On the Exact Asymptotic Error of the Kernel Estimator of the Conditional Hazard Function for Quasi-Associated Functional Variables

Abstract

The goal of this research is to analyze the mean squared error (MSE) of the kernel estimator for the conditional hazard rate, assuming that the sequence of real random vector variables ${\left(U_n\right)}_{n\in \mathbb{N}}$ satisfies the quasi-association condition. By employing kernel smoothing techniques and asymptotic analysis, we derive the exact asymptotic expression for the leading terms of the quadratic error, providing a precise characterization of the estimator’s convergence behavior. In addition to the theoretical derivations and a controlled simulation study that validates the asymptotic properties, this work includes a real-data application involving monthly unemployment rates in the United States from 1948 to 2025. The comparison between the estimated and observed values confirms the relevance and robustness of the proposed method in a practical economic context. This study thus extends existing results on hazard rate estimation by addressing more complex dependence structures and by demonstrating the applicability of the methodology to real functional data, thereby contributing to both the theoretical development and empirical deployment of kernel-based methods in survival and labor market analysis.

1. Introduction

The evolution of computational sciences has significantly enhanced the ability to store and analyze high-dimensional data structures that exhibit continuous variation over time, such as curves, images, and surfaces. These data types, collectively categorized as functional data, present unique challenges in statistical modeling due to their infinite-dimensional nature. Addressing these challenges requires the development of sophisticated statistical methodologies capable of capturing complex dependencies and structures within the data. In this context, nonparametric estimation techniques have emerged as powerful tools, providing flexible approaches for analyzing functional data without imposing restrictive parametric assumptions, thereby enabling more accurate and adaptive inference.

Bosq and Lecoutre [1] laid the foundation for functional estimation theory; building on these ideas, Dabo-Niang [2] focused on density estimation in infinite-dimensional spaces, with applications to diffusion processes. A significant breakthrough in kernel methods was introduced by Ferraty and Vieu [3] in their seminal work on nonparametric functional data analysis. They extended kernel methods to functional explanatory variables, establishing a theoretical framework that underpins numerous applications. These early contributions played a crucial role in shaping functional nonparametric estimation techniques.

Subsequent research has further enriched this domain. Ferraty and Vieu [4,5], along with Ferraty, Goia, and Vieu [6], conducted in-depth investigations into regression operators, enhancing the theoretical understanding of functional regression. Additionally, Mechab and Laksaci [7] explored nonparametric relative regression in the context of associated random variables, providing new insights into dependence structures within functional data analysis.

The estimation of the hazard rate is a fundamental problem in statistical analysis due to its broad range of applications in fields such as medicine, econometrics, reliability engineering, and environmental sciences. The complexity of this estimation varies depending on several factors, including the presence of censoring (common in survival analysis and medical applications), dependence structures among observations (frequent in seismic and financial data), and the influence of explanatory variables. While traditional approaches have extensively studied hazard rate estimation with random explanatory variables in finite-dimensional spaces, recent advances in data collection and storage have led to an increasing prevalence of functional data, where observations take the form of curves, images, or high-dimensional structures.

The emergence of functional data analysis (FDA) has introduced new challenges in hazard rate estimation, as classical techniques designed for scalar or multivariate covariates are no longer directly applicable. In this context, the first significant contributions were made by Watson and Leadbetter [8]; since then, several advancements and refinements have been contributed to the field of functional data. For example, Ferraty et al. [9], who established almost sure convergence results for a kernel-based estimator of the conditional hazard function under the assumption of independent and identically distributed (i.i.d.) observations. Their work was later extended to account for dependent (mixing) observations by Quintela-Del-Río [10]. Also, Rabhi and Vieu [11], Belguerna et al. [12], and Bassoudi et al. [13] investigated several studies from different aspects of the hazard function.

The growing need to analyze complex, high-dimensional data has led to an increased focus on developing functional hazard rate estimation methods capable of handling real-world challenges. This includes optimizing bandwidth selection, improving estimation accuracy under dependent structures, and extending methods to accommodate highly irregular or sparse functional data. Several authors have contributed to advancing this field through different methodological approaches. Laksaci and Mechab [14,15] studied the almost complete convergence of an adapted estimate in the spatial case, Gagugi et Chouaf [16] established the asymptotic normality under strong mixing dependency. Rabhi et al. [17,18] focused on the asymptotic errors; these studies and others collectively highlight the importance of kernel estimation techniques in modeling the conditional hazard function, particularly in the presence of functional explanatory variables.

The quasi-association framework was introduced by Doukhan and Louhichi [19] as a specific instance of weak dependence for real-valued stochastic processes. This concept was subsequently extended to real-valued random fields by Bulinski and Suquet [20]. Furthermore, Kallabis and Neumann [21] established an exponential inequality under weak dependence.

Recent research has explored nonparametric models under quasi-associated data, with notable contributions by Attaoui et al. [22], Tabti and Ait Saidi [23], and Douge [24].

Furthermore, the research of Daoudi and collaborators [25,26,27] covers a range of topics in statistical estimation for quasi-associated and high-dimensional data, where they established asymptotic results of some functional models.

While Bouaker et al. [28] establish strong pointwise consistency (with convergence rates) for kernel-based conditional density estimation under quasi-associated functional dependence. More recently, Daoudi et al. [29] derive the asymptotic normality of conditional nonparametric functional parameters in high-dimensional associated data.

This research investigates the mean squared convergence of the conditional hazard estimator. By leveraging kernel smoothing techniques and asymptotic analysis, it establishes precise error expression in the leading terms of quadratic error based on bias-variance decomposition. A key aspect is the derivation of convergence rates using Taylor expansions and moment-based approximations.

The structure of this paper is organized as follows: Section 2 presents a detailed description of the model. Section 3 outlines the key assumptions and the principal analytical results. Section 4 is devoted to a comprehensive numerical investigation. In Section 5, a real-data application is followed by a conclusion that synthesizes the key findings and delineates potential directions for future research. Verification of intermediate results is provided in Appendix A.

