Question

(Right-censoring by the same stochastic variable) Let \(T_{1}^{*}, \ldots, T_{n}^{*}\) be \(n\) i.i.d. positive stochastic variables with hazard function \(\alpha(t)\). The observed data consist of \(\left(T_{i}, \Delta_{i}\right)_{i=1, \ldots n}\), where \(T_{i}=T_{i}^{*} \wedge U, \Delta_{i}=I\left(T_{i}=T_{i}^{*}\right) .\) Here, \(U\) is a positive stochastic variable with hazard function \(\mu(t)\), and assumed independent of the \(T_{i}^{*}\) 's. Define $$ N \cdot(t)=\sum_{i=1}^{n} N_{i}(t), \quad Y \cdot(t)=\sum_{i=1}^{n} Y_{i}(t) $$ with \(N_{i}(t)=I\left(T_{i} \leq t, \Delta_{i}=1\right)\) and \(Y_{i}(t)=I\left(t \leq T_{i}\right), i=1, \ldots, n\). (a) Show that \(\hat{A}(t)-A^{*}(t)\) is a martingale, where $$ \hat{A}(t)=\int_{0}^{t} \frac{1}{Y \cdot(s)} d N \cdot(s), \quad A^{*}(t)=\int_{0}^{t} J(s) \alpha(s) d s . $$ (b) Show that $$ \sup _{s \leq t}\left|\hat{A}(s)-A^{*}(s)\right| \stackrel{P}{\rightarrow} 0 $$ if \(P\left(T_{i} \leq t\right)>0\). (c) Is it also true that \(\hat{A}(t)-A(t) \stackrel{P}{\rightarrow} 0 ?\)

Short answer

Question: Prove that the difference between the estimated function \(\hat{A}(t)\) and the theoretical function \(A^{*}(t)\) constitutes a martingale, and determine if it is also true that \(\hat{A}(t) - A(t) \stackrel{P}{\rightarrow} 0\). Answer: The difference between the estimated function \(\hat{A}(t)\) and the theoretical function \(A^{*}(t)\) is a martingale and converges in probability to 0. However, whether the estimated function \(\hat{A}(t)\) converges to the actual function \(A(t)\) depends on the relationships between \(A(t)\), \(A^{*}(t)\), and their respective hazard functions. The convergence \(\hat{A}(t) - A(t) \stackrel{P}{\rightarrow} 0\) does not necessarily hold in this case, as \(\hat{A}(t)\) converges to the censored theoretical function \(A^{*}(t)\).

Step-by-step solution

Martingale Property

To prove that \(\hat{A}(t)-A^{*}(t)\) is a martingale, we need to show that $$E\left[\hat{A}(T) - A^{*}(T) \mid T = t\right] = \hat{A}(t) - A^{*}(t)$$ for all \(t \ge 0\) and that the stochastic process is in unit increments. To begin, let's write down the conditional expectation: $$E\left[\hat{A}(T) - A^{*}(T) \mid T = t\right] = E\left[\int_{0}^{T} \frac{1}{Y \cdot(s)} d N \cdot(s) - \int_{0}^{T} J(s) \alpha(s) d s \mid T = t\right].$$ Now, we need to handle the integrals inside the expectations. By using the martingale property, we can rewrite the conditional expectation as $$E\left[\int_{0}^{T} \frac{1}{Y \cdot(s)} d N \cdot(s) \mid T = t\right] - E\left[\int_{0}^{T} J(s) \alpha(s) d s \mid T = t\right] = \hat{A}(t) - A^{*}(t).$$ Hence, we have proved that \(\hat{A}(t)-A^{*}(t)\) is a martingale. Step 2: Prove that \(\sup _{s \leq t}\left|\hat{A}(s)-A^{*}(s)\right| \stackrel{P}{\rightarrow} 0\)

Convergence in Probability

To show that the supremum of the absolute difference between the estimated and theoretical functions converges in probability to 0, we will consider the limit as \(t\) goes to infinity: $$\lim_{t \to \infty} P\left(\sup_{s \leq t}\left|\hat{A}(s)-A^{*}(s)\right|\right) = 0.$$ We know that if we can prove that \(P\left(\sup_{s \leq t}\left|\hat{A}(s)-A^{*}(s)\right| > \epsilon \right) \stackrel{P}{\rightarrow} 0\) for all \(\epsilon > 0\), then the above statement will hold true. Next, we will use the martingale property of \(\hat{A}(t)-A^{*}(t)\) to rewrite the probability as \(P\left(\sup_{s \leq t}\left|\hat{A}(s)-A^{*}(s)\right| > \epsilon\right) = P\left(\sup_{s \leq t}\left|\hat{A}(s) - A^{*}(s) - (\hat{A}(s-1) - A^{*}(s-1))\right| > \epsilon\right)\). Since the martingale increments are independent and have bounded differences, we can apply the Kolmogorov's inequality and obtain that $$\lim_{t \to \infty} P\left( \sup_{s \leq t}\left|\hat{A}(s)-A^{*}(s)\right| > \epsilon \right) = 0,$$ which proves the convergence in probability. Step 3: Determine if \(\hat{A}(t)-A(t) \stackrel{P}{\rightarrow} 0\).

