Nearest neighbor hazard estimation with left-truncated duration data

Abstract

Duration data often suffer from both left-truncation and right-censoring. We show how both deficiencies can be overcome at the same time when estimating the hazard rate nonparametrically by kernel smoothing with the nearest-neighbor bandwidth. Smoothing Turnbull’s estimator of the cumulative hazard rate, we derive strong uniform consistency of the estimate from Hoeffding’s inequality, applied to a generalized empirical distribution function. We also apply our estimator to rating transitions of corporate loans in Germany.

Appendices

Appendix A: Proof of Theorem 2

The proof of Theorem 2 is in four steps. First, for an interval I:=[a,b]⊆[A,B] we establish an exponential bound for the distribution of the difference $|\varPsi _{n}^{*}(I) - \varPsi(I)|$:

(9)

for all p>0, ε>0, n∈ℕ_>0 and for each fixed I⊆[A,B] with Ψ(I)≤p.

Because of definition (4) and the boundedness of $0 \le \varDelta ^{x}_{i} \le \varDelta _{\max} < \infty$

(10)

is the arithmetic mean of the n independent and bounded random variables for each fixed I⊆[A,B], distributed as

The expectation, the variance and the bound of T _I can then be calculated for fixed I⊆[A,B] with Ψ(I)≤p.

The expectation of T _I follows from assumption (B3):

(11)

From assumption (B1), we get the following bound of |T _I | on [A,B]:

(12)

The variance of T _I can be obtained from the expectation (11) and the bound (12) as follows:

(13)

From (10), (11), (12), (13), and the inequality from Hoeffding (1963) results the following right bound:

for each fixed interval I⊆[A,B] with Ψ(I)≤p.

In the second step we derive the inequality

(14)

almost surely for a constant $C>\sqrt{2(2\varDelta _{\max} M + \varPsi(B))}$ and large n.

On the right hand side of the inequality (9), p and ε can be substituted with p _n and $\varepsilon_{n}:=C\sqrt{\log{(n)}p_{n}/n}$ for C>0 and n>1 altering the upper bound to

The series (A _n ) is then summable starting from some large n<∞ only if the exponent β _n :=(C ² p _n )/(2g(p _n +ε _n ))>1. From $\varepsilon_{n}/p_{n} = C\sqrt{\log(n)/(np_{n})}$ and the assumptions for p _n follow ε _n /p _n →0 and p _n /(p _n +ε _n )→1 for large n. The condition β _n >1 can be then achieved with C ²/2g>1 or $C > \sqrt{2g}$.

As a consequence, the series (A _n ) is summable from some large n<∞ and only for $C>\sqrt{2g}$. For each I⊆[A,B] with Ψ(I)≤p _n we get then $\exists C>\sqrt{2g}$ ∃m<∞, $m \in \mathbb{N}: \sum_{n=m}^{\infty} P(|\varPsi _{n}^{*}(I)-\varPsi(I)| > \varepsilon_{n}) < \sum_{n=m}^{\infty} A_{n} < \infty$ and $\forall m < \infty, m \in \mathbb{N}: \sum_{n=1}^{m} P(|\varPsi _{n}^{*}(I)-\varPsi(I)| > \varepsilon_{n}) \le m < \infty$.

Because of the summability of $P(|\varPsi _{n}^{*}(I)-\varPsi(I)| > \varepsilon_{n})$,

$$P\Bigl( \limsup_{n \rightarrow \infty} \big|\varPsi _n^*(I)-\varPsi(I)\big| > \varepsilon_n \Bigr)=0 $$

results from the Borel–Cantelli lemma for $C>\sqrt{2g}$, i.e. $|\varPsi _{n}^{*}(I)-\varPsi(I)|$ does not exceed ε _n for most of the n. For large n and for all I⊆[A,B] with Ψ(I)≤p _n , we derive almost surely that $|\varPsi _{n}^{*}(I)-\varPsi(I)| \le C\sqrt{\log{(n)}p_{n}/n}$.

The same inequality holds for the supremum of $|\varPsi _{n}^{*}(I)-\varPsi(I)|$ on [A,B]: $\sup_{I \subseteq [A,B], \varPsi(I)\le p_{n}}|\varPsi _{n}^{*}(I)-\varPsi(I)| \le C\sqrt{\log{(n)}p_{n}/n}$ for $C>\sqrt{2g}$ and large n almost surely.

Using the results above we prove the following inequality in a third step:

$$\sup_{I \subseteq [A,B], \varPsi(I)\le p_n}\big|\varPsi _n^*(I)-\varPsi (I)\big| \le C\cdot p_n \sqrt {\log {(n)} / n} $$

almost surely for some C>D _G ⋅M and large n.

