In many problems related to reliability and survival analysis, the evaluation of uncertainty and information associated with system lifetimes must be performed conditional on the current age (survival condition) of the system. >The aim of this thesis is to study, analyze, and organize the existing knowledge on generalizations of dynamic cumulative information measures and their applications in reliability and model selection. To this end, we first extract the basic definitions of cumulative information measures (including cumulative residual entropy, cumulative residual Fisher information, and past-based versions) and review their analytical relationships with hazard rate and mean residual lifetime functions. Then, the theoretical properties of these measures are compared in specific models (equilibrium and proportional hazards). Next, nonparametric estimation methods (particularly kernel-based estimators) and Monte Carlo simulation results for various distributions are collected and evaluated. Finally, the practical application of dynamic cumulative information measures in selecting competing lifetime distributions (with emphasis on real reliability data) is described, and their performance is compared with classical criteria such as AIC and BIC.

   Abstract The problem of variable selection is a crucial and difficult aspect of statisticai modeling. Choosing the wrong predictor variables for the final model can result in either underfitting or overfitting. It is crucial to control the rate of false discovery during the inference stage, in variable selection methods. A false discovery rate is said in proportion to auxiliary variables that are improperly chosen among all selected variables. False discovery rate control has been studied for a long time, and several approaches have been developed to achieve an optimal solution. Recently, a new family of FDR control methods called “Knockoff filters” has been introduced. This thesis addresses the issue of variable selection in regression models error false discovery rate control whit various knockoff versions, including fixed-X knockoffs and X-model knockoffs. These knockoff methods can be suitable structures to imitate the existing variables and control the false discovery rate to acceptable limits. The performance of each type of knockoff method will be reviewed in comparison whit each other and some common methods to control the false discovery rate. The efficiency of the presented models has been studied using the data obtained from the simulation.       Keywords: variable selection, sparse regression, knockoff filter, false discovery rate.   

  Clustering longitudinal data is a complex task that requires taking into account the similarity of individual trajectories despite scattered and irregular observed times. Clustering is a widely used statistical technique in various fields, such as unsupervised machine learning, data mining, pattern recognition, image analysis, and bioinformatics. This thesis reviews and studies several multivariate longitudinal data clustering algorithms as well as introduces a new clustering algorithm called ClusterMLD. This new method shows promise in identifying meaningful patterns in high-dimensional longitudinal data. The algorithms have been compared using simulation studies and real data to evaluate their performance.

   In this research, heavy metal removal from wastewater was investigated using a combination of electro-Fenton and membrane filtration methods.The integration of these methods was done with the aim of increasing the purification efficiency, and the effect of operating parameters, reaction time, current density, solution acidity (pH), volume ratio of hydrogen peroxide to wastewater, molar ratio of hydrogen peroxide to ferrous ion (Fe2+), nanoparticle concentration and concentration of the input feed was evaluated on the removal percentage of this pollutant. In order to optimize the operating parameters with the aim of maximizing the removal percentage of this pollutant, two optimization methods, the D-optimal criterion which is a real valued function of the value according to Fisher's information matrix and the combined method of artificial neural network-genetic algorithm have been used. The aim of   comparison of statistical analysis for these methods is finding an objective function with the lowest mean squared error.

to the importance of its usage. One of the most important challenges in discussing theprogressive Type-II censored data is to determine the removal scheme. The removal schemecan be fixed or randomly selected according to a discrete probability distribution. Thisthesis considers the estimation problem for the two-parameter Pareto distribution underprogressive Type-II censoring with random removals, where the number of units removedat each failure time has a binomial distribution. The main focus of this study is on theBayesian estimates of Pareto distribution using Jeffery’s non-information and InformativePower Gamma distributions as priors for the unknown parameters under the squared errorand absolute error loss functions. Furthermore, the statistical performances of the obtainedestimators are compared with each other and with the maximum likelihood estimators.The comparisons have been done by Monte Carlo simulation. Finally, the E-Bayesian andhierarchical Bayesian estimations of the parameter derived from Pareto distribution arestudied and compared under different loss functions.  

