Hello. I really need some help. I am reading "Survival Models and Their Estimations" 2nd edition by Dick London and my question concerns with the chapter 4 of this book. It was staightforward that I cannot understand how did the author arrive at the formula. Please help me...

let t* = t + 0.5 represent the midpoint of the interval (t, t + 1]. It is assumed that T is exponentially distributed over (t, t+1].

It was mentioned that the estimator L(t*) can be written as

L(t*) = - ln (N_(t+1) / n_t), which is a biased estimator (why? I don't get it)

It says: the random variable L(t*) is a natural log function of the binomial random variable N_(t+1), so the variance of L(t*), conditional on n_t, can be approximated by the method of statistical differentials using this formula:

Var{g(X)} = [g'(m)]^2 Var(X), where E[X] = m.

and so, Var[L(t*) / n_t] = q_t / (p_t n_t). (how did that arrive as that?)

let t* = t + 0.5 represent the midpoint of the interval (t, t + 1]. It is assumed that T is exponentially distributed over (t, t+1].

It was mentioned that the estimator L(t*) can be written as

L(t*) = - ln (N_(t+1) / n_t), which is a biased estimator (why? I don't get it)

It says: the random variable L(t*) is a natural log function of the binomial random variable N_(t+1), so the variance of L(t*), conditional on n_t, can be approximated by the method of statistical differentials using this formula:

Var{g(X)} = [g'(m)]^2 Var(X), where E[X] = m.

and so, Var[L(t*) / n_t] = q_t / (p_t n_t). (how did that arrive as that?)

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