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

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

INFERENCIA ESTADÍSTICA Introduction 3

INFERENCIA ESTADÍSTICA Point estimation 4

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

6 Efficiency STATISTICAL INFERENCE Properties of estimators

7 Mean squared error STATISTICAL INFERENCE Properties of estimators

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

Standard error 9 STATISTICAL INFERENCE

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

11 Asymptotically normal is an asymptotically normal estimator with parameters if STATISTICAL INFERENCE Asymptotic properties

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.

13 Properties: (i)Consistency Let be a method of moments estimator of Then STATISTICAL INFERENCE Construction of estimators: method of moments

14 (ii) Asymptotic normality STATISTICAL INFERENCE Construction of estimators: method of moments

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:

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.

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

18 Properties (iii)Asymptotic normality (iv)Asymptotic efficiency The variance of is minimum. STATISTICAL INFERENCE Construction of estimators: maximum likelihood

Construction of estimators: maximum likelihood 19 INFERENCIA ESTADÍSTICA

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

21 distribution Let independent. Then We define and it verifies STATISTICAL INFERENCE Sampling in normal populations

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

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

24 Exact interval: Asymptotic interval: STATISTICAL INFERENCE Confidence intervals

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

26 Define such that Then where STATISTICAL INFERENCE Confidence intervals : asymptotic intervals

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

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

29 i. i. d. sample (i)Confidence interval for  with known  0 2. Then STATISTICAL INFERENCE Intervals for normal populations

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

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

32 Let Then STATISTICAL INFERENCE Intervals for normal populations

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

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

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

36 and then STATISTICAL INFERENCE Intervals for normal populations

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