Student Notes To Accompany MS4214: STATISTICAL INFERENCE

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Three main topics will be covered : Estimation Unbiasedness, mean square error, consistency, relative efficiency, suffi- ciency, minimum variance. Fisher information for a function of a parameter, Cramér-Rao lower bound, efficiency. Fitting standard distributions to discrete and continuous data. Method of moments. Maximum likelihood estimation: finding estimators analytically and numerically, invariance, censored data. Hypothesis testing Simple and composite hypotheses, types of error, power, op- erating characteristic curves, p-value. Neuman-Pearson method. Generalised likelhood ratio test. Use of asympotic results to construct tests. Central limit theorem, asymptotic distributions of maximum likelihood estimator and generalised ratio test statistic. Confidence intervals and sets Random intervals and sets. Use of pivotal qnatities. Relationship between tests and confidence intervals. Use of asymptotic results. 7
Chapter 2 The Theory of Estimation 2.1 The Frequentist Philosophy The dominant philosophy of inference is based on the frequentist theory of probability. According to the frequentist theory probability statements can only be made regarding events associated with a random experiment. A random experiment is an experiment which has a well defined set of possible outcomes S. In addition, we must be able to envisage an infinite sequence of independent repetitions of the experiment with the actual outcome of each repetition being some unpredictable element of S. A random variable is a numerical quantity associated with each possible outcome in S. A random vector is a collection of numerical quantities associated with each possible outcome in S. In performing the experiment we determine which element of S has occurred and thereby the observed values of all random variables or random vectors of interest. Since the outcome of the experiment is unpredictable so too is the value of any random variable or random vector. Since we can envisage an infinite sequence of independent repetitions of the experiment, we can envisage an infinite sequence of independent determinations of the value of a random variable (or vector). The purpose of a statistical model is to describe the unpredictability of such a sequence of determinations. Consider the random experiment which consists of picking someone at random from the 2007 electoral register for Limerick. The outcome of such an experiment will be a human being and the set S consists of all human beings whose names are on the register. We can clearly envisage an infinite sequence of independent repetitions of such an experiment. Consider the random variable X where X = 0 if the outcome of the experiment is a male and X = 1 if the outcome of the experiment is a female. When we say that P (X = 1) = 0.54 we are taken to mean that in an infinite sequence of independent repetitions of the experiment exactly 54% of the outcomes will produce a value of X = 1. 8
Page 1 and 2: Student Notes To Accompany MS4214:
Page 3 and 4: 4.8 Worked Problems . . . . . . . .
Page 5 and 6: Suppose that, in order to learn som
Page 7: Example 1.4 (Blood pressure). We wi
Page 11 and 12: Thus if we carried out an infinite
Page 13 and 14: Problem 2.1. Let X have a binomial
Page 15 and 16: Problem 2.6. Let X1, . . . , Xn be
Page 17 and 18: Theorem 2.8 (Cramér Rao lower boun
Page 19 and 20: Let us propose ˆµ = ¯ X as an es
Page 21 and 22: Example 2.2 (Bernoulli Trials). Con
Page 23 and 24: Relative Likelihood 0.0 0.2 0.4 0.6
Page 25 and 26: Example 2.7 (Exponential distributi
Page 27 and 28: ℓ(µ) = −m ln µ − 1 m� ti
Page 29 and 30: 2.5 Multi-parameter Estimation Supp
Page 31 and 32: Lemma 2.9 (Joint distribution of th
Page 33 and 34: So, if | ˆ θ − θ0| is small, w
Page 35 and 36: Then the log-likelihood function is
Page 37 and 38: A regression of the empirical distr
Page 39 and 40: 2.7 The Invariance Principle How do
Page 41 and 42: Event Probability Set 0 0 0 (1 −
Page 43 and 44: 2.10 Worked Problems The Problems 1
Page 45 and 46: (a) Show that ˆµ is an unbiased e
Page 47 and 48: 4. E(X2 ) = � ∞ 0 x2f(x)dx =
Page 49 and 50: Student Questions 1. Let X1, X2, .
Page 51 and 52: Chapter 3 The Theory of Confidence
Page 53 and 54: Suppose that we have data X1, X2, .
Page 55 and 56: Lemma 3.2 (The Student t-distributi
Page 57 and 58: Example 3.6. Suppose that we have d
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3.3 Approximate Confidence Interval
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3.4 Worked Problems The Problems 1.
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Student Questions 1. Let X1, X2, .
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Chapter 4 The Theory of Hypothesis
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Example 4.3 (The power function). S
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(d) Suppose Θ0 = {(µ, σ) : −
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(c) Θ0 = {(µ, µ,σ) : −∞ <
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The Score Test Statistic: This test
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4.5 The Neyman-Pearson Lemma Suppos
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According to the Neyman-Pearson lem
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would consider to be an unusually l
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pendent we would expect the proport
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3. Explain what is meant by the pow
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For the null hypothesis θ = 1, the
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0.1820. Similarly P (X = 3) = 0.218
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Appendix A Review of Probability A.
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where Xi ∼ Bernoulli(θ) for i =
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The expected value of the jth momen
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A.3 Continuous Random Variables A.3
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A.3.4 Gaussian Distribution A rando
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A.3.5 Weibull Distribution The Weib
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Next, let g(y) be a monotone decrea
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That is, X + Y ∼ Pois(θ + λ). =
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since � ∞ −∞ e−αu2 du =
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A.4.4 The Bivariate Normal Distribu
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A.5 Generating Functions Denote the
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Density p.g.f. m.g.f. ch.f. c.g.f B
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Uniform U(a, b) Continuous Distribu
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Student Notes To Accompany MS4214: STATISTICAL INFERENCE

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