Download Likelihood Function - Statistics - Exam and more Exams Statistics in PDF only on Docsity!
LANCASTER UNIVERSITY
2010 EXAMINATIONS
PART II (Second Year)
MATHEMATICS & STATISTICS 2 hours
Math 235: Statistics
You should answer ALL Section A questions and TWO 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 40. There are statistical tables at the end of this exam paper.
SECTION A
A1. Seven support requests have been submitted to the ‘Geek Squad’, the technical support service of a computer store, on a particular day. It is assumed that the number of support requests on any given workday follow a Poisson distribution with mean θ.
f (x|θ) = e
−θ (^) θx x! with^ θ^ ≥^ 0 and^ x^ = 0,^1 ,^2 ,... (a) Write down the likelihood function of θ, L(θ), and make a rough sketch of it. [5] (b) Write down the log-likelihood function, l(θ). [3] (c) Determine the maximum likelihood estimator for θ. Make sure to justify your answer. [4]
A2. A random survey of 3 members of a statistics department (total size 6) reported that three are in favour of merging with the mathematics department. Let n denote the number in favour in the whole department.
(a) Determine the likelihood function for n and determine the possible values for n. [4] (b) Sketch the likelihood function of n up to 7. [3] (c) Find the score function and the maximum likelihood estimator of n. [4] (d) Determine the maximum likelihood estimator for φ = n^4. Make sure to justify your answer. [2] please turn over
SECTION A continued
A3. The following table displays the results of an experiment to determine the optimal yield of sodium in a factorial experiment, where C denotes the presence or absence of a given catalyst. C = 1 10. 1 9. 2 3. 7 4. 8 C = 0 6. 1 5. 2 4. 2 4. 9 The four measurements on each row are replicates. The observations are ordered according to the vector y = (10. 1 , 9. 2 , 3. 7 , 4. 8 , 6. 1 , 5. 2 , 4. 2 , 4 .9)′^. The suggested model is that the expected yield when C = 0 is α and this increases by an amount β if C = 1.
(a) Write down the linear model for the observation y 3 and for the observation yi. [4] (b) Find the design matrix X for this model. [4] (c) Evaluate X′X. [4]
please turn over
SECTION B
B1. The independent random variables X 1 , X 2 ,... , Xn have a distribution defined by
fXi (x|β) = (^) (i −^1 1)!βi xi−^1 e−^ x β 0 ≤ x < ∞, β > 0
with E(Xi) = iβ and V (Xi) = iβ^2.
(a) Define what is meant by a sufficient statistic. [3] (b) Write down the likelihood function, L(β), and the log-likelihood function, l(β). [3] (c) Find the sufficient statistics for β, clearly stating your reasons. [3] (d) Find the maximum likelihood estimator for β. [4] (e) Show that the observed information is n( 2 nβˆ+1) 2. [3] (f) Find the Fisher information. [4]
B2. Let X 1 , X 2 ,... , Xn be a random sample of X which has pdf
f (x|θ) = 2 θ x exp(−θx^2 ), for x > 0.
(a) Write down the likelihood function, L(θ), and the log-likelihood function, l(θ). [3] (b) Find the maximum likelihood estimator for θ. [4]
An experiment with n = 10 observations yields that ∑ni=1 x^2 i = 27.3. Use these data to answer the following questions.
(c) Find the asymptotic distribution of the maximum likelihood estimator and use it to construct an approximate 90% confidence interval for θ. (Recall that the relevant value for a 90% region from the Normal distribution is 1.645.) [6] (d) Sketch the deviance function D(θ) over the interval [0. 15 , 0 .75]. Find an approximate 90% confidence interval for θ using the approximate chi-squared distribution of the deviance, D. [5] (e) Compare the two answers in (c) and (d). [2]
please turn over
SECTION B continued
B3. The standard linear regression model for observations in y is Y = Xθ + Z where X is the design matrix and Z ∼ N( 0 , σ^2 I).
(a) Write down the least squares estimate, ˆθ, of the regression coefficient. [2] (b) Quoting any results needed for evaluating the mean and variance of linear transforms of random vectors, show that ˆθ has variance σ^2 (X′X)−^1. [6] (c) The hat matrix P is defined in terms of the design matrix X by P = X(XT^ X)−^1 XT^. Show that P has the properties: P 2 = P , P X = X, and P T^ = P. [4] (d) Express P y in terms of the fitted values, and the residuals r in terms of y and P. Show that X′r = 0. [4] (e) What implication does (d) have for the statistical analysis of residuals? [4]
B4. A batch of steel rods contains a calcium impurity, in an unknown proportion β. To estimate the value of β, 4 rods are chosen, and the weights xi, i = 1,... , 4, are recorded. The rods are then sent to a laboratory for assaying, giving the measured impurity yi. The measurement process is such that the error Zi has mean 0 but variance σ^2 xi, depending on the weight xi. The value of σ^2 is known. Back at the foundry, the statistician decides to estimate the unknown proportion from the lab information (xi, yi), i = 1, ..., 4, using a linear model and weighted least squares estimation.
