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Thread: recognising uniform distibution

  1. #1
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    recognising uniform distibution

    I am going through an example where one needs to find score and information for a random sample of size n from the logistic distribution with a density function

    $\displaystyle f_Y(y;\mu)=\frac{e^{y-\mu}}{[1+e^{(e-\mu)}]^2}$

    and the score is $\displaystyle s_Y(Y;\mu)=\Sigma_{i=1}^n(-1+2F_Y(y_i;\mu))$

    then the explanation goes, 'by recognising that $\displaystyle F_Y(Y;\mu)\sim Unif[0;1]$ we can calculate information... and they use the formula for Var of a Uniform distribution.

    Can someone help me to see that $\displaystyle F_Y(Y;\mu)\sim Unif[0;1]$?
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  2. #2
    Moo
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    Hello,

    That's a standard property, if Fy has an inverse (it's the case in particular if the rv has a continuous density).

    Indeed : $\displaystyle P(F_Y(Y)\leq x)=P(Y\leq F_Y^{-1}(x))=F_Y(F_Y^{-1}(x))=x$, for $\displaystyle x\in[0,1]$ since $\displaystyle F_Y$ is a probability and is hence taking values between 0 and 1.
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  3. #3
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    Quote Originally Posted by Volga View Post
    I am going through an example where one needs to find score and information for a random sample of size n from the logistic distribution with a density function

    $\displaystyle f_Y(y;\mu)=\frac{e^{y-\mu}}{[1+e^{(e-\mu)}]^2}$

    and the score is $\displaystyle s_Y(Y;\mu)=\Sigma_{i=1}^n(-1+2F_Y(y_i;\mu))$

    then the explanation goes, 'by recognising that $\displaystyle F_Y(Y;\mu)\sim Unif[0;1]$ we can calculate information... and they use the formula for Var of a Uniform distribution.

    Can someone help me to see that $\displaystyle F_Y(Y;\mu)\sim Unif[0;1]$?
    There are a number of ways of looking at this, one is the usual transformation method where if $\displaystyle $$U$ has pdf $\displaystyle $$f_U(u)$ we determine the distribution of $\displaystyle $$V=h(U)$ where $\displaystyle $$h(.)$ is an increasing function.

    $\displaystyle f_V(v)=\dfrac{f_U(u)}{h'(u)}$

    But if $\displaystyle h(u)=F_U(u)$ then $\displaystyle h'(u)=f_U(u)$ so $\displaystyle f_V(v)=1$ in the range of $\displaystyle $$h(u)$ (which is $\displaystyle $$[0,1]$).

    CB
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  4. #4
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    Nice one, I didn't know that, thank you!
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