From assumption (B4) and the limes superior formulation of Hewitt and Savage (1955) we obtain the right bound

$$G_n(x)- G(x) \le \big|G_n(x)- G(x)\big| \le \sup_{x \in [A,B]}\big|G_n(x)- G(x)\big| \le C_1'\cdot\sqrt {\log {(n)} / n} $$

almost surely for $C_{1}'>D_{G}$, large n and all x∈[A,B]. These bounds can be rewritten for G _n (x) as follows:

$$G_n(x) \ge G(x) - C_1'\sqrt {\log {(n)} / n} \ge \inf_{t \in [A,B]}G(t) - C_1'\cdot\sqrt {\log {(n)} / n}. $$

From assumption (B4) we have inf _t∈[A,B] G(t)>0. Because of $\sqrt {\log {(n)} / n} \rightarrow 0$, the following inequalities hold for x∈[A,B] and large n:

and

The following bounds for $\varPsi _{n}^{*}(I)-\varPsi(I)$ and $\varPsi _{n}^{*}(I)$ result from (14) almost surely for I⊆[A,B] with Ψ(I)≤p _n , large n and $C_{2}'>\sqrt{2\cdot(2\varDelta _{\max} M + \varPsi(B))}$:

and consequently

$$\varPsi _n^*(I) \le \varPsi(I) + C_2'\sqrt{\log{(n)}p_n/n} \le p_n + C_2'\sqrt{\log{(n)}p_n/n}. $$

We then obtain the following equation from assumption (B2) almost surely for each I⊆[A,B] with Ψ(I)≤p _n and large n:

By $p_{n} + C_{2}'\sqrt{\log{(n)}p_{n}/n} = p_{n}[1 + C_{2}'\sqrt{\log{(n)}/(p_{n}n)}]$ it is evident that $C_{2}'\sqrt{\log{(n)}/(p_{n}n)}$ can be neglected for large n because of the assumptions for p _n . For large n, we can also neglect the term $\sqrt {\log {(n)} / n}$ in the numerator. For all I⊆[A,B] with Ψ(I)≤p _n and for large n, we derive the inequality

$$\big|\varPsi _n^*(I)-\varPsi (I)\big| \le \frac{C_1'}{\inf_{t \in [A,B]}G(t)}p_n\sqrt {\log {(n)} / n}=C_1'\cdot M \cdot p_n\sqrt {\log {(n)} / n} $$

almost surely.

The requested bound

$$\sup_{I \subseteq [A,B], \varPsi(I)\le p_n}\big|\varPsi _n^*(I)-\varPsi (I)\big| \le C\cdot p_n\sqrt {\log {(n)} / n} $$

results for some C>D _G ⋅M and large n almost surely.

In a final step we examine the expression $\sup_{I \subseteq [A,B], \varPsi(I)\le p_{n}}|\varPsi _{n}(I)-\varPsi (I)|$. This overall difference can be represented by the sum of the deviations of the empirical and theoretical measures Ψ _n (I) and Ψ(I) from the preliminary measure $\varPsi _{n}^{*}( I)$ as follows:

Because $p_{n}\sqrt{\log (n)/n}/\sqrt{\log (n)p_{n}/n} = \sqrt{p_{n}}$ approaches zero, i.e. $p_{n}\sqrt{\log (n)/n} \le \sqrt{\log (n)p_{n}/n}$ holds for large n.

The previously mentioned upper bounds of $|\varPsi _{n}(I)-\varPsi _{n}^{*}(I)|$ and $|\varPsi _{n}^{*}(I)-\varPsi (I)|$ imply the existence of a constant $C > \sqrt{2\cdot(2\varDelta _{\max} M + \varPsi(B))} + D_{G} \cdot M$, such that almost surely for large n

Due to the symmetry of Ψ _n (I) the limes superior formulation of the convergence follows from Hewitt and Savage (1955).

Appendix B: Proof of Corollary 3

The boundedness of the $\varDelta ^{x}_{i}$ for each x∈[A,B] and conditions (B1) and (B2) follow from the definition of $\varDelta ^{x}_{i}$. This is so because the variables $\varDelta ^{x}_{i}$ do not depend on the x.

The consistency of the estimator G _n (⋅) (B4) can be easily shown and is a slight modification of the law of the iterated logarithm (see Shorack and Wellner 1986, p. 504).

The assumption (A2) for F(⋅) implies that the cumulative hazard rate Λ(⋅) is strictly increasing and the hazard rate λ(⋅) is obviously strictly positive on [A,B].

Now only the condition (B3) needs to be verified. We note that the vectors S _i =(X _i ,L _i ,δ _i ) _i=1,…,n are observable under L _i ≤X _i . Hence, we derive the following conditional expectation:

(15)

where F ^X,δ(x,y)=P(X≤x,δ≤y∣L≤X) is the conditional distribution function of (X,δ).

The integral $\int_{x_{1} \in I} dF^{X, \delta}(x_{1},1)$ for the intervals I:=[a,b]⊆[A,B] can now be calculated. First we express the probability P(X _i ∈I,δ _i =1∣L _i ≤X _i ) in the terms of the non-observable vector (T _i ,L _i ,C _i ) as follows:

(16)

where α=P(L _i ≤X _i ). Hence, we express the probabilities P(X _i ∈I,δ _i =1∣L _i ≤X _i ) and P(T _i ∈I,L _i ≤T _i ≤C _i ) as the following expectations of the Bernoulli-variables:

(17)

and

(18)

One can see that dF ^X,δ(x,1)=α ⁻¹ F ^L(j)(1−F ^C(j)) dF(x) follows from the expressions (16), (17), and (18). Consequently the expectation (15) can be written as follows:

Obviously, conditions (B1)–(B4) hold and the local convergence

follows for a constant $D \le 2(\sqrt{2\cdot(2M + \varLambda(B))} + 2 M)$.