Mixing
Struggling to bridge understanding from Probability Theory to Hypothesis Testing Statistics
$begingroup$
I have recently done some Probability Theory and am struggling to come to terms with our new chapter: Statistics. This misunderstanding specifically pertains to hypothesis testing.
Let $(Omega, mathcal{F},( P_{vartheta},varthetaintheta))$ be a statistical model
I know that the general idea of a test $F in mathcal{F}$ for $H_{0}subseteq ( P_{vartheta},varthetaintheta)$ is determining a significance level $alpha in [0,1]$ such that $sup_{P_{vartheta} in H_{0}}P_{vartheta}(F)leq alpha$. We set $alpha$ appropriately low so that so that
$P_{vartheta}(F)leq alpha, forall P_{vartheta} in H_{0}$ is extremely unlikely.
Next, I get confused by the following:
If we observe $omega in F$(!) ($1.$Question: Should it not be $omega in mathcal{F}$?) then we should discard $H_{0}$, otherwise $H_{0}$ is kept.
$2.$ Question: Why are we looking at "singletons" $omega$?
If we, for example are on a continuous probability space $(Omega, mathcal{F},P_{vartheta})$ then $P({w})=0leq alpha$, so no $H_{0}$ would fit. Or are statistical models simply always discrete, and therefore looking at singletons makes sense?
$3.$ I may be missing some key intuition as to the differences between the Probability Theory and Statistics. Any intuitions are greatly appreciated.
probability probability-theory statistics hypothesis-testing
asked Jan 18 at 22:24
SABOYSABOY
656311
$endgroup$
|
show 1 more comment
$begingroup$
I have recently done some Probability Theory and am struggling to come to terms with our new chapter: Statistics. This misunderstanding specifically pertains to hypothesis testing.
Let $(Omega, mathcal{F},( P_{vartheta},varthetaintheta))$ be a statistical model
I know that the general idea of a test $F in mathcal{F}$ for $H_{0}subseteq ( P_{vartheta},varthetaintheta)$ is determining a significance level $alpha in [0,1]$ such that $sup_{P_{vartheta} in H_{0}}P_{vartheta}(F)leq alpha$. We set $alpha$ appropriately low so that so that
$P_{vartheta}(F)leq alpha, forall P_{vartheta} in H_{0}$ is extremely unlikely.
Next, I get confused by the following:
If we observe $omega in F$(!) ($1.$Question: Should it not be $omega in mathcal{F}$?) then we should discard $H_{0}$, otherwise $H_{0}$ is kept.
$2.$ Question: Why are we looking at "singletons" $omega$?
If we, for example are on a continuous probability space $(Omega, mathcal{F},P_{vartheta})$ then $P({w})=0leq alpha$, so no $H_{0}$ would fit. Or are statistical models simply always discrete, and therefore looking at singletons makes sense?
$3.$ I may be missing some key intuition as to the differences between the Probability Theory and Statistics. Any intuitions are greatly appreciated.
probability probability-theory statistics hypothesis-testing
asked Jan 18 at 22:24
SABOYSABOY
656311
$endgroup$
$begingroup$
1. The notation "$omega in mathcal{F}$" does not make sense. 2. We are looking at a particular outcome from $Omega$ (e.g. a sample drawn from $P_{vartheta}$). Based on that outcome we make a decision. Since the likelihood of getting $omegain F$ is small, we reject $H_0$ in favor to the alternative.
$endgroup$
– d.k.o.
Jan 19 at 4:39
$begingroup$
@d.k.o. In that case, should we not say $P_{vartheta}(omega)leq alpha$ rather than $P_{vartheta}(F)leq alpha$? So is the difference between the probability model $(Omega, mathcal{F}, P)$ and the statistical model $(Omega, mathcal{F}, P_{vartheta})$ is that in the probability model $F in mathcal{F}$ represents a particular event, while in statistics $F in mathcal{F}$ represents taking a sample. This leads me to believe however that all statiscal models are discrete. Is this true?
$endgroup$
– SABOY
Jan 20 at 11:10
$begingroup$
$F$ represents an event in both cases.
$endgroup$
– d.k.o.
Jan 20 at 18:23
$begingroup$
@d.k.o. An event in a statistical sense would stand for a sample however, correct?
