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1 ênúo (15-16 ïä)) wðï Page 1 of c

2 Introduction to Mathematical Statistics Fifth Edition Authors: Robert V. Hogg and Allen T. Craig Page 2 of 100

3 Page 3 of 100

4 Assistant: Zhang Juan (q ) URL: Page 4 of 100

5 Assignments Consulting References Page 5 of 100

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31 Ch. 6. Introduction to Statistical Inference 6.1. Point Estimation The first five chapters: Page 31 of 100

32 1. Concepts and probability theory. Random variable (r.v.) Distribution of r.v. Characteristic Moment Page 32 of 100

33 2. Statistical distribution. Discrete: Binomial, Poisson, Parskal, Discrete-uniform, Geometric, Hypergeometric, Multinomial, etc. Page 33 of 100

34 Continuous: Normal, Uniform, χ 2, t, F, Gamma, Exponential, Cauchy, Beta, Lognormal, Logistic, Multinormal, etc. Page 34 of 100

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45 D.f. and c.d.f. Random representation Expectation and variance Median and MAD Skewness and kurtosis R or SPLUS language (data, programme, picture, etc.) Page 45 of 100

46 Convergence of random variable sequence The Law of Large Numbers Central Limit Theorem Page 46 of 100

47 A study of some problems in statistics A sample space C of outcomes and the space A of one or more random variables defined on C. A random variable X as the outcome of a random experiment Page 47 of 100

48 Call X the numerical outcome. Once the experiment has been performed and it is found that X = x, we shall call x the experimental value of X. Page 48 of 100

49 Let a random experiment be repeated n independent times and under identical conditions. Page 49 of 100

50 Random variables (a random sample) X 1, X 2,, X n with the observations. It is found that X 1 = x 1, X 2 = x 2,, X n = x n, we shall refer to x 1, x 2,, x n as the experimental values of X 1, X 2,, X n or as the sample data. Page 50 of 100

51 Let a r.v. X have a p.d.f. that is of known functional form but depends upon an unknown parameter θ: {f(x; θ), θ Ω} Ω: parameter space. Page 51 of 100

52 Not with one, but with a family of distributions. To each value of θ, θ Ω, there corresponds one member of the family. Page 52 of 100

53 Normal distribution family: {N(θ, 1) : θ Ω}, where Ω is the set < θ <. Binomial distribution family: {B(n, p) : p Ω}, where Ω is the set 0 p 1. Page 53 of 100

54 Need to select precisely one member of the family as being the p.d.f. of his random variable: point estimate of θ. Page 54 of 100

55 Let X 1, X 2,, X n denote a random sample from the family {f(x; θ), θ Ω}. Define a statistic Y 1 = u 1 (X 1, X 2,, X n ) as an estimate of θ. Page 55 of 100

56 If x 1, x 2,, x n are the observed experimental values of X 1, X 2,, X n, then the number y 1 = u 1 (x 1, x 2,, x n ) will be a good point estimate of θ. Page 56 of 100

57 It should not depend on θ. One principle that is often used in finding point estimates. Page 57 of 100

58 Example 1. Let X 1, X 2,, X n denote a random sample from the distribution with p.d.f. f(x) = where 0 θ 1. θ x (1 θ) (1 x) x = 0, 1 0 elsewhere Page 58 of 100

59 The probability that X 1 = x 1, X 2 = x 2,, X n = x n is the joint p.d.f. θ x 1 (1 θ) (1 x 1) θ x 2 (1 θ) 1 x 2 θ x n (1 θ) 1 x n = θ x i (1 θ) n x i Page 59 of 100 where x i equals zero or 1, i = 1, 2,, n.

60 This probability,which is the joint p.d.f. of X 1, X 2,, X n, may be regarded as a function of θ and denoted it by L(θ) and called the likelihood function. That is, L(θ) = θ x i (1 θ) n x i Page 60 of θ 1.

