= Solution
Put $U=H(T)$ and $V=H(C)$, so $Y=\min(U,V)$ and $0\le\mathbb EY\le\mathbb EU=1$. Its <expectation> is not generally one and depends on censoring. Under <independent censoring>, condition on $V=v$: the <exponential distribution> gives
$$
\mathbb E(Y\mid V=v)=\int_0^v e^{-u}\,du=1-e^{-v}=\mathbb P(U\le V\mid V=v).
$$
Writing $\Delta=\mathbf1_{T\le C}$ for the observed event indicator, it follows that $\mathbb EY=\mathbb E\Delta$. The <exponential mean imputation under independent censoring> therefore gives \b[a <modified Cox–Snell residual> with known mean]:
$$
\boxed{Y^*=Y+(1-\Delta),\qquad\mathbb EY^*=1.}
$$
On censoring, the added one is the expected remaining transformed lifetime, by <memorylessness of the exponential distribution>. This gives a mean-one variable, not necessarily another exponential variable. Equivalently, $M=\Delta-Y=1-Y^*$ is a <martingale residual> with mean zero. <Independent censoring>, or its appropriate conditional version, is essential for these identities; arbitrary informative censoring does not justify the correction.
Fit a model using the relevant <covariates>, calculate $\widehat M_i=v_i-\widehat H_i(x_i)$, and plot these <martingale residuals> against a candidate explanatory variable or against an included variable's value. A smooth systematic trend away from zero can indicate an omitted effect or an unsuitable functional form, such as a nonlinear age effect. After fitting an appropriate effect, the residual trend should diminish. The equivalent plot of $\widehat Y_i^*$ has mean-one reference rather than zero.
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