Optimal designs have an important role in many applied areas such as medical, engineering, pharmaceutical and marketing studies. Using such as categories of designs designs can considerably reduce the cost of experiment. Finding an optimal design requires pre-specifying a criterion which needs to be optimized. For samples, these criteriaare chosen as functions of Fisher information matrix. The most popular criterion is D-optimal criterion which is determinant of Fisher information matrix. In a nonlinear model, dependency of this matrix onunknown parameters in an optimal design problem. A number of approaches such as Bayesian, Locally and Minimax optimal designs are suggested in order to solve dependency on the parameters.\\\\In this thesis we study the effect of the choice of different a priori distributions, such as the Uniform, Gamma and Lognormal distributions in obtaining the D-optimal designs for a non-linear model, when the errors present different correlation structures. we study the effect of the choice of different a priori distributions, such as the Uniform, Gamma and Lognormal distributions in obtaining the D-optimal designs for a non-linear model, when the errors present different correlation structures. In order to calculate these designs the Monte Carlo method is used and a general methodology is proposed that allows to find D-optimal designs for any type of non-linear model in the presence of correlated observations, later the designs found are compared by calculating the efficiencies taking as a reference design the one obtained with the a priori Uniform distribution, evidencing that depending on the selected correlation structure there is an a priori effect and finally through the information criteria AIC and BIC the best correlation structure is selected among the structures chosen for then make a simulation study with the purpose of checking and verifying from the point of view of the proposed statisticians.

The intuition of risk is based on two main concepts: the possibility of a negative outcome, i.e. a lo   and the variability in terms of an expected result, i.e. a deviation. Since the time when the modern theory of finance was accepted, the role of risk measurement has attracted attention. Initially, it was predominantly used as a dispersion measure, such as variance, which contemplates the second pillar of the intuition. More recently, the occurrence of critical events has turned the attention to tail-risk measurement, as is the case of well known Value at Risk (VaR) and Expected Shortfall (ES)measures, which contemplate the first pillar. In this Thesis, a risk measure is considered which contemplate both pillars of intuition on risk. These pillars include the possibility of negative results and variability over an expected result, as a single measure. This resulting composition, based on properties of the two components, is a coherent risk measure. Similar results for the cases of convex and co-monotone risk measures are exposed. Then, the eleven well-known risk measures consider from different classes. Finally, the empirical values of corresponding to loss, deviation and loss-deviation risk measures are obtained and compared using Monte Carlo and real data.   

In analyzing big data, time of computations is increased, so in data mining algorithms cannot use all the data. Therefore, using sampling methods in big data set is a good solution.\\\\In statistical studies of multivariate populations, obtaining information over all variation range of variables is very important. Since it is difficult or impossible to select all data, the required information can be obtained by survey a subpopulation as a sample. In such cases, the appropriate sample can be selected by LPM2-kdtree method.\\\\Also, in big data analysis, selection bias is very important. In this thesis, in order to decrease the bias by using importance sampling a method is explained. Finally, in a numerical study on two real populations, the spatially balance of LPM2-kdtree and decreasing selection bias of the sampling design that uses importance sampling are evaluated.\\\\ \\textbf{Keywords:} {Big Data}, {Clustering}, {Data Mining}, {Inverse sampling}, {Knowledge Discovery}, {Non-probability sampling}, {Selection bias} . \\end{latin}

Our research is applied in terms of purpose. Because the proposed model lays solutions to improve the premium determination and generally improve the performance of insurance companies.We offer model forecasting methods to determine the premium rate,That detects data exploration and modeling. Among these methods, the accelerated gradient is a method in composite Poisson model. Since the main variables and interaction effects used in the models are, therefore, a tree accelerated gradient algorithm with the name TDboost offer visited. Also for data with a large zero accumulation The methods will be provided to make the premium forecast possible . First, we will discuss the definitions and concepts required in insurance science . So we introduce and examine the accelerated gradient tree model .In the third chapter, we implement a model for the survey of the database composite Poisson with insurance studies data.In the fourth chapter, we will analyze and compare non-parametric models using data sets, And finally, we will conclude our suggestions and conclusions.  

گاهي اوقات ممكن است نمونه اي كه مشاهده مي كنيم نمونه اي اريب از جامعه باشد. به اين معنا كه تمام اعضا از شانس برابري براي انتخاب شدن در نمونه برخوردار نيستند. براي حل اين مشكل از نسخه اريب-طول كه نسخه ي وزني شده از متغير تصادفي اصلي جامعه است، استفاده مي شود. در نمونه گيري اريب-طول شانس حذف شدن هر واحد از نمونه نااريب، متناسب با طول عمر آن واحد مي باشد. حال آن كه در برخي حالات ممكن است شانس حذف شدن متناسب با طول عمر واحد تحت مطالعه نباشد، از اين رو براي حل اين مشكل مي توان از توزيع هاي وزني استفاده كرد. در اين پايان نامه توزيع اصلي جامعه را گسسته در نظر گرفتيم و سپس با استفاده از تابع ميانگين مانده عمر، توزيع پواسون بريده شده وزني شده و توزيع پواسون آماسيده در صفر بريده شده وزني شده را معرفي كرديم و ويژگي هاي آن ها را مورد بررسي قرار داديم، همچنين نشان داديم مشاهداتي كه ميانگين مانده عمر بزرگ تري دارند شانس بيشتري براي انتخاب شدن در نمونه را دارند. توزيع هاي وزني شده داراي كاربردهاي فراواني در مبحث تحليل بقا و قابليت اعتماد مي باشند، به همين دليل علاوه بر روش شبيه سازي، برخي موارد كاربرد آن با استفاده از داده هاي واقعي نيز تشريح شده است. 

