Show Cox regression estimating equation is unbiased Let $N(x) = I(X \leq x, \Delta=1)$ be the counting process for observed failure events, where $X = \min\{T,C\}$ and $\Delta = I(T \leq C)$, for censoring time $C$ and failure time $T$. Assume that $T$ and $C$ are independent.
Let $Z \in \mathbb{R}^p$ be a vector of covariates. Let subscripted variables denote i.i.d. draws. Define the estimating equation $$U(\beta) = \sum_j \int Z_j - \frac{\sum_i Z_i I(X_i \geq x) \mathrm{exp}(Z_i^T \beta)}{\sum_i I(X_i \geq x) \mathrm{exp}(Z_i^T \beta)} \,\mathrm{d}N_j(x).$$ How can it be directly shown that this vanishes in expectation?

The estimating/score equation is 0 (because $dN=0$) except at event times. Given that there is an event at time $x$, consider the expectation of $Z_j$ at that time. That expectation is over all the individuals at risk at that time.
With proportional hazards correctly modeled in terms of a baseline hazard $h_0(x)$, covariates $Z$, and associated coefficients $\beta$, the probability that individual $k$ has an event at time $x$ is $h_0(x)I(X_k \geq x)\mathrm{exp}(Z_k^T \beta)$.
The sum of such probabilities at time $x$ over all individuals $i$ is
$$ h_0(x)\sum_iI(X_i \geq x)\mathrm{exp}(Z_i^T \beta).$$
Given that an event occurs at $x$, the probability that the event happens to individual $k$ is:
$$\frac{I(X_k \geq x)\mathrm{exp}(Z_k^T \beta)}{\sum_iI(X_i \geq x)\mathrm{exp}(Z_i^T \beta)} ,$$
as the baseline hazard $h_0(x)$ cancels.
The expected value of $Z_j$ in the estimating/score equation at time $x$, based on those probabilities of each individual having an event at that time, is thus
$$\mathbb{E}(Z_j) =\frac{\sum_i Z_i I(X_i \geq x)\mathrm{exp}(Z_i^T \beta)}{\sum_iI(X_i \geq x)\mathrm{exp}(Z_i^T \beta)} $$
and the estimating/score equation
$$U(\beta) = \sum_j \int Z_j - \frac{\sum_i Z_i I(X_i \geq x) \mathrm{exp}(Z_i^T \beta)}{\sum_i I(X_i \geq x) \mathrm{exp}(Z_i^T \beta)} \,\mathrm{d}N_j(x)$$
vanishes in expectation at each event time $x$.
This holds for each of the $p$ components of $Z$. Furthermore, $Z$ and $\beta$ could vary with time as under proportional hazards only the values at event times $x$ matter.
