In analysis of variance (ANOVA) models, it is generally assumed that the distribution of the error terms is normal with mean zero and constant variance ${\sigma }^2.$ Traditionally, a least square (LS) method is used for estimating the unknown parameters and testing the null hypothesis. It is known that, when the normality assumption is not satisfied, LS estimators of the parameters and the test statistics based on them lose their efficiency, see Tukey. On the other hand, a non constant variance problem which is called heteroskedasticity is another serious issue for LS estimators. The LS estimators still remain unbiased but the estimated standard error (SE) is wrong. Because of this, confidence intervals and hypotheses tests cannot be relied on. Welch’s ANOVA is the most popular method to solve this problem. In this paper, we assume that the distribution of the error terms is skew-t with non constant variance in one-way ANOVA. We also propose a new test statistics based on the maximum likelihood (ML) estimators of skew-t distribution. A Monte Carlo simulation study is performed to compare traditional LS and Welch’s method with the proposed method in terms of Type I errors and the powers. At the end of this study, an example is given for the illustration of the methods.

Meteorological drought, exacerbated by global climate change, is a recurrent and complex catastrophe that disrupts water resources, agriculture, and economies worldwide. Forecasting future drought conditions is therefore of utmost importance in facilitating effective intervention and adaptation strategies. Artificial intelligence and machine learning techniques have gained attention owing to their potential in forecasting drought severity. However, they are often criticized for their black-box nature and lack of explainability and interpretability. To address this issue, this study explores the potential of local and global explainable artificial intelligence (XAI) tools, including SHAP and LIME, to extract intrinsic characteristics from different drought indices, including Standardized Precipitation Index (SPI) and Standardized Precipitation Evapotranspiration Index (SPEI) and climatic parameters (temperature, relative humidity, and wind speed). These techniques were applied to understand the driving forces behind drought events under different scenarios. SHAP was used to provide global model explanations, revealing how climatic parameters varied in importance across stations and scenarios, while LIME was employed to offer local explanations for specific predictions, highlighting the role of lag time and the significance of each parameter in individual forecasting scenarios. Using data from four meteorological stations in the Konya province of Türkiye, spanning 1965 to 2023, six ensemble-based machine learning models were employed to forecast SPI and SPEI values at 6-month and 12-month scales, resulting in a total of 768 scenarios. Widely used evaluation metrics, namely MSE, RMSE, MAE, MAPE and $$\:{R}^{2},$$ were employed to compare model performances. The prediction accuracy ranged from 0.513 to 0.864 $$\:{R}^{2}$$ values, and CatBoost emerged as the best-performing method in the scenario in which SPEI-12 (t), SPEI-12 (t-1), temperature (T), and wind speed (W) were used as inputs. The global explanation using the SHAP revealed that climatic parameters had varying degrees of impact across stations and scenarios, whereas the influence of lag time was significant in the predictive models. Alongside these considerations, LIME and SHAP methods were applied to investigate the status of the parameters at local and global levels. LIME and SHAP analyses also revealed that T and W parameters played less decisive roles at all stations than SPEI(t) and SPEI(t-1) parameters. The conclusion drawn from the current investigation may offer more explainability when working with machine learning algorithms in drought prediction studies, which could be critical for predicting future drought events.

In this article, we introduce some examples of cubic rank transmuted distributions proposed by Granzatto et al. (2017). The statistical aspects of the introduced distributions such as probability density functions, hazard rate functions and reliability functions are studied. The maximum likelihood estimation method is used in order to estimate the parameters of interest. Finally, real data examples are applied for the illustration of these distributions.

The pairwise comparisons or post-hoc methods are used for determining the source of the difference of group means in one-way ANOVA. These methods are mostly depend on normality assumption. However, nonnormal distributions are more prevalent than normal distribution. Therefore, robust estimation methods become very important tools in statistical analysis. In this paper, we assume that the distribution of the error terms is Azzalini's skew $t$ and obtain the robust estimators in order to make post-hoc tests in one-way ANOVA. We use maximum likelihood (ML) methodology and compare this methodology with some of robust estimators like M estimator, Wave estimator, trimmed mean and modified maximum likelihood (MML) methodology with Monte Carlo simulation study. Simulation results show that the proposed methodology is more preferable. We also compare power values of the test statistics and conclude that the test statistics based on the ML estimators are more powerful than the test statistics based on other methods.

The preference of the proper distribution for modelling data is often a substantial concern to researchers and practitioners. For this reason, new statistical distributions or the generalizations of well-known distributions have been proposed for exible modeling. Weighted distributions are one of the generalization methods for these distributions. In this article, area biased beta, Rayleigh and log-normal distributions are introduced. Some main statistical properties, like probability density functions, cumulative distribution functions, moments and the estimation of the parameters of these distributions are obtained. Real data examples are used for illustration of these distributions.