Question

Let \(T_{1}\) and \(T_{2}\) be independent lifetimes with hazard functions $$ \alpha(t), \quad \theta \alpha(t) $$ respectively, where \(\theta>0, \alpha(t) \geq 0, t \geq 0\) and \(A(t)=\int_{0}^{t} \alpha(s) d s<\infty\) for all \(t\). Let \(C_{1}\) and \(C_{2}\) be independent censoring variables inducing independent censoring, and let \(N_{j}(t)=I\left(T_{j} \wedge C_{j} \leq t, \Delta=1\right)\) with \(\Delta=I\left(T_{j} \leq C_{j}\right)\), \(j=1,2 .\) Assume we observe in \([0, \tau]\) with \(\tau\) a deterministic point. (a) Specify the intensities of \(N_{j}(t), j=1,2\). Assume now that we have \(n\) independent copies from the above generic model giving rise to \(N_{j i}(t), j=1,2, i=1, \ldots, n .\) Let \(\tilde{\theta}_{\tau}\) be the solution to $$ \theta=\tilde{N}_{2}(\tau) / \int_{0}^{\tau} \frac{\tilde{Y}_{2}(t)}{\tilde{Y}_{1}(t)+\theta \tilde{Y}_{2}(t)} d\left(\tilde{N}_{1}(t)+\tilde{N}_{2}(t)\right) $$ where \(\tilde{N}_{j}(t)=\sum_{i} N_{j i}(t), \tilde{Y}_{j}(t)=\sum_{i} Y_{j i}(t), Y_{j i}(t)=I\left(t \leq T_{j i} \wedge C_{j i}\right)\) \(j=1,2\). (b) Show, for \(n\) tending to infinity, that \(n^{1 / 2}\left(\tilde{\theta}_{\tau}-\theta\right)\) converges in distribution towards a zero-mean normal distribution and specify the variance (c) Let \(\beta=\log (\theta)\) and put \(\hat{\theta}=\exp (\hat{\beta})\), where \(\hat{\beta}\) is the maximum partial likelihood estimator. Compare the asymptotic variance of \(n^{1 / 2}(\hat{\theta}-\theta)\) with that obtained in (c). Is \(\hat{\theta}\) different from \(\tilde{\theta}_{\tau} ?\)

Short answer

Question: Calculate the intensities of \(N_{j}(t), j=1,2\) and show that as \(n\) tends to infinity, \(n^{1/2}(\tilde{\theta}_{\tau}-\theta)\) converges in distribution towards a zero-mean normal distribution. Additionally, compare the asymptotic variance of \(\hat{\theta}\) with that obtained in (c) and determine if it is different from \(\tilde{\theta}_{\tau}\). Answer: The intensities of \(N_{j}(t), j=1,2\) are given by \(\lambda_{1}(t) = \alpha(t)Y_{1}(t)\) and \(\lambda_{2}(t) = (\theta \alpha(t))Y_{2}(t)\). When \(n\) tends to infinity, \(n^{1/2}(\tilde{\theta}_{\tau}-\theta)\) converges in distribution towards a zero-mean normal distribution under conditions specified in the Central Limit Theorem. In general, the asymptotic variance of \(\hat{\theta}\) is different from that of \(\tilde{\theta}_{\tau}\) because the likelihood function and censoring mechanism handle the data differently, and the estimators react differently to the variation introduced by their respective constraints.

Step-by-step solution

Calculate the Intensities of \(N_{j}(t), j=1,2\)

The intensities for \(N_{j}(t), j=1,2\), are given by the expected value of their hazard functions times their population at risk, which is given by the probability functions (\(Y_{j}(t)\)) that the event \(T_{j}\) has not occurred before time \(t\). Hence, their intensities are: $$ \lambda_{1}(t) = \alpha(t)Y_{1}(t) \\ \lambda_{2}(t) = (\theta \alpha(t))Y_{2}(t) $$ Next, we will address part (b) by showing that, for \(n\) tending to infinity, \(n^{1/2}(\tilde{\theta}_{\tau}-\theta)\) converges in distribution towards a zero-mean normal distribution and specifying the variance.

Prove Convergence in Distribution and Find Variance

To show that for \(n\) tending to infinity, \(n^{1/2}(\tilde{\theta}_{\tau}-\theta)\) converges in distribution towards a zero-mean normal distribution, we will make use of the asymptotic properties of independent, identically distributed random variables, such as the Central Limit Theorem. We must verify the conditions for the Central Limit Theorem, which are that the random variables are independent, identically distributed, and both their means and variances are finite. Then, it can be shown that the sum of these random variables, scaled by \(n^{1/2}\), converges in distribution towards a zero-mean normal distribution as \(n\) tends to infinity. By writing down the population moments, we can find the variance of the random variables. This can be obtained by asymptotic theory using Taylor expansions and calculating the variance of the terms that have \(\theta\) involved. Now let's answer part (c) by defining \(\beta=\log(\theta)\) and putting \(\hat{\theta}=\exp(\hat{\beta})\), where \(\hat{\beta}\) is the maximum partial likelihood estimator. We will compare the asymptotic variance of \(n^{1/2}(\hat{\theta}-\theta)\) with that obtained in (c) and determine if \(\hat{\theta}\) is different from \(\tilde{\theta}_{\tau}\).

Compare Asymptotic Variances and Determine if \(\hat{\theta}\) is different from \(\tilde{\theta}_{\tau}\).

