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 Jan 25 at 22:21
kjetil b halvorsen
31.1k983222
asked Jan 25 at 19:38
knrumseyknrumsey
1,673416
$endgroup$
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 Jan 25 at 22:21
kjetil b halvorsen
31.1k983222
asked Jan 25 at 19:38
knrumseyknrumsey
1,673416
$endgroup$
1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
Jan 25 at 21:11
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
Jan 25 at 21:14
$begingroup$
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
Jan 25 at 22:20
$begingroup$
@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
Jan 25 at 22:22
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
Jan 26 at 6:01
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 Jan 25 at 22:21
kjetil b halvorsen
31.1k983222
asked Jan 25 at 19:38
knrumseyknrumsey
1,673416
$endgroup$
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 Jan 25 at 22:21
kjetil b halvorsen
31.1k983222
asked Jan 25 at 19:38
knrumseyknrumsey
1,673416
edited Jan 25 at 22:21
kjetil b halvorsen
31.1k983222
asked Jan 25 at 19:38
knrumseyknrumsey
1,673416
edited Jan 25 at 22:21
kjetil b halvorsen
31.1k983222
edited Jan 25 at 22:21
kjetil b halvorsen
31.1k983222
edited Jan 25 at 22:21
kjetil b halvorsen
31.1k983222
31.1k983222
asked Jan 25 at 19:38
knrumseyknrumsey
1,673416
asked Jan 25 at 19:38
knrumseyknrumsey
1,673416
asked Jan 25 at 19:38
knrumseyknrumsey
1,673416
1,673416
1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
Jan 25 at 21:11
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
Jan 25 at 21:14
$begingroup$
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
Jan 25 at 22:20
$begingroup$
@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
Jan 25 at 22:22
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
Jan 26 at 6:01
add a comment |
1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
Jan 25 at 21:11
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
Jan 25 at 21:14
$begingroup$
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
Jan 25 at 22:20
$begingroup$
@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
Jan 25 at 22:22
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
Jan 26 at 6:01
1
1
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
Jan 25 at 21:11
$begingroup$
That could be called a U-formed hazard function. See books.google.no/…
$endgroup$
– kjetil b halvorsen
Jan 25 at 21:11
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
Jan 25 at 21:14
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
Jan 25 at 21:14
$begingroup$
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
Jan 25 at 22:20
$begingroup$
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
Jan 25 at 22:20
$begingroup$
@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
Jan 25 at 22:22
$begingroup$
@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
Jan 25 at 22:22
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
Jan 26 at 6:01
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
Jan 26 at 6:01
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 Jan 25 at 22:42
kjetil b halvorsenkjetil b halvorsen
31.1k983222
$endgroup$
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♦
Jan 26 at 5:55
add a comment |
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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 Jan 25 at 22:42
kjetil b halvorsenkjetil b halvorsen
31.1k983222
$endgroup$
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♦
Jan 26 at 5:55
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 Jan 25 at 22:42
kjetil b halvorsenkjetil b halvorsen
31.1k983222
$endgroup$
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♦
Jan 26 at 5:55
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 Jan 25 at 22:42
kjetil b halvorsenkjetil b halvorsen
31.1k983222
$endgroup$
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 Jan 25 at 22:42
kjetil b halvorsenkjetil b halvorsen
31.1k983222
edited Jan 26 at 22:31
answered Jan 25 at 22:42
kjetil b halvorsenkjetil b halvorsen
31.1k983222
answered Jan 25 at 22:42
kjetil b halvorsenkjetil b halvorsen
31.1k983222
answered Jan 25 at 22:42
kjetil b halvorsenkjetil b halvorsen
31.1k983222
31.1k983222
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♦
Jan 26 at 5:55
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♦
Jan 26 at 5:55
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♦
Jan 26 at 5:55
$begingroup$
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♦
Jan 26 at 5:55
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
Jan 25 at 21:11
$begingroup$
@kjetilbhalvorsen Yep that looks right. Do you know of any parametric distributions with this type of Hazard function?
$endgroup$
– knrumsey
Jan 25 at 21:14
$begingroup$
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
Jan 25 at 22:20
$begingroup$
@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
Jan 25 at 22:22
$begingroup$
Many questions on site about such hazard functions ... e.g. stats.stackexchange.com/search?q=bathtub+hazard
$endgroup$
– Glen_b♦
Jan 26 at 6:01