1.令Y1 <Y2 <Y3 <Y4是来自分布的大小为n = 4的随机样本的阶统计

（a）计算P（Y1 <0：1）
（b）计算P（Y2 <0：2）。
（c）通过使用R进行仿真来近似（b）中的概率；至少进行2000次模拟运行；

（d）简述如何才能确定模拟运行的次数，以便

2.让X1; :::; Xn是i.i.d.在间隔上统一[; 0]; <0。
（a）找到的最大潜能。
（b）在（a）部分中确定估算器的MSE。
（c）在（a）中修改MLE，使其成为的无偏估计量；然后确定其MSE。
（c）当n = 5时，在同一张图上绘制两个MSE。
3.订单统计的应用。给定以下25个观察值，确定

n + 1）（或1（
ķ
n + 1），如果您对观测值rst）进行排序，然后
（a）计算样本相关系数；
（b）获得正态概率图；即成对的散点图（1（

n + 1）; Xk）。
2.86 0.39 -1.98 -0.47 6.67 -1.21 3.56 -0.19 0.63 1.54 1.18 -0.01 -0.36 3.40 -1.84 2.28 0.81 2.58
1.74 -2.16 3.36 1.32 4.31 -2.14 4.41
1（）是标准正态分布的CDF的倒数。
4.假设fX1; X2; :::; Xng是来自连续分布的大小为n的随机样本，

X1; X2;的订单统计:::; Xn。令Xn + 1是来自同一点的另一个观察结果

（a）找出P（Yk> Xn + 1）; 1 k n：
（b）假设独立观测值取自标准正态分布
n = 10，通过使用R的仿真估计（a）中的概率；至少使用2000次模拟

（c）假设独立意见是取自有关资料的不正当分布─

5.令Y1 <Y2 <::: <Yn是分布中大小为n的随机样本的阶次统计量
pdf f（x）= 1时； 0 <x <1;零在其他地方。证明k阶统计量Yk具有参数= k和= n k + 1的beta分布：有关beta的定义，请参见第181页

Homework 12
Read Section 4.4 on Order Statistics (pages 262 – 263) and the lecture notes before you answer
problems 1 – 8.
1. Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 from distri-
bution with probability density function f(x) = 2(1 x); 0 < x < 1, and zero elsewhere.
(a) Compute P(Y1 < 0:1)
(b) Compute P(Y2 < 0:2).
(c) Approximate the probability in (b) by simulation using R; use least 2000 simulation runs;
compare the result with the theoretical answer you obtained in (b).
(d) Describe brie y how can we determine the number of simulation runs such that the ap-
proximate value is within 0:05 of the true probability?
2. Let X1; :::;Xn be i.i.d. uniform on the interval [; 0];  < 0.
(a) Find the mle of .
(b) Determine the MSE of the estimator in part (a).
(c) Modify the MLE in (a) so it becomes an unbiased estimator of ; then determine its MSE.
(c) Plot both MSEs on the same graph when n = 5.
3. An application of order statistics. Given the following 25 observed values, determine the
normal scores,  1(
rank(Xk)
n+1 ) (or  1(
k
n+1 ) if you sort the observed values rst), and then
(a) compute the sample correlation coecient;
(b) obtain the Normal Probability Plot; i.e., the scatter plot of the pairs ( 1(
rank(Xk)
n+1 );Xk).
2.86 0.39 -1.98 -0.47 6.67 -1.21 3.56 -0.19 0.63 1.54 1.18 -0.01 -0.36 3.40 -1.84 2.28 0.81 2.58
1.74 -2.16 3.36 1.32 4.31 -2.14 4.41
Here  1() is the inverse of the CDF of standard normal distribution.
4. Suppose that fX1;X2; :::;Xng is a random sample of size n from a continuous distribution,
with probability density function f(x), on the interval ( 1;1). Let Y1 < Y2 < ::: < Yn be
the order statistics of X1;X2; :::;Xn. Let Xn+1 be an additional observation from the same
continuous distribution which is independent of the other n observations X1;X2; :::;Xn.
(a) Find P(Yk > Xn+1); 1  k  n:
(b) Assume the independent observations are taken from the standard normal distribution
and n = 10, estimate the probability in (a) by simulation using R; use least 2000 simulation
runs; compare the result with the theoretical answer in (a).
(c) Assume the independent observations are taken from the unform distribution on the in-
terval [0; 1] and n = 10, estimate the probability in (a) by simulation using R; use least 2000
simulation runs; compare the result with the theoretical answer in (a) and the simulated
result in (b).

5. Let Y1 < Y2 < ::: < Yn be the order statistics of a random sample of size n from a distribution
with pdf f(x) = 1; 0 < x < 1; zero elsewhere. Show that the kth order statistic Yk has a betadistribution with parameters = k and = n k+1: See page 181 for the de nition of beta
distribution.

EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

E-mail: easydue@outlook.com  微信:easydue

EasyDue™是一个服务全球中国留学生的专业代写公司