Non-monotone hazard functions
$begingroup$
I should start with the caveat that I am relatively new to Survival analysis. I was watching a Hulu documentary about Crocodiles last night, and they mentioned that baby crocodiles have a low chance of survival when they are young, but "with each passing day they have fewer predators". It seems that this should be true for most (if not all animals) including Humans (maybe to a lesser extent).
It seems that this early stage of life could be modeled with a monotonically decreasing hazard function such as this one from a $Gamma(1/2, 1)$ distribution. 
Of course if we want to know the hazard function for the duration of the Crocodiles life, the Hazard function should eventually increase due to old age. All of the common parametric models that I have looked at (weibull, pareto, gamma, etc) are monotone, with the exception of Lognormal which is concave down.
Are there any simple parametric distributions which have a concave up (bowl shaped) Hazard function?
survival parametric hazard demography
edited 2 days ago
kjetil b halvorsen
29.6k980216
asked 2 days ago
knrumseyknrumsey
1,416315
$endgroup$
add a comment |
$begingroup$
I should start with the caveat that I am relatively new to Survival analysis. I was watching a Hulu documentary about Crocodiles last night, and they mentioned that baby crocodiles have a low chance of survival when they are young, but "with each passing day they have fewer predators". It seems that this should be true for most (if not all animals) including Humans (maybe to a lesser extent).
It seems that this early stage of life could be modeled with a monotonically decreasing hazard function such as this one from a $Gamma(1/2, 1)$ distribution.
Of course if we want to know the hazard function for the duration of the Crocodiles life, the Hazard function should eventually increase due to old age. All of the common parametric models that I have looked at (weibull, pareto, gamma, etc) are monotone, with the exception of Lognormal which is concave down.
Are there any simple parametric distributions which have a concave up (bowl shaped) Hazard function?
survival parametric hazard demography
edited 2 days ago
kjetil b halvorsen
29.6k980216
asked 2 days ago
knrumseyknrumsey
1,416315
$endgroup$
1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
2 days ago
$begingroup$
Its also known as a bathtube function! See Wikipedia and references there. Specifically Gompertz-Makeham. Many more hits on google, one is researchgate.net/publication/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen That's what I'm looking for! If you want to quickly add this as an answer I will accept it. Thanks!
$endgroup$
– knrumsey
2 days ago
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
yesterday
add a comment |
$begingroup$
I should start with the caveat that I am relatively new to Survival analysis. I was watching a Hulu documentary about Crocodiles last night, and they mentioned that baby crocodiles have a low chance of survival when they are young, but "with each passing day they have fewer predators". It seems that this should be true for most (if not all animals) including Humans (maybe to a lesser extent).
It seems that this early stage of life could be modeled with a monotonically decreasing hazard function such as this one from a $Gamma(1/2, 1)$ distribution.
Of course if we want to know the hazard function for the duration of the Crocodiles life, the Hazard function should eventually increase due to old age. All of the common parametric models that I have looked at (weibull, pareto, gamma, etc) are monotone, with the exception of Lognormal which is concave down.
Are there any simple parametric distributions which have a concave up (bowl shaped) Hazard function?
survival parametric hazard demography
edited 2 days ago
kjetil b halvorsen
29.6k980216
asked 2 days ago
knrumseyknrumsey
1,416315
$endgroup$
I should start with the caveat that I am relatively new to Survival analysis. I was watching a Hulu documentary about Crocodiles last night, and they mentioned that baby crocodiles have a low chance of survival when they are young, but "with each passing day they have fewer predators". It seems that this should be true for most (if not all animals) including Humans (maybe to a lesser extent).
It seems that this early stage of life could be modeled with a monotonically decreasing hazard function such as this one from a $Gamma(1/2, 1)$ distribution.
Of course if we want to know the hazard function for the duration of the Crocodiles life, the Hazard function should eventually increase due to old age. All of the common parametric models that I have looked at (weibull, pareto, gamma, etc) are monotone, with the exception of Lognormal which is concave down.
Are there any simple parametric distributions which have a concave up (bowl shaped) Hazard function?
survival parametric hazard demography
survival parametric hazard demography
edited 2 days ago
kjetil b halvorsen
29.6k980216
asked 2 days ago
knrumseyknrumsey
1,416315
edited 2 days ago
kjetil b halvorsen
29.6k980216
asked 2 days ago
knrumseyknrumsey
1,416315
edited 2 days ago
kjetil b halvorsen
29.6k980216
edited 2 days ago
kjetil b halvorsen
29.6k980216
edited 2 days ago
kjetil b halvorsen
29.6k980216
29.6k980216
asked 2 days ago
knrumseyknrumsey
1,416315
asked 2 days ago
knrumseyknrumsey
1,416315
asked 2 days ago
knrumseyknrumsey
1,416315
1,416315
1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
2 days ago
$begingroup$
Its also known as a bathtube function! See Wikipedia and references there. Specifically Gompertz-Makeham. Many more hits on google, one is researchgate.net/publication/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen That's what I'm looking for! If you want to quickly add this as an answer I will accept it. Thanks!
