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Interval Estimation and Confidence Intervals in Parameter Estimation

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Understand maximum likelihood estimators, likelihood functions, and confidence intervals for unknown parameters in various distributions.

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Interval Estimation and Confidence Intervals in Parameter Estimation

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  1. Parameter Estimation Chapter 7 Interval estimation & Confidence interval

  2. Parameters • Let X1, X2, X3, …Xn be a random sample from a distribution Fθ that is specified up to a vector of unknown parameter θ. • E.g., this sample derived from a normal distribution with unknown mean μ and variance σ2; from a Poisson distribution with unknown λ; from a binomial distribution with unknown p, etc.

  3. Maximum likelihood estimators • Suppose the i.i.d. random variables X1, X2, …Xn, whose joint distribution is assumed given except for an unknown parameter θ, are to be observed and constituted a random sample. • The problem of interest is to use the observed values to estimate the unknown θ. • f(x1,x2,…,xn)=f(x1)f(x2)…f(xn), • if f(xi) is the exponential distribution, then f(x1,x2,…,xn) • = (1/θ)n exp{-(x1+x2+…xn)/ θ} (a function of θ) • The value of likelihood function f(x1,x2,…,xn/θ) will be determined by the observed sample (x1,x2,…,xn)if the true value of θ couldalso befound.

  4. The reasoning of maximum likelihood for a point estimator

  5. Maximum Likelihood estimator in the normal distribution

  6. Maximum Likelihood estimator in the Poisson distribution

  7. Confidence interval • Specify an interval for which we obtain a certain degree of confidence that a specific parameter lies within. • Suppose that X1,X2,…Xn is a sample from a normal population having unknown μ and known σ.

  8. The lower or upper 95% confidence interval • One-side upper confidence interval: (, ∞) • One-side lower confidence interval: (-∞,)

  9. General forms of confidence interval for the normal mean μ

  10. Confidence interval for the normal mean μ given an unknown σ

  11. Confidence interval for the normal variance

  12. Confidence interval for the difference between two normal populations

  13. Confidence interval for the difference between two normal populations (cont.)

  14. Approximate confidence interval for the mean of a Bernoulli random variable

  15. Homework #6 • Problem 1,11,28,41,47,54

  16. Optional homework • 某一醉漢踉蹌於左右各寬12步，長約100步的碼頭，原本欲至頂端處搭船，竟在途中墜落海中，保險公司聲稱該醉漢為自殺而非意外，拒絕支付理賠金。 • 根據觀察，一般醉漢踉蹌實為左右隨機，並沒有傾向任一方的現象。 • 雖有目擊者看到該墜海的醉漢最初是走在碼頭的中央線，但家屬仍聲稱隨機機率並不能代表特定的意外事件，就像擲銅板10次，也不能保証正反兩面各有5次，因而請求法院仲裁。你身為法官，應該如何判決?

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