61 What value of θ would maximize the probability L(θ) of obtaining this particular sample x 1, x 2,, x n? Page 61 of 100

62 The maximizing value of θ would seemingly be a good estimate of θ because it would provide the largest probability of this particular sample. Page 62 of 100

63 Either L(θ) or ln L(θ) can be used, since the likelihood function L(θ) and its logarithm, ln L(θ), are maximized for the same value θ. Page 63 of 100

64 Here ln L(θ) = ( n x i ) ln θ+(n n x i ) ln(1 θ); we have d ln L(θ) dθ 1 = xi θ provided that θ 0, 1. 1 n x i 1 θ = 0 Page 64 of 100

65 This is equivalent to the equation n n (1 θ) x i = θ(n x i ). 1 The solution for θ is n 1 x i/n, it maximizes L(θ) and ln L(θ) can be easily checked ( even for all of x 1, x 2,, x n = 0 or 1). 1 Page 65 of 100

66 Then n 1 x i/n is the value of θ that maximizes L(θ). The corresponding statistic, ˆθ = 1 n X i = n X i=1 is called the maximum likelihood estimator (MLE) of θ. Page 66 of 100

67 The observed value of ˆθ, namely n 1 x i/n, is called the value of maximum likelihood estimator (MLE) of θ. For example, suppose that n = 3, and x 1 = 1, x 2 = 0, x 3 = 1, then L(θ) = θ 2 (1 θ) and the observed ˆθ = 2 3 is the MLE of θ. Page 67 of 100

68 The principle of maximum likelihood: Consider a random sample X 1, X 2,, X n f(x, θ), θ Ω. The joint p.d.f. of X 1, X 2,, X n is f(x 1 ; θ)f(x 2 ; θ) f(x n ; θ). Page 68 of 100

69 This joint p.d.f. may be regarded as a function of θ. It is called the likelihood function L of the random sample. Write for θ Ω, L(θ; x 1, x 2,, x n ) = f(x 1 ; θ)f(x 2 ; θ) f(x n ; θ). Page 69 of 100

70 Suppose that u(x 1, x 2,, x n ) is a nontrivial function such that, when θ is replaced by u(x 1, x 2,, x n ), the likelihood function L is maximized: L[u(x 1, x 2,, x n ); x 1, x 2,, x n ] = max θ Ω L(θ; x 1, x 2,, x n ). Page 70 of 100

71 Then the statistic u(x 1, x 2,, x n ) will be called a maximum likelihood estimator(mle) of θ and will be denoted by the symbol ˆθ = u(x 1, x 2,, x n ). Page 71 of 100

72 In many instances there will be a unique MLE ˆθ of a parameter θ, It may be obtained by the process of differentiation. Page 72 of 100

73 Example 2. Let X 1, X 2,, X n N(θ, 1), < θ <. Here L(θ; x 1, x 2,, x n ) 1 n = ( ) n (x i θ) 2 exp[ ] 2π 2 1 Page 73 of 100

74 This function L can be maximized by setting the first derivative of L, with respect to θ for ln L. It may be easier to solve d ln L(θ; x 1, x 2,, x n ) dθ = 0. Page 74 of 100

75 d ln L(θ; x 1, x 2,, x n ) dθ = n (x i θ) = 0, the solution for the parameter θ is n u(x 1, x 2,, x n ) = x i /n. That n 1 x i/n actually maximizes L. 1 1 Page 75 of 100

76 Thus the statistic ˆθ = u(x 1, X 2,, X n ) = 1 n is the unique MLE of the mean θ. n X i = X 1 Page 76 of 100

77 From Examples 1, 2, E(ˆθ) = θ. The expected value of the estimator is equal to the corresponding parameter. Page 77 of 100

78 Definition: Any statistic whose mathematical expectation is equal to a parameter θ is called an unbiased estimator of the parameter θ. Otherwise, the statistic is said to be biased. Page 78 of 100

79 Example 3. Let 1 θ 0 < x θ, 0 < θ <, f(x; θ) = 0 elsewhere X 1, X 2,, X n denote a random sample from f(x; θ). Note that 0 < x θ. Page 79 of 100

80 We have L(θ; x 1, x 2,, x n ) = 1 θ n, 0 < x i θ Contents which is an ever-decreasing function of θ. The maximum of such functions cannot be found by differentiation. Page 80 of 100

81 L can be made no larger than 1 [max(x i )] n and the unique MLE ˆθ of θ is the nth order statistic max(x i ). E[max(X i )] = nθ/(n + 1). The MLE of the parameter θ is biased. Page 81 of 100

Probability Distributions Columns (a) through (d)

Discrete Probability Distributions Columns (a) through (d) Probability Mass Distribution Description Notes Notation or Density Function --------------------(PMF or PDF)-------------------- (a) (b) (c)