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The Consistency of the Distribution Function Conditional Estimate and Application on the Consistency and Asymptotic Normality of the Conditional Hazard Function Estimate for High Dimensional Quasi-Associated Data

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Hamza Daoudi^*,

Zouaoui Chikr Elmezouar

Fatimah Alshahrani

Hamza Daoudi^*,

Zouaoui Chikr Elmezouar

Fatimah Alshahrani

This version is not peer-reviewed

Abstract

The objective of this study is to examine a nonparametric estimate , using the kernel approach, of the conditional distribution function of a scalar response variable that is given a random variable whose values take place in a separable real Hilbert space. The observations will be dependent on one another in a quasi-associated fashion. The pointwise practically perfect consistencies with rates of this estimator are established by us under some broad conditions. The study’s major objective is to investigate the convergence rate of the proposed estimator and its application in the convergence rate and asymptotic normality of the hazard function. The asymptotic normality of the developed estimator is established precisely. Simulation studies were conducted to investigate the behavior of the asymptotic property in the context of finite sample data.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

Mathematics and statistical analysis techniques have been shown to be of considerable relevance in a variety of scientific sectors in recent years, including engineering, the economics, clinical medicine, and healthcare. In particular, the application of methods of mathematical and statistical analysis to engineering, the economics, healthcare, and clinical medicine, as well as demonstrating how these approaches may assist in such vital areas as comprehension, prediction, correlation, diagnosis, therapy, and data processing.

Functional data is the subject of this research. [3] excerpt: Statistics’ functional data analysis (FDA) analyzes infinite-dimensional variables including curves, sets, and pictures. The "Big Data" revolution has spurred its rapid expansion over the past 20 years.

This may be demonstrated by researching the topic’s past (for an example, see [1]). In [24], the topics of density and mode estimation for normed vector space data are discussed. In addition to that, he discussed the problem of excessive dimensionality in functional data and offered potential remedies. Nonparametric models were investigated for use in regression estimation in [22].

The treatment of functional data today typically involves contemporary theory. For example, the reference [18] presented the consistency rates of a variety of conditional distribution functionals, such as the regression function, conditional cumulative distribution, and conditional density, uniformly over a subset of the explanatory variable. These functionals include the conditional cumulative distribution and conditional density. The conditional cumulative distribution as well as the conditional density are both included in these functionals.

Additional examples include the conditional cumulative distribution and the conditional density. Uniformly in bandwidth (UIB) consistency was extended to the ergodic scenario, and the rates of consistency for different functional nonparametric models were investigated in [26]. The regression function was a part of these models, a conditional hazard function, conditional distribution, and conditional density.

In the field of statistical mathematics, in recent years, there has been a surge of curiosity on the statistical analysis of functional data. These numbers are used in econometrics, medicine, environmental science, and many other fields. In the statistical functional, [22] made the first attempt to estimate the conditional density function and its derivatives . In addition, they were the ones who were the first to do so in the scientific community.

These authors reached an extent of convergence in the case i.i.d. that was really near to being finished. Since this work was published, much more research has been done on estimating the conditional density and its derivatives, especially for computing the conditional mode. Since the publication of this work, the conditional mode has been the subject of a significant amount of research.

In point of fact, [19] demonstrated that a kernel estimator of the conditional mode will almost certainly converge to the true value. They did this by taking into account data that included a-mixing. The conditional density is maximized by the random variable that corresponds to this mode, which is defined as.

The point that nullifies the kernel density estimator derivative was used by [16,17] to estimate the conditional mode. This was done in order to determine the conditional mode. The outcomes were comparable using this approach.

The latter put more of an emphasis on the estimator’s asymptotic normality, which was provided in both the iid and mixed circumstances respectively. Both scenarios included mixing. [27] was able to identify with what level of accuracy the terms that dominate the quadratic error that is produced by the kernel density estimator.

We suggest that the reader check out [28] for further information on the topic of the smoothing parameter that should be used in the process of estimating the conditional density in relation to the functional explanatory variable.

