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Let $t=T_F$. Conditional on the number of occurences $N=n$, the arrival times $t_1,t_2,\dots,t_N$ are known to have the same distribution as the order statstics of $n$ iid unif$(0,t)$ random variables. Hence, the likelihood becomes \begin{align} L(\lambda,t) &= P(N=n) f(t_1,t_2,\dots,t_N|N=n) \\ &= \frac{e^{-\lambda t}(\lambda t)^n}{n!}\frac{n!}{t^n} \\ &= e^{-\lambda t}\lambda^n. \end{align} for $t\ge t_n$ and zero elsewhere. This is maximised for $\hat t=t_n$ and $\hat\lambda=n/t_n$. These MLEs don't exist if there are no occurrences $N=0$, however. Conditional on $N=n$, again using the fact that $t_n$ can be viewed as an order statistic (the maximum) of $n$ iid unif$(0,t)$ random variables, $E(t_N|N=n)=\frac n{n+1} t$. Hence, the estimator $t^*=\frac {n+1}n t_n$ is unbiased for $t$ conditional on $N=n$ and hence also conditional on $N\ge 1$. A reasonable frequentist estimator of $\lambda$ might be $\lambda^* = n/t^* = \frac{n^2}{(n+1)t_n}$ but this does not have finite expectation when $N=1$ so assessing its bias is even more troublesome.

Bayesian inference using independent, non-informative scale priors on $\lambda$ and $t$ on the other hand leads to a posterior $$ f(\lambda,t|t_1,\dots,t_N) \propto e^{-\lambda t}\lambda^{n-1}t^{-1}. $$ for $t>t_n,\lambda>0$. Integrating out $\lambda$, the marginal distribution of $t$ becomes $$ f(t|t_1,\dots,t_N) = \frac{n t_n^n}{t^{n+1}}, t>t_n, $$ and the posterior mean $E(t|t_1,\dots,t_N)=\frac n{n-1} t_n$. A $(1-\alpha)$-credible interval for $t$ is given by $\left(\frac{t_n}{(1-\alpha/2)^{1/n}}, \frac{t_n}{(\alpha/2)^{1/n}}\right)$.

The marginal posterior of $\lambda$, \begin{align} f(\lambda|t_1,\dots,t_N) &\propto \int_{t_\text{max}}^\infty e^{-\lambda t}\lambda^{n-1}t^{-1} dt \\ &= \lambda^{n-1}\Gamma(0,\lambda t_n) \end{align} where $\Gamma$ is the incomplete gamma function.

Let $t=T_F$. Conditional on the number of occurences $N=n$, the arrival times $t_1,t_2,\dots,t_N$ are known to have the same distribution as the order statstics of $n$ iid unif$(0,t)$ random variables. Hence, the likelihood becomes \begin{align} L(\lambda,t) &= P(N=n) f(t_1,t_2,\dots,t_N|N=n) \\ &= \frac{e^{-\lambda t}(\lambda t)^n}{n!}\frac{n!}{t^n} \\ &= e^{-\lambda t}\lambda^n. \end{align} for $t\ge t_n$ and zero elsewhere. This is maximised for $\hat t=t_n$ and $\hat\lambda=n/t_n$. These MLEs don't exist if there are no occurrences $N=0$, however. Conditional on $N=n$, again using the fact that $t_n$ can be viewed as an order statistic (the maximum) of $n$ iid unif$(0,t)$ random variables, $E(t_N|N=n)=\frac n{n+1} t$. Hence, the estimator $t^*=\frac {n+1}n t_n$ is unbiased for $t$ conditional on $N=n$ and hence also conditional on $N\ge 1$. A reasonable frequentist estimator of $\lambda$ might be $\lambda^* = n/t^* = \frac{n^2}{(n+1)t_n}$ but this does not have finite expectation when $N=1$ so assessing its bias is even more troublesome.

Bayesian inference using independent, non-informative scale priors on $\lambda$ and $t$ on the other hand leads to a posterior $$ f(\lambda,t|t_1,\dots,t_N) \propto e^{-\lambda t}\lambda^{n-1}t^{-1}. $$ for $t>t_n,\lambda>0$. Integrating out $\lambda$, the marginal posterior of $t$ becomes $$ f(t|t_1,\dots,t_N) = \frac{n t_n^n}{t^{n+1}}, t>t_n, $$ and the posterior mean $E(t|t_1,\dots,t_N)=\frac n{n-1} t_n$. A $(1-\alpha)$-credible interval for $t$ is given by $\left(\frac{t_n}{(1-\alpha/2)^{1/n}}, \frac{t_n}{(\alpha/2)^{1/n}}\right)$.

The marginal posterior of $\lambda$, \begin{align} f(\lambda|t_1,\dots,t_N) &\propto \int_{t_\text{max}}^\infty e^{-\lambda t}\lambda^{n-1}t^{-1} dt \\ &= \lambda^{n-1}\Gamma(0,\lambda t_n) \end{align} where $\Gamma$ is the incomplete gamma function.