$endgroup$
– SABOY
Jan 20 at 21:45
$begingroup$
Consider again the example below. If $X(omega)=omega$, then $X$ is a random variable having the exponential distribution with parameter $vartheta$. A sample here would correspond to a single realization of $X$, i.e. to a single outcome $omega$. An example of an event is ${omega:X(omega)>x}$.
$endgroup$
– d.k.o.
Jan 20 at 22:55
|
show 1 more comment
$begingroup$
I have recently done some Probability Theory and am struggling to come to terms with our new chapter: Statistics. This misunderstanding specifically pertains to hypothesis testing.
Let $(Omega, mathcal{F},( P_{vartheta},varthetaintheta))$ be a statistical model
I know that the general idea of a test $F in mathcal{F}$ for $H_{0}subseteq ( P_{vartheta},varthetaintheta)$ is determining a significance level $alpha in [0,1]$ such that $sup_{P_{vartheta} in H_{0}}P_{vartheta}(F)leq alpha$. We set $alpha$ appropriately low so that so that
$P_{vartheta}(F)leq alpha, forall P_{vartheta} in H_{0}$ is extremely unlikely.
Next, I get confused by the following:
If we observe $omega in F$(!) ($1.$Question: Should it not be $omega in mathcal{F}$?) then we should discard $H_{0}$, otherwise $H_{0}$ is kept.
$2.$ Question: Why are we looking at "singletons" $omega$?
If we, for example are on a continuous probability space $(Omega, mathcal{F},P_{vartheta})$ then $P({w})=0leq alpha$, so no $H_{0}$ would fit. Or are statistical models simply always discrete, and therefore looking at singletons makes sense?
$3.$ I may be missing some key intuition as to the differences between the Probability Theory and Statistics. Any intuitions are greatly appreciated.
probability probability-theory statistics hypothesis-testing
asked Jan 18 at 22:24
SABOYSABOY
656311
$endgroup$
I have recently done some Probability Theory and am struggling to come to terms with our new chapter: Statistics. This misunderstanding specifically pertains to hypothesis testing.
Let $(Omega, mathcal{F},( P_{vartheta},varthetaintheta))$ be a statistical model
I know that the general idea of a test $F in mathcal{F}$ for $H_{0}subseteq ( P_{vartheta},varthetaintheta)$ is determining a significance level $alpha in [0,1]$ such that $sup_{P_{vartheta} in H_{0}}P_{vartheta}(F)leq alpha$. We set $alpha$ appropriately low so that so that
$P_{vartheta}(F)leq alpha, forall P_{vartheta} in H_{0}$ is extremely unlikely.
Next, I get confused by the following:
If we observe $omega in F$(!) ($1.$Question: Should it not be $omega in mathcal{F}$?) then we should discard $H_{0}$, otherwise $H_{0}$ is kept.
$2.$ Question: Why are we looking at "singletons" $omega$?
If we, for example are on a continuous probability space $(Omega, mathcal{F},P_{vartheta})$ then $P({w})=0leq alpha$, so no $H_{0}$ would fit. Or are statistical models simply always discrete, and therefore looking at singletons makes sense?
$3.$ I may be missing some key intuition as to the differences between the Probability Theory and Statistics. Any intuitions are greatly appreciated.
probability probability-theory statistics hypothesis-testing
probability probability-theory statistics hypothesis-testing
asked Jan 18 at 22:24
SABOYSABOY
656311
asked Jan 18 at 22:24
SABOYSABOY
656311
asked Jan 18 at 22:24
SABOYSABOY
656311
asked Jan 18 at 22:24
SABOYSABOY
656311
asked Jan 18 at 22:24
SABOYSABOY
656311
656311
$begingroup$
1. The notation "$omega in mathcal{F}$" does not make sense. 2. We are looking at a particular outcome from $Omega$ (e.g. a sample drawn from $P_{vartheta}$). Based on that outcome we make a decision. Since the likelihood of getting $omegain F$ is small, we reject $H_0$ in favor to the alternative.
$endgroup$
– d.k.o.
Jan 19 at 4:39
$begingroup$
@d.k.o. In that case, should we not say $P_{vartheta}(omega)leq alpha$ rather than $P_{vartheta}(F)leq alpha$? So is the difference between the probability model $(Omega, mathcal{F}, P)$ and the statistical model $(Omega, mathcal{F}, P_{vartheta})$ is that in the probability model $F in mathcal{F}$ represents a particular event, while in statistics $F in mathcal{F}$ represents taking a sample. This leads me to believe however that all statiscal models are discrete. Is this true?
