Value-at-Risk

Chapter 4, Page 170
Exercise 2

Consider the data of Exhibit 4.3:

i	x^[i]	i	x^[i]	i	x^[i]	i	x^[i]	i	x^[i]
1	9.91	11	3.86	21	8.00	31	12.51	41	3.15
2	11.41	12	9.65	22	11.68	32	3.51	42	6.86
3	7.34	13	8.59	23	10.25	33	1.61	43	5.18
4	6.23	14	4.74	24	8.30	34	0.84	44	9.28
5	0.64	15	6.30	25	12.06	35	5.69	45	8.88
6	2.82	16	7.36	26	9.19	36	1.52	46	3.82
7	6.35	17	6.57	27	0.42	37	11.47	47	9.66
8	5.91	18	6.26	28	5.29	38	0.21	48	2.59
9	12.48	19	13.69	29	7.55	39	1.39	49	7.99
10	14.66	20	4.71	30	12.58	40	3.82	50	11.25

Exhibit 4.3 Data for Exercise 4.2.

Treat the data as a realization of a sample {X^[1], X^[2], … , X^[50]}, where X^[i] ~ U(0,). We wish to estimate the unknown parameter . Consider two estimators:

	[4.23]
	[4.24]

a. Make sure you understand both estimators. Describe in your own words why each is reasonable.

b. Using the data, estimate based upon each of the estimators.

c. In light of the fact that x^[10] = 14.66, did both estimators produce reasonable estimates for the upper bound of the interval (0, )?

d. Are the estimators biased or unbiased?

Solution

a. The first estimator is based upon the fact that, out of 50 realizations drawn from a U(0,) distribution, we would expect the largest of those realizations to fall near the upper bound . The second estimator is based upon the fact that the mean of a U(0,) distribution is /2. Accordingly, multiplying the sample mean by 2 provides an estimate of .

b. estimates as 14.66. estimates as 13.84.

c. No. Estimator produced an estimate that is clearly impossible. Since realization 14.66 was drawn from the interval (0,), it is impossible that equals 13.84.

d. is biased. It will always underestimate . is unbiased.

Comment

This exercise illustrates the tradeoffs we often face in selecting a statistical estimator. In this case, estimator is biased. It will always underestimate . On the other hand, estimator is unbiased, but it can produce estimates that are clearly incorrect.