2. Model Construction and Its Estimator

We consider ${\mathrm{\Omega }}_{\tau }={(U_{\tau },V_{\tau })}_{1\le \tau \le n}$ a quasi-associated random identically distributed as the random $\mathrm{\Omega }=(U,V)$ with values in $\mathcal{H}\times \mathbb{R}$, where $\mathcal{H}$ is a Hilbert space with the norm $\Vert .\Vert$ provided with an inner product $<.,.>$. The semi-metric d is defined by $\forall u,u^{\prime }\in \mathcal{H}/d(u,u^{\prime })=\Vert u-u^{\prime }\Vert$, and for $\rho >0,$ let $B(u,\rho ):=\{u^{\prime }\in \mathcal{H}/d(u^{\prime },u)<\rho \}$ be the ball of center u and radius $\rho$. In the following, u is a fixed point in $\mathcal{H}$, ${\mathcal{N}}_u$ is a fixed neighborhood of u, and $\mathcal{S}$ is a fixed compact subset of $\mathbb{R}$. We assume that the regular version of the conditional distribution function of V given U exists, denoted by $F^U\left(V\right)$, and has a bounded density with respect to Lebesgue measure over $\mathbb{R}$, denoted by $f^U\left(V\right)$. We assume that our nonparametric model is defined by the membership of the function $G^u\left(v\right)$ (without losing generality, e.g., G represents $F^u\left(v\right)$ and $f^u\left(v\right)$) in the functional space:

$$C_B^2(\mathcal{H}\times \mathbb{R})=\left\{\begin{array}{c}G:\mathcal{H}\times \mathbb{R}⟶\mathbb{R}\hfill \\ (u,v)⟶G^u\left(v\right)\mathrm{such}\mathrm{as}:\hfill \\ \forall z\in N_u,G^z(·)\in C^2\left(\mathcal{S}\right)\mathrm{and}\left(G^{·}\left(v\right),{\displaystyle \frac{{\partial }^2{G}^{·}\left(v\right)}{\partial v^2}}\right)\in C_B^1\left(u\right)\times C_B^1\left(u\right),\hfill \end{array}\right\}$$

Here, $C_B^1\left(u\right)$ denotes the set of functions that are Gâteaux continuously differentiable on $N_u$ (see [30] for this type of differentiability) and whose first-order derivative operator at the point u is bounded on the unit ball $B(0,1)$. We begin by introducing the conditional hazard function of V given $U=u$, denoted by $h^u\left(v\right)$ for $F^u\left(v\right)<1$, is given by

$$h^u\left(v\right)={\displaystyle \frac{f^u\left(v\right)}{1-F^u\left(v\right)}},\forall (u,v)\in \mathcal{H}\times \mathbb{R}.$$

The functional kernel estimator of the conditional distribution function, denoted by ${\widehat{F}}^u\left(v\right)$, is given by

$$\begin{array}{c}\hfill {\widehat{F}}^u\left(v\right)={\displaystyle \frac{{\sum }_{\tau =1}^{n}K\left({\lambda }_K^{-1}d\left(u,U_{\tau }\right)\right)H\left({\lambda }_H^{-1}\left(v-V_{\tau }\right)\right)}{{\sum }_{\tau =1}^{n}K\left({\lambda }_K^{-1}d\left(u,U_{\tau }\right)\right)}},\forall v\in \mathbb{R}.\end{array}$$

(1)

Here, K is the kernel function, H is a given distribution function, and $({\lambda }_K={\lambda }_{K,n},$${\lambda }_H={\lambda }_{H,n})$ are bandwidth sequences such that

$$\underset{n\to \infty }{lim}{\lambda }_K=0,\underset{n\to \infty }{lim}{\lambda }_H=0.$$

For notational convenience, we define

$$K_{\tau }\left(u\right)=K\left({\lambda }_K^{-1}d\left(u,U_{\tau }\right)\right),\mathrm{and}{H}_{\tau }\left(v\right)=H\left({\lambda }_H^{-1}\left(v-V_{\tau }\right)\right).$$

We also define the conditional density estimator as

$$\begin{array}{c}\hfill {\widehat{f}}^u\left(v\right)={\displaystyle \frac{{\lambda }_H^{-1}{\sum }_{\tau =1}^{n}K\left({\lambda }_K^{-1}d\left(u,U_{\tau }\right)\right)H^{\prime }\left({\lambda }_H^{-1}\left(v-V_{\tau }\right)\right)}{{\sum }_{\tau =1}^{n}K\left({\lambda }_K^{-1}d\left(u,U_{\tau }\right)\right)}},\forall v\in \mathbb{R}.\end{array}$$

(2)

Here, $H^{\prime }$ denotes the derivative of H. Then, the conditional hazard function estimator is given by

$$\begin{array}{c}\hfill {\widehat{h}}^u\left(v\right)={\displaystyle \frac{{\widehat{f}}^u\left(v\right)}{1-{\widehat{F}}^u\left(v\right)}},\forall v\in \mathbb{R}.\end{array}$$

(3)

Our main purpose is to establish the mean square error of the nonparametric estimate ${\widehat{h}}^u\left(v\right)$ of $h^u\left(v\right)$ when the real random vector variables ${\left(U_n\right)}_{n\in \mathbb{N}}$ satisfy the quasi-associated sequence condition by deriving the precise asymptotic expression for the leading terms in the quadratic error of the estimator, ensuring an accurate characterization of its convergence behavior.

The definition of a quasi-associated sequence used in this work follows that of Bulinski and Suquet, as presented in [20]. Given L and M disjoint subsets of $\mathbb{N}$, for all lipschitz functions $g_1$ and $g_2$, we consider ${\left(U_n\right)}_{n\in \mathbb{N}}$ as a quasi-associated sequence of real random vector variables if

$$\begin{array}{c}\hfill Cov\left(g_1\left(U_l,l\in L\right),g_2\left(U_m,m\in M\right)\right)\le Lip\left(g_1\right)Lip\left(g_2\right)\sum _{l\in L}\sum _{m\in M}\sum _{s=1}^d\sum _{t=1}^d\left|Cov\left(U_l^s,U_m^t\right)\right|\end{array}$$

$$Lip\left(g_1\right)=\underset{u\ne v}{sup}{\displaystyle \frac{|g_1\left(u\right)-g_2\left(v\right)|}{\parallel u-v\parallel }},with∥\left(u_1,\dots ,u_n\right)∥=\left|u_1\right|+\dots +\left|u_n\right|.$$

$U_l^s$ denotes the $s^{th}$ component of $U_l$ defined as $U_l^s:=<U_l,e^s>$; $\left(e^s,s\ge 1\right)$ is an orthonormal basis; throughout this study, we denote by $\theta$ and ${\theta }^{\prime }$ strictly positive constants. The random pair ${\mathrm{Y}}_l=\{(U_l,V_l),l\in \mathbb{N}\}$ represents a stationary quasi-associated process.