$endgroup$
– knrumsey
2 days ago
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
yesterday
add a comment |
1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
2 days ago
$begingroup$
Its also known as a bathtube function! See Wikipedia and references there. Specifically Gompertz-Makeham. Many more hits on google, one is researchgate.net/publication/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen That's what I'm looking for! If you want to quickly add this as an answer I will accept it. Thanks!
$endgroup$
– knrumsey
2 days ago
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
yesterday
1
1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
2 days ago
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
2 days ago
$begingroup$
Its also known as a bathtube function! See Wikipedia and references there. Specifically Gompertz-Makeham. Many more hits on google, one is researchgate.net/publication/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
Its also known as a bathtube function! See Wikipedia and references there. Specifically Gompertz-Makeham. Many more hits on google, one is researchgate.net/publication/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen That's what I'm looking for! If you want to quickly add this as an answer I will accept it. Thanks!
$endgroup$
– knrumsey
2 days ago
$begingroup$
@kjetilbhalvorsen That's what I'm looking for! If you want to quickly add this as an answer I will accept it. Thanks!
$endgroup$
– knrumsey
2 days ago
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
yesterday
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
yesterday
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
What you search for is called a U-formed hazard function or bathtub function (and references in those links). One specific case is the Gompertz-Makeham law from demography. An example is the hazard function of humans, high but falling hazard first few years of life, a minimum around 9-10 years of life, then slowly increasing.
Googling with those terms will lead to much information. Much of interest here
EDIT
Some more information. This paper is a good starting point. They discuss a new extension of the Weibull, which they call EMWE (Exponentiated Modified Weibull Extension distribution) with four parameters, which permits bathtube shaped hazard with form close to hazard functions seen in practice. A plot from that paper is
with pdf's on the left and corresponding hazard rates on the right.
For reference I will give the cdf and pdf functions:
$$
f(x;alpha,beta,lambda,gamma)=lambdabetagamma(x/alpha)^{beta-1}expleft{(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta} right}cdot left{1-e^{lambdaalpha(1-e^{(x/alpha)^beta}}right}^{gamma-1}\
F(x;alpha,beta,lambda,gamma)=left{1-exp[lambdaalpha(1-e^{(x/alpha)^beta}]right}^gamma
$$
and the hazard rate is
$$
h(x;alpha,beta,lambda,gamma)=frac{lambdabetagamma(x/alpha)^{beta-1}exp[(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta})]}{[1-exp[lambdaalpha(1-e^{(x/alpha)^beta})]]^{1-gamma}+expleft{ lambdaalpha(1-e^{(x/alpha)^beta})right} -1}
$$
Estimation can be done with maximum likelihood.
answered 2 days ago
kjetil b halvorsenkjetil b halvorsen
29.6k980216
$endgroup$
2
$begingroup$
Gompertz-Makeham hazard ($h(x)=alpha e^{beta x}+lambda$) is monotonic-increasing, not U-shaped (the plot in that article is not a plot of a Gompertz-Makeham hazard function; it's actual human mortality rates -- which do decrease and then increase, but Gompertz-Makeham only works as an approximation from somewhere in the 30s to about 80 or so, give or take)
$endgroup$
– Glen_b♦
yesterday
add a comment |
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1 Answer
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1 Answer
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$begingroup$
What you search for is called a U-formed hazard function or bathtub function (and references in those links). One specific case is the Gompertz-Makeham law from demography. An example is the hazard function of humans, high but falling hazard first few years of life, a minimum around 9-10 years of life, then slowly increasing.
Googling with those terms will lead to much information. Much of interest here
EDIT
Some more information. This paper is a good starting point. They discuss a new extension of the Weibull, which they call EMWE (Exponentiated Modified Weibull Extension distribution) with four parameters, which permits bathtube shaped hazard with form close to hazard functions seen in practice. A plot from that paper is
with pdf's on the left and corresponding hazard rates on the right.