Let $t=T_F$. Conditional on the number of occurences $N=n$, the arrival times $t_1,t_2,\dots,t_N$ are known to have the same distribution as the order statstics of $n$ iid unif$(0,t)$ random variables. Hence, the likelihood becomes \begin{align} L(\lambda,t) &= P(N=n) f(t_1,t_2,\dots,t_N|N=n) \\ &= \frac{e^{-\lambda t}(\lambda t)^n}{n!}\frac{n!}{t^n} \\ &= e^{-\lambda t}\lambda^n. \end{align} for $t\ge t_n$ and zero elsewhere. This is maximised for $\hat t=t_n$ and $\hat\lambda=n/t_n$. These MLEs don't exist if there are no occurrences $N=0$, however. Conditional on $N=n$, again using the fact that $t_n$ can be viewed as the maximum of $n$ iid unif$(0,t)$ random variables, $E(t_N|N=n)=\frac n{n+1} t$. Hence, the estimator $t^*=\frac {n+1}n t_n$ is unbiased conditional on $N=n$ and hence also conditional on $N\ge 1$. A reasonable frequentist estimator of $\lambda$ might be $\lambda^* = n/t^* = \frac{n^2}{(n+1)t_n}$ but this does not have finite expectation when $N=1$ so assessing its bias is even more troublesome.

Bayesian inference using independent, non-informative scale priors on $\lambda$ and $t$ on the other hand leads to a posterior $$ f(\lambda,t|t_1,\dots,t_N) \propto e^{-\lambda t}\lambda^{n-1}t^{-1}. $$ for $t>t_n,\lambda>0$. Integrating out $\lambda$, the marginal distribution of $t$ becomes $$ f(t|t_1,\dots,t_N) = \frac{n t_n^n}{t^{n+1}}, t>t_n, $$ and the posterior mean $E(t|t_1,\dots,t_N)=\frac n{n-1} t_n$. A $(1-\alpha)$-credible interval for $t$ is given by $\left(\frac{t_n}{(1-\alpha/2)^{1/n}}, \frac{t_n}{(\alpha/2)^{1/n}}\right)$.

The marginal posterior of $\lambda$, \begin{align} f(\lambda|t_1,\dots,t_N) &\propto \int_{t_\text{max}}^\infty e^{-\lambda t}\lambda^{n-1}t^{-1} dt \\ &= \lambda^{n-1}\Gamma(0,\lambda t_n) \end{align} where $\Gamma$ is the incomplete gamma function.

Let $t=T_F$. Conditional on the number of occurences $N=n$, the arrival times $t_1,t_2,\dots,t_N$ are known to have the same distribution as the order statstics of $n$ iid unif$(0,t)$ random variables. Hence, the likelihood becomes \begin{align} L(\lambda,t) &= P(N=n) f(t_1,t_2,\dots,t_N|N=n) \\ &= \frac{e^{-\lambda t}(\lambda t)^n}{n!}\frac{n!}{t^n} \\ &= e^{-\lambda t}\lambda^n. \end{align} for $t\ge t_n$ and zero elsewhere. This is maximised for $\hat t=t_n$ and $\hat\lambda=n/t_n$. These MLEs don't exist if there are no occurrences $N=0$, however. Conditional on $N=n$, again using the fact that $t_n$ can be viewed as an order statistic (the maximum) of $n$ iid unif$(0,t)$ random variables, $E(t_N|N=n)=\frac n{n+1} t$. Hence, the estimator $t^*=\frac {n+1}n t_n$ is unbiased for $t$ conditional on $N=n$ and hence also conditional on $N\ge 1$. A reasonable frequentist estimator of $\lambda$ might be $\lambda^* = n/t^* = \frac{n^2}{(n+1)t_n}$ but this does not have finite expectation when $N=1$ so assessing its bias is even more troublesome.

Bayesian inference using independent, non-informative scale priors on $\lambda$ and $t$ on the other hand leads to a posterior $$ f(\lambda,t|t_1,\dots,t_N) \propto e^{-\lambda t}\lambda^{n-1}t^{-1}. $$ for $t>t_n,\lambda>0$. Integrating out $\lambda$, the marginal distribution of $t$ becomes $$ f(t|t_1,\dots,t_N) = \frac{n t_n^n}{t^{n+1}}, t>t_n, $$ and the posterior mean $E(t|t_1,\dots,t_N)=\frac n{n-1} t_n$. A $(1-\alpha)$-credible interval for $t$ is given by $\left(\frac{t_n}{(1-\alpha/2)^{1/n}}, \frac{t_n}{(\alpha/2)^{1/n}}\right)$.

The marginal posterior of $\lambda$, \begin{align} f(\lambda|t_1,\dots,t_N) &\propto \int_{t_\text{max}}^\infty e^{-\lambda t}\lambda^{n-1}t^{-1} dt \\ &= \lambda^{n-1}\Gamma(0,\lambda t_n) \end{align} where $\Gamma$ is the incomplete gamma function.