$endgroup$
– SABOY
Jan 20 at 11:10
$begingroup$
$F$ represents an event in both cases.
$endgroup$
– d.k.o.
Jan 20 at 18:23
$begingroup$
@d.k.o. An event in a statistical sense would stand for a sample however, correct?
$endgroup$
– SABOY
Jan 20 at 21:45
$begingroup$
Consider again the example below. If $X(omega)=omega$, then $X$ is a random variable having the exponential distribution with parameter $vartheta$. A sample here would correspond to a single realization of $X$, i.e. to a single outcome $omega$. An example of an event is ${omega:X(omega)>x}$.
$endgroup$
– d.k.o.
Jan 20 at 22:55
|
show 1 more comment
$begingroup$
1. The notation "$omega in mathcal{F}$" does not make sense. 2. We are looking at a particular outcome from $Omega$ (e.g. a sample drawn from $P_{vartheta}$). Based on that outcome we make a decision. Since the likelihood of getting $omegain F$ is small, we reject $H_0$ in favor to the alternative.
$endgroup$
– d.k.o.
Jan 19 at 4:39
$begingroup$
@d.k.o. In that case, should we not say $P_{vartheta}(omega)leq alpha$ rather than $P_{vartheta}(F)leq alpha$? So is the difference between the probability model $(Omega, mathcal{F}, P)$ and the statistical model $(Omega, mathcal{F}, P_{vartheta})$ is that in the probability model $F in mathcal{F}$ represents a particular event, while in statistics $F in mathcal{F}$ represents taking a sample. This leads me to believe however that all statiscal models are discrete. Is this true?
$endgroup$
– SABOY
Jan 20 at 11:10
$begingroup$
$F$ represents an event in both cases.
$endgroup$
– d.k.o.
Jan 20 at 18:23
$begingroup$
@d.k.o. An event in a statistical sense would stand for a sample however, correct?
$endgroup$
– SABOY
Jan 20 at 21:45
$begingroup$
Consider again the example below. If $X(omega)=omega$, then $X$ is a random variable having the exponential distribution with parameter $vartheta$. A sample here would correspond to a single realization of $X$, i.e. to a single outcome $omega$. An example of an event is ${omega:X(omega)>x}$.
$endgroup$
– d.k.o.
Jan 20 at 22:55
$begingroup$
1. The notation "$omega in mathcal{F}$" does not make sense. 2. We are looking at a particular outcome from $Omega$ (e.g. a sample drawn from $P_{vartheta}$). Based on that outcome we make a decision. Since the likelihood of getting $omegain F$ is small, we reject $H_0$ in favor to the alternative.
$endgroup$
– d.k.o.
Jan 19 at 4:39
$begingroup$
1. The notation "$omega in mathcal{F}$" does not make sense. 2. We are looking at a particular outcome from $Omega$ (e.g. a sample drawn from $P_{vartheta}$). Based on that outcome we make a decision. Since the likelihood of getting $omegain F$ is small, we reject $H_0$ in favor to the alternative.
$endgroup$
– d.k.o.
Jan 19 at 4:39
$begingroup$
@d.k.o. In that case, should we not say $P_{vartheta}(omega)leq alpha$ rather than $P_{vartheta}(F)leq alpha$? So is the difference between the probability model $(Omega, mathcal{F}, P)$ and the statistical model $(Omega, mathcal{F}, P_{vartheta})$ is that in the probability model $F in mathcal{F}$ represents a particular event, while in statistics $F in mathcal{F}$ represents taking a sample. This leads me to believe however that all statiscal models are discrete. Is this true?
$endgroup$
– SABOY
Jan 20 at 11:10
$begingroup$
@d.k.o. In that case, should we not say $P_{vartheta}(omega)leq alpha$ rather than $P_{vartheta}(F)leq alpha$? So is the difference between the probability model $(Omega, mathcal{F}, P)$ and the statistical model $(Omega, mathcal{F}, P_{vartheta})$ is that in the probability model $F in mathcal{F}$ represents a particular event, while in statistics $F in mathcal{F}$ represents taking a sample. This leads me to believe however that all statiscal models are discrete. Is this true?
