Question

3. Let Xi, . . . , Xn be random samples of X and X(1), . . . , X(n) ordered random samples of X which are obtained from a rearrangement of X1,... , Xn such that (a) Show that the empirical distribution functions of Xi,..., Xn and Xo),..., X(n) coincide. (b) Consider the samples taken from X ~ F. Use (a) to compute A-2), F,,(-1)Ћ,(1.8),Ћ,(25) (c) (Continued from (b)) Plot A,(z) over-2 4.

Answer 1

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ANSWER::-

(a) By definition

n Fn(r)
number of observations among $(X_1,X_2,..,X_n)$ which are at most x., where
Fn(r)
is the empirical cdf.

Since,  the number of observations which are at most x does not change if we consider the full set of order statistics, the empirical distribution functions of $(X_1,X_2,..,X_n)$ and the full set of order statistics coincide.

(b) We arrange the given observations from smaller to larger to get the ordered values:

-2, -1, -.5, 0, 0.8, 1, 1.5, 2, 2.4, 3

Then n=10,
0Fn (1.8)
number of observations in the above which are less than or equal to 1.8

Then
10F,(1.8) 7
and $F_n(1.8)=7/10$

10Fn (2.5)
number of observations in the above which are less than or equal to 2.5

Then
10F (2.5)-9
and
E,(2.5)-9/10

10Fn(-1)-
number of observations in the above which are less than or equal to -1

Then
10F,(-1) = 2
and
F,i (-1)-2/10

10F-2)
number of observations in the above which are less than or equal to -2

Then $10F_n(-2)=1$ and
Fn(-2)1/10

(c)  

ecdf(x) -2 2 4

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