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Instructions: Solutions to problems 1 and 2 are to be submitted on Blackboard (PDF lesstrongly preferred) { the deadline is 11:59pm on March 16. You are strongly encouraged todo problems 3 through 6 but these are not to be submitted for grading.Problems to hand in:1. Suppose that D1; ;Dn are random directions { we can think of these random variablesas coming from a distribution on the unit circle f(x;y) : x2 + y2 = 1g and represent eachobservation as an angle so that D1; ;Dn come from a distribution on [0;2 ).(Directional or circular data, which indicate direction or cyclical time, can be of great interestto biologists, geographers, geologists, and social scientists. The de ning characteristic of suchdata is that the beginning and end of their scales meet so, for example, a direction of 5 degreesis closer to 355 degrees than it is to 40 degrees.)A simple family of distributions for these circular data is the von Mises distribution whosedensity on [0;2 ) iswhere 0 0 and 0 0 and> 0, use the results of your simulation to estimate the value of a and . (Hint: If dVar(b n)is a Monte Carlo estimate of the variance of the half sample mode for a sample size n, thenln(dVar(b n)) ln(a) ln(n);which should allow you to estimate a and .)(c) It can be shown that Xn is independent of b n Xn where b n is the half sample mode.(This is a property of the sample mean of the normal distribution { you will learn this inSTA452/453.) Show that we can use this fact to estimate the density function of b n fromthe Monte Carlo data by the kernel&" estimatorwhere 2 = Var(Xi), (t) is the N(0;1) density, and D1; ;DN are the values of b n Xn from the Monte Carlo simulation. Use the function density (with the appropriatebandwidth) to give a density estimate for the half sample mode for n = 100.Supplemental problems (not to hand in):3. Suppose that X1; ;Xn are independent random variables with density or mass func-tion f(x; ) and suppose that we estimate using the maximum likelihood estimator b ; weestimate its standard error using the observed Fisher information estimatorwhere ‘0(x; );‘00(x; ) are the rst two partial derivatives of lnf(x; ) with respect to .Alternatively, we could use the jackknife to estimate the standard error of b ; if our modelis correct then we would expect (hope) that the two estimates are similar. In order toinvestigate this, we need to be able to get a good approximation to the leave-one-out&"estimators fb ig.(a) Show that b i satis es the equation(b) Expand the right hand side in (a), in a Taylor series around b to show that(You should try to think about the magnitude of the approximation error but a rigorousproof is not required.)(c) Use the results of part (b) to derive an approximation for the jackknife estimator of thestandard error. Comment on the di erences between the two estimators - in particular, whyis there a di erence? (Hint: What type of model { parametric or non-parametric { are weassuming for the two standard error estimators?)(d) For the air conditioning data considered in Assignment #1, compute the two standarderror estimates for the parameter in the Exponential model (f(x; ) = exp( x) forx 0). Do these two estimates tell you anything about how well the Exponential model tsthe data?4. Suppose that X1; ;Xn are independent continuous random variables with densityf(x; ) where is real-valued. We are often not able to observe the Xi’s exactly rather onlyif they belong to some region Bk (k = 1; ;m); an example of this is interval censoring insurvival analysis where we are unable to observe an exact failure time but know that thefailure occurs in some nite time interval. Intuitively, we should be able to estimate moree ciently with the actual values of fXig; in this problem, we will show that this is true (atleast) for MLEs.Assume that B1; ;Bm are disjoint sets such that P(Xi2[mk=1Bk) = 1. De ne independentdiscrete random variables Y1; ;Yn where Yi = k if Xi2Bk; the probability mass functionof Yi isUnder the standard MLE regularly conditions, the MLE of based on X1; ;Xn will havevariance approximately 1=fnIX( )g while the MLE based on Y1; ;Yn will have varianceapproximately 1=fnIY ( )g.(a) Assume the usual regularity conditions for f(x; ), in particular, that f(x; ) can bedi erentiated with respect to inside integral signs with impunity! Show that IX( ) IY ( )and indicate under what conditions there will be strict inequality.Hints: (i) f(x; )=p(k; ) is a density function on Bk.(ii) For any function g,g(x)f(x; )p(k; ) dx:(iii) For any random variable U, E(U2) [E(U)]2 with strict inequality unless U is constant.(b) Under what conditions on B1; ;Bm will IX( ) IY ( )?5. In seismology, the Gutenberg-Richter law states that, in a given region, the number ofearthquakes N greater than a certain magnitude m satis es the relationshiplog10(N) = a b mfor some constants a and b; the parameter b is called the b-value and characterizes the seismicactivity in a region. The Gutenberg-Richter law can be used to predict the probability oflarge earthquakes although this is a very crude instrument. On Blackboard, there is a lecontaining earthquakes magnitudes for 433 earthquakes in California of magnitude (roundedto the nearest tenth) of 5.0 and greater from 1932{1992.(a) If we have earthquakes of (exact) magnitudes M1; ;Mn greater than some known m0,the Gutenberg-Richter law suggests that M1; ;Mn can be modeled as independent randomvariables with densityf(x; ) = exp( (x m0)) for x m0.where = b ln(10). However, if the magnitudes are rounded to the nearest then theyare e ectively discrete random variables taking values xk = m0 + =2 +k for k = 0;1;2; with probability mass functionp(xk; ) = P(m0 +k M 0 and > 0 are hyperparameters. What is the posterior distribution of givenXAA = x1, XAa = x2, and Xaa = x3本团队核心人员组成主要包括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：[email protected] 微信：codehelp QQ：99515681 或邮箱：[email protected] 微信：codehelp