The concept of a quasi-association variable refers to a variable that exhibits some degree of association with another variable, Examples of research that processed data under both positive and negative dependent random variables are [34], [32], and [33]. [14] were the first people to offer the concept of quasi-association for the purpose of conducting an analysis of real-valued stochastic occurrences. This is a very striking illustration of the idea of weak reliance. It was used by [7] to real valued random fields, and it provides a unified technique for the study of families of positively dependent as well as negatively dependent random variable families.

According to our knowledge, the nonparametric estimation of quasi-associated random variables is addressed in a vanishingly small number of published papers. The study conducted by [13] focuses on a limit theorem for quasi associated Hilbertian random variables. Meanwhile, the research conducted by [5] explores asymptotic results for an M-estimator of the regression function for quasi-associated processes; and [36], who investigated both quasi-associated processes and asymptotic results. [9] investigated the asymptotic normality of this final estimator as part of their study, which focused on the single-index structure of the conditional hazard function.

To solve relative regression, [31] explored the nonparametric estimate for linked random variables. it was found to be significant by [10] and [10]. Both of these results were found to be significant. Both Daoudi and Mechab were responsible for conducting these investigations in their own separate ways.

It is important to keep in mind that the phenomena that our results are connected to the model’s functional space in some manner, just as they are related to every previous asymptotic statistics functional nonparametric finding.

In this research, we examine the conditional distribution function within the context of the quasi-associated situation, focusing on its asymptotic bias and dispersion. This study examined pointwise consistency for nonparametric conditional density function estimates.

This objective was successfully accomplished through the conduct of research on the rate of the almost-complete convergence (a.co.) for nonparametric estimates of the conditional distribution function.

The following is the structure of our paper: In the following section, we will describe our model. In the part 3, we present the hypotheses and remarks, along with a few notations, followed by a discussion of supplementary findings and supporting in paragraph 4. The Main Results are found in Section 5. The part 6 covers the application on consistency rates and the user is referring to the concepts of consistency and asymptotic normality in the context of estimating a conditional hazard function. he establishment of confidence bands for estimations in part 7. In Section 8, we will analyze and assess the behavior of our asymptotic normality results, on finite sample data. In conclusion, Auxiliary results and proofs are presented in Section 9.

2. Model and Estimator

To commence, we provide a precise delineation of quasi-association pertaining to random variables that possess values within a separable Hilbert space.

Consider a separable Hilbert space

(H, < ., . >)

furnished with an orthonormal basis. The sequence of elements is represented by

e_{k}, k \geq 1

. Let

{(R_{n})}_{n \in N}

be a sequence of real random variables with values in

H

. The statement is made that this sequence exhibits quasi-association in relation to the basis. The user defines

e_{k}

as a variable. The term "quasi-associated" is used to describe the sequence

e_{k}

if, for any positive integer d, the d-dimensional sequence

{(< R_{i}, e_{j 1} >, . . ., < R_{i}, e_{j d} >), i \in N}

is also quasi-associated.

In this analysis, we will examine a group of n quasi-associated random variables, which we will represent as

W_{i} = {(R_{i}, S_{i})}_{1 \leq i \leq n}

. The random variables mentioned exhibit the same distribution as the random variable

W = (R, S)

, which represents values in a separable real Hilbert space denoted as

H \times R

. This Hilbert space is equipped with an inner product denoted as

< ., . >

, which generates the norm. The user’s text does not contain any meaningful information. The semi-metric d is considered, as given by.

\forall r, r^{'} \in H / d (x, x^{'}) = ∥ r - r^{'} ∥

. A point r in the

H

space, its fixed neighborhood

N_{r}

, and a compact subset of

R

all have the notation

S

For each

r \in N_{r}

, there is a S such that

R = r

. Using a sample of n dependent observations from random variables with the same distribution as

W : = (R, S)

, the conditional distribution function

F^{r} (s)

is estimated.

