2 Quantile

Question 1

(S) QF-(p) / QF+(p)

Answer 1

1. QF-(p) := min{t ∈ R | F(t) ≥ p} is the smallest p–quantile.
2. QF+(p) := max{t ∈ R | F(t-) ≤ p} is the largest p–quantile.

Question 2

(D) Quantilfunktion

Answer 2

F CDF. Define QF-, QF+ : (0, 1) -> R by
QF-(p) = min{t ∈ R | F(t) ≥ p} , QF+(p) = max{t ∈ R | F(t-) ≤ p}.

QF- is called the quantile function of F.

Question 3

(B) Properties of quantile function

Answer 3

Obviously QF- ≤ QF+, and QF-,QF+ monotonic. Reflection principle suggests:

1. QF- left continuous, QF+ right continuous
2. F continuous <==> QF-,QF+ strictly monotonic
3. F strictly monotonic <==>. QF-,QF+ continuous

Question 4

(L) Inversion lemma (2)

Answer 4

F CDF, p ∈ (0, 1), t ∈ R.

1. We have
   F(t) ≥ p <==> t ≥ QF-(p), F(t-) ≤ p <==> t ≤ QF+(p).
2. If F is piecewise constant, then we have additionally
   F(t) > p <==> t ≥ QF+(p), F(t-) < p <==> t ≤ QF-(p).

Question 5

(D) Boxplot (3)

Answer 5

1. fette Linie in der Box: Median
2. oberes/unteres Ende der Box: erstes/drittes Quartil
3. oberste/unterste Linie: Maximum/Minimum

Question 6

(D) p-quantile, quartiles, median

Answer 6

F CDF, p\in (0,1). A real number xp is called p-quantile of F if

F(xp-) ≤ p ≤ F(xp),

where F(t-) = lim_s>t(nachoben) F(s) the left limit. x0.25, x0.5, x0.75 are called quartiles, x0.5 median of F.