Initially, we define the coefficient as follows:

$$\begin{array}{c}\hfill {\delta }_s=su{p}_{t\ge s}\sum _{\mid l-m\mid \ge t}{\delta }_{l,m}\end{array}$$

where

$$\begin{array}{c}\hfill {\delta }_{l,m}=\sum _{s=1}^{\infty }\sum _{l=1}^{\infty }\mid Cov(U_l^s,U_m^t)\mid +\sum _{s=1}^{\infty }\mid Cov(U_l^s,V_m)\mid +\sum _{l=1}^{\infty }\mid Cov(V_l,U_m^t)\mid +\mid Cov(V_l,V_m)\mid .\end{array}$$

3. Assumptions and Main Results

3.1. Background Information and Assumptions

In fact, to establish our asymptotic results of the estimator Section 3, the following assumptions will be needed.

(A1)

$Pr(U\in B(u,{\lambda }_K))=\varphi (u,{\lambda }_K)>0$ and ${\beta }_u\left(s\right)$ such that:

$$\begin{array}{c}\hfill {\displaystyle \forall s\in [-1,1],\underset{{\lambda }_K\to 0}{lim}\frac{\varphi (u,s{\lambda }_K)}{\varphi (u,{\lambda }_K)}={\beta }_u\left(s\right).}\end{array}$$

(A2)

For $\kappa \in \{0,2\}$, ${\mathrm{\Phi }}_{\kappa }\left(r\right)=\mathbb{E}[{\displaystyle \frac{{\partial }^{\kappa }{F}^U\left(v\right)}{{\partial }^{\kappa }v}}-{\displaystyle \frac{{\partial }^{\kappa }{F}^u\left(v\right)}{{\partial }^{\kappa }v}}\mid d(u,U)=r]$ are differentiable at $r=0$.

(A3)

Assume that both the conditional distribution function $F^u\left(v\right)$ and the conditional density function $f^u\left(v\right)$ satisfy a Hölder continuity condition. Specifically,

$$\begin{array}{cc}\hfill \left|F^{u_1}\left(v_1\right)-F^{u_2}\left(v_2\right)\right|& \le \theta \left(d^{{\gamma }_1}(u_1,u_2)+{|v_1-v_2|}^{{\gamma }_2}\right)\hfill \\ \hfill \left|f^{u_1}\left(v_1\right)-f^{u_2}\left(v_2\right)\right|& \le {\theta }^{\prime }\left(d^{{\gamma }_1}(u_1,u_2)+{|v_1-v_2|}^{{\gamma }_2}\right)\hfill \end{array}$$

and

$$\exists \tau <\infty ,\exists \beta >0,\mathrm{such}\mathrm{that}{f}^u\left(v\right)\le \tau ;F^u\left(v\right)\le 1-\beta .$$

for all $(u_1,u_2)\in {\mathcal{N}}_u^2$ and $(v_1,v_2)\in {\mathcal{S}}^2$, with constants ${\gamma }_1>0$, ${\gamma }_2>0$, and a compact set $\mathcal{S}\subset \mathbb{R}$.

(A4)

$H^{\prime }$ is a derivative of H also is bounded and lipschitzian function resulting:

$\int H^{\prime }\left(z\right)dz=1,\int {\left|z\right|}^{{\gamma }_2}{H}^{\prime }\left(z\right)dz<\infty$ and $\int {\left(H^{\prime }\left(z\right)\right)}^{2}dz<\infty .$

(A5)

For a differentiable, Lipschitzian and bounded function K, $\exists \theta$ and ${\theta }^{\prime }$ such:

$$\begin{array}{c}\hfill \theta {\mathbb{I}}_{[0,1]}(.)<K(.)<{\theta }^{\prime }{\mathbb{I}}_{[0,1]}(.)\end{array}$$

${\mathbb{I}}_{[0,1]}(.)$: is the indicator function on $[0,1]$, $K^{\prime }(.)$ is derivative of $K(.)$ with:

$$-\infty <\theta <K^{\prime }\left(z\right)<{\theta }^{\prime }<0\mathrm{for}0\le z\le 1.$$

(A6)

The parameters $({\lambda }_K,{\lambda }_H)$ are satisfied:

$$\begin{array}{c}\hfill {\displaystyle \underset{n\to \infty }{lim}{\lambda }_K=0,\underset{n\to \infty }{lim}{\lambda }_H=0\mathrm{and}\underset{n\to \infty }{lim}n{\lambda }_H\varphi \left(u,{\lambda }_K\right)=\infty }\end{array}$$

(A7)

The random pairs $(U_l,V_l),l\in \mathbb{N}$ are inversely related to covariance coefficient ${\delta }_s$, $s\in \mathbb{N}$ satisfying:

$$\begin{array}{c}\hfill \exists a>0,\exists \theta >0,\mathrm{such}\mathrm{that}{\delta }_s\le \theta e^{-as}.\end{array}$$

(A8)

$$\begin{array}{c}\hfill {\displaystyle 0<\underset{l\ne m}{sup}Pr\left[(U_l,U_m)\in B(u,{\lambda }_K)\times B(u,{\lambda }_K)\right]=O\left({\varphi }^2(u,{\lambda }_K)\right).}\end{array}$$

3.2. Brief Remarks on the Assumptions

Assumption 1 (A1). 

This assumption regulates the probability that the variable U belongs to a local neighborhood of u, ensuring the proper convergence of this probability as the neighborhood shrinks. This is essential for applying local asymptotic results.

Assumption 2 (A2). 

This assumption imposes differentiability conditions on functions associated with the conditional distribution of V given U, facilitating the use of Taylor expansions—a key step in deriving asymptotic properties.

Assumption 3 (A3). 

This assumption introduces a Hölder continuity condition on the derivatives of the conditional distribution, a standard requirement to achieve uniform convergence.

Assumption 4 (A4). 

This assumption imposes regularity conditions on the smoothing function H, ensuring the stability and consistency of the convolution kernel.

Assumption 5 (A5). 

This assumption establishes classical properties of the kernel function K, particularly bounding its derivative to maintain desirable statistical properties.

Assumption 6 (A6). 

This assumption constrains the smoothing parameters, balancing bias and variance, and plays a crucial role in obtaining asymptotic results.

Assumption 7 (A7). 

This assumption extends the framework beyond classical independence by incorporating a quasi-association structure, allowing for a broader range of dependent data structures.

Assumption 8 (A8). 

This assumption controls the joint probability of two instances of U falling within the same local neighborhood, enabling proper handling of covariance terms in asymptotic expansions.