(a) Consider the first observation. Write down the linear model for Y 1 , relating to the measured impurity y 1 , in terms of β, x 1 , and Z 1 , where Z 1 is the error with mean 0 and the variance given above. [2] (b) The contribution to the ordinary sum of squares from the first observation is z^21. Express this in terms of observation (x 1 , y 1 ). [2] (c) Similarly express the weighted contribution z^21 / var (Z 1 ). [2] (d) Write down the weighted sum of squares to minimise as function of β from all 4 rods. [4] (e) Show that the weighted least squares estimates of β is βˆ = (
i xi)−^1
i yi.^ [4] (f) Find an estimate of the variance of βˆ in (e). [6]
end of exam
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.
Table 4: The F distribution Values of f for which P (F > f ) = 0.05 (upper values) and P (F > f ) = 0.01 (lower values) where F has an F -distribution with r and s degrees of freedom. r
- 0.20 0.10 0.05 0.01 0. p - 1 3.078 6.314 12.706 63.657 636. - 2 1.886 2.920 4.303 9.925 31. - 3 1.638 2.353 3.182 5.841 12. - 4 1.533 2.132 2.776 4.604 8. - 5 1.476 2.015 2.571 4.032 6. - 6 1.440 1.943 2.447 3.707 5. - 7 1.415 1.895 2.365 3.499 5. - 8 1.397 1.860 2.306 3.355 5. - 9 1.383 1.833 2.262 3.250 4. - 10 1.372 1.812 2.228 3.169 4. - 11 1.363 1.796 2.201 3.106 4. - 12 1.356 1.782 2.179 3.055 4. - 13 1.350 1.771 2.160 3.012 4. - 14 1.345 1.761 2.145 2.977 4. - 15 1.341 1.753 2.131 2.947 4. - 16 1.337 1.746 2.120 2.921 4.
- r 17 1.333 1.740 2.110 2.898 3. - 18 1.330 1.734 2.101 2.878 3. - 19 1.328 1.729 2.093 2.861 3. - 20 1.325 1.725 2.086 2.845 3. - 21 1.323 1.721 2.080 2.831 3. - 22 1.321 1.717 2.074 2.819 3. - 23 1.319 1.714 2.069 2.807 3. - 24 1.318 1.711 2.064 2.797 3. - 25 1.316 1.708 2.060 2.787 3. - 26 1.315 1.706 2.056 2.779 3. - 27 1.314 1.703 2.052 2.771 3. - 28 1.313 1.701 2.048 2.763 3. - 29 1.311 1.699 2.045 2.756 3. - 30 1.310 1.697 2.042 2.750 3. - 40 1.303 1.684 2.021 2.704 3. - 50 1.299 1.676 2.009 2.678 3. - 60 1.296 1.671 2.000 2.660 3. - 70 1.294 1.667 1.994 2.648 3. - 80 1.292 1.664 1.990 2.639 3. - 90 1.291 1.662 1.987 2.632 3.
- 100 1.290 1.66 1.984 2.626 3.
- ∞ 1.282 1.645 1.960 2.576 3.
- 3 10.13 9.55 9.28 9.12 9.01 8.94 8.85 8.79 8. 1 2 3 4 5 6 8 10 ∞
- 34.12 30.82 29.46 28.71 28.24 27.91 27.49 27.23 26.
- 4 7.71 6.94 6.59 6.39 6.26 6.16 6.04 5.96 5.
- 21.20 18.00 16.69 15.98 15.52 15.21 14.80 14.55 13.
- 5 6.61 5.79 5.41 5.19 5.05 4.95 4.82 4.74 4.
- 16.26 13.27 12.06 11.39 10.97 10.67 10.29 10.05 9.
- 6 5.99 5.14 4.76 4.53 4.39 4.28 4.15 4.06 3.
- 13.75 10.92 9.78 9.15 8.75 8.47 8.10 7.87 6.
- 8 5.32 4.46 4.07 3.84 3.69 3.58 3.44 3.35 2.
- 11.26 8.65 7.59 7.01 6.63 6.37 6.03 5.81 4.
- 10 4.96 4.10 3.71 3.48 3.33 3.22 3.07 2.98 2. - 10.04 7.56 6.55 5.99 5.64 5.39 5.06 4.85 3.
- s 15 4.54 3.68 3.29 3.06 2.90 2.79 2.64 2.54 2. - 8.68 6.36 5.42 4.89 4.56 4.32 4.00 3.80 2. - 20 4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.35 1. - 8.10 5.85 4.94 4.43 4.10 3.87 3.56 3.37 2. - 25 4.24 3.39 2.99 2.76 2.60 2.49 2.34 2.24 1. - 7.77 5.57 4.68 4.18 3.85 3.63 3.32 3.13 2. - 30 4.17 3.32 2.92 2.69 2.53 2.42 2.27 2.16 1. - 7.56 5.39 4.51 4.02 3.70 3.47 3.17 2.98 2. - 40 4.08 3.23 2.84 2.61 2.45 2.34 2.18 2.08 1. - 7.31 5.18 4.31 3.83 3.51 3.29 2.99 2.80 1. - 60 4.00 3.15 2.76 2.53 2.37 2.25 2.10 1.99 1. - 7.08 4.98 4.13 3.65 3.34 3.12 2.82 2.63 1.
- 120 3.92 3.07 2.68 2.45 2.29 2.18 2.02 1.91 1. - 6.85 4.79 3.95 3.48 3.17 2.96 2.66 2.47 1.
- ∞ 3.84 3.00 2.60 2.37 2.21 2.10 1.94 1.83 1. - 6.64 4.61 3.78 3.32 3.02 2.80 2.51 2.32 1.
