题意：

概率论与数理统计中的几个小题

解析：

问题1：使用R语言统计数据生成图像；假设服从正态分布，计算参数的极大似然估计；绘制密度直方图；绘制QQ图

一般求解极大似然估计步骤写出似然函数，再对似然函数取对数并整理，然后求导数，最后解似然方程 。

问题2：按照给出的离散随机变量计算：a.计算期望和方差、使用矩估计计算θ和计算标准差，b.求解F1、F2、F3的似然函数、最大似然估计和θ估计

离散随机变量的期望公式为E(X)=()，方差公式为var(X)=E()-，矩估计思想就是如果总体中有K个未知参数，可以用前 K阶样本矩估计相应的前k阶总体矩，然后利用未知参数与总体矩的函数关系，求出参数的估计量

问题5：(a)：计算四个估计中哪些是µ的无偏估计；(b)：无偏估计的最小方差

按照无偏估计定义计算，估计量的数学期望等于被估计参数的真实值，无偏估计的最小方差mse()=E[]

涉及知识点：

R，随机变量，希望与方差，矩估计，极大似然估计，无偏估计

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MAST20005/MAST90058: Assignment 1

Due date: 11am, Friday 30 August 2019

Instructions: Questions labelled with ‘(R)’ require use of R. Please provide appropriate R commands and their output, along with sufficient explanation and interpretation of the output to demonstrate your understanding. Such R output should be presented in an integrated form together with your explanations; do not attach them as separate sheets. All other questions should be completed without reference to any R commands or output, except for looking up quantiles of distributions where necessary. Make sure you give enough explanation so your tutor can follow your reasoning if you happen to make a mistake. Please also try to be as succinct as possible. Each assignment will include marks for good presentation and for attempting all problems.

Problems:

1. (R) Let X be a random variable representing distance travelled (in kilometers) until a tire is worn out. The following are 16 observations of X:

41300 40300 43200 41100 39300 42100 42700 41300

38900 41200 44600 42300 40700 43500 39800 40400

(a) Give basic summary statistics for these data and produce a box plot. Briefly comment on center, spread and shape of the distribution.

(b) Assuming a normal distribution, compute maximum likelihood estimates for the parameters.

(c) Draw a density histogram and superimpose a pdf for a normal distribution using the estimated parameters.

(d) Draw a QQ plot to compare the data against the fitted normal distribution. Include a reference line. Comment on the fit of the model to the data.

2. A discrete random variable X has the following pmf:

A random sample of size n = 20 produced the following observations:

1, 1, 2, 3, 1, 2, 1, 3, 2, 2, 2, 1, 3, 1, 3, 1, 1, 2, 1, 2.

(a) i. Find E(X) and var(X).

ii. Find the method of moments estimator and estimate of θ.

iii. Find the standard error of this estimate.

(b) Let F1, F2 and F3 denote the sample frequencies of 1, 2 and 3, respectively.

i. Find the likelihood function in terms of F1, F2 and F3.

ii. Find that the maximum likelihood estimator and estimate of θ.

iii. Find the variance of this estimator.

(Hint: write the estimator in terms of the sample mean.)

3. Let X ∼ Unif(0, θ), a uniform distribution with an unknown endpoint θ.

(a) Suppose we have a single observation on X.

i. Find the method of moments estimator (MME) for θ and derive its mean and variance.

ii. Find the maximum likelihood estimator (MLE) for θ and derive its mean and variance.

(b) The mean square error (MSE) of an estimator is defined as MSE()=E

i. Let bias=E-.Show that，=

ii. Compare the MME and MLE from above in terms of their mean square errors.

iii. Find an estimator with smaller MSE than either of the above estimators.

(c) Suppose we have a random sample of size n from X.

i. Find the MME and derive its mean, variance and MSE.

ii. Find the MLE and derive its mean, variance and MSE.

iii. Consider the estimator awhere is the MLE. Find a that minimises the MSE.Some information that might be useful:

,,,

4. Let X1, . . . , Xn be a random sample from the lognormal distribution, Lognormal(µ, λ),whose pdf is:

(a) Show that the MLE of µ and λ are and 

(b) It is known that ln Xi ∼ N(µ, λ). Derive a 100 · (1 − α)% CI for λ.

(c) (R) Consider the following dataset:

0.27, 3.30, 4.58, 2.61, 0.38, 3.77, 1.11, 1.15, 4.11, 2.10,

0.07, 1.74, 2.11, 12.79, 1.85, 0.30, 0.34, 1.31, 0.14, 0.74

i. Assuming a lognormal distribution is an appropriate model for these data, compute the maximum likelihood estimate of λ and give a 95% CI.

ii. Draw a QQ plot to compare these data to the fitted lognormal distribution, Lognormal(ˆµ, λˆ). Is this model appropriate for these data?

Hint: Quantiles of the lognormal distribution can be computed using the qlnorm() function.

5. Let , , ,  be iid rvs with E(Xi) = µ and var(Xi) =  > 0, for i = 1, 2, 3, 4.

Consider the following four estimators of µ:

T1 =( + ) + ( + ) ，T2=(+2+3+4)

T3=(+++)，T4=(++)+

(a) Which of these estimates are unbiased? Show your working.

(b) Among the unbiased estimators, which one has the smallest variance