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Part A Statistics HT 2014

Problem Sheet 2

1. Independent random variablesX1, . . . , X n have common probability density functionf(x; )depending on the unknown parameter. Define the termslikelihood functionfor,maximum

likelihood estimator for , Fishers information for .

What is the connection between Fishers information and the asymptotic distribution of themaximum likelihood estimator?

Suppose X1, . . . , X n are independent, each with density

f(x; ) = 1

[1 + (x )2], x R

with parameter R. Derive an equation which must be satisfied by the maximum likelihoodestimator

.

Prove that Fishers information for is 12

n. Explain how to calculate an approximate 95%confidence interval for .

2. If S2 is the variance of a random sample of size n from a normal distribution N(, 2),thencS2 has a standard distribution for a particular value of the constant c. Write down thedistribution and specify the constant c.

How many pairs of positive constants a, b exist such that P(a < cS2 < b) = 0.95? For anysuch pair, deduce that the random interval

(n 1)S2

b ,

(n 1)S2

a

will contain the variance2 for approximately 95% of a large number of such random samples.

3. The following data are time intervals in days between earthquakes which either registeredmagnitudes greater than 7.5 on the Richter scale or produced over 1,000 fatalities. Recordingstarts on 16 December, 1902 and ends on 4 March, 1977, a total period of 27,107 days. Therewere 63 earthquakes in all, and therefore 62 recorded time intervals.

840 1901 40 139 246 157 695 1336 780 1617145 294 335 203 638 44 562 1354 436 937

33 721 454 30 735 121 76 36 384 38150 710 667 129 365 280 46 40 9 92

434 402 209 82 736 194 99 599 220 584759 556 304 83 887 319 375 832 263 460567 328

Assuming the data to be from a random sample X1, . . . , X n drawn from an exponentialdistribution with parameter, obtain the maximum likelihood estimatorof and calculatethe maximum likelihood estimate.

Given that the moment generating function of a gamma distribution with parameters (n, )is

Mn(t) =

tn

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show thatY =n

i=1 Xi has a gamma distribution. Show that a

nx,

b

nx

is an exact 95% central confidence interval for if

a0

yn1ey

(n) dy=

b

yn1ey

(n) dy= 0.025.

Obtain Fishers information for and use it to show that (0.0018, 0.0029) is an approximate95% confidence interval for .

4. LetX1 . . . , X nbe a random sample from a normal distribution with mean and variance2.

What statistics, based on this sample, have (i) a standard normal distribution, (ii) a 2(n1)distribution, (iii) a t(n 1) distribution?

What is meant by a 100(1 )% confidence interval for an unknown parameter of a dis-tribution? Construct a central confidence interval (i.e. an equal-tail confidence interval) for

the mean of a normal distribution with unknown variance 2, based on a random sample ofsizen.

Let the confidence interval constructed above have width W. Show that there exists aconstantc such that cW2/2 has a 2-distribution and findc in terms ofn and .

5. Let X1, . . . , X m and Y1, . . . , Y n be independent random samples from normal distributionsN(1,

2) and N(2, 2), respectively, where the parameters 1, 2,

2 are unknown. Let

S2 = (m + n 2)1 m

i=1

(Xi X)2 +

nj=1

(Yj Y)2

.

Determine the distributions of both

(m + n 2)S2/2 and X Y (1 2)

S2( 1m

+ 1n

).

Show how to construct a confidence interval for 1 2.

6. The elastic modulus of a material is 1.183103 (in appropriate SI units), this being accuratelymeasured on sophisticated equipment. In an attempt to find a quicker, cheaper method ofmeasurement, the elastic modulus was measured using a simpler method and the followingresults were obtained (all 103 SI units): 1.5, 1.0, 1.4, 1.7, 1.2, 1.3, 1.3, 1.2.

(i) Assuming the data arise from a normal distribution, how would you test whether thisdistribution has the correct mean? State the appropriate null and alternative hypothe-ses, and any assumptions you need to make for the hypothesis test to be appropriate.

(ii) Carry out the test you suggested in (i) and state your conclusions.

(iii) Modify your test to test whether the data are from a distribution with a mean value

higherthan the true value and re-state your conclusions.

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7. (Optional, using R.) Read in the earthquake data from question 3 and try an exponentialQ-Q plot:

x

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