examples/pawitan2001all/2.md

In edxu96/MatrixTSA: Tidy Multivariate (Non)Linear Dynamic Systems

Exercises in Chapter 2 on the Book pawitan2001all

Exercise 2.2

runners participate in a marathon and they are each assigned a number from 1 to . From one location we record the following participants:

218 88 264 33 368 236 294 116 9

Make a reasonable assumption about the data and draw the likelihood of .

Exercise 2.5

The following shows the heart rate (in beats/minute) of a person, measured throughout the day:

73 75 84 76 93 79 85 80 76 78 80

Assume the data are an iid sample from , where is known at the observed sample variance. Denote the ordered values by . Draw and compare the likelihood of if

(a) the whole data are reported.

(b) only the sample mean is reported.

(c) only the sample median is reported.

(d) only the minimum and maximum are reported.

(e) only the lowest two values and are reported.

Exercise 2.6

Given the following data

0.5 -0.32 -0.55 -0.76 -0.07 0.44 -0.48

draw the likelihood of based on each of the following models:

1. The data are an iid sample from a uniform distribution on .
2. The data are an iid sample from a uniform distribution on .
3. The data are an iid sample from .

Exercise 2.11

Ten light bulbs are left on for 30 days. One fails after 6 days, another one after 28 days, but the others are still working after 30 days. Assume the lifetime is exponential with density .

(a) Given , what is the probability of a light bulb working more than 30 days?

(b) Derive and draw the likelihood of . (Hint: only the first two order statistics are reported.)

(c) Derive the MLE of .

To get the MLE, the score function and the score equation is used.

(d) Estimate how long it takes before 90% of the light bulbs will fail.

(e) Suppose the only information available is that two have failed by 30 days, but not their exact failure times. Draw the likelihood of A and compare with (b).

edxu96/MatrixTSA documentation built on Feb. 5, 2021, 11:30 p.m.