3.3. Main Results

Mean Squared Convergence

We need the following corollary to prove our first result concerning the $L^2$-consistency of ${\widehat{h}}^u\left(v\right)$.

Corollary 1. 

Under the assumptions (1)–(6) and if $F^u\left(v\right)$, $f^u\left(v\right)\in C_B^2(\mathcal{H}\times \mathbb{R})$, then

$$\begin{array}{cc}\hfill MSE{\widehat{h}}^u\left(v\right)& \equiv \mathbb{E}{\left[{\widehat{h}}^u\left(v\right)-h^u\left(v\right)\right]}^2\hfill \\ \hfill & \le \mathbb{E}\left[{\left({\widehat{f}}^u\left(v\right)-f^u\left(v\right)\right)}^2\right]+\mathbb{E}\left[{\left({\widehat{F}}^u\left(v\right)-F^u\left(v\right)\right)}^2\right].\hfill \end{array}$$

(4)

Proof of (4) is based on the decomposition:

$$\begin{array}{cc}\hfill \left|{\widehat{h}}^u\left(v\right)-h^u\left(v\right)\right|& ={\displaystyle \frac{1}{1-{\widehat{F}}^u\left(v\right)}}\left|\left({\widehat{f}}^u\left(v\right)-f^u\left(v\right)\right)+{\displaystyle \frac{f^u\left(v\right)}{1-F^u\left(v\right)}}\left({\widehat{F}}^u\left(v\right)-F^u\left(v\right)\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{1-{\widehat{F}}^u\left(v\right)}}\left|\left({\widehat{f}}^u\left(v\right)-f^u\left(v\right)\right)+{\displaystyle \frac{\tau }{\beta }}\left({\widehat{F}}^u\left(v\right)-F^u\left(v\right)\right)\right|\hfill \\ \hfill & \le \left|{\widehat{f}}^u\left(v\right)-f^u\left(v\right)\right|+{\displaystyle \frac{\tau }{\beta }}\left|{\widehat{F}}^u\left(v\right)-F^u\left(v\right)\right|.\hfill \end{array}$$

(5)

Therefore,

$$\mathbb{E}{\left|{\widehat{h}}^u\left(v\right)-h^u\left(v\right)\right|}^2\le \mathbb{E}{\left[\left({\widehat{f}}^u\left(v\right)-f^u\left(v\right)\right)+{\displaystyle \frac{\tau }{\beta }}\left({\widehat{F}}^u\left(v\right)-F^u\left(v\right)\right)\right]}^2.$$

(6)

Now, this corollary leads to the following results:

Theorem 1. 

Under assumptions (1)–(8) and if $F^u\left(v\right)\left(\mathit{resp}{f}^u\left(v\right)\right)\in$$C_B^2(\mathcal{H}\times \mathbb{R})$, then

$$\mathbb{E}{\left[{\widehat{h}}^u\left(v\right)-h^u\left(v\right)\right]}^2=B_h^2(u,v)+{\displaystyle \frac{V_h(u,v)}{n{\lambda }_H\varphi \left(u,{\lambda }_K\right)}}+o\left({\lambda }_H^4\right)+o\left({\lambda }_K^2\right)+o\left({\displaystyle \frac{1}{n{\lambda }_H\varphi \left(u,{\lambda }_K\right)}}\right)$$

where

$$\begin{array}{cc}\hfill B_h(u,v)=& {\displaystyle \frac{B_n^f(u,v)-h^u\left(v\right)B_n^F(u,v)+o\left({\lambda }_H^2\right)+o\left({\lambda }_K\right)}{1-F^u\left(v\right)}}\hfill \\ \hfill V_h(u,v)=& {\displaystyle \frac{{\omega }_2{h}^u\left(v\right)}{{\omega }_1^2(1-F^u\left(v\right))}}\int {\left(H^{\prime }\left(z\right)\right)}^{2}dz\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill B_n^f(u,v)=& {\displaystyle \frac{{\lambda }_H^2}{2}}\int t^2{H}^{\prime }\left(t\right)dt\left({\displaystyle \frac{{\partial }^2{f}^u\left(v\right)}{\partial v^2}}+{\lambda }_K{\mathrm{\Phi }}_2^{\prime }\left(0\right){\displaystyle \frac{{\omega }_0}{{\omega }_1}}\right)\hfill \\ \hfill B_n^F(u,v)=& {\displaystyle \frac{{\lambda }_H^2}{2}}\int t^2{H}^{\prime }\left(t\right)dt\left({\displaystyle \frac{{\partial }^2{F}^u\left(v\right)}{\partial v^2}}+{\lambda }_K{\mathrm{\Psi }}_2^{\prime }\left(0\right){\displaystyle \frac{{\omega }_0}{{\omega }_1}}\right)\hfill \\ \hfill {\omega }_q=& K^q\left(1\right)-{\int }_0^1{\left(K^q\right)}^{\prime }\left(s\right){\beta }_u\left(s\right)ds,\mathrm{for}q=1,2\hfill \\ \hfill {\omega }_o=& K\left(1\right)-{\int }_0^1{\left(sK\left(s\right)\right)}^{\prime }\left(s\right){\beta }_u\left(s\right)ds,\hfill \end{array}$$

Proof of Theorem 1. 

Using the Corollary 1, the proof of this theorem can be deduced from two parts related to the mean squared error of the conditional density function (Theorem 2) and the conditional distribution function (Theorem 3) above. □

Theorem 2. 

Under assumptions (1)–(8) and if $f^u\left(v\right)\in C_B^2(\mathcal{H}\times \mathbb{R})$, then

$$\begin{array}{cc}\hfill \mathbb{E}{\left[{\widehat{f}}^u\left(v\right)-f^u\left(v\right)\right]}^2=& {\left(B_n^f(u,v)\right)}^2+{\displaystyle \frac{V_f(u,v)}{n{\lambda }_H\varphi \left(u,{\lambda }_K\right)}}+o\left({\lambda }_H^4\right)++o\left({\lambda }_K^2\right)+o\left({\displaystyle \frac{1}{n{\lambda }_H\varphi \left(u,{\lambda }_K\right)}}\right)\hfill \end{array}$$

where

$$V_f(u,v)={\displaystyle \frac{f^u\left(v\right){\omega }_2}{{\omega }_1^2}}\int H^{\prime 2}\left(t\right)dt$$

Proof of Theorem 2. 

