題意：

概率論與數(shù)理統(tǒng)計中的幾個小題

解析：

問題1：使用R語言統(tǒng)計數(shù)據(jù)生成圖像；假設(shè)服從正態(tài)分布芽世，計算參數(shù)的極大似然估計斜脂；繪制密度直方圖；繪制QQ圖

一般求解極大似然估計步驟寫出似然函數(shù)燕差，再對似然函數(shù)取對數(shù)并整理遭笋，然后求導(dǎo)數(shù)，最后解似然方程 徒探。

問題2：按照給出的離散隨機(jī)變量計算：a.計算期望和方差瓦呼、使用矩估計計算θ和計算標(biāo)準(zhǔn)差，b.求解F1、F2央串、F3的似然函數(shù)磨澡、最大似然估計和θ估計

離散隨機(jī)變量的期望公式為E(X)=()，方差公式為var(X)=E()-质和，矩估計思想就是如果總體中有K個未知參數(shù)稳摄，可以用前 K階樣本矩估計相應(yīng)的前k階總體矩，然后利用未知參數(shù)與總體矩的函數(shù)關(guān)系饲宿，求出參數(shù)的估計量

問題5：(a)：計算四個估計中哪些是μ的無偏估計厦酬；(b)：無偏估計的最小方差

按照無偏估計定義計算，估計量的數(shù)學(xué)期望等于被估計參數(shù)的真實(shí)值瘫想，無偏估計的最小方差mse()=E[]

涉及知識點(diǎn)：

R仗阅，隨機(jī)變量，希望與方差国夜，矩估計减噪，極大似然估計，無偏估計

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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车吹，=

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(?μ, λ?). 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 =(?+ ) + ( + ) 筹裕，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