defined by We say that a measurable real function is a density of X, if for all Borel measurable subsets or, equivalently, if
Two-Parameter Logistic-Exponential Distribution: Some New Properties and Estimation Methods
Published in American Journal of Mathematical and Management Sciences, 2020
Sajid Ali, Sanku Dey, Muhammad Hussain Tahir, Muhammad Mansoor
Lan and Leemis (2008) pointed out that the moments of the LE distribution are finite but cannot be expressed in closed form. Therefore, the basic summaries, such as the mean, variance, skewness, and kurtosis are computed numerically. The graphical depiction of these different measures is given in Figure 2. It is clear from the figure that the behavior of the mean, variance, skewness, and kurtosis become the same (except scale difference) as λ increases (for fixed ). The quantile function of a distribution has many uses, both in statistical theory and applications. The quantile function of X is obtained by inverting Equation (1), which is given as where