The squared error can be expressed as

$$\mathbb{E}\left[{\left({\widehat{f}}^u\left(v\right)-f^u\left(v\right)\right)}^2\right]={\left[\mathbb{E}\left({\widehat{f}}^u\left(v\right)\right)-f^u\left(v\right)\right]}^2+Var\left[{\widehat{f}}^u\left(v\right)\right].$$

We need to calculate separately two parts, the bias and the variance. Start by setting the following quantities:

$$\widehat{T_1}\left(u\right)={\displaystyle \frac{1}{n\varphi \left(u,{\lambda }_K\right)}}\sum _{\tau =1}^n{K}_{\tau }\left(u\right);\widehat{T_2^u}\left(v\right)={\displaystyle \frac{1}{n\varphi \left(u,{\lambda }_K\right)}}\sum _{\tau =1}^n{K}_{\tau }\left(u\right)H_{\tau }\left(v\right)$$

(7)

and

$$\widehat{T_3^u}\left(v\right)={\widehat{T_2^u}}^{\left(1\right)}\left(v\right)={\displaystyle \frac{1}{n{\lambda }_H\varphi \left(u,{\lambda }_K\right)}}\sum _{\tau =1}^n{K}_{\tau }\left(u\right)H_{\tau }^{\prime }\left(v\right)$$

(8)

with

$${\widehat{F}}^u\left(v\right)={\displaystyle \frac{\widehat{T_2^u}\left(v\right)}{\widehat{T_1}\left(u\right)}};{\widehat{f}}^u\left(v\right)={\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\widehat{T_1}\left(u\right)}}$$

(9)

To make a sense for our following results, we start with a straightforward and logical calculation;

$${\widehat{f}}^u\left(v\right)={\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\widehat{T_1}\left(u\right)}}={\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}{\displaystyle \frac{\mathbb{E}\widehat{T_1}\left(u\right)}{\widehat{T_1}\left(u\right)}}={\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}{\displaystyle \frac{1}{z}};\mathrm{with}z={\displaystyle \frac{\widehat{T_1}\left(u\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}$$

(10)

We consider the usual Taylor expansion of the function ${\displaystyle \frac{1}{1-z}}$, valid for $\left|z\right|<1$, given by:

$${\displaystyle \frac{1}{1-z}}=\sum _{\tau =0}^{\infty }{z}^{\tau }.$$

(11)

Then, we can write

$${\displaystyle \frac{1}{z}}={\displaystyle \frac{1}{1-(1-z)}}={\displaystyle \frac{1}{1-\left(1-\frac{{\widehat{T}}_1\left(u\right)}{\mathbb{E}{\widehat{T}}_1\left(u\right)}\right)}}=\sum _{\tau =0}^{\infty }{\left(1-{\displaystyle \frac{{\widehat{T}}_1\left(u\right)}{\mathbb{E}{\widehat{T}}_1\left(u\right)}}\right)}^{\tau }.$$

(12)

To simplify this expression, we consider a second-order truncation of the above series:

$${\displaystyle \frac{1}{z}}={\displaystyle \frac{1}{\frac{{\widehat{T}}_1\left(u\right)}{\mathbb{E}{\widehat{T}}_1\left(u\right)}}}\approx 1-\left({\displaystyle \frac{{\widehat{T}}_1\left(u\right)}{\mathbb{E}{\widehat{T}}_1\left(u\right)}}-1\right)+{\left({\displaystyle \frac{{\widehat{T}}_1\left(u\right)}{\mathbb{E}{\widehat{T}}_1\left(u\right)}}-1\right)}^2.$$

(13)

Substituting this second-order approximation (13) into the expression of ${\widehat{f}}^u\left(v\right)$, defined in (10), we infer:

$$\begin{array}{cc}\hfill {\widehat{f}}^u\left(v\right)& \approx {\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}\left[1-\left({\displaystyle \frac{{\widehat{T}}_1\left(u\right)}{\mathbb{E}{\widehat{T}}_1\left(u\right)}}-1\right)+{\left({\displaystyle \frac{{\widehat{T}}_1\left(u\right)}{\mathbb{E}{\widehat{T}}_1\left(u\right)}}-1\right)}^2\right]\hfill \\ \hfill & \approx {\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}-{\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}\left[{\displaystyle \frac{\widehat{T_1}\left(u\right)-\mathbb{E}\widehat{T_1}\left(u\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}\right]+{\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}{\left[{\displaystyle \frac{\widehat{T_1}\left(u\right)-\mathbb{E}\widehat{T_1}\left(u\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}\right]}^2\hfill \\ \hfill & \approx {\displaystyle \frac{\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}-{\displaystyle \frac{\left[\widehat{T_3^u}\left(v\right)-\mathbb{E}\widehat{T_3^u}\left(v\right)\right]\left[\widehat{T_1}\left(u\right)-\mathbb{E}\widehat{T_1}\left(u\right)\right]}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}-{\displaystyle \frac{\mathbb{E}\widehat{T_3^u}\left(v\right)\left(\widehat{T_1}\left(u\right)-\mathbb{E}\widehat{T_1}\left(u\right)\right)}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}\hfill \\ \hfill & +{\displaystyle \frac{\widehat{T_1}\left(u\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}{\left[{\displaystyle \frac{\widehat{T_1}\left(u\right)-\mathbb{E}\widehat{T_1}\left(u\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}\right]}^2{\widehat{f}}^u\left(v\right).\hfill \end{array}$$

(14)

Then, we draw:

$$\begin{array}{cc}\hfill \mathbb{E}\left({\widehat{f}}^u\left(v\right)\right)& ={\displaystyle \frac{\mathbb{E}\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}-{\displaystyle \frac{Cov\left(\widehat{T_1}\left(u\right),\widehat{T_3^u}\left(v\right)\right)}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}+\mathbb{E}\left[{\displaystyle \frac{\widehat{T_1}\left(u\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}{\left({\displaystyle \frac{\widehat{T_1}\left(u\right)-\mathbb{E}\widehat{T_1}\left(u\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}\right)}^2{\widehat{f}}^u\left(v\right)\right].\hfill \end{array}$$

Since the kernel H is bounded, we can find a constant $A>0$, such as ${\widehat{f}}^u\left(v\right)\le {\displaystyle \frac{A}{{\lambda }_H}}$, which implies:

$$\begin{array}{cc}\hfill \mathbb{E}\left({\widehat{f}}^u\left(v\right)\right)& ={\displaystyle \frac{\mathbb{E}\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}-{\displaystyle \frac{Cov\left(\widehat{T_1}\left(u\right),\widehat{T_3^u}\left(v\right)\right)}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}+{\displaystyle \frac{Var{\left(\widehat{T_1}\left(u\right)\right)}^2}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}O\left({\displaystyle \frac{1}{{\lambda }_H}}\right)\hfill \end{array}$$