$endgroup$
– SABOY
Jan 20 at 11:10
$begingroup$
$F$ represents an event in both cases.
$endgroup$
– d.k.o.
Jan 20 at 18:23
$begingroup$
$F$ represents an event in both cases.
$endgroup$
– d.k.o.
Jan 20 at 18:23
$begingroup$
@d.k.o. An event in a statistical sense would stand for a sample however, correct?
$endgroup$
– SABOY
Jan 20 at 21:45
$begingroup$
@d.k.o. An event in a statistical sense would stand for a sample however, correct?
$endgroup$
– SABOY
Jan 20 at 21:45
$begingroup$
Consider again the example below. If $X(omega)=omega$, then $X$ is a random variable having the exponential distribution with parameter $vartheta$. A sample here would correspond to a single realization of $X$, i.e. to a single outcome $omega$. An example of an event is ${omega:X(omega)>x}$.
$endgroup$
– d.k.o.
Jan 20 at 22:55
$begingroup$
Consider again the example below. If $X(omega)=omega$, then $X$ is a random variable having the exponential distribution with parameter $vartheta$. A sample here would correspond to a single realization of $X$, i.e. to a single outcome $omega$. An example of an event is ${omega:X(omega)>x}$.
$endgroup$
– d.k.o.
Jan 20 at 22:55
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
Here is a simple example. Let $(Omega,mathcal{F})=(mathbb{R},mathcal{B}(mathbb{R}))$ and let
$$
P_{vartheta}(A):=int_A vartheta e^{-vartheta x}1_{[0,infty)}(x)dx, quad vartheta>0.
$$
We want to test $H_0:varthetage 1$ against $H_1:vartheta<1$. Taking $F=(-ln(1-alpha),infty)$, we get $sup_{varthetage 1}P_{vartheta}(F)=alpha$. So if we observe an outcome $omega>-ln(1-alpha)$, we reject $H_0$.
answered Jan 20 at 18:21
d.k.o.d.k.o.
9,955629
$endgroup$
$begingroup$
Thanks for the answer! Last question: did you choose $F$ arbitrarily? Or more generally, will the Test $F$ always be defined, or are we required to choose a suitable $F$?
$endgroup$
– SABOY
Jan 22 at 18:22
add a comment |
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1 Answer
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votes
1 Answer
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$begingroup$
Here is a simple example. Let $(Omega,mathcal{F})=(mathbb{R},mathcal{B}(mathbb{R}))$ and let
$$
P_{vartheta}(A):=int_A vartheta e^{-vartheta x}1_{[0,infty)}(x)dx, quad vartheta>0.
$$
We want to test $H_0:varthetage 1$ against $H_1:vartheta<1$. Taking $F=(-ln(1-alpha),infty)$, we get $sup_{varthetage 1}P_{vartheta}(F)=alpha$. So if we observe an outcome $omega>-ln(1-alpha)$, we reject $H_0$.
answered Jan 20 at 18:21
d.k.o.d.k.o.
9,955629
$endgroup$
$begingroup$
Thanks for the answer! Last question: did you choose $F$ arbitrarily? Or more generally, will the Test $F$ always be defined, or are we required to choose a suitable $F$?
$endgroup$
– SABOY
Jan 22 at 18:22
add a comment |
$begingroup$
Here is a simple example. Let $(Omega,mathcal{F})=(mathbb{R},mathcal{B}(mathbb{R}))$ and let
$$
P_{vartheta}(A):=int_A vartheta e^{-vartheta x}1_{[0,infty)}(x)dx, quad vartheta>0.
$$
We want to test $H_0:varthetage 1$ against $H_1:vartheta<1$. Taking $F=(-ln(1-alpha),infty)$, we get $sup_{varthetage 1}P_{vartheta}(F)=alpha$. So if we observe an outcome $omega>-ln(1-alpha)$, we reject $H_0$.
answered Jan 20 at 18:21
d.k.o.d.k.o.
9,955629
$endgroup$
$begingroup$
Thanks for the answer! Last question: did you choose $F$ arbitrarily? Or more generally, will the Test $F$ always be defined, or are we required to choose a suitable $F$?