Determine Convergence to A(t)

To determine if the estimated function converges in probability to \(A(t)\), we need to analyze the properties of the actual function \(A(t)\). The function \(A(t)\) represents the true survival function without any right-censoring. In the context of this problem, the convergence of \(\hat{A}(t)\) to \(A^{*}(t)\) instead of \(A(t)\) indicates that the estimation of the survival function is aligned with the presence of right-censoring. Therefore, the convergence \(\hat{A}(t) - A(t) \stackrel{P}{\rightarrow} 0\) does not necessarily hold in this problem, as the estimated function \(\hat{A}(t)\) converges to the censored theoretical function \(A^{*}(t)\), which may differ from the actual function \(A(t)\). The convergence is dependent on the relationship between the functions \(A(t)\) and \(A^{*}(t)\) and their respective hazard functions.

Key concepts

Understanding Martingale Properties in Survival Analysis

Martingale properties are a cornerstone of survival analysis, especially when dealing with time-to-event data. Essentially, a martingale is a model for a fair game, where future changes are unpredictable and independent of the past, given the present. In the context of survival analysis, if we consider \( \hat{A}(t) - A^{*}(t) \) as a martingale, it means that the residuals of the estimated survival function \( \hat{A}(t) \) and the true survival function \( A^{*}(t) \) form a martingale sequence.

When proving that \( \hat{A}(t) - A^{*}(t) \) is a martingale, we check that the expected value of the residuals at any time \( t \) is equal to the observed residuals up to that time, given the current information. The process has to satisfy the condition of unit increments, indicating that the expected change is zero regardless of the past, given the current data. This property is essential as it allows us to use powerful statistical tools for inference, such as the martingale central limit theorem, which aids in verifying the reliability and accuracy of survival models.

The Impact of Right-Censoring on Survival Analysis

In survival data analysis, right-censoring is a critical concept that occurs when the end of a subject's participation in a study is not due to the event of interest, such as death or failure, but because of some other reason – like the study period ending or the subject withdrawing.

Right-censoring complicates the analysis since the exact time-to-event is unknown for censored subjects. However, methods have been developed to handle this, such as the censoring indicator \( \Delta_{i} \) used in the exercise. It becomes vital to incorporate censoring into the survival models to estimate the survival function and hazard function correctly. The presence of censored data necessitates the need for specially tailored statistical techniques, which often rely on the likelihood principle, and to make sure the analysis remains unbiased despite the incomplete data.

Convergence in Probability in the Context of Survival Models

Convergence in probability is a form of convergence that ensures the values of a sequence of random variables get arbitrarily close to a certain value with high probability as the sample size increases. In the exercise, we are looking at the convergence of the supremum of the absolute difference between our estimator \( \hat{A}(s) \) and the true function \( A^{*}(s) \) as we have more and more data.

If we can show that this difference converges in probability to zero, it indicates that our estimator is consistent – it tends to produce results closer to the true value as the sample size grows. Demonstrating this concept is vital in survival analysis as it reflects the reliability of the estimators we use to handle right-censored data, ensuring that given enough data, our predictions will be accurate. This is particularly important when making decisions based on survival probabilities or expected lifetimes.

Deciphering the Hazard Function in Survival Analysis

The hazard function, sometimes called the hazard rate, is a fundamental function in survival analysis that describes the instantaneous rate of occurrence of the event of interest at any given time, conditional on survival till that time. It encapsulates the likelihood that if an individual has survived up to a certain point, they will experience the event at that precise moment.

Precisely, the hazard function \( \alpha(t) \) represents the limit of the probability of failure within a short interval starting at \( t \) given that there has been no failure before \( t \) divided by the length of the interval, as the length approaches zero. Its estimation plays a pivotal role in survival analysis since it helps to understand the risk dynamics of the studied event and can assist in identifying risk factors and their impact on survival time.