Let $t=T_F$. Conditional on the number of occurences $N=n$, the arrival times $t_1,t_2,\dots,t_N$ are known to have the same distribution as the order statstics of $n$ iid unif$(0,t)$ random variables. Hence, the likelihood becomes \begin{align} L(\lambda,t) &= P(N=n) f(t_1,t_2,\dots,t_N|N=n) \\ &= \frac{e^{-\lambda t}(\lambda t)^n}{n!}\frac{n!}{t^n} \\ &= e^{-\lambda t}\lambda^n. \end{align} for $t\ge t_n$ and zero elsewhere. This is maximised for $\hat t=t_n$ and $\hat\lambda=n/t_n$. These MLEs don't exist if there are no occurrences $N=0$, however. Conditional on $N=n$, again using the fact that $t_n$ can be viewed as the maximum of $n$ iid unif$(0,t)$ random variables, $E(t_N|N=n)=\frac n{n+1} t$. Hence, the estimator $t^*=\frac {n+1}n t_n$ is unbiased conditional on $N=n$ and hence also conditional on $N\ge 1$. A reasonable frequentist estimator of $\lambda$ might be $\lambda^* = n/t^* = \frac{n^2}{(n+1)t_n}$ but this does not have finite expectation when $N=1$ so assessing its bias is even more troublesome.

Bayesian inference using independent, non-informative scale priors on $\lambda$ and $t$ on the other hand leads to a posterior $$ f(\lambda,t|t_1,\dots,t_N) \propto e^{-\lambda t}\lambda^{n-1}t^{-1}. $$ for $t>t_n,\lambda>0$. Integrating out $\lambda$, the marginal distribution of $t$ becomes $$ f(t|t_1,\dots,t_N) = \frac{n t_n^n}{t^{n+1}}, t>t_n. $$ with posterior mean $E(t|t_1,\dots,t_N)=\frac n{n-1} t_n$. A $(1-\alpha)$-credible interval for $t$ is given by $\left(\frac{t_n}{(1-\alpha/2)^{1/n}}, \frac{t_n}{(\alpha/2)^{1/n}}\right)$.

The marginal posterior of $\lambda$, \begin{align} f(\lambda|t_1,\dots,t_N) &\propto \int_{t_\text{max}}^\infty e^{-\lambda t}\lambda^{n-1}t^{-1} dt \\ &= \lambda^{n-1}\Gamma(0,\lambda t_n) \end{align} where $\Gamma$ is the incomplete gamma function.

Let $t=T_F$. Conditional on the number of occurences $N=n$, the arrival times $t_1,t_2,\dots,t_N$ are known to have the same distribution as the order statstics of $n$ iid unif$(0,t)$ random variables. Hence, the likelihood becomes \begin{align} L(\lambda,t) &= P(N=n) f(t_1,t_2,\dots,t_N|N=n) \\ &= \frac{e^{-\lambda t}(\lambda t)^n}{n!}\frac{n!}{t^n} \\ &= e^{-\lambda t}\lambda^n. \end{align} for $t\ge t_n$ and zero elsewhere. This is maximised for $\hat t=t_n$ and $\hat\lambda=n/t_n$. These MLEs don't exist if there are no occurrences $N=0$, however. Conditional on $N=n$, again using the fact that $t_n$ can be viewed as the maximum of $n$ iid unif$(0,t)$ random variables, $E(t_N|N=n)=\frac n{n+1} t$. Hence, the estimator $t^*=\frac {n+1}n t_n$ is unbiased conditional on $N=n$ and hence also conditional on $N\ge 1$. A reasonable frequentist estimator of $\lambda$ might be $\lambda^* = n/t^* = \frac{n^2}{(n+1)t_n}$ but this does not have finite expectation when $N=1$ so assessing its bias is even more troublesome.

Bayesian inference using independent, non-informative scale priors on $\lambda$ and $t$ on the other hand leads to a posterior $$ f(\lambda,t|t_1,\dots,t_N) \propto e^{-\lambda t}\lambda^{n-1}t^{-1}. $$ for $t>t_n,\lambda>0$. Integrating out $\lambda$, the marginal distribution of $t$ becomes $$ f(t|t_1,\dots,t_N) = \frac{n t_n^n}{t^{n+1}}, t>t_n, $$ and the posterior mean $E(t|t_1,\dots,t_N)=\frac n{n-1} t_n$. A $(1-\alpha)$-credible interval for $t$ is given by $\left(\frac{t_n}{(1-\alpha/2)^{1/n}}, \frac{t_n}{(\alpha/2)^{1/n}}\right)$.

The marginal posterior of $\lambda$, \begin{align} f(\lambda|t_1,\dots,t_N) &\propto \int_{t_\text{max}}^\infty e^{-\lambda t}\lambda^{n-1}t^{-1} dt \\ &= \lambda^{n-1}\Gamma(0,\lambda t_n) \end{align} where $\Gamma$ is the incomplete gamma function.