  In this distribution after introducing the concept of expected shortfall (ES) as a financial risk measure we briefly discuss the coherence properties of this measure. This statistic risk measure arises in a natural way from the estimation of the average of the 100p percent worst losses in a sample of returns to a portfolio where p is some fixed confidence level. Then we comprehensively review several know parametric method for estimating expected shortfall.  

Financial institutions are always exposed to investment risk.

  Determining the suitable premium for an insurer is one of the most important categories in the insurance industry. Inmostbonus-malussystems premiumbasedonthenumberofclaims Theclaimamountsarenot taken into accont. In this case, policyholders who had accidents with small or large claims are penalized unfairlyinthesameway. Ateventhepolicyholdersmayleavetheinsurancecompanytogetridoftheirbad history claims. In this thesis, in addition to the number of claims, the amount of claim is also considered. Also, in the malus zone, relative premiums softened by introducing and applying deductible  

  We all have experienced the discomfort of waiting the queue.traffic or for paying tolls and ... .Themain reason for queuing is that demand for service is more than the service facilitties. Queuetheory is linked to factors such as queuing time,queue length,etc.With the expected properties ofthe flow of input to the system and service practices,it proposes an optimal system to reduce thedamage caused by queuing.In this thesis,after presenting the introduction and initial concepts, an invistigation of an unlimitedsource system and its prominent features,including the number of applicants in queue andsystem,waiting time,and so on will be done.In this model inputs have poisson distribution and servicetime exponential distribution.We then examine the bayesian queues M/M/1 and M/M/s,andwe see that there are some queuing features that are not mathematical expectation for them.Thenwe obtain the point estimate,distance and hypothesis test for Bayes systems and finally,we willsimulate what we have been studying at the end.

  The research on optimal reinsurance design has a long history for academicians and practitioners. Because it is an effective risk management tool for insurers. Depending on the chosen objective and constraints , there are many ways for optimal design of reinsurance.The primary objective of the thesis is to examine theoretically sound and yet practical solution in the quest for optimal reinsurance designs. In order to achieve such an objective, this thesis is divided into two parts. In the first part, a numberof reinsurance models are examined and their optimal reinsurance treaties are derived. This part focuses on the risk measure minimization reinsurance models and discusses the optimal reinsurance treaties by exploiting two of the mostcommon risk measures known as the Value-at-Risk (VaR) and the Conditional Tail Expectation (CTE). Some additional important economic factors such as the reinsurance premium budget, the insurer’s profitability are also considered. The second part proposes an innovative method in formulating the reinsurance models, which we refer as the empirical approach since it exploits explicitly the insurer’s empirical loss data. The empirical approach has the advantage that it is practical and intuitively appealing. This approach is motivated by the difficulty that the reinsurance models are often infinite dimensional optimization problems and hence the explicit solutions are achievable only in some special cases. The empirical approach effectively reformulates the optimal reinsurance problem into a finite dimensional optimization problem. Furthermore, we demonstrate that the second-order conic programming can be used to obtain the optimal solutions for a wide range of reinsurance models formulated by the empirical approach.

  When determining the locating of the units with the largest number of variables studied (Hot spots), determining the most efficient  sampling method is important. Some ofthe methods employed in spatial communities are systematic sampling, stratifiedsampling method, and adaptive sampling method. In this thesis, theeffectiveness of various sampling methods including simple random sampling, systematicsampling, stratified sampling and cluster adaptive sampling for locating Hot spots have been studied.  

  Measuring and comparing risk in the insurance management process is essential, because of the risk measurement in banks, insurance companies and financial.The basis for desicion making is to allocate their resources.One way, the comparison based on some of the measure of the important risks and compare the risk with the stochastic order.In this dissertation, we propose a generalization of the increasing convex order to the multivariate setting to compare vectors of risks that accounts for both the marginal impacts and the dependence structures ofthe vectors. This generalization is suitable for comparing vectors with heterogeneous components andextends some well-known properties of the univariate increasing convex order. For example, comparisonsof vectors with the same copula can be characterized in terms of the multivariate tail conditionalexpectations introduced by Cousin and Di Bernardino. Also the multivariate extensions of the risk measures, tail conditional expectation and value at risk are presented and their invariance with respect to the univariate case are characterized.

