Censored piecewise exponential maximum likelihood
$begingroup$
Let $0=a_0 < a_1 < dots < a_m = infty$ be given real numbers and let $lambda_1, dots, lambda_m$ be positive real numbers. Let $T_i$ have hazard
$$
h(t) = lambda_j, quad text{if $t in (a_{j-1}, a_j]$}
$$
Let $(t_1, delta_1), dots, (t_n, delta_n)$ be $n$ independent observations of failure times with the given hazard and indicators of right-censoring (that happens independently of the failure times). Estimate $lambda_1, dots, lambda_m$ using maximum likelihood estimation.
My attempt at solution
Since the censoring is independent of the failure times, we have
$$
L(lambda_1, dots, lambda_m) propto prod_{i=1}^n h(t_i)^{delta_i} S(t_i)
$$
where $S$ is the survival function corresponding to the hazard function, $h$. Defining $Delta_j = a_j-a_{j-1}$ for $j=1, dots, m$, we get that
$$
S(t) = expleft(-lambda_j (t-a_{j-1}) - sum_{i<j}lambda_i Delta_i right), quad text{if $t in (a_{j-1},a_j]$}
$$
I'm not quite sure how to rewrite the likelihood function from here. Any help?
statistics maximum-likelihood
asked Dec 1 '18 at 14:52
LundborgLundborg
746414
$endgroup$
add a comment |
$begingroup$
Let $0=a_0 < a_1 < dots < a_m = infty$ be given real numbers and let $lambda_1, dots, lambda_m$ be positive real numbers. Let $T_i$ have hazard
$$
h(t) = lambda_j, quad text{if $t in (a_{j-1}, a_j]$}
$$
Let $(t_1, delta_1), dots, (t_n, delta_n)$ be $n$ independent observations of failure times with the given hazard and indicators of right-censoring (that happens independently of the failure times). Estimate $lambda_1, dots, lambda_m$ using maximum likelihood estimation.
My attempt at solution
Since the censoring is independent of the failure times, we have
$$
L(lambda_1, dots, lambda_m) propto prod_{i=1}^n h(t_i)^{delta_i} S(t_i)
$$
where $S$ is the survival function corresponding to the hazard function, $h$. Defining $Delta_j = a_j-a_{j-1}$ for $j=1, dots, m$, we get that
$$
S(t) = expleft(-lambda_j (t-a_{j-1}) - sum_{i<j}lambda_i Delta_i right), quad text{if $t in (a_{j-1},a_j]$}
$$
I'm not quite sure how to rewrite the likelihood function from here. Any help?
statistics maximum-likelihood
asked Dec 1 '18 at 14:52
LundborgLundborg
746414
$endgroup$
add a comment |
$begingroup$
Let $0=a_0 < a_1 < dots < a_m = infty$ be given real numbers and let $lambda_1, dots, lambda_m$ be positive real numbers. Let $T_i$ have hazard
$$
h(t) = lambda_j, quad text{if $t in (a_{j-1}, a_j]$}
$$
Let $(t_1, delta_1), dots, (t_n, delta_n)$ be $n$ independent observations of failure times with the given hazard and indicators of right-censoring (that happens independently of the failure times). Estimate $lambda_1, dots, lambda_m$ using maximum likelihood estimation.
My attempt at solution
Since the censoring is independent of the failure times, we have
$$
L(lambda_1, dots, lambda_m) propto prod_{i=1}^n h(t_i)^{delta_i} S(t_i)
$$
where $S$ is the survival function corresponding to the hazard function, $h$. Defining $Delta_j = a_j-a_{j-1}$ for $j=1, dots, m$, we get that
$$
S(t) = expleft(-lambda_j (t-a_{j-1}) - sum_{i<j}lambda_i Delta_i right), quad text{if $t in (a_{j-1},a_j]$}
$$
I'm not quite sure how to rewrite the likelihood function from here. Any help?
statistics maximum-likelihood
asked Dec 1 '18 at 14:52
LundborgLundborg
746414
$endgroup$
Let $0=a_0 < a_1 < dots < a_m = infty$ be given real numbers and let $lambda_1, dots, lambda_m$ be positive real numbers. Let $T_i$ have hazard
$$
h(t) = lambda_j, quad text{if $t in (a_{j-1}, a_j]$}
$$
Let $(t_1, delta_1), dots, (t_n, delta_n)$ be $n$ independent observations of failure times with the given hazard and indicators of right-censoring (that happens independently of the failure times). Estimate $lambda_1, dots, lambda_m$ using maximum likelihood estimation.
My attempt at solution
Since the censoring is independent of the failure times, we have
$$
L(lambda_1, dots, lambda_m) propto prod_{i=1}^n h(t_i)^{delta_i} S(t_i)
$$
where $S$ is the survival function corresponding to the hazard function, $h$. Defining $Delta_j = a_j-a_{j-1}$ for $j=1, dots, m$, we get that
$$
S(t) = expleft(-lambda_j (t-a_{j-1}) - sum_{i<j}lambda_i Delta_i right), quad text{if $t in (a_{j-1},a_j]$}
$$
I'm not quite sure how to rewrite the likelihood function from here. Any help?
statistics maximum-likelihood
statistics maximum-likelihood
asked Dec 1 '18 at 14:52
LundborgLundborg
746414
asked Dec 1 '18 at 14:52
LundborgLundborg
746414
asked Dec 1 '18 at 14:52
LundborgLundborg
746414
asked Dec 1 '18 at 14:52
LundborgLundborg
746414
asked Dec 1 '18 at 14:52
LundborgLundborg
746414
746414
add a comment |
add a comment |
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