Is this function concave or can it be made concave?

Question

I am working with a point process with an event arrival rate of:

$$ \lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$$

where $ t_1,..t_n $ are the event arrival times.

The log likelihood function is therefore:

$$ - t_n \mu + \frac{\alpha}{\beta} \sum{( e^{-\beta(t_n-t_i)}-1 )} + \sum\limits_{i<j}{\ln(\mu+\alpha e^{-\beta(t_j-t_i)})} $$

To obtain the maximum likelihood estimate (MLE) I need to maximize this log likelihood function under the restrictions that $\mu, \alpha, \beta > 0$ and $\beta > \alpha$.

1. Is the log likelihood function concave? The parameters are $\mu, \alpha, \beta$.
2. If not, is there a reparameterization that would make it concave?

In R code the log likelihood is

l.loglik <- function(params, data, opt=TRUE) {
  mu <- params[1]
  alpha <- params[2]
  beta <- params[3]
  t <- sort(data)
  r <- rep(0,length(t))
  for(i in 2:length(t)) {
    r[i] <- exp(-beta*(t[i]-t[i-1]))*(1+r[i-1])
  }
  loglik <- -tail(t,1)*mu
  loglik <- loglik+alpha/beta*sum(exp(-beta*(tail(t,1)-t))-1)
  loglik <- loglik+sum(log(mu+alpha*r))
  if(!opt) {
    return(list(negloglik=-loglik, mu=mu, alpha=alpha, beta=beta, t=t,
                r=r))
  }
  else {
    return(loglik)
  }
}

Answer 1

If you fix $\beta$ as a constant, the function is jointly concave in $(\mu, \alpha)$. That is because $\log(\mu + \alpha c)$ is jointly concave in $(\mu, \alpha)$. So you could fix $\delta>0$ and consider $\beta \in \{\delta, 2\delta, 3\delta, \ldots, K\delta\}$ for some value $K>0$. Then for each $k \in \{1, \ldots,K\}$ do:

---

(i) Fix $\beta = k\delta$ and solve the concave maximization over $(\mu, \alpha)$ subject to $0 < \alpha < \beta$ and $0<\mu$. Let the resulting value be $s_k$.

(ii) If $k==1$ then define $s_{best} = s_1$, $\mu_{best} = \mu$, $\alpha_{best} = \alpha$, $\beta_{best} = k\delta$.

(iii) If $k>1$, then if $s_k>s_{max}$ define $s_{max}=s_k$, $\mu_{best} = \mu$, $\alpha_{best} = \alpha$, $\beta_{best} = k\delta$.

---

The resulting $s_{best},\mu_{best}, \alpha_{best}, \beta_{best}$ values are a good approximation to the optimal answer (depending on how small your discretization unit $\delta$ is, and assuming $K\delta$ is larger than the optimal value of $\beta$). Overall, assuming the concave maximization is easy, this turns the 3-dimensional search for the best $\alpha, \mu, \beta$ into a 1-dimensional search for the best $\beta$.