In the literature on Statistical Reliability Theory and Stochastic ORDERs, several results based on theory of TP < sub>2/RR2 functions have been extensively used in establishing various properties. In this paper, we provide a review of some useful results in this direction and highlight connections between them.

Introduction Fail-safe systems ((n 􀀀,1)-out-of-n Systems) are commonly used in many day-to-day applied structures. A fail-safe is a special design feature that will respond when a failure occurs so that no harm happens to the system itself. The brake system in a train is an excellent example of a fail-safe system in which the brakes are held in off-position by air pressure. If a brake line splits or a carriage becomes sepaRATEd, the air pressure will be lost,in that case, the brakes will be applied by a local air reservoir. However, another classic example of a fail-safe system is an elevator in which brakes are held off brake pads by tension, and if the tension gets lost, the brakes latch on the rails in the shaft, thus preventing the elevator from falling off. There are many other such fail-safe systems in common use. Balakrishnan et al. (2015) established necessary and sufficient conditions for comparing two fail-safe systems with independent homogeneous exponential components in terms of mean residual life, dispersive, HAZARD RATE and likelihood ratio ORDERs. Their results specifically showed how an (n−, ١, )-out-of-n system consisting of heterogeneous components with exponential lifetimes could be compared with any (m−, ١, )-out-of-m system consisting of homogeneous components with exponential lifetimes. Similarly, Zhang et al. (2019) presented sufficient (and necessary) conditions on the lifetimes of components and their survival probabilities from random shocks for comparing the lifetimes of two fail-safe systems in terms of standard stochastic, HAZARD RATE and likelihood ratio ORDERs. Cai et al. (2017) compared the HAZARD RATE ORDER of second-ORDER statistics arising from two sets of independent multiple-outlier proportional HAZARD RATEs (PHR) samples. Material and Methods The comparison of essential characteristics associated with lifetimes of technical systems is an exciting topic in reliability theory since it usually enables us to approximate complex systems with simpler systems and subsequently obtain various bounds for important agreeing characteristics of the complex system. A convenient tool for this purpose is the theory of stochastic ORDERings. Results and Discussion This paper discusses the HAZARD RATE ORDER of (n􀀀, 1)-out-of-n systems arising from two sets of independent multiple-outlier modified proportional HAZARD RATEs components. Under certain conditions on the parameters and the submajorization ORDER between the sample size vectors, the HAZARD RATE ORDER between the (n􀀀, 1)-out-of-n systems from multiple-outlier modified proportional HAZARD RATEs is established. Conclusion In this paper, we have presented sufficient conditions for the HAZARD RATE ORDER between fail-safe systems. It will be of great interest to generalize the current work from lifetimes of fail-safe systems to those of k-out-of-n. Another problem of interest will be to consider the setting of general systems with several subsystems having dependent components and extend the results established here to this general case. We are currently working on these problems and hope to report those findings in a future paper.

In some applied problems we need to choose a population from the given populations and estimate the parameter of the selected population. Suppose k random samples are chosen from k populations with proportional HAZARD RATE model or proportional reversed HAZARD RATE model. According to a specified selection rule, it is desired to estimate the parameter of the best (worst) selected population. In this paper, under the entropy loss function we obtain the uniformly minimum risk unbiased (UMRU) estimator of the parameters of the selected population, and derived sufficient conditions for minimaxity of a given estimator.Then we find the class of admissible and inadmissible linear estimators of the parameters of the selected population and determine the class of dominators of a given estimator. We show that the UMRU estimator is inadmissible and compare the obtained estimators by plotting their risk functions.

Introduction Convolutions of independent random variables often arise in many applied areas, including applied probability, reliability theory, actuarial science, nonparametric goodness-of-fit testing, and operations research. Since the distribution theory is quite complicated when the convolution involves independent and non-identical random variables, it is of great interest to investigate the stochastic properties of convolutions and derive bounds and approximations on some characteristics of interest in this setup. The results in this work give responses that under the new condition, the convolution of two random variables is ORDERed according to some stochastic ORDERs such as likelihood ratio ORDER, HAZARD RATE ORDER and reversed HAZARD RATE ORDER. In general cases, let X1, X2 and X,1,X,2 be independent random variables that X1 , lr X2 and X,1 , lr X,2. Then it is not necessarily true that X1 + X2 , lr X,1 + X,2. However, if these random variables have log-concave densities, then it is true. This paper compared the random variables from scale models according to likelihood ratio ORDER, HAZARD RATE ORDER and reversed HAZARD RATE ORDER. Random variable X be said to belong to the scale family of distributions if it has the distribution function. The density function F(, x) and , f(, x), respectively, where F is an absolutely continuous distribution function with density function f and ,is the scale parameter. Material and Methods The comparison of essential characteristics associated with lifetimes of technical systems is an exciting topic in reliability theory since it usually enables us to approximate complex systems with simpler designs and subsequently obtain various bounds for important ageing characteristics of the complex system. A convenient tool for this purpose is the theory of stochastic ORDERings. Results and Discussion This paper deals with some stochastic comparisons of convolution of random variables comprising scale variables. Sufficient conditions are established for these convolutions’,likelihood ratio ORDERing and HAZARD RATE ORDERing. The results established in this paper generalize some known results in the literature. Several examples are also presented for more illustrations. Conclusion Convolutions of independent random variables occur quite frequently in probability and statistics, stochastic activity networks, optics, acoustics, electrical engineering, physics, the area of digital signal and insurance mathematics. Therefore, their stochastic properties are essential and have been discussed extensively in the literature. We obtained sufficient conditions to compare the convolution of random variables from the scale model concerning likelihood ratio and HAZARD RATE ORDER. Recently Amini-Seresht and Barmalzan (٢, ٠, ٢, ٠, ) have studied ORDERing properties of parallel and series systems consisting of outlier scale components. They provided some sufficient conditions on the parameter vectors for the likelihood ratio, HAZARD RATE, reversed HAZARD RATE and mean residual lifetime ORDERs between the lifetimes of the series and parallel systems, respectively. Therefore, a generalization of the present work to the case random variables with an outlier scale model framework will be of interest. We are working on this problem and hope to report these findings in a forthcoming paper.

