PEP-PMMA Training Session

Statistical inference

Lima, Peru

Abdelkrim Araar / Jean-Yves Duclos

9-10 June 2007

Why statistical inference?

• Distributive estimates obtained from surveys are not exact population values.

• The estimates normally follow a known asymptotic distribution. The parameters of that distribution can be estimated using sample information (including sampling design).

• Statistically, we can then peform hypothesis tests and draw confidence intervals.

Statistical inference

Assume that our statistic of interest is simply average income, and its estimator follows a normal distribution:

20ˆ ˆ~ N( , )

Statistical inference

A centred and normalised distribution can be obtained:

0ˆz ~ N(0,1)

ˆ

Hypothesis testing

There are three types of hypotheses that can be tested:

1. An index is equal to a given value: • Difference in poverty equals 0• Inequality equals to 20%

2. An index is higher than a given value: • Inequality has increased between two periods.

3. An index is lower than a given value: • Poverty has increased between two periods.

The interest of the statistical inferences

• The outcome of an hypothesis test is a statistical decision

• The conclusion of the test will either be to reject a null hypothesis, H0 in favour of an alternative, H1, or to fail to reject it.

• Most hypothesis tests involving an unknown true population parameter fall into three special cases:

1. H0 : μ = μ0 against H1: μ ≠ μ0

2. H0 : μ ≤ μ0 against H1: μ > μ0

3. H0 : μ ≥ μ0 against H1: μ < μ0

The interest of the statistical inferences

The ultimate statistical decision may be correct or incorrect. Two types of error can occur:

• Type I error, occurs when we reject H0 when it is in fact true;

• Type II error, occurs when we fail to reject H0 when H0 is in fact false.

• Power of the test of an hypothesis H0 versus H1 is the probability of rejecting H0 in favour of H1 when H1 is true.

• P-value is the smallest significance level for which H0 would be rejected in favour of some H1.

Hypothesis tests

0 0 1 / 2 0 0 1 / 2ˆ ˆˆ ˆz or z

Reject H0: μ = μ0 versus H1: μ ≠ μ0 if and only if :

Hypothesis tests

0 0 1 / 2ˆˆ z

Reject H0: μ ≤ μ0 versus H1: μ > μ0 if and only if :

Hypothesis tests

0 0 0 1ˆ ˆˆ ˆz z

Reject H0: μ > μ0 versus H1: μ ≤ μ0 if and only if :

Confidence intervals

• Loosely speaking, a confidence interval contains all of the values that “cannot be rejected” in a null hypothesis.

• Three types of confidence intervals can be drawn: