Let { X n≥  1} be a stationary sequence of pair wise negative quadrant dependent random variables with survival function  F( x)=P[X >x]. The empirical survival function Fn(x) based on X1, X2,… X n is proposed as an estimator for Fn ( x). STRONG CONSISTENCY and point wise as well as UNIFORM of Fn (x) are discussed.    

In this paper, we prove the STRONG UNIFORM CONSISTENCY and asymptotic normality of the kernel density estimator proposed by Jones for length-biased data. The approach is based on the invariance principle for the empirical processes proved by Horvath. All simulations are drawn for different cases to demonstrate both, CONSISTENCY and asymptotic normality and the method is illustrated by real automobile brake pads data.

Introduction: Flood discharge is of high importance in studies regarding water resource exploitation, flood control, construction of dams, basin management, and hydrologic studies (Alzahrani et al, 2017). Therefore, to a large extent the accuracy of these studies and the safety of water constructions depend on flood study methods. Flood is a natural phenomenon that threatens the life and properties of a large number of people all over the world, and it is impossible to manage water resources in basins without the accurate determination of the peak flood discharge (Badri et al, 2017). The advances in flood estimation techniques have made it possible to use rainfall-runoff models to assess the hydrographic properties of the flood in watersheds and decrease the risks of the flood. Therefore, this study was carried out to compare the SCS unit hydrograph and UNIFORM methods in determining the peak flood discharge with WMS model in Amughin basin of Ardabil province. Methodology: Amughin basin with an area of approximately 78 km2 is located in the northwest of Iran. The physiographic features were extracted using the basin map (scale: 1: 25000) and WMS model. This study applied Arc GIS 9. 2 and Idrisi32 software to obtain the properties of the studied basin using DEM (Digital Elevation Map) of the National Cartographic Center, NCC. Remote sensing methodology was utilized to study the geographical land use changes occurred during the study period. Landsat images of TM and ETM+ of Amughin basin area were collected from the USGS Earth Explorer web site. After image preprocessing, un-supervised and supervised image classification were performed to classify the images into different land use categories. In general, soil hydrologic groups were divided into three subgroups of B, C, and D and CN value of 78. 7 was estimated for the Amughin basin based on the geological examination, permeability, vegetation, and hydrologic conditions of the basin soil. Results: The model calibration results showed that the simulated peak discharge and flow volume were in good correspondence with the observed values (RE%= 7. 17, RMSE= 0. 44). Thus, the calibration results were used for optimum values of parameters. The model was validated using two rainfall events and the model performance indices were acceptable in both cases (RE%= 2. 51, RMSE= 0. 0042) in SCS method. To evaluate and test model validation, two rainfall events, were used. That the model performance indices were acceptable. Distribution of CN amount in the area showed that the upstream flow had higher CN values and consequently increased flood volume in these areas. Based on the values of obtained CN, the amount of peak flood discharge was calculated for return periods of 25, 50, and 100 years. Discussion and conclusion: According to the results, the SCS model has good agreement with experimental results among the different methods used for estimating flood discharge in the northwest of Iran. In fact, this model requires calibration in the study region. In small watersheds in the northwest of Iran, the SCS model yields better results than the UNIFORM method because the conditions required for using this model are satisfied in these basins. Moreover, the results obtained from this method can be closer to actual values provided that the watershed concentration time is calculated more accurately. Our results also showed that the SCS model has a high sensitivity to rainfall distribution across the region and that the rainfall across the region needs to be analyzed to obtain desirable results. Besides, the rainfall distribution and its time distribution should be close to the corresponding values in the region. A comparison between the obtained results of peak discharge from the SCS and UNIFORM methods in return periods of 25, 50, and 100 years revealed that the average estimates of the UNIFORM were approximately 5% higher than the SCS method. According to paired T-test, the difference between UNIFORM and SCS values were not significant at a confidence level of 0. 01. Overall, the results obtained from this method can be closer to actual values if the watershed lag time is calculated more accurately using the floods occurred in the studied basin.

We apply Preuss' concept of E-connectedness to the categories of lattice-valued UNIFORM convergence spaces and of lattice-valued UNIFORM spaces. A space is UNIFORMly E-connected if the only UNIFORMly continuous mappings from the space to a space in the class E are the constant mappings. We develop the basic theory for E-connected sets, including the product theorem. Furthermore, we define and study UNIFORM local E-connectedness, generalizing a classical definition from the theory of UNIFORM convergence spaces to the lattice-valued case. In particular it is shown that if the underlying lattice is completely distributive, the quotient space of a UNIFORMly locally E-connected space and products of locally UNIFORMly E-connected spaces are locally UNIFORMly E-connected.

‎Let $\{Y_i; i = 1,\ldots,n \}$ be a length-biased sample from a population with cumulative distribution function $F(\cdot)$‎. If the probability of an item selected in the sample is proportional to its length‎, ‎then the distribution of the observed length is known as the length-biased distribution‎.‎We consider the kernel-type estimator $F_n^s(\cdot)$ of $F(\cdot)$‎. ‎Under suitable conditions‎, ‎the extended Glivenko-Cantelli theorem for $F_n^s(\cdot)$ is proved‎.

Kernel density estimators are the basic tools for density estimation in non-parametric statistics. The k-nearest neighbor kernel estimators represent a special form of kernel density estimators, in which the bandwidth is varied depending on the location of the sample points. In this paper, we initially introduce the k-nearest neighbor kernel density estimator in the random left-truncation model, and then prove some of its asymptotic behaviors, such as STRONG UNIFORM CONSISTENCY and asymptotic normality. In particular, we show that the proposed estimator has truncation-free variance. Simulations are presented to illustrate the results and show how the estimator behaves for finite samples. Moreover, the proposed estimator is used to estimate the density function of a real data set.

In this paper, we define the concepts of compatibility between two fuzzy subsets on Q the set of states of a max- min general fuzzy automaton and transitivity in a max-min general fuzzy automaton. By using these notions, we construct a UNIFORM structure on Q, and also introduce a topology on it. Then we define the concept of semi-UNIFORM structure on a nonempty set X and by using this notion, we construct a semi-UNIFORM structure on the set of states of a general fuzzy automaton. Also, we construct a semi-UNIFORM structure on S*the set of all finite words on S, the set of input symbols of a general fuzzy automaton. Moreover by using these semi-UNIFORM structures, we construct two topologies on Q and S* and obtain some properties of these topologies.