Let $X_1, X2,dots, X_n$ be iid variables with cdf $F(x)$. Define the empirical cdf as
Let $X_1, X_2,dots, X_n$ be iid random variables with CDF $F(x)$.
Define the empirical CDF as
$$
F_n(x) ={1over n} sum_{i=1}^n I(X_i le x)quad -infty < x < +infty
$$
where $I(.)$ is the indicator function.
Fix the value of $x$.
Find $Ebig(F_n(x)big)$ and $mathrm{Var}big(F_n(x)big)$
Show that $F_n(x)$ converges in probability to $F(x)$. [ Here they are asking to show that the empirical CDF converges to the CDF $F(x)$]
Let $Y_n = sqrt{n}big(F_n(x) - F(x)big)$. Find the asymptotic distribution (limiting distribution) of $Y_n$.
I was able to figure out that $I(X_i leq x)$ is a Bernoulli random variable and we need to use Central Limit theorem and might as well the delta method for the third question.
Can you all please guide me in solving the problem?
probability-theory convergence central-limit-theorem order-statistics delta-method
edited Dec 7 at 9:16
Daniele Tampieri
1,6331619
asked Dec 7 at 8:28
Captain Erwin
1
add a comment |
Let $X_1, X_2,dots, X_n$ be iid random variables with CDF $F(x)$.
Define the empirical CDF as
$$
F_n(x) ={1over n} sum_{i=1}^n I(X_i le x)quad -infty < x < +infty
$$
where $I(.)$ is the indicator function.
Fix the value of $x$.
Find $Ebig(F_n(x)big)$ and $mathrm{Var}big(F_n(x)big)$
Show that $F_n(x)$ converges in probability to $F(x)$. [ Here they are asking to show that the empirical CDF converges to the CDF $F(x)$]
Let $Y_n = sqrt{n}big(F_n(x) - F(x)big)$. Find the asymptotic distribution (limiting distribution) of $Y_n$.
I was able to figure out that $I(X_i leq x)$ is a Bernoulli random variable and we need to use Central Limit theorem and might as well the delta method for the third question.
Can you all please guide me in solving the problem?
probability-theory convergence central-limit-theorem order-statistics delta-method
edited Dec 7 at 9:16
Daniele Tampieri
1,6331619
asked Dec 7 at 8:28
Captain Erwin
1
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 7 at 8:40
Terrible title. Body with no personal input.
– Did
Dec 7 at 9:25
add a comment |
Let $X_1, X_2,dots, X_n$ be iid random variables with CDF $F(x)$.
Define the empirical CDF as
$$
F_n(x) ={1over n} sum_{i=1}^n I(X_i le x)quad -infty < x < +infty
$$
where $I(.)$ is the indicator function.
Fix the value of $x$.
Find $Ebig(F_n(x)big)$ and $mathrm{Var}big(F_n(x)big)$
Show that $F_n(x)$ converges in probability to $F(x)$. [ Here they are asking to show that the empirical CDF converges to the CDF $F(x)$]
Let $Y_n = sqrt{n}big(F_n(x) - F(x)big)$. Find the asymptotic distribution (limiting distribution) of $Y_n$.
I was able to figure out that $I(X_i leq x)$ is a Bernoulli random variable and we need to use Central Limit theorem and might as well the delta method for the third question.
Can you all please guide me in solving the problem?
probability-theory convergence central-limit-theorem order-statistics delta-method
edited Dec 7 at 9:16
Daniele Tampieri
1,6331619
asked Dec 7 at 8:28
Captain Erwin
1
Let $X_1, X_2,dots, X_n$ be iid random variables with CDF $F(x)$.
Define the empirical CDF as
$$
F_n(x) ={1over n} sum_{i=1}^n I(X_i le x)quad -infty < x < +infty
$$
where $I(.)$ is the indicator function.
Fix the value of $x$.
Find $Ebig(F_n(x)big)$ and $mathrm{Var}big(F_n(x)big)$
Show that $F_n(x)$ converges in probability to $F(x)$. [ Here they are asking to show that the empirical CDF converges to the CDF $F(x)$]
Let $Y_n = sqrt{n}big(F_n(x) - F(x)big)$. Find the asymptotic distribution (limiting distribution) of $Y_n$.
I was able to figure out that $I(X_i leq x)$ is a Bernoulli random variable and we need to use Central Limit theorem and might as well the delta method for the third question.
Can you all please guide me in solving the problem?
probability-theory convergence central-limit-theorem order-statistics delta-method
probability-theory convergence central-limit-theorem order-statistics delta-method
edited Dec 7 at 9:16
Daniele Tampieri
1,6331619
asked Dec 7 at 8:28
Captain Erwin
1
edited Dec 7 at 9:16
Daniele Tampieri
1,6331619
asked Dec 7 at 8:28
Captain Erwin
1
edited Dec 7 at 9:16
Daniele Tampieri
1,6331619
edited Dec 7 at 9:16
Daniele Tampieri
1,6331619
edited Dec 7 at 9:16
Daniele Tampieri
1,6331619
1,6331619
asked Dec 7 at 8:28
Captain Erwin
1
asked Dec 7 at 8:28
Captain Erwin
1
asked Dec 7 at 8:28
Captain Erwin
1
1
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 7 at 8:40
Terrible title. Body with no personal input.
– Did
Dec 7 at 9:25
add a comment |
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 7 at 8:40
Terrible title. Body with no personal input.
– Did
Dec 7 at 9:25
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 7 at 8:40
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 7 at 8:40
Terrible title. Body with no personal input.
– Did
Dec 7 at 9:25
Terrible title. Body with no personal input.
– Did
Dec 7 at 9:25
add a comment |
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Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Dec 7 at 8:40
Terrible title. Body with no personal input.
– Did
Dec 7 at 9:25