Statistical inference form observational data Parameter estimation: Method of moments Use the data you have to calculate first and second moment To fit a certain distribution, use “relation to moments” formulae Method of maximum likelihood (too difficult) Interval estimation – confidence interval

Method of moments Suppose you have 10 data about x 0.3, 4, 5, 1, 1.3, 6.5, 0.85, 2.5, 4.56, 3.14 After calculation, mean = 2.915, var =

Method of moments Suppose we want to fit with uniform, Now Solving, b = , a = f X (x) x b a

Method of moments Suppose we want to fit with normal, Now E(X) = = μ Var(X) = = σ 2 N (2.915, 2.07) is suitable Try Lognormal yourself

Confidence interval of μ To calculate confidence interval, you need to know 1) One sided / two sided? 2) (true) variance known / unknown? Normal student-t

Confidence interval of μ- one sided Suppose you have 25 samples, sample mean = 9, sample s.d. = 2. Assume sample s.d. = true s.d. (why confidence interval?) P (True mean) < 10?

Confidence interval of μ- one sided true mean is smaller than a certain value with probability 0.98? 0.98 Φ (0.98) =

0.95 Φ (0.975) = Confidence interval of μ- two sided

Confidence interval of μ Compare k 0.02 k k 1-α k 1- α/2 α=0.02 α=0.05 k depends on 1) Confidence level α you want 2) One sided / two sided

Confidence interval of μ- student-t When the true variance is unknown, we use t and sample variance Suppose you have 25 samples, sample mean = 9, sample s.d. = 2 You keep everything the same but just check on another table! To check t, you need 1) confidence level, 2) d.o.f.

Confidence interval of μ- student-t true mean is smaller than a certain value with probability 0.98? T depends on 1) Confidence level α you want 2) One sided / two sided 3) Degree of freedom

Confidence interval of μ Compare k 0.02 k T 0.02, 24 k 1-α k 1- α/2 T 1- α, 24 α=0.02 α=0.05 α=0.02 compare 1.96 and

As you only have limited data points, your sample variance will also subject to variation JUST AS variation of sample mean Example DO data: n = 30, s 2 = 4.2 Variance of variance? To check chi-square, you need 1) Probability level α 2) d.o.f. We usually construct one-sided confidence interval of variance (why?)

Probability Paper (old) Chi-square test (χ 2 ) (common) Kolmogorov-Smirnov test (K-S) (difficult to use) Chi-square test ei No. of parameters in the model Goodness of fit test of distribution