Efforts to find solutions to prevent the risk and damage caused by accidents have been one of the most overwhelming human issues throughout history. Insurance is one of the most efficient industries or methods that can help them reduce the harmful effects of risk. One of the most important and practical aspects of insurance statistics is the investigation of claims distributions that follow a heavy tail distribution in heavy economic losses that are more economically important. These heavy tailings include Elliptic, Laplace, T-student, Normal, Paratou,Logistic,... distributions. The various risk measures, including tail variance, conditional variance, value at risk, conditional value at risk,... under these distributions, are designed to minimize risk and increase returns. We will benefit from these conditions. In this thesis, we focus more on tail variance and tail conditional expected and their application.

Determination of optimal design for testing procedures is one of the major topics of interest to most statisticians. These designs are commonly obtained by optimizing some functions of information matrix to acquire the most appropriate parameter estimates. But many results on optimal designs of experiments are derived under the assumption that the statistical model is known at the design stage. Thus, the purpose of an experiment should be dual: to determine which of more rival models is the more adequate and then to estimate the parameters of the chosen model. In the present thesis, study DKL and DT optimality

 In this thesis‎, ‎we the first study point process and their models and summary functions‎. ‎Then‎, ‎we discussion global envelope test for spatial processes‎. ‎Finally‎, ‎we compare power of stated test with other test for simulated and real data‎.  In this thesis‎, ‎we the first study point process and their models and summary functions‎. ‎Then‎, ‎we discussion global envelope test for spatial processes‎. ‎Finally‎, ‎we compare power of stated test with other test for simulated and real data‎.

In many from life ‎testing and reliability ‎studies, the experimenter may not always obtain complete information on failure times for all experimental units. for Example, individuals in a clinical trial may drop out of the study, may have to be terminated for lack of funds or in an industrial experiment, units may break accidentally. ‎Therefore, one has to remove some units prior to failure for saving time and cost associated with testing. Data obtained from such experiments are called censored. The most common censoring schemes are type I and type II. The Type I and Type II censoring schemes have major deficiency in that they only do not allow removal of units at points other than the terminal point of the experiment. Due to experimenter use a versatile scheme of censoring called progressive.‎This thesis has been focused on the scenario of progressive Type II censoring. A problem associated with this scheme is that the total testing duration might be unacceptably long. To address this issue, a hybrid variant of the ‎progressive ‎‎censoring scheme was proposed in which imposing a time limit T on the test. Although this hybrid ‎progressive ‎‎censoring scheme controls the total testing duration not larger than T, it is possible that the effective sample size is very small or even zero in which usual efficient statistical inference may not be feasible. To strike a balance between the ‎total ‎testing ‎time ‎and ‎the ‎efficiency ‎in ‎statistical ‎inference, ‎Ng ‎et.al ‎(2009)‎ proposed an adaptive Type II progressive censoring scheme.In this thesis influential methods for progressive Type II censoring and adaptive Type II progressive censoring under some statistical distributions are studied. We ‏‎obtain‎ both maximum likelihood estimators, approximate maximum likelihood estimators and observed information matrix for the unknown parameters. We simulated values of the estimates parameters, a comparison of the values variance and covariance of the estimators with those obtained from the corresponding observed information matrix and coverage probabilities for pivotal quantities based estimators. Various interval estimation methods for the unknown parameters such as asymptotic confidence intervals with both observed information matrix and fisher information matrix, percentile bootstrap and bootstrap-t confidence intervals are obtained. Then these methods are compared in terms of their expected lengths and coverage probabilities using simulation.

 Comparing risks play an important role in insurance statistics. One of the ways to compare risks is by using the measures of risk. In actuarial literature, considering tail variabilities of risks which are   low frequency and   high severity losses is vital. While in many cases, comparing based on different risk measures will be followed various results as well as using a risk measure in different cases. Furthermore, we cannot also propose an explicit expressions for risk measure under a special statistical distribution.   Because of these limitations, actuaries should use stochastic orders for ordering risks. Therefore, comparison of random risks with using the   functions of probability   distributions as Tail ,   top Loss,   excess-Mean functions and etc is more helpful than comparing based on some   umerical criteria associated distributions. The comparison of the Random risks with using mentioned functions which usually produce partial orders among probability distributions is called “Stochastic Orders.”In this thesis, firstly, The risk concept, some measures of risk and type of stochastic orders are introduced which allow us to compare variabilites between random variables. Then, the relationships between the excess wealth order and other familiar stochastic variability orders (Dispersive Order, Stop Loss, convex, star and mean-excess) has been studied. In the reminder of this thesis, some characterizations of variability stochastic orders are showed by using the usual and distortion risk measures.