A region of 390 square kilometers of Neyshabour was selected for flood zoning. This region is divided into the city and hydrographic networks. This confined area includes three geo-morphological part namely mountain, plain head (along alluvial fans) and lowland. A part of the city is located in Plain geomorphology unit and the other part is laid in the mountains area that covered surface of alluvial fan. The study on the desired area was conducted as follows: The study area is too wide therefore, it is divided into 3 sub-basins; then, the effect of urban development on the basin was estimated through topographic maps, aerial photographs and field visits. The maximum flood event of every basin is calculated by a 100- year return period. The purpose of this calculation is estimating the maximum flood volume which is entered into the urban of the mentioned basins. By providing the required layers for both the city and the basin area of the study, the weight of each layer is determined based on its importance in the occurrence of floods. (Importance of each layer was determined based on the opinions of experts in various organizations of Neyshabour). After the final weighing, the layers were compared with each other and the resultant matrix of these comparisons is used in Idrisi and eventually final coefficients were determined for each layer. Finally, by applying these coefficients through Raster calculator in ArcGis, maps of flood HAZARD zoning in the basin were determined and the boundaries of Neyshabour were specified sepaRATEly. Thus, the critical boundaries of urban area against flood were defined and flood damages in the form of the maps of the damage assessment were presented.

Using quantitative models can be a good tool for planning or evaluation. In this regard, given the multifaceted transit (economic, social, political, transportation), estimating the market size and the share of Iran's corridors is the starting point for planning. In this paper, initially, the transit potential is estimated. For prediction, it is applied the gravity model. It is based on examining the trade trends between the economic blocs in the region and beyond. In the next step, implemented the Logit model, for estimating Iran's share of the transit market. The results show that current transit demand is 92 million tones, which is expected to reach 118 million tones based on economic assumptions. Travel time through Iran's corridors without the China-Turkmenistan block is potentially reduced by up to 70%. The likelihood of TurkeyTurkmenistan, India-Ukraine, India-Caucasian economic blocks being attracted to Iran Corridors is 75%, Turkey-India, Turkey-UAE, China-Ukraine, India-Russia 73% and attracting trade between China-Turkmenistan countries and Russia-UAE is estimated at 20%.

Social discount RATE is an important variable for the Cost-Benefit analysis. Commonly used discount factor in applied works is exponential discount factor. However, exponential factor has some problems: 1- Determining factors of discount RATE and their impacts are varying in time. 2- Empirical and experimental evidences suggest that different individuals and groups of people discount far future with RATEs lower than early future. 3- If the discount RATE were uncertain, it can be shown that the discount RATE is decreasing in time. In this paper, we show that the problems can be mitigated by relating the discount RATE with HAZARD RATE. That relation can be established using project life probability distribution or its beneficiary’s mortality RATE. In this paper, the discount factors of different statistical distribution of projects life are introduced and applied to some countries and Iran. The results show the HAZARD RATE in Iran and India are higher and mean project life are lower than other countries in the sample. 

In this paper, we consider series-parallel and parallel-series systems with independent subsystems consisting of dependent homogeneous components whose joint lifetimes are modeled by an Archimedean copula. Then, by considering two such systems with different numbers of components within each subsystem, we establish HAZARD RATE and reversed HAZARD RATE ORDERings between the two system lifetimes, and also discuss how these systems age relative to each other in terms of HAZARD RATE and reversed HAZARD RATE functions.

In this paper we specify the conditions on the parameters of pairs of gOS’s under which the corresponding generalized ORDER statistics are ORDERed according to usual stochastic ORDERing, HAZARD RATE ORDERing, likelihood ratio ORDERing and dispersive ORDERing. We consider this problem in one-sample as well as two-sample problems. We show that some of the results obtained by Franco et al. [Probab.Engrg. Inform. Sci. 2002, 16, 471-484] and Belzunce et al. [Probab.Engrg. Inform. Sci. (2004), to appear] for stochastic ORDERings of gOS’s are contained in our new results.