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本次美国代写主要为数学统计的homework

1.对于两个因素(淀粉或含糖的)和绿色基叶或白色基叶|以下
观察到自体受精杂合子的后代计数(Fisher 1958):
类型计数
淀粉绿1997
淀粉白906
甜白904
甜绿32
根据遗传理论,细胞概率为(2+)= 4; (1)= 4; (1)= 4;和= 4;
其中(0 <<1)是与因子链接有关的参数。
(a)找出的最大均方根及其渐近方差。
(b)根据(a)部分形成大约95%的置信区间。
2.让X1; :::; Xn是具有参数的泊松分布中大小为n = 10的随机样本
。请考虑以下两种情况:
(a)假设参数空间=全部> 0的集合(即。可以是任何正数
数),确定的最大似然估计量的采样分布。
(b)假设参数空间=所有可能值= f1的集合; 1:5g,确定
的最大似然估计。
(c)找出(b)中的最大似然估计等于真实值的概率
的价值。您必须提供特定的数学表达式,但无需计算
最终数值。
(d)在(c)部分中找到概率的正态近似值。
3.假设X1; X2; :::; Xn是i.i.d.普通均值= 0的正常随机变量
和共同方差2。
(a)设= 2。找到的MLE,然后表明它没有偏见。还要找出方差
MLE。提示:使用第192页定理3.4.1的结果和推论的结果
第180页的3.3.1。
(b)找到Cramer-Rao下界,然后验证部分mle的方差
(a)达到下限。因此,这种估计是有效的。即,它具有最小的方差
在的所有无偏估计量中
(c)令b为(a)部分的估计量。的渐近分布是什么
pn(b)?
4.问题6.2.14,第376页。在模拟中取= 3。
(d)[其他部分]:重复(c)部分至少2000次,然后确定模拟的
可信度(即模拟的覆盖率)。
5.问题6.4.1,第394页。首先阅读示例6.4.5,在第391页。提示:使用表达式(4.2.7)
在第241页。

6.问题5.2.18,第340页。阅读有关增量法或定理5.2.9的讲义,第340页。
335。
(b)[附加部分]:模拟分配
pn(
pXn 1)至少进行2000次模拟
运行。考虑两种情况:n = 10和n = 40。
(c)[其他部分]:获取直方图,然后覆盖限制分布的pdf-
(a)部分获得的结果。
7.请参阅上面的问题3。当2 = 1 =时模拟MLE的值
p2和n = 40
至少3000次获取直方图,然后使用
从问题3获得的方差。
8.考虑与上述问题5类似的情况。考虑4单元多项式分布
n = 200,单元格概率为p1; p2; p3和p4:令b pi为pi的MLE; i = 1; 2; 3; 4:
当p1 = 60 = 200时,模拟这些MLE的值至少5000次。 p2 = 45 = 200; p3 =
70 = 200;并且p4 = 25 = 200:
(a)获得(b p1; b p2)的散点图?模拟协方差是否与理论相符
第391页的示例6.4.5中给出的协方差?
(b)获得(b p3; b p4)的散点图?模拟的协方差是否与理论相符
第391页的示例6.4.5中给出的协方差?
(c)将(b p1; b p2)的值作为t的合适的二维分布是什么?
解释。
(d)将(b p3; b p4)的值作为t的合适的二维分布是什么?
解释。

1. For two factors (starchy or sugary), and green base leaf or white base leaf|the following
counts for the progeny of self-fertilized heterozygotes were observed (Fisher 1958):
Type Count
Starchy green 1997
Starchy white 906
Sugary white 904
Sugary green 32
According to genetic theory, the cell probabilities are (2+ )=4; (1 )=4; (1 )=4; and =4;
where (0 <  < 1) is a parameter related to the linkage of the factors.
(a) Find the mle of  and its asymptotic variance.
(b) Form an approximate 95% con dence interval for  based on part (a).
2. Let X1; :::;Xn be a random sample of size n = 10 from a Poisson distribution with parameter
. Consider the following two cases:
(a) Assume the parameter space  = the set of all  > 0 (i.e., . could be any positive
number), determine the sampling distribution of the maximum likelihood estimator of .
(b) Assume the parameter space  = the set of all possible values of  = f1; 1:5g, determine
the maximum likelihood estimator of .
(c) Find the probabilities that the maximum likelihood estimator in (b) equals to the true
value of . You must provide speci c mathematical expressions but don’t need to compute
the nal numerical values.
(d) Find normal approximations to the probabilities in part (c).
3. Suppose that X1;X2; :::;Xn are i.i.d. normal random variables with common mean  = 0
and common variance 2.
(a) Let  = 2. Find the MLE of  and then show that it is unbiased. Also nd the variance
of the MLE. Hint: Use the result of Theorem 3.4.1 on page 192 and the result of Corollary
3.3.1 on page 180.
(b) Find the Cramer-Rao lower bound and then verify that the variance of the mle in part
(a) attains the lower bound. Hence this estimator is ecient; i.e., it has smallest variance
among all unbiased estimators of .
(c) Let b  be the estimator in part (a). What is the asymptotic distribution of
pn(b  )?
4. Problem 6.2.14, page 376. Take  = 3 in the simulation.
(d) [Additional part]: Repeat part (c) at least 2000 times and then determine the simulated
con dence coecient (i.e, the simulated coverage probability).
5. Problem 6.4.1, page 394. First read Example 6.4.5 on page 391. Hint: Use expression (4.2.7)
on page 241.

6. Problem 5.2.18, page 340. Read the lecture notes on delta-method or Theorem 5.2.9 on page
335.
(b) [additional part]: Simulate the distribution of
pn(
pXn 1) with at least 2000 simulation
runs. Consider two cases: n = 10 and n = 40.
(c) [additional part]: Obtain the histogram and then overlay the pdf of the limiting distribu-
tion obtained in part (a).
7. Refer to Problem 3 above. Simulate the values of the MLE when 2 = 1=
p2 and n = 40 for
at least 3000 times. Obtain the histogram and then overlay the approximate normal pdf with
variance obtained from Problem 3.
8. Consider a similar situation to Problem 5 above. Consider a 4-cell multinomial distribution
with n = 200 and cell probabilities p1; p2; p3 and p4: Let b pi be the MLEs of pi; i = 1; 2; 3; 4:
Simulate the values of these MLEs for at least 5000 times when p1 = 60=200; p2 = 45=200; p3 =
70=200; and p4 = 25=200:
(a) Obtain the scatter plot of ( b p1; b p2)? Does the simulated covariance match the theoretical
covariance given in Example 6.4.5 on page 391?
(b) Obtain the scatter plot of ( b p3; b p4)? Does the simulated covariance match the theoretical
covariance given in Example 6.4.5 on page 391?
(c) What would be an appropriate 2-dimensional distribution to t the values of ( b p1; b p2)?
Explain.
(d) What would be an appropriate 2-dimensional distribution to t the values of ( b p3; b p4)?
Explain.


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