The asymptotic variance of \(n^{1/2}(\hat{\theta}-\theta)\) can be found using the Delta Method. The Delta Method states that if a sequence of random variables \(n^{1/2}(X_n - \mu)\) converges in distribution to a normal distribution and if a smooth function \(g(X)\) exists, then the following holds: \[ n^{1/2}(g(X_n) - g(\mu)) \overset{d}{\to} N(0, g'(\mu)^2 Var(X_n)) \] The maximum partial likelihood estimator \(\hat{\beta}\) is the solution to the score equation for the likelihood function which is asymptotically linear. Therefore, we can assume that the asymptotic variance of \(n^{1/2}(\hat{\theta}-\theta)\) will be different from the variance obtained in (b). This is because the likelihood function behaves differently under the change of variable \(\beta=\log(\theta)\), and the maximum partial likelihood estimator takes into account the likelihood of observing the censored data. Comparing the variances obtained in (b) and (c), in general, we can say that \(\hat{\theta}\) is different from \(\tilde{\theta}_{\tau}\) for most cases. This is due to the fact that the likelihood function and the censoring mechanism handle the data differently, and, in turn, the estimators react differently to the variation introduced by the different constraints in the problem.

Key concepts

Hazard Functions

Hazard functions are at the heart of survival data analysis, representing the instant rate at which events occur over time. They quantify the risk of an event, such as a system failure or a person's death, at a particular time given that the event has not occurred before that time.

In the context of the given exercise, the hazard functions for two independent lifetimes are \( \alpha(t) \) and \( \theta \alpha(t) \) respectively. It is important to note that \( \theta \) is a parameter that modifies the baseline hazard \( \alpha(t) \) for the second lifetime. By knowing these hazard functions, we can derive other important quantities in survival analysis such as survival probabilities and intensities.

Censoring Variables

Censoring occurs in survival data analysis when we have incomplete information about the time to event for subjects in a study. Censoring variables, in this case \( C_{1} \) and \( C_{2} \) for two independent lifetimes, represent the time beyond which we cannot observe the event. If an individual's event time is beyond the censoring time, we only know that the event occurred after the censoring time, not the exact time.

This concept is essential because it helps manage incomplete data without introducing bias. The independent censoring assumption, meaning that the lifetimes and censoring variables are independent, simplifies the analysis significantly. Observing how censoring variables play a role in the calculations helps students understand the practical challenges in dealing with real-world survival data.

Intensities

Intensities, in survival analysis, correspond to the expected number of events per unit time at time \( t \) among subjects at risk. For our case, the intensities \( \lambda_{1}(t) \) and \( \lambda_{2}(t) \) for the lifetimes with hazard functions \( \alpha(t) \) and \( \theta \alpha(t) \) and censoring are derived by multiplying the hazard functions by the population at risk at time \( t \) - mathematically represented as \( Y_{j}(t) \) for \( j = 1, 2 \).

The concept of intensities allows students to see the connection between the mathematical representations of hazard functions and the actual observed data, providing a clear understanding of how these calculated values feed into the analysis of survival data.

Asymptotic Properties

Asymptotic properties are characteristics of estimators or statistical procedures as the sample size \( n \) becomes large. In survival analysis, these properties become particularly important when dealing with estimators such as \( \tilde{\theta}_{\tau} \).

The exercise explores the behavior of \( \tilde{\theta}_{\tau} \) as \( n \) tends to infinity, utilizing the fact that with large sample sizes, certain distributions tend to become normal due to the Central Limit Theorem. Understanding these properties gives students insights into why large samples are often desirable in survival data analysis and how they contribute to the robustness of statistical conclusions.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics that describes the distribution of sample means. It states that, under certain conditions, the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed, regardless of the original distribution of the variables.

In the given problem, the CLT plays a crucial role. It justifies using normal approximations for the distribution of \( n^{1/2}(\tilde{\theta}_{\tau}-\theta) \) as \( n \) becomes large. The theorem is an essential component in understanding how the normal distribution can be a powerful tool for inference in survival data analysis, particularly in hypothesis testing and confidence interval estimation.

Maximum Partial Likelihood Estimator

The maximum partial likelihood estimator (MPLE) \( \hat{\beta} \) is employed in the presence of censored data within survival analysis. It maximizes the likelihood function that incorporates observed event times and censored observations, providing an estimate of the underlying parameters governing the survival process.

In the context of this exercise, \( \hat{\beta} \) is used to estimate \( \beta = \log(\theta) \) and subsequently to compute the estimate \( \hat{\theta} = \exp(\hat{\beta}) \). By comparing \( \hat{\theta} \) with \( \tilde{\theta}_{\tau} \) in terms of their respective asymptotic variances, students can appreciate the difference that the estimation method makes in the analysis of survival data, particularly when dealing with censored data.

Delta Method

The Delta Method is a technique in statistics used to derive the asymptotic distribution of a function of an estimator. It is particularly useful when applied to the function of a sequence of random variables that converges in distribution to a normal distribution.

In this exercise, the Delta Method is crucial for comparing asymptotic variances in part (c) by helping to determine the variance of \( n^{1/2}(\hat{\theta}-\theta) \) based on the known variance of \( n^{1/2}(\hat{\beta}-\log(\theta)) \) and the transformation \( \hat{\theta}=\exp(\hat{\beta}) \). This method enables students to see the implications of nonlinear transformations on the distribution of estimators, deepening their understanding of asymptotic analysis in survival data.