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BatWannaBe
BatWannaBe
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I want to bound the difference between two variables sampled from two different populations. When it's just one population with paired measurements a.k.a. "paired samples" (e.g. before treatment vs after treatment), I simply subtract the paired measurements and use univariate statistical inference. Is there a reason I cannot do the same for independent samples? Say I have two i.i.d. samples $X_1,X_2,...,X_n$ and $Y_1,Y_2,...,Y_m$ with respective cumulative distribution functions $F_X$ and $F_Y$. Of course, if I construct a set from all possible pairwise differences, the set is not i.i.d.; $X_1-Y_1$ and $X_1-Y_2$ are dependent, for example. However, if I construct a set where each random variable from the original two samples only appears in at most one difference, such as ${X_1-Y_1,X_2-Y_2,...,X_{min(n, m)}-Y_{min(n, m)}}$, is that set i.i.d. with cumulative distribution function $F_{X-Y}$? If so, can I then apply univariate statistics to that?
UPDATE: Based on the two methods of constructing a set from two independent samples, I simulated 10,000 confidence intervals for the median bound by order statistics and 10,000 tolerance intervals bound by order statistics for the difference between two normal random variables. I stuck to normal variables because I could derive the median and cdf to test if a particular simulated interval really did contain the median or population proportion. The dependent nXm pairwise differences method yielded intervals that contained the median/proportion far less frequently than the intervals should have for iid samples, which is expected for a non-iid sample, and the min(n,m) differences method contained the median/proportion about as frequently as the interval should have for iid samples. I'm working on a way to do the same for non-normal populations next, but it's clear from this that the n*m non-iid sample wouldn't work for order statistics-based intervals.

I want to bound the difference between two variables sampled from two different populations. When it's just one population with paired measurements a.k.a. "paired samples" (e.g. before treatment vs after treatment), I simply subtract the paired measurements and use univariate statistical inference. Is there a reason I cannot do the same for independent samples? Say I have two i.i.d. samples $X_1,X_2,...,X_n$ and $Y_1,Y_2,...,Y_m$ with respective cumulative distribution functions $F_X$ and $F_Y$. Of course, if I construct a set from all possible pairwise differences, the set is not i.i.d.; $X_1-Y_1$ and $X_1-Y_2$ are dependent, for example. However, if I construct a set where each random variable from the original two samples only appears in at most one difference, such as ${X_1-Y_1,X_2-Y_2,...,X_{min(n, m)}-Y_{min(n, m)}}$, is that set i.i.d. with cumulative distribution function $F_{X-Y}$? If so, can I then apply univariate statistics to that?

I want to bound the difference between two variables sampled from two different populations. When it's just one population with paired measurements a.k.a. "paired samples" (e.g. before treatment vs after treatment), I simply subtract the paired measurements and use univariate statistical inference. Is there a reason I cannot do the same for independent samples? Say I have two i.i.d. samples $X_1,X_2,...,X_n$ and $Y_1,Y_2,...,Y_m$ with respective cumulative distribution functions $F_X$ and $F_Y$. Of course, if I construct a set from all possible pairwise differences, the set is not i.i.d.; $X_1-Y_1$ and $X_1-Y_2$ are dependent, for example. However, if I construct a set where each random variable from the original two samples only appears in at most one difference, such as ${X_1-Y_1,X_2-Y_2,...,X_{min(n, m)}-Y_{min(n, m)}}$, is that set i.i.d. with cumulative distribution function $F_{X-Y}$? If so, can I then apply univariate statistics to that?
UPDATE: Based on the two methods of constructing a set from two independent samples, I simulated 10,000 confidence intervals for the median bound by order statistics and 10,000 tolerance intervals bound by order statistics for the difference between two normal random variables. I stuck to normal variables because I could derive the median and cdf to test if a particular simulated interval really did contain the median or population proportion. The dependent nXm pairwise differences method yielded intervals that contained the median/proportion far less frequently than the intervals should have for iid samples, which is expected for a non-iid sample, and the min(n,m) differences method contained the median/proportion about as frequently as the interval should have for iid samples. I'm working on a way to do the same for non-normal populations next, but it's clear from this that the n*m non-iid sample wouldn't work for order statistics-based intervals.

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BatWannaBe
BatWannaBe
  • 270
  • 1
  • 8

Is Function of Random Samples of Two Independent Populations I.I.D.?

I want to bound the difference between two variables sampled from two different populations. When it's just one population with paired measurements a.k.a. "paired samples" (e.g. before treatment vs after treatment), I simply subtract the paired measurements and use univariate statistical inference. Is there a reason I cannot do the same for independent samples? Say I have two i.i.d. samples $X_1,X_2,...,X_n$ and $Y_1,Y_2,...,Y_m$ with respective cumulative distribution functions $F_X$ and $F_Y$. Of course, if I construct a set from all possible pairwise differences, the set is not i.i.d.; $X_1-Y_1$ and $X_1-Y_2$ are dependent, for example. However, if I construct a set where each random variable from the original two samples only appears in at most one difference, such as ${X_1-Y_1,X_2-Y_2,...,X_{min(n, m)}-Y_{min(n, m)}}$, is that set i.i.d. with cumulative distribution function $F_{X-Y}$? If so, can I then apply univariate statistics to that?