Synthesis of optimal experiment designs under the conditions of Generalized Lambda-distribution of errors for models with errors in variables

Рассмотрена задача планирования эксперимента в условиях появления ошибок в объясняющих переменных. Сформулировано и доказано утверждение о способе вычисления элементов информационной матрицы Фишера с использованием обобщенного лямбда-распределения, доказано следствие о способе вычисления функции эффективности плана эксперимента. Сравнение результатов вычисления функции эффективности с использованием выведенного в следствии соотношения и с помощью известного соотношения для нормального распределения ошибок показало, что результаты совпадают. Построены оптимальные планы эксперимента для различных распределений случайных компонент. The problem of experimental design under conditions of errors in the explanatory variables is considered. The proposition of the method for calculating the Fisher information matrix elements using the Generalized Lambda-distribution is formulated and proved, the consequence of the method for calculating the efficiency function of the experimental design is proved. This method of calculating the Fisher information matrix takes into account the heterogeneity of the errors in random distribution throughout the planning area. In this paper, studies of the synthesis of optimal experimental designs using proven proposition and consequence under various conditions of computational experiments are presented. The results of calculating the efficiency function using the obtained relation and using the known relation for the normal distribution of errors are compared, it is found that the results coincide. Optimal experimental designs are constructed for various distributions of random components. The results of the synthesis of optimal experimental design showed that when function of efficiency is constant throughout the planning area then the optimal experimental design is equilibrium plan. When there are differences in the values of the efficiency function in the planning area, the optimal plan ceases to be equilibrium

The distribution of extreme share return in different Malaysian economic circumstances

This study evaluated the performance of probability distribution in various financial periods by investigating the effect of economic cycle on extreme stock return activity. Malaysian stock price KLCI data from 1994–2008 were split into three economy periods correspond to the growth, financial crisis, and the recovery. Four prevalent distributions specifically generalized lambda distribution (GLD), generalized extreme value (GEV), generalized logistic (GLO), and generalized pareto (GPA) were employed to model weekly and monthly maximum and minimum Kuala Lumpur Composite Index (KLCI) share returns. The L-moment approach was used to estimate the parameter while k-sample Anderson darling (k-ad) test was applied to measure the goodness of fit estimation. In conclusion, GLD is the most appropriate distribution representing a weekly maximum minimum return for overall three economic scenarios in Malaysia.

A Runs Test for the Hypothesis of Symmetry with one Sided Alternative

We propose a trimmed runs test for the hypothesis of symmetry with one sided alternative, in samples coming from the Generalized Lambda Distribution (GLD) with unknown median. We provide a method to calculate the exact distribution, showing that it is symmetric around zero and we give arguments to justify the approximation by means of the normal distribution. The size of the proposed test is calibrated with four symmetrical cases of the GLD and the empirical power is compared with that of some other tests for the same hypothesis, using eight asymmetrical cases of the GLD. The results show that the proposed test is unbiased in the cases used for calibration, and that the empirical power of the proposed test overtakes the empirical power of all compared tests, excepting one of them in two specific cases. Some hints are given concerning how to optimize the empirical power according to the size of the tails of the sampled distributions.

Adaptive rank tests for location with generalizaed Lambda distribution scores

We propose adaptive rank tests for the location alternative in one sample, using as score function the percentile function of the Generalized Lambda Distribution (GLD ). We give expressions for its eciency as functions of the kurtosis parameters of the distribution used for the score function and those of the sampled distribution. A simulation study shows that the proposed tests maintain its nominal size and that this test using scores functions with small kurtosis parameter, are very ecient for samples coming from distributions with large kurtosis, overtaking the sign test and the Wilcoxon test. Reciprocally, tests which use scores from GLD distributions with large kurtosis are more ecient when the sample comes from GLD distributions with small kurtosis.