Define the tilted probability measure $Q$ by $dQ=ZdP$; this is normalized because $\mathbb{E}Z=1$. ThenLet ${\mathbb{E}}_i$ average only coordinate $i$, keeping $X^{(i)}$ fixed. The marginal density of $Q_{X^{(i)}}$ relative to $P_{X^{(i)}}$ is ${\mathbb{E}}_{i}Z$, and henceMoreover,Substituting Han's inequality for relative entropy and rearranging gives the tensorization of entropy

For $X^{(i)}=X_{1:n}\setminus X_i$, Han's inequality for relative entropy states that, when $P={⨂}_{i=1}^n{P}_i$,

First prove the chain rule for relative entropy. For two coordinates,Taking expectation under $Q$ givesand iteration proves the chain rule for any finite product.

Write $Q_i(\cdot \mid x_{<i})$ for the successive conditional distributions. Since $P$ is a product, the chain rule givesFor a fixed omitted coordinate $j$, applying the chain rule in the remaining coordinate order givesThe convexity of Kullback-Leibler divergence implies that removing $X_j$ from the conditioning can only decrease each averaged conditional divergence. Summing over $j$, each full conditional increment occurs for exactly the $n-1$ indices $j\ne i$, and thereforewhich is the claimed inequality. Equivalently, this is Han's entropy inequality after expanding each divergence: the product-reference cross-entropy terms cancel because they are modular.

Put $x=\sqrt{2\nu t}+ct$ and $y=c^{2}t/\nu$. ThenConsequentlyThe tail bound from part c therefore has exponent $-(\nu /c^2)y=-t$, and

The Chernoff bound gives, for $0<\lambda <1/c$,Writing $u=ct/\nu$, elementary differentiation shows that the exponent is minimized atSubstitution givesThus $h$ is the Legendre transform generated by the sub-Gamma cumulant bound after its natural rescaling.

Independence makes cumulant-generating functions additive:SetFor $0<\lambda <1/c$, every denominator is positive and $1-c_i\lambda \ge 1-c\lambda$, soAlso $\mathbb{E}Z=0$, and hence $Z\in {\Gamma }_{+}(\nu ,c)$.

Because the moment-generating function is finite in a neighborhood to the right of zero and $\mathbb{E}X=0$, its cumulant-generating function satisfiesDivide the defining Sub-Gamma random variable in the right tail inequality by ${\lambda }^2/2$ and let $\lambda \downarrow 0$. The right side converges to $\nu$, proving $\mathrm{Var}(X)\le \nu$.

A period survival analysis estimates survival under the recent mortality conditions operating during a fixed calendar period. For calendar year 2022, each cancer patient's contribution begins at the later of diagnosis and 1 January 2022, which treats earlier diagnoses as left-truncated at the start of the year. Follow-up ends at death, loss to follow-up, or 31 December 2022, so surviving records are right-censored. Combining these risk-set contributions, commonly through a Kaplan–Meier estimator, constructs the 2022 period survivor function; records from several diagnosis cohorts may contribute different segments of follow-up.

Empirical likelihood assigns unknown probability masses $p_j\ge 0$ to data-supported event times or intervals, imposes ${\sum }_j{p}_j=1$, and maximizes the product of each observation's probability. For event-time data, write $F(t)=\mathbb{P}(T\le t)$ and $S(t)=1-F(t)$ for the survivor function. Exact, right-censored, left-censored, interval-censored, and truncated observations contribute the probability of their respective compatible sets.

The maximization uses that $S$ is nonincreasing and right-continuous, $S(0)=1$, and $S(t)\to 0$ as $t\to \infty$. Probability mass need only be placed at endpoints that change an observation's compatible set; moving mass within any observationally indistinguishable interval leaves the likelihood unchanged. Maximizing over those masses gives the nonparametric maximum-likelihood estimator of the survivor function.

A semiparametric proportional hazards model specifieswith finite-dimensional parameter $\beta$ but an unspecified baseline hazard $h_0$. A partial likelihood uses a component of the data likelihood that depends on $\beta$ while eliminating the nuisance function. In the Cox proportional-hazards model, conditioning on which member of each risk set experiences the event produces the Cox partial likelihood.

Choose a parametric baseline hazard $h_0(t;\eta )$ and fitUnder independent right censoring, the full survival likelihood isEstimate $(\eta ,\beta )$ by maximum likelihood estimation and test $H_0:\beta =0$ with a likelihood-ratio test, Wald test, or score test. Equality of the two event-time distributions is exactly $\beta =0$ within this model.