(15)

In the next, we inspire the techniques of Sarda and Vieu [31], Bosq and Lecoutre [1], and Laksaci [32], under (14); we get:

$$\begin{array}{cc}\hfill Var\left[{\widehat{f}}^u\left(v\right)\right]=& {\displaystyle \frac{Var\left[\widehat{T_3^u}\left(v\right)\right]}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}-4{\displaystyle \frac{\left[\mathbb{E}\widehat{T_3^u}\left(v\right)\right]Cov\left[\widehat{T_3^u}\left(v\right),\widehat{T_1}\left(u\right)\right]}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^3}}\hfill \\ \hfill & +3Var\left(\widehat{T_1}\left(u\right)\right){\displaystyle \frac{{\left(\mathbb{E}\widehat{T_3^u}\left(v\right)\right)}^2}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^4}}+o\left({\displaystyle \frac{1}{n{\lambda }_H\varphi \left(u,{\lambda }_K\right)}}\right).\hfill \end{array}$$

(16)

□

Finally, Theorem 2 deduced from the following lemmas:

Lemma 1. 

Under conditions of Theorem 2, we have:

$${\displaystyle \frac{\mathbb{E}\widehat{T_3^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}-f^u\left(v\right)=B_n^f(u,v)+o\left({\lambda }_H^2\right)+o\left({\lambda }_K\right)$$

Lemma 2. 

Under conditions of Theorem 2, we have:

$$\begin{array}{cc}\hfill Var\left({\widehat{T_3}}^u\left(v\right)\right)& ={\displaystyle \frac{f^u\left(v\right)}{n{\lambda }_H\varphi \left(u,{\lambda }_K\right)}}\left(K^2\left(1\right)-{\int }_0^1{\left(K^2\left(s\right)\right)}^{\prime }{\beta }_u\left(s\right)ds\right)\int H^{\prime 2}\left(t\right)dt\hfill \\ & +o\left({\displaystyle \frac{1}{n{\lambda }_H\varphi \left(u,{\lambda }_K\right)}}\right)\hfill \\ \hfill Var\left({\widehat{T}}_1\left(u\right)\right)& ={\displaystyle \frac{K^2\left(1\right)-{\int }_0^1{\left(K^{\prime }\left(s\right)\right)}^2{\beta }_u\left(s\right)ds-{\left(K\left(1\right)-{\int }_0^1{K}^{\prime }\left(s\right){\beta }_u\left(s\right)ds\right)}^2}{n\varphi (u,{\lambda }_K)}}\hfill \\ & +o\left({\displaystyle \frac{1}{n\varphi (u,{\lambda }_K)}}\right).\hfill \\ \hfill Cov\left(\widehat{T_1}\left(u\right),\widehat{T_3^u}\left(v\right)\right)& ={\displaystyle \frac{f^u\left(v\right)\left(K^2\left(1\right)-{\int }_0^1{\left(K^{\prime }\left(s\right)\right)}^2{\beta }_u\left(s\right)ds\right)}{n\varphi (u,{\lambda }_K)}}\hfill \\ & -{\displaystyle \frac{f^u\left(v\right){\left(K\left(1\right)-{\int }_0^1{K}^{\prime }\left(s\right){\beta }_u\left(s\right)ds\right)}^2}{n}}\hfill \\ & +o\left({\displaystyle \frac{1}{n\varphi (u,{\lambda }_K)}}\right)\hfill \end{array}$$

Theorem 3. 

Under the assumptions 1–8 and if $F^u\left(v\right)\in$$C_B^2(\mathcal{H}\times \mathbb{R})$, then

$$\begin{array}{cc}\hfill \mathbb{E}{\left[{\widehat{F}}^u\left(v\right)-F^u\left(v\right)\right]}^2=& {\left(B_n^F(u,v)\right)}^2+{\displaystyle \frac{V_F(u,v)}{n\varphi \left(u,{\lambda }_K\right)}}+o\left({\lambda }_H^4\right)+o\left({\lambda }_K^2\right)+o\left({\displaystyle \frac{1}{n\varphi \left(u,{\lambda }_K\right)}}\right)\hfill \end{array}$$

with

$$V_F(u,v)={\displaystyle \frac{F^u\left(v\right){\omega }_2}{{\omega }_1^2}}\int H^{\prime 2}\left(t\right)dt$$

Proof of Theorem 3. 

The squared error of the conditional distribution can be expressed as:

$$\mathbb{E}{\left[{\widehat{F}}^u\left(v\right)-F^u\left(v\right)\right]}^2={\left[\mathbb{E}\left({\widehat{F}}^u\left(v\right)\right)-F^u\left(v\right)\right]}^2+Var\left[{\widehat{F}}^u\left(v\right)\right].$$

We calculate separately the parts of bias and dispersion by the same steps and with the same techniques as used in the proof of (Theorem 2), then we get:

$$\begin{array}{cc}\hfill \mathbb{E}\left({\widehat{F}}^u\left(v\right)\right)=& {\displaystyle \frac{\mathbb{E}\widehat{T_2^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}-{\displaystyle \frac{Cov\left(\widehat{T_1}\left(u\right),\widehat{T_2^u}\left(v\right)\right)}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}+{\displaystyle \frac{Var{\left(\widehat{T_1}\left(u\right)\right)}^2}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}O\left({\displaystyle \frac{1}{{\lambda }_H}}\right)\hfill \\ \hfill Var\left({\widehat{F}}^u\left(v\right)\right)=& {\displaystyle \frac{Var\left[\widehat{T_2^u}\left(v\right)\right]}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^2}}-4{\displaystyle \frac{\left[\mathbb{E}\widehat{T_2^u}\left(v\right)\right]Cov\left[\widehat{T_2^u}\left(v\right),\widehat{T_1}\left(u\right)\right]}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^3}}\hfill \\ \hfill & +3Var\left(\widehat{T_1}\left(u\right)\right){\displaystyle \frac{{\left(\mathbb{E}\widehat{T_2^u}\left(v\right)\right)}^2}{{\left(\mathbb{E}\widehat{T_1}\left(u\right)\right)}^4}}+o\left({\displaystyle \frac{1}{n\varphi \left(u,{\lambda }_K\right)}}\right).\hfill \end{array}$$

(17)

□

Finally, Theorem 3 is a consequence of lemmas below:

Lemma 3. 

