Definition of Statistic
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I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the random sample, after each random variable of the random sample has taken on a specific value?
$$(1) S=f(X_1,X_2...X_n)$$
$$(2) s=f(x_1,x_2...x_n)$$
I haven't been able to get any clarification for this and I've seen the term statistic describe both situations
estimation sampling inference random-variable interpretation
asked Mar 11 at 21:03
Colin HicksColin Hicks
1353
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add a comment |
$begingroup$
I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the random sample, after each random variable of the random sample has taken on a specific value?
$$(1) S=f(X_1,X_2...X_n)$$
$$(2) s=f(x_1,x_2...x_n)$$
I haven't been able to get any clarification for this and I've seen the term statistic describe both situations
estimation sampling inference random-variable interpretation
asked Mar 11 at 21:03
Colin HicksColin Hicks
1353
$endgroup$
$begingroup$
It seems that there are also plenty of conflicting views on estimator and estimate which go hand in hand with this. As an estimator is supposed to be a kind of statistic, whether an estimator is a random variable or not would also clarify some things. There seems to be a lot of conflicting definitions on this site though.
$endgroup$
– Colin Hicks
Mar 11 at 21:22
add a comment |
$begingroup$
I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the random sample, after each random variable of the random sample has taken on a specific value?
$$(1) S=f(X_1,X_2...X_n)$$
$$(2) s=f(x_1,x_2...x_n)$$
I haven't been able to get any clarification for this and I've seen the term statistic describe both situations
estimation sampling inference random-variable interpretation
asked Mar 11 at 21:03
Colin HicksColin Hicks
1353
$endgroup$
I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the random sample, after each random variable of the random sample has taken on a specific value?
$$(1) S=f(X_1,X_2...X_n)$$
$$(2) s=f(x_1,x_2...x_n)$$
I haven't been able to get any clarification for this and I've seen the term statistic describe both situations
estimation sampling inference random-variable interpretation
estimation sampling inference random-variable interpretation
asked Mar 11 at 21:03
Colin HicksColin Hicks
1353
asked Mar 11 at 21:03
Colin HicksColin Hicks
1353
asked Mar 11 at 21:03
Colin HicksColin Hicks
1353
asked Mar 11 at 21:03
Colin HicksColin Hicks
1353
asked Mar 11 at 21:03
Colin HicksColin Hicks
1353
1353
$begingroup$
It seems that there are also plenty of conflicting views on estimator and estimate which go hand in hand with this. As an estimator is supposed to be a kind of statistic, whether an estimator is a random variable or not would also clarify some things. There seems to be a lot of conflicting definitions on this site though.
$endgroup$
– Colin Hicks
Mar 11 at 21:22
add a comment |
$begingroup$
It seems that there are also plenty of conflicting views on estimator and estimate which go hand in hand with this. As an estimator is supposed to be a kind of statistic, whether an estimator is a random variable or not would also clarify some things. There seems to be a lot of conflicting definitions on this site though.
$endgroup$
– Colin Hicks
Mar 11 at 21:22
$begingroup$
It seems that there are also plenty of conflicting views on estimator and estimate which go hand in hand with this. As an estimator is supposed to be a kind of statistic, whether an estimator is a random variable or not would also clarify some things. There seems to be a lot of conflicting definitions on this site though.
$endgroup$
– Colin Hicks
Mar 11 at 21:22
$begingroup$
It seems that there are also plenty of conflicting views on estimator and estimate which go hand in hand with this. As an estimator is supposed to be a kind of statistic, whether an estimator is a random variable or not would also clarify some things. There seems to be a lot of conflicting definitions on this site though.
$endgroup$
– Colin Hicks
Mar 11 at 21:22
add a comment |
1 Answer
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A statistic is a function that maps from the set of outcomes of the observable values to a real number. Thus, with $n$ data points, a statistic will be a function $s: mathbb{R}^nrightarrow mathbb{R}$ as in your second form. However, it is also possible to view the statistic in its random sense by taking the appropriate composition of function with the original random variables. (Remember that each random variable $X_i: Omega rightarrow mathbb{R}$ is a measurable function that maps from the sample space to the real numbers.) That is, you can form the random variable $S: Omega rightarrow mathbb{R}$ as:
$$S(omega) = s(X_1(omega), ..., X_n(omega)).$$
The random variable $S$ is the random version of the statistic $s$. Both are often referred to as "statistics", but it is important to bear in mind that $S$ is a composition with the functions for the observable random variables.
answered Mar 11 at 21:28
BenBen
26.8k230124
$endgroup$
$begingroup$
that was very helpful. A lot of this notation is really confusing and seems at time almost conflicting as in this case where the term can be used in both contexts.
