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2. Point and interval estimation

2. Point and interval estimation. Introduction Properties of estimators Finite sample size Asymptotic properties Construction methods Method of moments Maximum likelihood estimation Sampling in normal populations. 1.

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2. Point and interval estimation

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  1. 2. Point and interval estimation Introduction Properties of estimators Finite sample size Asymptotic properties Construction methods Method of moments Maximum likelihood estimation Sampling in normal populations 1

  2. 2. Point and interval estimation Interval estimation Asymptotic intervals Intervals for normal populations 2

  3. Introduction 3 INFERENCIA ESTADÍSTICA

  4. Point estimation 4 INFERENCIA ESTADÍSTICA

  5. Propertiesofestimators Unbiased estimator is an unbiased estimator of if (bias of ) The bias of an unbiased estimator is zero: 5 STATISTICAL INFERENCE

  6. Propertiesofestimators Efficiency 6 STATISTICAL INFERENCE

  7. Propertiesofestimators Mean squared error 7 STATISTICAL INFERENCE

  8. Propertiesofestimators Mean squared error If the estimator is unbiased, then and the best one is chosen in terms of variance. The global criterion to select between two estimators is: is preferred to if 8 STATISTICAL INFERENCE

  9. Standarderror 9 STATISTICAL INFERENCE

  10. Asymptotic behavior Properties of estimators when Consistency is a consistent estimator for parameter if (weak consistency) is strongly consistent for if 10 STATISTICAL INFERENCE

  11. Asymptoticproperties Asymptotically normal is an asymptotically normal estimator with parameters if 11 STATISTICAL INFERENCE

  12. Construction of estimators: methodofmoments • X with or and we have a sample • The kthmoment is • Method of moments: • Equal population moments to sample moments. • (ii) Solve for the parameters. 12 STATISTICAL INFERENCE

  13. Construction of estimators: methodofmoments • Properties: • Consistency • Let be a method of moments estimator of • Then 13 STATISTICAL INFERENCE

  14. Construction of estimators: methodofmoments (ii) Asymptotic normality 14 STATISTICAL INFERENCE

  15. Construction of estimators: maximumlikelihood X; i.i.d. sample The likelihood function is the probability density function or the probability mass functionof the sample: 15 STATISTICAL INFERENCE

  16. Construction of estimators: maximumlikelihood The maximum likelihood estimator of is the value of making the observed sample most likely. is the maximum likelihood estimator of if 16 STATISTICAL INFERENCE

  17. Construction of estimators: maximumlikelihood • Properties • Consistency • Let be a maximum likelihood estimator of . • Then • Invariance • If is a maximum likelihood estimator of , • then is a maximum likelihood estimator • of 17 STATISTICAL INFERENCE

  18. Construction of estimators: maximumlikelihood • Properties • Asymptotic normality • Asymptotic efficiency • The variance of is minimum. 18 STATISTICAL INFERENCE

  19. Construction of estimators: maximum likelihood 19 INFERENCIA ESTADÍSTICA

  20. Samplinginnormalpopulations: Fisher’s lemma Let Given the i. i. d. sample let Then: (i) (ii) (iii) are independent. 20 STATISTICAL INFERENCE

  21. Samplinginnormalpopulations distribution Let independent. Then We define and it verifies 21 STATISTICAL INFERENCE

  22. Samplinginnormalpopulations If the population is normal, the distribution of the estimators is exactly known for any sample size. 22 STATISTICAL INFERENCE

  23. Confidence intervals Let , and the sample Construct an interval with such that is the confidence coefficient. 23 STATISTICAL INFERENCE

  24. Confidence intervals Exact interval: Asymptotic interval: 24 STATISTICAL INFERENCE

  25. Confidence intervals: asymptoticintervals an asymptotically normal estimator of Then 25 STATISTICAL INFERENCE

  26. Confidence intervals: asymptoticintervals Define such that Then where 26 STATISTICAL INFERENCE

  27. Confidence intervals: asymptoticintervals Then, the confidence interval for is 27 STATISTICAL INFERENCE

  28. Intervalestimation: Asymptoticintervals Remark: For large samples, we can obtain asymptotic confidence intervals. For small samples, we can obtain exact confidence intervals if the population is normal. 28 STATISTICAL INFERENCE

  29. Intervalsfornormalpopulations • i. i. d. sample • Confidence interval for  with known 02. • Then 29 STATISTICAL INFERENCE

  30. Intervals for normal populations • (ii) Confidence intervals for  with unknown 2. • 2 is unknown: we estimate it. 30 STATISTICAL INFERENCE

  31. Intervals for normal populations Student t distribution Let be independent. Then 31 STATISTICAL INFERENCE

  32. Intervals for normal populations Let Then 32 STATISTICAL INFERENCE

  33. Intervals for normal populations The confidence interval is thus 33 STATISTICAL INFERENCE

  34. Intervals for normal populations We change from an expression with 2 andN(0,1) to another expression with S2n-1 and tn-1 34 INFERENCIA ESTADÍSTICA

  35. Intervals for normal populations • (iii) Confidence interval for 2 with known 0. • Each satisfies: • and for the whole sample: 35 STATISTICAL INFERENCE

  36. Intervals for normal populations and then 36 STATISTICAL INFERENCE

  37. Intervals for normal populations • (iv) Confidence interval for 2 with unknown . • If , then applying Fisher’s Lemma: • The confidence interval is: 37 STATISTICAL INFERENCE

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