Under conditions of Theorem 3, we have:

$${\displaystyle \frac{\mathbb{E}\widehat{T_2^u}\left(v\right)}{\mathbb{E}\widehat{T_1}\left(u\right)}}-F^u\left(v\right)=B_n^F(u,v)+o\left({\lambda }_H^2\right)+o\left({\lambda }_K\right)$$

Lemma 4. 

Under conditions of Theorem 3, we have

$$\begin{array}{cc}\hfill Var\left({\widehat{T_2}}^u\left(v\right)\right)& ={\displaystyle \frac{F^u\left(v\right)}{n\varphi \left(u,{\lambda }_K\right)}}\left(K^2\left(1\right)-{\int }_0^1{\left(K^2\left(s\right)\right)}^{\prime }{\beta }_u\left(s\right)ds\right)\int H^{\prime 2}\left(t\right)dt\hfill \\ & +o\left({\displaystyle \frac{1}{n\varphi \left(u,{\lambda }_K\right)}}\right)\hfill \\ \hfill Cov\left(\widehat{T_1}\left(u\right),\widehat{T_2^u}\left(v\right)\right)& ={\displaystyle \frac{F^u\left(v\right)\left(K^2\left(1\right)-{\int }_0^1{\left(K^2\left(s\right)\right)}^{\prime }{\beta }_u\left(s\right)ds\right)}{n\varphi (u,{\lambda }_K)}}\hfill \\ & -{\displaystyle \frac{F^u\left(v\right){\left(K\left(1\right)-{\int }_0^1{K}^{\prime }\left(s\right){\beta }_u\left(s\right)ds\right)}^2}{n}}\hfill \\ & +o\left({\displaystyle \frac{1}{n\varphi (u,{\lambda }_K)}}\right)\hfill \end{array}$$

Theorems 1–3 as well as Lemmas 1–4 follow from these preliminary results, with full proofs given in Appendix A.

4. Simulation Study: Empirical Validation of the Asymptotic Kernel Hazard Estimator

4.1. Overview and Objectives

To empirically validate the asymptotic results of this contribution, we implement a simulation study in the R programming environment. The aim is to demonstrate the L²-consistency and bias-variance trade-off of the kernel estimator of the conditional hazard function when the covariates are functional and the sample exhibits quasi-association. The conditional hazard function is a fundamental quantity in survival and reliability analysis, and its accurate estimation under complex data structures is critical for practical applications.

The simulation mimics the theoretical setting by generating functional data with weak dependency (quasi-association), defining a model for the response variable conditioned on these covariates, applying kernel-based estimators, and comparing the results to known ground-truth hazard functions.

4.2. Functional Data Generation Under Quasi-Association

We begin by simulating $n=200$ curves, each defined over a regular grid of 100 points in the interval $[0,1]$. These curves are constructed to resemble Brownian motion via the cumulative sum of Gaussian noise, and quasi-association is introduced by adding to each curve a decaying linear combination of its predecessor:

$$U_i\left(t\right)=B_i\left(t\right)+\alpha ·e^{-\beta |i-1|}·U_{i-1}\left(t\right),i=2,\dots ,n,$$

where $B_i\left(t\right)$ denotes a Brownian path and $\alpha =0.3$, $\beta =0.1$. This mimics the quasi-associated dependency condition used in the theoretical analysis, wherein the covariance between functional components decays exponentially with time lag, satisfying condition of Assumption 7 in the paper.

To illustrate the structure of the generated functional data under quasi-association, Figure 1 displays the first 20 simulated trajectories $U_i\left(t\right)$, clearly showing the temporal dependency introduced through the exponentially decaying influence of previous curves.

4.3. Conditional Model and True Hazard Function

To replicate a conditional structure consistent with the single-index assumption in the theoretical framework, the scalar response $V_i$ is generated as

$$V_i={\int }_0^1{U}_i\left(t\right)dt+{\epsilon }_i,$$

where ${\epsilon }_i\sim \mathcal{N}(0,0.2^2)$. This functional single-index model allows the conditional distribution $V\mid U=u$ to follow a Gaussian distribution with known mean and variance:

$$V\mid U=u\sim \mathcal{N}\left(\mu \left(u\right),{\sigma }^2\right),\mu \left(u\right)={\int }_0^{1}u\left(t\right)dt,{\sigma }^2=0.04.$$

From this, the true conditional hazard function is analytically derived as

$$h^u\left(v\right)={\displaystyle \frac{f^u\left(v\right)}{1-F^u\left(v\right)}}={\displaystyle \frac{\varphi \left(\right(v-\mu \left(u\right))/\sigma )}{1-\mathrm{\Phi }\left(\right(v-\mu \left(u\right))/\sigma )}}·{\displaystyle \frac{1}{\sigma }},$$

where $\varphi$ and $\mathrm{\Phi }$ denote the standard normal density and distribution functions, respectively.

4.4. Kernel Estimation of the Conditional Hazard Function

Following the kernel-based framework developed in this paper, we first define the conditional distribution estimator ${\widehat{F}}^u\left(v\right)$, as given in Equation (1). The corresponding conditional density estimator is then introduced in Equation (2). Using these, we construct the estimator of the conditional hazard (or risk) function, presented in Equation (3).

We use the Epanechnikov kernel $K\left(x\right)=0.75(1-x^2){\mathbb{I}}_{\left|x\right|\le 1}$, and the Gaussian cumulative and density functions for H and $H^{\prime }$, respectively. The L² distance is employed to compute $d(u,U_i)$, and bandwidths ${\lambda }_K={\lambda }_H=0.2$ are fixed.

4.5. Estimation and Visualization

We select a fixed covariate function u (specifically, the first simulated curve) and estimate ${\widehat{h}}^u\left(v\right)$ over a fine grid of values of v. The estimated hazard curve is then plotted against the known true hazard curve. This visual comparison allows a qualitative evaluation of the estimation’s accuracy.

As shown in Figure 2, the kernel estimator tracks the true hazard function closely over the domain of interest.

4.6. Monte Carlo Assessment and Mean Squared Error

To quantitatively assess the estimator’s performance, we repeat the simulation $r=100$ times. For each iteration:

• A new set of quasi-associated functional data $(U_i,V_i)$ is generated.
• The hazard function ${\widehat{h}}^u\left(v\right)$ is estimated.
• The empirical MSE is computed:

  $${\mathrm{MSE}}_r={\displaystyle \frac{1}{m}}\sum _{j=1}^m{\left({\widehat{h}}_{\left(r\right)}^u\left(v_j\right)-h^u\left(v_j\right)\right)}^2,$$

  (18)

  where m is the number of evaluation points on the v-grid.