$endgroup$
– SABOY
Jan 22 at 18:22
add a comment |
$begingroup$
Here is a simple example. Let $(Omega,mathcal{F})=(mathbb{R},mathcal{B}(mathbb{R}))$ and let
$$
P_{vartheta}(A):=int_A vartheta e^{-vartheta x}1_{[0,infty)}(x)dx, quad vartheta>0.
$$
We want to test $H_0:varthetage 1$ against $H_1:vartheta<1$. Taking $F=(-ln(1-alpha),infty)$, we get $sup_{varthetage 1}P_{vartheta}(F)=alpha$. So if we observe an outcome $omega>-ln(1-alpha)$, we reject $H_0$.
answered Jan 20 at 18:21
d.k.o.d.k.o.
9,955629
$endgroup$
Here is a simple example. Let $(Omega,mathcal{F})=(mathbb{R},mathcal{B}(mathbb{R}))$ and let
$$
P_{vartheta}(A):=int_A vartheta e^{-vartheta x}1_{[0,infty)}(x)dx, quad vartheta>0.
$$
We want to test $H_0:varthetage 1$ against $H_1:vartheta<1$. Taking $F=(-ln(1-alpha),infty)$, we get $sup_{varthetage 1}P_{vartheta}(F)=alpha$. So if we observe an outcome $omega>-ln(1-alpha)$, we reject $H_0$.
answered Jan 20 at 18:21
d.k.o.d.k.o.
9,955629
edited Jan 20 at 19:13
answered Jan 20 at 18:21
d.k.o.d.k.o.
9,955629
answered Jan 20 at 18:21
d.k.o.d.k.o.
9,955629
answered Jan 20 at 18:21
d.k.o.d.k.o.
9,955629
9,955629
$begingroup$
Thanks for the answer! Last question: did you choose $F$ arbitrarily? Or more generally, will the Test $F$ always be defined, or are we required to choose a suitable $F$?
$endgroup$
– SABOY
Jan 22 at 18:22
add a comment |
$begingroup$
Thanks for the answer! Last question: did you choose $F$ arbitrarily? Or more generally, will the Test $F$ always be defined, or are we required to choose a suitable $F$?
$endgroup$
– SABOY
Jan 22 at 18:22
$begingroup$
Thanks for the answer! Last question: did you choose $F$ arbitrarily? Or more generally, will the Test $F$ always be defined, or are we required to choose a suitable $F$?
$endgroup$
– SABOY
Jan 22 at 18:22
$begingroup$
Thanks for the answer! Last question: did you choose $F$ arbitrarily? Or more generally, will the Test $F$ always be defined, or are we required to choose a suitable $F$?
$endgroup$
– SABOY
Jan 22 at 18:22
add a comment |
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$begingroup$
1. The notation "$omega in mathcal{F}$" does not make sense. 2. We are looking at a particular outcome from $Omega$ (e.g. a sample drawn from $P_{vartheta}$). Based on that outcome we make a decision. Since the likelihood of getting $omegain F$ is small, we reject $H_0$ in favor to the alternative.
$endgroup$
– d.k.o.
Jan 19 at 4:39
$begingroup$
@d.k.o. In that case, should we not say $P_{vartheta}(omega)leq alpha$ rather than $P_{vartheta}(F)leq alpha$? So is the difference between the probability model $(Omega, mathcal{F}, P)$ and the statistical model $(Omega, mathcal{F}, P_{vartheta})$ is that in the probability model $F in mathcal{F}$ represents a particular event, while in statistics $F in mathcal{F}$ represents taking a sample. This leads me to believe however that all statiscal models are discrete. Is this true?
$endgroup$
– SABOY
Jan 20 at 11:10
$begingroup$
$F$ represents an event in both cases.
$endgroup$
– d.k.o.
Jan 20 at 18:23
$begingroup$
@d.k.o. An event in a statistical sense would stand for a sample however, correct?
$endgroup$
– SABOY
Jan 20 at 21:45
$begingroup$
Consider again the example below. If $X(omega)=omega$, then $X$ is a random variable having the exponential distribution with parameter $vartheta$. A sample here would correspond to a single realization of $X$, i.e. to a single outcome $omega$. An example of an event is ${omega:X(omega)>x}$.
$endgroup$
– d.k.o.
Jan 20 at 22:55