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Problem sheet 5The next question is optional, yet it will be a bonus to your final score.2. If the density function is compactly supported, standard kernel density estimators are invalid/inconsistent at ornear boundary points. Below we introduce a new estimator that automatically adapts to the (possibly unknown)boundaries of the support of the density without requiring specific data modification or additional tuning parameters.Let be a random sample, where is a continuous random variable with a smooth CDF over its possiblyunknown support for . If and , then is supported on the whole real line. Thedensity function is , where the derivative is interpreted as a one-sided derivative at a boundarypoint of . Let be the empirical distribution function. For every , solvewhere is a bandwidth, and denotes a nonnegative, symmetric and continuous kernel function supported on. The proposed boundary adaptive density estimator is then defined as.X1,…, Xn Xi[a, b] a f (x) = dd x(Xi ≤ x)[a, b] F(x) = (1/n)∑ni=1I(Xi ≤ x) x ∈= argminβ=(β0,β1,β2)(Xi) β0β1(Xi x) β2(Xi x)2}2K(Xi xh ),h > 0 K[1,1]f ba(x) = β?1(x)1(a) Write an R function to implement the above estimator, with as an input parameter. Use the triangularkernel , .(b) Conduct simulation study based on exponential distribution with density function if and if . This density function is then supported on . We want to estimate the density at, where is the boundary, is a near-boundary point and is an interior point. Take the samplesize . For various choices of the bandwidth , conduct 5000 Monte Carlo repetitions, and report theaverage estimation error and the corresponding standard deviation at .(c) Use 5-fold cross-validation to select the bandwidth for the above estimator. Again, for exponential distribution,take and conduct 5000 Monte Carlo repetitions to compare the performance of this estimator withstandard kernel density estimator (bandwidth chosen by default) at .h > 0K(u) = max{0,1 |u |} u ∈f (x) = e?x x ≥ 0f (x) = 0 x x ∈ {0,0.5,1} 0 0.5 1n = 1000 h| f ba(x) f (x)| x ∈ {0,0.5,1}n = 1000x ∈ {0,0.5,1}本团队核心人员组成主要包括硅谷工程师、BAT一线工程师，精通德英语！我们主要业务范围是代做编程大作业、课程设计等等。我们的方向领域：window编程 数值算法 AI人工智能 金融统计 计量分析 大数据 网络编程 WEB编程 通讯编程 游戏编程多媒体linux 外挂编程 程序API图像处理 嵌入式/单片机 数据库编程 控制台 进程与线程 网络安全 汇编语言 硬件编程 软件设计 工程标准规等。其中代写编程、代写程序、代写留学生程序作业语言或工具包括但不限于以下范围:C/C++/C#代写Java代写IT代写Python代写辅导编程作业Matlab代写Haskell代写Processing代写Linux环境搭建Rust代写Data Structure Assginment 数据结构代写MIPS代写Machine Learning 作业 代写Oracle/SQL/PostgreSQL/Pig 数据库代写/代做/辅导Web开发、网站开发、网站作业ASP.NET网站开发Finance Insurace Statistics统计、回归、迭代Prolog代写Computer Computational method代做因为专业，所以值得信赖。如有需要，请加QQ：99515681 或邮箱：99515681@qq.com 微信：codehelp