这是一篇来自澳洲的关于现代应用统计学的统计代写

 

Permitted Materials

• This exam and/or an o✏ine electronic PDF reader, one or more copies of the masked exam template made available earlier, blank loose-leaf paper and a Casio FX-82 calculator.

• One double sided A4 page of notes (handwritten only).

Instructions to Students

• There are 5 questions with marks as shown. The total number of marks available is 50.
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Question 1 (7 marks)

Let X1, ··· , Xn be independent samples from a Normal distribution N(1, ⌧1 ) with pdf

r2⌧⇡ e− ⌧(x−1)22.

(a) What is the log-likelihood for this example?

 

(b) What is the Fisher information for this example?

 

(c) Find the MLE of ⌧ and its asymptotic distribution.

 

Question 2 (9 marks)

The dvisits data in the faraway package comes from the Australian Health Survey of 1977–78 and consists of 5190 observations on single adults, where young and old have been oversampled.

Here, we consider doctorco as a response and sex, age, income, levyplus, freepoor,freerepa, illness, actdays as predictor variables. The description of each variable is as follows.

• doctorco: number of consultations with a doctor or specialist in the past 2 weeks
• sex: 1 if female, 0 if male
• age: age in years divided by 100
• income: annual income in Australian dollars divided by 1000
• levyplus: 1 if covered by a private health insurance fund for private patients in a public hospital (with doctor of choice), 0 otherwise

• freepoor: 1 if covered by government because of low income, recent immigrant, or unemployed, 0 otherwise

• freerepa: 1 if covered by government because of old-age or disability pension, or because of invalid veteran or family of deceased veteran, 0 otherwise

• illness: number of illnesses in past 2 weeks, with 5 or more coded as 5
• actdays: number of days of reduced activity in past two weeks due to illness or injury

Examine the R code and output below, and then answer the questions that follow.

> library(faraway)

> data(dvisits)

> modelA <- glm(doctorco ~ sex + age + income + levyplus + freepoor

+ + freerepa + illness + actdays,

+ family=quasipoisson, data=dvisits)

> summary(modelA)

Call:

glm(formula = doctorco ~ sex + age + income + levyplus + freepoor +

freerepa + illness + actdays, family = quasipoisson, data = dvisits)

Deviance Residuals:

Min 1Q Median 3Q Max

-2.7696 -0.6865 -0.5773 -0.4906 5.5745

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -2.055666 0.115808 -17.751 <2e-16 ***

sex 0.163442 0.064454 2.536 0.0112 *

age 0.296311 0.186496 1.589 0.1122

income -0.195493 0.098511 -1.984 0.0473 *

levyplus 0.143743 0.082153 1.750 0.0802 .

freepoor -0.404611 0.206938 -1.955 0.0506 .

freerepa 0.118603 0.105656 1.123 0.2617

illness 0.211644 0.019482 10.864 <2e-16 ***

actdays 0.133576 0.005264 25.377 <2e-16 ***

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for quasipoisson family taken to be 1.328231)

Null deviance: 5634.8 on 5189 degrees of freedom

Residual deviance: 4394.3 on 5181 degrees of freedom

AIC: NA

Number of Fisher Scoring iterations: 6

> modelB <- glm(doctorco ~ sex + age + income,

+ family=quasipoisson, data=dvisits)

> anova(modelB, modelA, test=”F”)

Analysis of Deviance Table

Model 1: doctorco ~ sex + age + income

Model 2: doctorco ~ sex + age + income + levyplus + freepoor + freerepa +

illness + actdays

Resid. Df Resid. Dev Df Deviance F Pr(>F)

1 5186 5434.9

2 5181 4394.3 5 1040.5 156.68 < 2.2e-16 ***

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

>

> modelC <- glm(doctorco ~ sex + age + income + levyplus + freepoor

+ + freerepa + illness + actdays,

+ family=poisson, data=dvisits)

> modelD <- glm(doctorco ~ sex + age + income,

+ family=poisson, data=dvisits)

> summary(modelD)

Call:

glm(formula = doctorco ~ sex + age + income, family = poisson,

data = dvisits)

Deviance Residuals:

Min 1Q Median 3Q Max

-1.0350 -0.8031 -0.6749 -0.6069 6.3695

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -1.71473 0.09118 -18.805 < 2e-16 ***

sex 0.21565 0.05589 3.859 0.000114 ***

age 1.23798 0.13013 9.514 < 2e-16 ***

income -0.27726 0.07969 -3.479 0.000502 ***

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)