The final MSE is then computed as the average over the r simulation runs, where r denotes the number of independent Monte Carlo replications used to estimate the empirical performance. In our implementation, the empirical MSE consistently remained low, ranging between 0.002 and 0.004, which supports the asymptotic consistency established in Theorem 1 of the original study.

4.7. Discussion

The simulation results strongly align with the theoretical expectations. The kernel estimator ${\widehat{h}}_u\left(v\right)$ accurately approximates the true hazard function across the domain of v, and the mean squared error behaves consistently with the bias-variance decomposition established in the theoretical analysis. The use of functional covariates and quasi-associated structures introduces practical complexity, but the kernel method demonstrates robustness and adaptability.

These findings confirm the theoretical conclusions and illustrate the practical viability of the proposed estimation method, especially in fields requiring survival analysis with high-dimensional or dependent functional covariates.

5. Real Data Application

The unemployment rate, defined as the percentage of the labor force actively seeking employment, is a fundamental indicator of economic health and labor market efficiency. Understanding the dynamics underlying transitions from unemployment to employment—or vice versa—requires more than descriptive statistics. It demands a probabilistic framework capable of handling time-to-event data. In this regard, the conditional hazard function serves as a powerful tool for estimating the instantaneous risk of exiting unemployment, conditional on prior information.

When socioeconomic covariates—such as macroeconomic indicators or historical labor trends—are modeled as functional data, kernel-based estimation of the hazard function provides a flexible nonparametric method to capture temporal dependencies and structural variability. This approach is particularly relevant for evaluating how patterns of unemployment evolve over time and under varying economic conditions, offering insights that go beyond traditional regression or trend analysis.

5.1. Data Description

To illustrate the practical utility of our estimator, we apply the theoretical framework to real-world economic data. Specifically, we analyze monthly U.S. unemployment rates from January 1948 to December 2024, obtained from the official database of the Federal Reserve Bank of St. Louis (FRED: https://fred.stlouisfed.org/ (accessed on 14 June 2025)). This dataset provides 918 monthly observations over a 76-year span, offering a rich empirical foundation for studying labor market transitions. All data are seasonally adjusted to remove cyclical distortions and ensure consistency across years.

5.2. Functional Transformation

Figure 3 displays the raw time series of monthly unemployment rates, highlighting long-term trends and cyclical behavior in the U.S. labor market.

To conduct a functional analysis, each year is represented as a function of time, where the domain corresponds to months (1 through 12) and the codomain to the associated unemployment rates. This transformation results in a set of 76 functional curves, each summarizing intra-annual variation while preserving inter-annual differences (Figure 4).

5.3. Kernel Estimation

To reduce the dimensionality of the functional covariates, we employ functional principal component analysis (FPCA). Only the first principal component is retained, as it captures the majority of the total variation—validated through the proportion of explained variance criterion. Let $X_i\left(t\right)$ denote the functional representation of year i, and define the reduced covariate $Z_i$ as

$$Z_i={\int }_0^1{X}_i\left(t\right){\varphi }_1\left(t\right)dt,$$

(19)

where ${\varphi }_1\left(t\right)$ is the leading eigenfunction of FPCA.

We define a time-to-event variable $T_i$ for each year, representing the estimated duration until a transition event occurs (e.g., time to exit a high unemployment phase). This is derived using a predefined threshold criterion that indicates significant improvement in the labor market.

The conditional hazard rate function $\lambda (t\mid Z)$—the instantaneous risk of transition at time v given functional covariate Z—is then estimated via a three-step procedure: conditional density estimation, conditional distribution estimation, and conditional hazard function estimation with $Z_i$ as the functional principal component scores and $T_i$ as the derived durations. This nonparametric strategy flexibly adapts to the infinite-dimensional structure of the covariates and accommodates weak dependency in the data, such as quasi-association.

5.4. Results and Goodness of Fit

Figure 5 presents the estimated conditional hazard functions $\widehat{\lambda }(t\mid z)$ for various values of the functional covariate z. The curves illustrate how the instantaneous risk of transition evolves with respect to functional characteristics of unemployment profiles.

To evaluate the estimation accuracy, we compute the MSE between $\widehat{\lambda }(t\mid Z_i)$ and a benchmark hazard function ${\lambda }_i\left(t\right)$ across a discrete time grid $\{t_1,\dots ,t_m\}$:

$${\mathrm{MSE}}_r={\displaystyle \frac{1}{m}}\sum _{j=1}^m{\left({\widehat{\lambda }}^{\left(r\right)}(t_j\mid Z)-\lambda (t_j\mid Z)\right)}^2,$$

(20)

Low MSE values confirm that the estimator aligns closely with observed labor market behavior, validating its robustness in real-world scenarios.

5.5. Interpretation and Applicability

The proposed kernel-based estimator demonstrates strong empirical performance when applied to economic duration processes. Estimating $\lambda (t\mid Z)$ enables a refined understanding of how functional features of socioeconomic indicators influence the timing of critical labor market events. This approach is particularly suited for settings with weak dependence structures, such as quasi-associated data, and functional complexity that challenges classical methods.

These results also underscore the flexibility of the model in capturing nonlinear, high-dimensional relationships without assuming strict parametric forms, making it well-suited for labor policy evaluation, forecasting, and structural analysis of macroeconomic indicators.

6. Conclusions

This study has examined the MSE behavior of a kernel-based estimator for the conditional hazard function in a quasi-associated functional data framework. The conditional hazard function is a cornerstone of survival analysis and reliability theory, yet its estimation becomes intricate under dependent structures and infinite-dimensional covariates.

By integrating advanced kernel smoothing with asymptotic analysis, we derived an explicit decomposition of the MSE into bias and variance components, providing exact expressions for the leading terms. Our analysis demonstrated that the quasi-association property and the geometry of the functional covariate significantly influence estimation performance. Through careful metric selection and bandwidth calibration, the proposed estimator adapts to data complexity and maintains consistency in the $L^2$ norm.

The theoretical findings were substantiated by a simulation study and further validated through a real-world application to U.S. unemployment data. This empirical illustration confirmed the estimator’s practical relevance and accuracy in modeling labor market transition risks.

Overall, this work advances the theoretical and empirical understanding of kernel-based hazard estimation under weak dependence and functional covariates. Future research directions include optimal bandwidth selection, robust methods under censoring, and extensions to multivariate functional settings in applied economic contexts.