Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 44 / 2

Assume the incubation times are independent observations from the same upper-truncated distribution. Conditional on lying below $M$, the gamma distribution density is $g(t;a,s)/G(M;a,s)$ on its support. With $t_{(n)}={\max }_i{t}_i$, the likelihood function isIn particular, the normalizing factor $G(M;a,s{)}^{-n}$ cannot be omitted. On the admissible region the log-likelihood isThe profile likelihood eliminates the nuisance parameters $a,s$ by fitting them separately at each candidate endpoint:For fixed $a,s$ and $M>t_{(n)}$,Equivalently, for $t_{(n)}<M_1<M_2$, every nuisance-parameter fit has $L(a,s,M_1)>L(a,s,M_2)$. Taking suprema shows that $L_p$ is nonincreasing. If the nuisance maximum at $M_2$ is attained and finite, evaluating its maximizing parameters at $M_1$ shows strict decrease there as well. Values below the largest observation are inadmissible. Thus upper-endpoint estimation in a truncated distribution gives the usual boundary estimateThere is a support-convention qualification: the literal strict support $t<M$ excludes $M=t_{(n)}$, so it gives a supremum approached as $M\downarrow t_{(n)}$, rather than an attained maximum-likelihood estimator. Using the equivalent closed-endpoint density convention for a continuous distribution yields the usual boundary estimate above. The argument presumes a meaningful finite nuisance fit; degenerate samples can instead have an unbounded density likelihood. Since $M$ changes the support, regular interior-parameter asymptotic formulas are not automatically applicable.

For the log-transformed location-scale model, let $F_W$ denote the distribution function associated with $f$. The change-of-variables formula for a probability density gives the untruncated density and distribution functionConsequently its upper-truncated likelihood is ${\prod }_{i}p(t_i)/F(M{)}^n$ on the observed support, and the profile likelihood maximizes this over $\alpha ,\sigma ,q$. Restricting to the corresponding subfamilies gives the gamma distribution and lognormal distribution profiles. Write the plotted likelihood ratio asIt measures how well each endpoint fits relative to that model's own best endpoint, after nuisance refitting. It is not a probability density function for $M$ and need not integrate to one. Endpoints whose ratios are near one are relatively compatible with the observations; a rapidly falling profile indicates more information favoring endpoints near the maximum.

All three profiles favor the smallest admissible endpoint, approximately 14 days; their information about larger endpoints differs substantially. Reading the actual PDF legend and curves, the dotted gamma distribution profile falls initially but is nearly flat around $0.60$ from about 18 days onward. The solid generalized log-gamma profile declines much further, to roughly $0.20$ at 19 days and $0.18$ at 21 days. The dashed lognormal distribution profile falls most strongly, to roughly $0.14$ at 18 days and $0.08$ at 21 days. These are graphical readings, not recomputed fits.

Thus the gamma distribution fit gives weak discrimination between moderate and very large endpoints: much of its untruncated mass can already be below these endpoints, making further changes to the truncation have little effect. Over the displayed range, the lognormal distribution fit gives stronger relative evidence against large endpoints, and the more flexible generalized log-gamma fit is intermediate. The solid curve remains close to the dashed curve near the maximum but has appreciably more relative support farther out. The plot shows slowing declines rather than a sharply determined finite upper endpoint; it cannot establish a biological maximum.

Normalization is essential to this comparison. Although the generalized family includes the two special families and therefore has at least as large a raw maximized likelihood at any fixed $M$, its normalization constant can also be larger. Its normalized profile need not lie above the normalized profiles of its submodels. Their individual maxima have all been replaced by one, so this plot alone cannot compare overall goodness of fit or select a family while accounting for extra fitted parameters.

A set $\{M:R(M)\ge c\}$ is a relative-likelihood support set once $c$ is chosen. Turning it into a calibrated confidence interval requires an appropriate sampling argument, simulation or other justified method. In particular, a usual Wilks theorem chi-squared cutoff cannot simply be assumed for this changing-support endpoint problem. The original incubation observations are not supplied here, so numerical refitting or calibrated limits cannot be recovered from the graph alone.

Quarantine restricts contacts of exposed people whose infection status is uncertain and monitors them for illness, reducing opportunities for transmission before infection is recognized. Recognized cases are managed by isolation rather than treated as merely uncertain exposures. Duration must be measured from the last possible infection-producing exposure, or otherwise account for uncertainty in exposure time. The mathematical calculations below concern the incubation period among infected people who develop symptoms, not a complete infectiousness model.

With a credible hard endpoint $M$, the truncated distribution hasA duration $q\ge M$ then covers all symptom-onset times under that model, provided the exposure time is correctly specified. If a nonzero late-onset probability $\epsilon$ is accepted, solveThe calculation is appropriate when external biological evidence genuinely supports a hard maximum, or when a deliberately bounded model is justified for the population being considered. Truncation caused by a study's ascertainment window instead requires an observation correction: it does not prove that the disease has a hard biological endpoint. Nor does $\hat{M}=t_{(n)}$ guarantee that future cases will develop symptoms by the current sample maximum. Endpoint and nuisance-parameter uncertainty need to enter a conservative duration calculation.

For an untruncated gamma distribution, lognormal distribution or other model with an unbounded upper tail, no finite duration eliminates the modeled late-onset probability. Choose a tail tolerance and solveThis is appropriate when a hard endpoint is unsupported and rare long incubation times remain possible. One can estimate a high quantile, assess its uncertainty and balance the modeled residual probability against the costs of longer restrictions. Different tail families can give very different high quantiles even when they fit the central observations similarly, which makes the model dependence seen in part b operationally relevant.

If an individual has already remained symptom-free for $a$ time units after a known exposure, a conditional additional duration $b$ instead uses $S(a+b)/S(a)\le \epsilon$, where $S$ is the relevant survivor function and $S(a)>0$. This conditions on being an infected eventual symptomatic case. Incubation data alone do not describe asymptomatic infection, the timing of infectiousness or compliance with restrictions; symptom surveillance, testing where informative and a transmission model may therefore be needed to translate late onset into an actual release-risk calculation. Use a hard bound only when it is supported; otherwise quantify and allow for the upper-tail uncertainty.