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LANCASTER UNIVERSITY
2007 EXAMINATIONS
PART II (Second year)
MATHEMATICS & STATISTICS 2 hours
Math 235: Statistics
You should answer ALL Section A questions and THREE Section B questions.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40.
There are statistical tables at the end of this exam paper.
SECTION A
A1. In a coin throwing experiment the trials are independent and θ is the probability of a head on any throw. The coin is tossed ten times and seven heads and three tails were observed. (a) Write down the likelihood function L(θ) and make a rough sketch of it; [5] (b) Write down the log-likelihood function `(θ); [3] (c) Determine the maximum likelihood estimate of θ. [4] A2. X 1 , X 2 ,... , Xn are independent and identically distributed exponential random variables with mean 1/θ, each having probability density function
f (x|θ) = θ exp(−θ x), x > 0.
(a) Determine the log-likelihood function `(θ); [4] (b) Find the score function and the maximum likelihood estimator of θ; [5] (c) Determine a maximum likelihood estimator of the mean 1/θ. [4]
please turn over
SECTION A continued
A3. Observations y 1 , y 2 , y 3 and y 4 are realised values from the normal linear model for independent measurements Yi = θ xi + Zi for i = 1, 2 , 3 , 4, where x 1 = 1, x 2 = 2, x 3 = 3, x 4 = 4 and E Zi = 0, Var Zi = σ^2. (a) Write down the response vector y and the (single column) design matrix X for this linear model. [3] (b) Show that the least squares estimate of θ is θˆ = (y 1 + 2 y 2 + 3 y 3 + 4 y 4 )/30. [3] You may assume the formula (X′X)θˆ = X′y for the least squares equations, but must show all your working. (c) Show directly, that this formula for θˆ, considered as a function of the random variables Yi, has the properties (i) E θˆ = θ and [3] (ii) Var θˆ = σ^2 /30. [4] You may assume, without stating, the standard properties of expectations and variances of linear combinations of independent random variables.
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SECTION B
B1. (a)^ X 1 ,^ X 2 ,... , Xn are independent and identically distributed Poisson(θ) random variables. (i) Write down the likelihood function L(θ) and show that ∑^ xi is a sufficient statistic for θ, stating clearly your reasons. [3] (b) The number of road accidents in a small town on any given day has a Poisson(θ) distri- bution. The numbers of accidents on distinct days are independent. (i) You are told that over a ten day period there is a total of five accidents in the town. Write down the likelihood function L(θ) and produce a rough sketch of it over the range 0 ≤ θ ≤ 1. Obtain the maximum likelihood estimate of θ and the value of the observed information. [6] (ii) You are now simply told that over a ten day period, on exactly six of the days there were no accidents in the town. Write down the likelihood function for θ corresponding to these data. Obtain the maximum likelihood estimate of θ and the value of the observed information. [8] (iii) Which of the data configurations described in (b)(i) and (b)(ii) do you think provides the more satisfactory basis for making inferences about θ? Explain your answer. [3]
B2. (a) The quantities^ X 1 ,^ X 2 ,... , Xn constitute a random sample drawn from a population with probability density function f (x|θ) = 3θx^2 exp(−θx^3 ), x > 0 , where θ is a parameter which is positive-valued. (i) Write down formulae for the likelihood and the true score; [2] (ii) Write down what you know about the distribution of the true score for this example when n is large. [4] (b) In a particular experiment, n = 10 and
∑^ n i=
x^3 i = 15.2. (i) Construct an approximate 95% confidence interval for θ, using the asymptotic dis- tribution of the maximum likelihood estimator; [6] (ii) Give a formula for the deviance D(θ) for this example and produce a rough sketch of it. Describe in detail how to use the deviance to develop an alternative approximate 95% confidence interval for θ; [6] (iii) Name any advantages of developing confidence intervals from the deviance. [2]
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SECTION B continued
B3. (a) Given two normal linear regression models (for the same response) y = Xθ + z and y = Aφ + z, (i) say what is meant by the design matrices X and A being equivalent. [3] (ii) given that X = AT for some non-singular T , show that φ = T θ. [1] (b) In an experiment to determine the acceleration due to gravity, the distance s that a body had fallen since its release, was measured at time t, each 1/10th of a second following its release. The measurements are given in the table below: Time t since release (in units of 1/10seconds) 1 2 3 4 5 Distance s fallen (cm) 5.1 19.9 44.1 78.8 122. The equation relating the measurement of s to t is assumed to be s = c + ut + 12 at^2 + error, where c, u and a are unknown coefficients, being respectively the initial position, the initial velocity and the acceleration due to gravity. Using the measurements (and units) in the table, write down the observed response vector y and design matrix X of the normal linear model that may be used to estimate the coefficients θ 1 = c, θ 2 = u and θ 3 = 12 a. [2] A new design A is constructed, with elements of the first column set to 1, of the second set to (t − 3) and the third to (t − 3)^2 − 2. (i) Calculate and write out the matrix A. [2] (ii) Calculate the matrix A T , where
T =
Show your working, and say how the result relates to X. [2] (iii) Show that φ 3 = θ 3 in the model (for the same response) with design matrix A and coefficient vector φ. [2] (iv) Show that the new design A is orthogonal. [2] Hence, assuming that the residual variance is σ^2 = 0.04: (v) write down the equation for the least squares estimate of φ 3 , using the above data, [2] (vi) obtain the least squares estimate of 12 a, [2] (vii) determine the standard error of this estimate. [2] please turn over
Formulae Sheet for Math
Standard Distributions
Here we list the basic properties of three standard distributions. Throughout X denotes either a continuous random variable with probability density function fX (x) or a discrete random variable with probability mass function pX (x).
Poisson distribution: if X ∼ Pois(θ), θ > 0, the probability mass function is
p(x|θ) = exp(−θ)^ θ
x x! for^ x^ = 0,^1 ,^2 ,...
Summary measures are: E(X) = θ and var (X) = θ.
Exponential Distribution: if X ∼ Exponential(β), with β > 0,
fX (x|β) =
β exp(−βx) x ≥ 0 , 0 otherwise,
Summary measures are: E(X) = β^1 and var (X) = (^) β^12.
Normal Distribution: if X ∼ N (μ, σ^2 ), with σ > 0, then
fX (x|μ, σ^2 ) = √^1 2 πσ^2
exp
− (x^ −^ μ)
2 2 σ^2
for − ∞ < x < ∞,
Summary measures are: E(X) = μ and var (X) = σ^2.
Table 2: The T -distribution
Values of t for which P (| T |> t) = p, where T has a T -distribution with r degrees of freedom.
