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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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Interval estimation Asymptotic intervals Intervals for normal populations 2 2. Point and interval estimation
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INFERENCIA ESTADÍSTICA Introduction 3
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INFERENCIA ESTADÍSTICA Point estimation 4
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STATISTICAL INFERENCE Properties of estimators 5 Unbiased estimator is an unbiased estimator of if (bias of ) The bias of an unbiased estimator is zero:
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6 Efficiency STATISTICAL INFERENCE Properties of estimators
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7 Mean squared error STATISTICAL INFERENCE Properties of estimators
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8 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 STATISTICAL INFERENCE Properties of estimators
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Standard error 9 STATISTICAL INFERENCE
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10 Properties of estimators when Consistency is a consistent estimator for parameter if STATISTICAL INFERENCE Asymptotic behavior (Weak consistency) is strongly consistent for if
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11 Asymptotically normal is an asymptotically normal estimator with parameters if STATISTICAL INFERENCE Asymptotic properties
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Construction of estimators: method of moments 12 STATISTICAL INFERENCE X with or and we have a sample The k th moment is Method of moments: (i)Equal population moments to sample moments. (ii) Solve for the parameters.
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13 Properties: (i)Consistency Let be a method of moments estimator of Then STATISTICAL INFERENCE Construction of estimators: method of moments
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14 (ii) Asymptotic normality STATISTICAL INFERENCE Construction of estimators: method of moments
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Construction of estimators: maximum likelihood 15 STATISTICAL INFERENCE X; i.i.d. sample The maximum likelihood function is the probability density function or the probability mass function of the sample:
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16 is the maximum likelihood estimator of if Construction of estimators: maximum likelihood STATISTICAL INFERENCE The maximum likelihood estimator of is the value of making the observed sample most likely.
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17 Properties (i)Consistency Let be a maximum likelihood estimator of. Then (ii)Invariance If is a maximum likelihood estimator of, then is a maximum likelihood estimator of Construction of estimators: maximum likelihood STATISTICAL INFERENCE
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18 Properties (iii)Asymptotic normality (iv)Asymptotic efficiency The variance of is minimum. STATISTICAL INFERENCE Construction of estimators: maximum likelihood
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Construction of estimators: maximum likelihood 19 INFERENCIA ESTADÍSTICA
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Sampling in normal populations: Fisher’s lemma 20 Let Given the i. i. d. sample let Then: (i) (ii) (iii) are independent. STATISTICAL INFERENCE
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21 distribution Let independent. Then We define and it verifies STATISTICAL INFERENCE Sampling in normal populations
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22 If the population is normal, the distribution of the estimators is exactly known for any sample size. Sampling in normal populations STATISTICAL INFERENCE
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Confidence intervals 23 Let, and the sample Construct an interval with such that is the confidence coefficient. STATISTICAL INFERENCE
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24 Exact interval: Asymptotic interval: STATISTICAL INFERENCE Confidence intervals
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Confidence intervals : asymptotic intervals 25 an asymptotically normal estimator of Then STATISTICAL INFERENCE
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26 Define such that Then where STATISTICAL INFERENCE Confidence intervals : asymptotic intervals
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27 Then, the confidence interval for is STATISTICAL INFERENCE Confidence intervals : asymptotic intervals
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28 Remark: For large samples, we can obtain asymptotic confidence intervals. For small samples, we can obtain exact confidence intervals if the population is normal. Interval estimation : Asymptotic intervals STATISTICAL INFERENCE
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29 i. i. d. sample (i)Confidence interval for with known 0 2. Then STATISTICAL INFERENCE Intervals for normal populations
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30 (ii) Confidence intervals for with unknown 2. 2 is unknown: we estimate it. STATISTICAL INFERENCE Intervals for normal populations
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31 Student t distribution Let be independent. Then STATISTICAL INFERENCE Intervals for normal populations
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32 Let Then STATISTICAL INFERENCE Intervals for normal populations
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33 The confidence interval is thus STATISTICAL INFERENCE Intervals for normal populations
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34 INFERENCIA ESTADÍSTICA We change from an expression with 2 and N(0,1) to another expression with S 2 n-1 and t n-1 Intervals for normal populations
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35 (iii) Confidence interval for 2 with known 0. Each satisfies: and for the whole sample: STATISTICAL INFERENCE Intervals for normal populations
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36 and then STATISTICAL INFERENCE Intervals for normal populations
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37 (iv) Confidence interval for 2 with unknown . If, then applying Fisher’s Lemma: The confidence interval is: STATISTICAL INFERENCE Intervals for normal populations
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