Definition of StatisticComplete sufficient statisticIs “test statistic” a value or a random...

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Definition of Statistic


Complete sufficient statisticIs “test statistic” a value or a random variable?Example of sample $X_1,X_2,ldots,X_n$Expected value of iid random variablesAnalytical properties of sample quantiles in statistical packages“Let random variables $X_1,dots, X_n$ be a iid random sample from $f(x)$” - what does it mean?Is $bar{X}_n = dfrac{(X_1 + X_2 + … + X_n)}{n}$ an estimator of the mean in general (for random variables with any distribution)?What is statistic in statistics?Statistics can't be a function of a parameter - but isn't the sample a function of the parameter?Sufficient statistics - how can the con. pdf not depend on θ when θ is in the equation?













2












$begingroup$


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










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Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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
    4 hours ago
















2












$begingroup$


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










share|cite|improve this question







New contributor




Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    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
    4 hours ago














2












2








2





$begingroup$


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










share|cite|improve this question







New contributor




Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




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






share|cite|improve this question







New contributor




Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







New contributor




Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question






New contributor




Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 5 hours ago









Colin HicksColin Hicks

1133




1133




New contributor




Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Colin Hicks is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    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
    4 hours ago


















  • $begingroup$
    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
    4 hours ago
















$begingroup$
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
4 hours ago




$begingroup$
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
4 hours ago










1 Answer
1






active

oldest

votes


















3












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
    4 hours ago











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
    4 hours ago
















3












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
    4 hours ago














3












3








3





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 4 hours ago









BenBen

26.5k229124




26.5k229124












  • $begingroup$
    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
    4 hours ago


















  • $begingroup$
    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
    4 hours ago
















$begingroup$
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
4 hours ago




$begingroup$
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
4 hours ago










Colin Hicks is a new contributor. Be nice, and check out our Code of Conduct.










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