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Is $leftlbrace 1 - Fleft( x_n right)^n rightrbrace_n$ increasing when $x_n$ is a fixed point of the Mills ratio?

1

$begingroup$

Let $X_1, ..., X_n$ be $n$ i.i.d. Let let $f$ be their log-concave PDF and $F$ be their CDF.

The nth order statistic $max_i=1...n X_i$ has for CDF $F_nleft(xright) = mathbbPleft( max_i=1...N X_i < x right) = Fleft( x right)^n$ and its PDF is $f_nleft(xright) = n fleft(xright) Fleft(xright)^n-1$

---

I am interested in the growth of $leftlbrace 1 - Fleft( x_n right)^n rightrbrace_n$ when the number of variables $n$ increases. Here, $x_n = frac1 - Fleft( x_n right)^nf_nleft( x_n right)$. Note that $x_n$ is the fixed point of the Mills ratio of $max_i=1...n X_i$.

I have run some simulations and it seems that $1 - Fleft( x_n+1 right)^n+1 > 1 - Fleft( x_n right)^n$ but I have failed to prove it.

Is $leftlbrace 1 - Fleft( x_n right)^n rightrbrace_n$ really increasing?

---

$f$ being log-concave, there are properties I have tried to exploit without success:

• 
  $f_n$ and $F_n$ are log concave


• 
  $frac1 - Fleft( x right)^nf_nleft( x right)$ is decreasing in $x$
  


• 
  $fracf_nleft( x right)Fleft( x right)^n$ is decreaing in $x$
  


• 
  $x_n$ is the unique fixed point of
  $frac1 - Fleft( x right)^nf_nleft( x right)$ because $frac1 - Fleft( x right)^nf_nleft( x right)$ is strictly decreasing


• 
  $leftlbrace x_n rightrbrace_n$ is increasing

Finally, if the $X_1, ..., X_n$ are uniformly distributed, $1 - Fleft( x_n right)^n$ is increasing and has an explicit form.

probability sequences-and-series order-statistics

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asked Mar 28 at 10:40

KuwiKuwi

61

New contributor

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

$endgroup$

add a comment | 

1

$begingroup$

Let $X_1, ..., X_n$ be $n$ i.i.d. Let let $f$ be their log-concave PDF and $F$ be their CDF.

The nth order statistic $max_i=1...n X_i$ has for CDF $F_nleft(xright) = mathbbPleft( max_i=1...N X_i < x right) = Fleft( x right)^n$ and its PDF is $f_nleft(xright) = n fleft(xright) Fleft(xright)^n-1$

---

I am interested in the growth of $leftlbrace 1 - Fleft( x_n right)^n rightrbrace_n$ when the number of variables $n$ increases. Here, $x_n = frac1 - Fleft( x_n right)^nf_nleft( x_n right)$. Note that $x_n$ is the fixed point of the Mills ratio of $max_i=1...n X_i$.

I have run some simulations and it seems that $1 - Fleft( x_n+1 right)^n+1 > 1 - Fleft( x_n right)^n$ but I have failed to prove it.

Is $leftlbrace 1 - Fleft( x_n right)^n rightrbrace_n$ really increasing?

---

$f$ being log-concave, there are properties I have tried to exploit without success:

• 
  $f_n$ and $F_n$ are log concave


• 
  $frac1 - Fleft( x right)^nf_nleft( x right)$ is decreasing in $x$
  


• 
  $fracf_nleft( x right)Fleft( x right)^n$ is decreaing in $x$
  


• 
  $x_n$ is the unique fixed point of
  $frac1 - Fleft( x right)^nf_nleft( x right)$ because $frac1 - Fleft( x right)^nf_nleft( x right)$ is strictly decreasing


• 
  $leftlbrace x_n rightrbrace_n$ is increasing

Finally, if the $X_1, ..., X_n$ are uniformly distributed, $1 - Fleft( x_n right)^n$ is increasing and has an explicit form.

probability sequences-and-series order-statistics

share|cite|improve this question

asked Mar 28 at 10:40

KuwiKuwi

61

New contributor

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

$endgroup$

add a comment | 

$begingroup$

Let $X_1, ..., X_n$ be $n$ i.i.d. Let let $f$ be their log-concave PDF and $F$ be their CDF.

The nth order statistic $max_i=1...n X_i$ has for CDF $F_nleft(xright) = mathbbPleft( max_i=1...N X_i < x right) = Fleft( x right)^n$ and its PDF is $f_nleft(xright) = n fleft(xright) Fleft(xright)^n-1$

---

I am interested in the growth of $leftlbrace 1 - Fleft( x_n right)^n rightrbrace_n$ when the number of variables $n$ increases. Here, $x_n = frac1 - Fleft( x_n right)^nf_nleft( x_n right)$. Note that $x_n$ is the fixed point of the Mills ratio of $max_i=1...n X_i$.

I have run some simulations and it seems that $1 - Fleft( x_n+1 right)^n+1 > 1 - Fleft( x_n right)^n$ but I have failed to prove it.

Is $leftlbrace 1 - Fleft( x_n right)^n rightrbrace_n$ really increasing?

---

$f$ being log-concave, there are properties I have tried to exploit without success:

• 
  $f_n$ and $F_n$ are log concave


• 
  $frac1 - Fleft( x right)^nf_nleft( x right)$ is decreasing in $x$
  


• 
  $fracf_nleft( x right)Fleft( x right)^n$ is decreaing in $x$
  


• 
  $x_n$ is the unique fixed point of
  $frac1 - Fleft( x right)^nf_nleft( x right)$ because $frac1 - Fleft( x right)^nf_nleft( x right)$ is strictly decreasing


• 
  $leftlbrace x_n rightrbrace_n$ is increasing

Finally, if the $X_1, ..., X_n$ are uniformly distributed, $1 - Fleft( x_n right)^n$ is increasing and has an explicit form.

probability sequences-and-series order-statistics

share|cite|improve this question

asked Mar 28 at 10:40

KuwiKuwi

61

New contributor

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

$endgroup$

share|cite|improve this question

asked Mar 28 at 10:40

KuwiKuwi

61

New contributor

Kuwi 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

asked Mar 28 at 10:40

KuwiKuwi

61

New contributor

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

asked Mar 28 at 10:40

KuwiKuwi

61

New contributor

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

asked Mar 28 at 10:40

KuwiKuwi

61