$endgroup$
– Colin Hicks
Mar 11 at 21:38
add a comment |
Your Answer
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1 Answer
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$begingroup$
A statistic is a function that maps from the set of outcomes of the observable values to a real number. Thus, with $n$ data points, a statistic will be a function $s: mathbb{R}^nrightarrow mathbb{R}$ as in your second form. However, it is also possible to view the statistic in its random sense by taking the appropriate composition of function with the original random variables. (Remember that each random variable $X_i: Omega rightarrow mathbb{R}$ is a measurable function that maps from the sample space to the real numbers.) That is, you can form the random variable $S: Omega rightarrow mathbb{R}$ as:
$$S(omega) = s(X_1(omega), ..., X_n(omega)).$$
The random variable $S$ is the random version of the statistic $s$. Both are often referred to as "statistics", but it is important to bear in mind that $S$ is a composition with the functions for the observable random variables.
answered Mar 11 at 21:28
BenBen
26.8k230124
$endgroup$
$begingroup$
that was very helpful. A lot of this notation is really confusing and seems at time almost conflicting as in this case where the term can be used in both contexts.
$endgroup$
– Colin Hicks
Mar 11 at 21:38
add a comment |
$begingroup$
A statistic is a function that maps from the set of outcomes of the observable values to a real number. Thus, with $n$ data points, a statistic will be a function $s: mathbb{R}^nrightarrow mathbb{R}$ as in your second form. However, it is also possible to view the statistic in its random sense by taking the appropriate composition of function with the original random variables. (Remember that each random variable $X_i: Omega rightarrow mathbb{R}$ is a measurable function that maps from the sample space to the real numbers.) That is, you can form the random variable $S: Omega rightarrow mathbb{R}$ as:
$$S(omega) = s(X_1(omega), ..., X_n(omega)).$$
The random variable $S$ is the random version of the statistic $s$. Both are often referred to as "statistics", but it is important to bear in mind that $S$ is a composition with the functions for the observable random variables.
answered Mar 11 at 21:28
BenBen
26.8k230124
$endgroup$
$begingroup$
that was very helpful. A lot of this notation is really confusing and seems at time almost conflicting as in this case where the term can be used in both contexts.
$endgroup$
– Colin Hicks
Mar 11 at 21:38
add a comment |
$begingroup$
A statistic is a function that maps from the set of outcomes of the observable values to a real number. Thus, with $n$ data points, a statistic will be a function $s: mathbb{R}^nrightarrow mathbb{R}$ as in your second form. However, it is also possible to view the statistic in its random sense by taking the appropriate composition of function with the original random variables. (Remember that each random variable $X_i: Omega rightarrow mathbb{R}$ is a measurable function that maps from the sample space to the real numbers.) That is, you can form the random variable $S: Omega rightarrow mathbb{R}$ as:
$$S(omega) = s(X_1(omega), ..., X_n(omega)).$$
The random variable $S$ is the random version of the statistic $s$. Both are often referred to as "statistics", but it is important to bear in mind that $S$ is a composition with the functions for the observable random variables.
answered Mar 11 at 21:28
BenBen
26.8k230124
$endgroup$
A statistic is a function that maps from the set of outcomes of the observable values to a real number. Thus, with $n$ data points, a statistic will be a function $s: mathbb{R}^nrightarrow mathbb{R}$ as in your second form. However, it is also possible to view the statistic in its random sense by taking the appropriate composition of function with the original random variables. (Remember that each random variable $X_i: Omega rightarrow mathbb{R}$ is a measurable function that maps from the sample space to the real numbers.) That is, you can form the random variable $S: Omega rightarrow mathbb{R}$ as:
$$S(omega) = s(X_1(omega), ..., X_n(omega)).$$
The random variable $S$ is the random version of the statistic $s$. Both are often referred to as "statistics", but it is important to bear in mind that $S$ is a composition with the functions for the observable random variables.
answered Mar 11 at 21:28
BenBen
26.8k230124
answered Mar 11 at 21:28
BenBen
26.8k230124
answered Mar 11 at 21:28
BenBen
26.8k230124
answered Mar 11 at 21:28
BenBen
26.8k230124
26.8k230124
$begingroup$
that was very helpful. A lot of this notation is really confusing and seems at time almost conflicting as in this case where the term can be used in both contexts.
$endgroup$
– Colin Hicks
Mar 11 at 21:38
add a comment |
$begingroup$
that was very helpful. A lot of this notation is really confusing and seems at time almost conflicting as in this case where the term can be used in both contexts.
$endgroup$
– Colin Hicks
Mar 11 at 21:38
$begingroup$
that was very helpful. A lot of this notation is really confusing and seems at time almost conflicting as in this case where the term can be used in both contexts.
$endgroup$
– Colin Hicks
Mar 11 at 21:38
$begingroup$
that was very helpful. A lot of this notation is really confusing and seems at time almost conflicting as in this case where the term can be used in both contexts.
$endgroup$
– Colin Hicks
Mar 11 at 21:38
add a comment |
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$begingroup$
It seems that there are also plenty of conflicting views on estimator and estimate which go hand in hand with this. As an estimator is supposed to be a kind of statistic, whether an estimator is a random variable or not would also clarify some things. There seems to be a lot of conflicting definitions on this site though.
$endgroup$
– Colin Hicks
Mar 11 at 21:22