Normal probability plots are widely used as a statistical tool for assessing whether an observed simple random sample is drawn from a normally distributed population. The users, however, have to judge subjectively, if no objective rule is provided, whether the plotted points fall close to a straight line. In this paper, we focus on how a normal probability plot can be augmented by intervals for all the points so that, if the population distribution is normal, then all the points should fall into the corresponding intervals simultaneously with probability 1-α. These simultaneous 1-α probability intervals provide therefore an objective mean to judge whether the plotted points fall close to the straight line: the plotted points fall close to the straight line if and only if all the points fall into the corresponding intervals. The powers of several normal probability plot based (graphical) tests and the most popular nongraphical Anderson-Darling and Shapiro-Wilk tests are compared by simulation. Based on this comparison, recommendations are given in Section 3 on which graphical tests should be used in what circumstances. An example is provided to illustrate the methods.
We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.
One of the basic graphical methods for assessing the validity of a distributional assumption is the Q-Q plot which compares quantiles of a sample against the quantiles of the distribution. In this paper, we focus on how a Q-Q plot can be augmented by intervals for all the points so that, if the population distribution is Weibull or exponential then all the points should fall inside the corresponding intervals simultaneously with probability 1 − α. These simultaneous 1 − α probability intervals provide therefore an objective mean to judge whether the plotted points fall close to the straight line: the plotted points fall close to the straight line if and only if all the points fall within the corresponding intervals. The powers of five Q-Q plot based graphical tests and the most popular non-graphical Anderson-Darling and Cramér-von-Mises tests are compared by simulation. Based on this power study, the tests that have better powers are identified and recommendations are given on which graphical tests should be used in what circumstances. Examples are provided to illustrate the methods.
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