For reference I will give the cdf and pdf functions:
$$
f(x;alpha,beta,lambda,gamma)=lambdabetagamma(x/alpha)^{beta-1}expleft{(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta} right}cdot left{1-e^{lambdaalpha(1-e^{(x/alpha)^beta}}right}^{gamma-1}\
F(x;alpha,beta,lambda,gamma)=left{1-exp[lambdaalpha(1-e^{(x/alpha)^beta}]right}^gamma
$$
and the hazard rate is
$$
h(x;alpha,beta,lambda,gamma)=frac{lambdabetagamma(x/alpha)^{beta-1}exp[(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta})]}{[1-exp[lambdaalpha(1-e^{(x/alpha)^beta})]]^{1-gamma}+expleft{ lambdaalpha(1-e^{(x/alpha)^beta})right} -1}
$$
Estimation can be done with maximum likelihood.
answered 2 days ago
kjetil b halvorsenkjetil b halvorsen
29.6k980216
$endgroup$
2
$begingroup$
Gompertz-Makeham hazard ($h(x)=alpha e^{beta x}+lambda$) is monotonic-increasing, not U-shaped (the plot in that article is not a plot of a Gompertz-Makeham hazard function; it's actual human mortality rates -- which do decrease and then increase, but Gompertz-Makeham only works as an approximation from somewhere in the 30s to about 80 or so, give or take)
$endgroup$
– Glen_b♦
yesterday
add a comment |
$begingroup$
What you search for is called a U-formed hazard function or bathtub function (and references in those links). One specific case is the Gompertz-Makeham law from demography. An example is the hazard function of humans, high but falling hazard first few years of life, a minimum around 9-10 years of life, then slowly increasing.
Googling with those terms will lead to much information. Much of interest here
EDIT
Some more information. This paper is a good starting point. They discuss a new extension of the Weibull, which they call EMWE (Exponentiated Modified Weibull Extension distribution) with four parameters, which permits bathtube shaped hazard with form close to hazard functions seen in practice. A plot from that paper is
with pdf's on the left and corresponding hazard rates on the right.
For reference I will give the cdf and pdf functions:
$$
f(x;alpha,beta,lambda,gamma)=lambdabetagamma(x/alpha)^{beta-1}expleft{(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta} right}cdot left{1-e^{lambdaalpha(1-e^{(x/alpha)^beta}}right}^{gamma-1}\
F(x;alpha,beta,lambda,gamma)=left{1-exp[lambdaalpha(1-e^{(x/alpha)^beta}]right}^gamma
$$
and the hazard rate is
$$
h(x;alpha,beta,lambda,gamma)=frac{lambdabetagamma(x/alpha)^{beta-1}exp[(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta})]}{[1-exp[lambdaalpha(1-e^{(x/alpha)^beta})]]^{1-gamma}+expleft{ lambdaalpha(1-e^{(x/alpha)^beta})right} -1}
$$
Estimation can be done with maximum likelihood.
answered 2 days ago
kjetil b halvorsenkjetil b halvorsen
29.6k980216
$endgroup$
2
$begingroup$
Gompertz-Makeham hazard ($h(x)=alpha e^{beta x}+lambda$) is monotonic-increasing, not U-shaped (the plot in that article is not a plot of a Gompertz-Makeham hazard function; it's actual human mortality rates -- which do decrease and then increase, but Gompertz-Makeham only works as an approximation from somewhere in the 30s to about 80 or so, give or take)
$endgroup$
– Glen_b♦
yesterday
add a comment |
$begingroup$
What you search for is called a U-formed hazard function or bathtub function (and references in those links). One specific case is the Gompertz-Makeham law from demography. An example is the hazard function of humans, high but falling hazard first few years of life, a minimum around 9-10 years of life, then slowly increasing.
Googling with those terms will lead to much information. Much of interest here
EDIT
Some more information. This paper is a good starting point. They discuss a new extension of the Weibull, which they call EMWE (Exponentiated Modified Weibull Extension distribution) with four parameters, which permits bathtube shaped hazard with form close to hazard functions seen in practice. A plot from that paper is
with pdf's on the left and corresponding hazard rates on the right.
For reference I will give the cdf and pdf functions:
$$
f(x;alpha,beta,lambda,gamma)=lambdabetagamma(x/alpha)^{beta-1}expleft{(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta} right}cdot left{1-e^{lambdaalpha(1-e^{(x/alpha)^beta}}right}^{gamma-1}\
F(x;alpha,beta,lambda,gamma)=left{1-exp[lambdaalpha(1-e^{(x/alpha)^beta}]right}^gamma
$$
and the hazard rate is
$$
h(x;alpha,beta,lambda,gamma)=frac{lambdabetagamma(x/alpha)^{beta-1}exp[(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta})]}{[1-exp[lambdaalpha(1-e^{(x/alpha)^beta})]]^{1-gamma}+expleft{ lambdaalpha(1-e^{(x/alpha)^beta})right} -1}
$$
Estimation can be done with maximum likelihood.
answered 2 days ago
kjetil b halvorsenkjetil b halvorsen
29.6k980216
$endgroup$
What you search for is called a U-formed hazard function or bathtub function (and references in those links). One specific case is the Gompertz-Makeham law from demography. An example is the hazard function of humans, high but falling hazard first few years of life, a minimum around 9-10 years of life, then slowly increasing.
