Maximum likelihood estimators for transition intensities: formula sheet

Key formulas

Survival probability, constant force

tp_x = e^(−μt)

Holds when μ is constant over the period of observation.

Density of death at time t

f(t) = μ × e^(−μt)

Used for a life observed to die at waiting time t.

Contribution of one life

L_i(μ) = μ^(d_i) × e^(−μv_i)

d_i = 1 if death observed, 0 if censored. v_i is the time observed in the alive state.

Full likelihood

L(μ) = μ^d × e^(−μv)

d = Σd_i is total deaths. v = Σv_i is total waiting time. Assumes lives are independent.

Log-likelihood

ln L(μ) = d ln μ − μv

Differentiate this to find the MLE in the next topic.

Maximum likelihood estimate

μ̂ = d ÷ v

Obtained by setting d ÷ μ − v = 0. Deaths divided by total time at risk.

Likelihood (constant μ, two-state)

L(μ) = μ^d × e^(−μv)

d = number of deaths, v = total waiting time (sum of all observed times). Constants not involving μ are dropped.

Log-likelihood

ln L = d ln μ − μv

Add a constant if you keep the full likelihood; it does not affect the maximum.

Maximum likelihood estimator

μ̂ = d ÷ v

Number of transitions divided by total time spent in the starting state.

Second-derivative check

d²(ln L)/dμ² = −d ÷ μ² < 0 (for d > 0)

Confirms a maximum. If d = 0, the likelihood is decreasing and the formula gives μ̂ = 0 at the boundary.

Multi-state version

μ̂_ij = d_ij ÷ v_i

d_ij = observed transitions from i to j; v_i = total time spent in state i by all lives.

Estimator as a random variable

μ̃ = D ÷ V

Use capital letters when treating the estimator as random; lower case for the estimate from observed data.

MLE of constant intensity

μ̂ = v ÷ T

v is observed number of transitions; T is total time spent in the starting state (central exposed to risk).

Log-likelihood

l(μ) = v ln μ − μT + constant

Likelihood is proportional to μ^v × e^(−μT).

Asymptotic distribution

μ̂ ≈ N(μ, μ² ÷ E[V])

Valid for large samples, with T treated as fixed.

Cramér-Rao lower bound

Var(μ̂) ≈ 1 ÷ I(μ), where I(μ) = −E[d²l/dμ²]

Here −d²l/dμ² = V ÷ μ², so I(μ) = E[V] ÷ μ².

Estimated variance

Var(μ̂) ≈ v ÷ T² = μ̂² ÷ v

Replace E[V] by v and μ by μ̂.

Approximate confidence interval

μ̂ ± z × √v ÷ T

For 95%, z = 1.96. This is the normal-approximation interval.

Likelihood for constant intensities

L ∝ Π over i≠j of exp(-μij Ti) × μij^Nij

Ti is total time in state i. Nij is the number of i→j transitions. Constants and the Markov assumption are required.

MLE of a transition intensity

μ̂ij = Nij ÷ Ti

Use the same Ti for every intensity leaving state i.

Log-likelihood

ln L = Σ (Nij ln μij - μij Ti) + constant

Differentiate with respect to each μij separately.

Variance of the estimator (asymptotic)

Var(μ̂ij) ≈ μij ÷ E[Ti], estimated by Nij ÷ Ti²

Based on the second derivative: -d²lnL/dμij² = Nij ÷ μij². It is an approximation for large samples.

Distribution of the estimator

μ̂ij is approximately Normal(μij, μij ÷ E[Ti])

Use for confidence intervals: μ̂ij ± 1.96 × √(Nij) ÷ Ti for 95%.

Total exit intensity

μi· = Σ over j≠i of μij, estimated by (Σj Nij) ÷ Ti

Waiting time in state i is exponential with this rate.

Quick revision

• Two-state model: alive to dead with constant intensity μ over the period of study.
• Data needed: number of deaths and the total time each life was observed.
• The total observed time is called the waiting time or central exposure.
• Likelihood: L(μ) ∝ μ^d × exp(−μ × v), where d = deaths and v = total waiting time.
• MLE: μ̂ = d ÷ v.
• Variance: approximately μ² ÷ E[D], estimated as μ̂² ÷ d.
• Large samples: μ̂ is approximately normal with mean μ and variance μ² ÷ E[D].
• Approximate 95% confidence interval: μ̂ ± 1.96 × μ̂ ÷ √d.
• Multi-state: for each transition i to j, μ̂ij = number of i to j transitions ÷ total time in state i.
• In a multi-state model, the likelihood factorises, so each intensity is estimated separately.
• State your assumptions: constant intensity, and independent lives.

Common mistakes

• Leaving censored lives out of the total waiting time. Fix: Include every life's time in v. Censored lives add to v but not to d.
• Using μe^(−μv) for every life, including censored ones. Fix: Use e^(−μv_i) for censored lives. The factor μ appears only for deaths.
• Using the number of lives at the start as the denominator instead of total time lived. Fix: Use total waiting time v. Add the time each life actually spent in the state, including time of survivors and those who died early.
• Leaving out the censored lives or their time. Fix: Censored lives contribute e^(−μv_i) to the likelihood. Their time counts in v, though they add nothing to d.
• Using the number of lives instead of the total time T as the denominator. Fix: The denominator is total time in the state, summed across all lives, including part-years.
• Writing the variance as μ ÷ v or v ÷ T instead of v ÷ T². Fix: Derive it: μ̂ = v ÷ T, so the variance is μ² ÷ E[V], which becomes v ÷ T² after substitution. Check units.
• Using only the time before a transition to j as the denominator for μij. Fix: Use total time in state i for every intensity out of i. Time in i before a move to k still counts.
• Counting censored lives as transitions. Fix: Add the censored life's time to Ti but leave Nij unchanged.

Exam tips

• Write the likelihood in terms of d and v first. Marks are usually given for this form before any calculus.
• Always show how each life contributes, one line for deaths and one for censored lives.
• Define your symbols: say what d, v and μ mean and in what units.
• Check the total waiting time by adding the deaths' times and the survivors' times separately.
• If the question says 'derive', show the log-likelihood, the derivative and the second-derivative check.
• Show the likelihood, the log-likelihood, the derivative and the second derivative in order. Each earns method marks in written questions.
• State the constant intensity assumption clearly at the start of the answer.
• In numerical questions, list d and v separately before dividing, so partial credit is possible if one is wrong.