Googling with those terms will lead to much information. Much of interest here
EDIT
Some more information. This paper is a good starting point. They discuss a new extension of the Weibull, which they call EMWE (Exponentiated Modified Weibull Extension distribution) with four parameters, which permits bathtube shaped hazard with form close to hazard functions seen in practice. A plot from that paper is
with pdf's on the left and corresponding hazard rates on the right.
For reference I will give the cdf and pdf functions:
$$
f(x;alpha,beta,lambda,gamma)=lambdabetagamma(x/alpha)^{beta-1}expleft{(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta} right}cdot left{1-e^{lambdaalpha(1-e^{(x/alpha)^beta}}right}^{gamma-1}\
F(x;alpha,beta,lambda,gamma)=left{1-exp[lambdaalpha(1-e^{(x/alpha)^beta}]right}^gamma
$$
and the hazard rate is
$$
h(x;alpha,beta,lambda,gamma)=frac{lambdabetagamma(x/alpha)^{beta-1}exp[(x/alpha)^beta+lambdaalpha(1-e^{(x/alpha)^beta})]}{[1-exp[lambdaalpha(1-e^{(x/alpha)^beta})]]^{1-gamma}+expleft{ lambdaalpha(1-e^{(x/alpha)^beta})right} -1}
$$
Estimation can be done with maximum likelihood.
answered 2 days ago
kjetil b halvorsenkjetil b halvorsen
29.6k980216
edited yesterday
answered 2 days ago
kjetil b halvorsenkjetil b halvorsen
29.6k980216
answered 2 days ago
kjetil b halvorsenkjetil b halvorsen
29.6k980216
answered 2 days ago
kjetil b halvorsenkjetil b halvorsen
29.6k980216
29.6k980216
2
$begingroup$
Gompertz-Makeham hazard ($h(x)=alpha e^{beta x}+lambda$) is monotonic-increasing, not U-shaped (the plot in that article is not a plot of a Gompertz-Makeham hazard function; it's actual human mortality rates -- which do decrease and then increase, but Gompertz-Makeham only works as an approximation from somewhere in the 30s to about 80 or so, give or take)
$endgroup$
– Glen_b♦
yesterday
add a comment |
2
$begingroup$
Gompertz-Makeham hazard ($h(x)=alpha e^{beta x}+lambda$) is monotonic-increasing, not U-shaped (the plot in that article is not a plot of a Gompertz-Makeham hazard function; it's actual human mortality rates -- which do decrease and then increase, but Gompertz-Makeham only works as an approximation from somewhere in the 30s to about 80 or so, give or take)
$endgroup$
– Glen_b♦
yesterday
2
2
$begingroup$
Gompertz-Makeham hazard ($h(x)=alpha e^{beta x}+lambda$) is monotonic-increasing, not U-shaped (the plot in that article is not a plot of a Gompertz-Makeham hazard function; it's actual human mortality rates -- which do decrease and then increase, but Gompertz-Makeham only works as an approximation from somewhere in the 30s to about 80 or so, give or take)
$endgroup$
– Glen_b♦
yesterday
$begingroup$
Gompertz-Makeham hazard ($h(x)=alpha e^{beta x}+lambda$) is monotonic-increasing, not U-shaped (the plot in that article is not a plot of a Gompertz-Makeham hazard function; it's actual human mortality rates -- which do decrease and then increase, but Gompertz-Makeham only works as an approximation from somewhere in the 30s to about 80 or so, give or take)
$endgroup$
– Glen_b♦
yesterday
add a comment |
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1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
2 days ago
$begingroup$
Its also known as a bathtube function! See Wikipedia and references there. Specifically Gompertz-Makeham. Many more hits on google, one is researchgate.net/publication/…
$endgroup$
– kjetil b halvorsen
2 days ago
$begingroup$
@kjetilbhalvorsen That's what I'm looking for! If you want to quickly add this as an answer I will accept it. Thanks!
$endgroup$
– knrumsey
2 days ago
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
yesterday