We present the

{\hat{F}}^{r}

estimator of

F^{r}

, a kernel type estimator, defined as:

{\hat{F}}^{r} (s) = \frac{\sum_{i = 1}^{n} K (h_{H}^{- 1} d (r, R_{i})) H (h_{H}^{- 1} (s - S_{i}))}{K (h_{H}^{- 1} d (r, R_{i}))} \forall s \in R

As n increases to infinity, the sequence of positive real integers

h_{K} = h_{K, n}

(resp.

h_{H} = h_{H, n}

) converges to zero.

Let K denote the kernel and H represent a particular distribution function.

3. Hypotheses and Notations

When there is no chance of misunderstanding, we shall designate any strictly positive generic constants along the actual paper by the notation l or/and

l^{^{'}}

. This will only happen when there is no risk of mistake. w in the process signifies a fixed point in

H

, and

N_{w}

stands for a fixed neighborhood of w. Taking into account the fact that the random couple

{(W_{i}, T_{i}), i \in N}

is a process that is stationary.

Let

λ_{k}

denote the covariance coefficient, which may be found by using the equation:

λ_{k} = s u p_{s \geq k} \sum_{| i - j | \geq s} λ_{i, j} .

Where

λ_{i, j} = \sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} | C o v (W_{i}^{k}, W_{j}^{l}) | + \sum_{k = 1}^{\infty} | C o v (W_{i}^{k}, T_{j}) | + \sum_{l = 1}^{\infty} | C o v (W_{i}, X_{j}^{l}) | + | C o v (T_{i}, T_{j}) | .

The

k^{t h}

component of

W_{i}

, as represented by

R_{i}^{k}

, where

R_{i}^{k} : = < W_{i}, e^{k} >

is the definition of

R k_{i}

. Let’s denote the ball as

B (w, h) : = {r^{'} \in H / d (w^{'}, w) < h}

. This represents the ball with a center of w and a radius

h,

when

h > 0

For the purpose of establishing the virtually full convergence of the estimator of the

{\hat{F}}^{r} (s)

, the following assumptions are required, and in order to have a functional explanation, we need to incorporate he following assumptions:

(P1): $P (R \in B (r, h)) = ϕ_{r} (h) > 0$ and the function $ϕ_{r} (h)$ is a differentiable at 0.
(P2): The conditional distribution function $F^{r} (s)$ satisfies the Holder condition, that is: $\forall (r_{1}, r_{2}) \in N_{r} \times N_{x}, \forall (s_{1}, s_{2}) \in S^{2}$

$| F^{r_{1}} (s_{1}) - F^{r_{2}} (s_{2}) | \leq l (d^{z_{1}} (r_{1}, r_{2}) + {| s_{1} - s_{2} |}^{z_{2}}), z_{1} > 0, z_{2} > 0 .$

$S$ is a fixed compact subset of $R$ .
(P3): The kernel, denoted by H, is a differentiable function, and its inverse, denoted by $H^{'}$ , is a positive, bounded, and Lipschitzian continuous function:

${\int | t |}^{z_{2}} H^{'} (t) d t < \infty a n d \int H^{' 2} (t) d t < \infty .$
(P4): For K, we have the necessary conditions for it to be a bounded continuous Lipschitz function, which are:

${l ‖}_{[0, 1]} (.) < K (.) < l^{^{'}} ‖_{[0, 1]} (.)$

$‖_{[0, 1]}$ is referred to as an indicator function.
(P5): A quasi-association exists between the sequence of random pairings $(R_{i}, S_{i})$ , where $i \in N$ , and the covariance coefficient $λ_{k}, k \in N$ , as long as the conditions are met.

$\exists a > 0, \exists l > 0, t h e n λ_{k} \leq l e^{- a k}$
(P6): The joint distribution functions are defined so that they hold for each and every pair $(i, j)$ .

$Ψ_{i, j} (h) = P [(R_{i}, R_{j}) \in B (r, h) \times B (r, h))]$

satisfy :

$sup_{i \neq j} Ψ_{i, j} (h) = O (ϕ_{r}^{2} (h_{k})) > 0 .$
(P7): The sequences of positive values that make up the bandwidths $h_{K}$ and $h_{H}$ fulfill the following equations when j is equal to 0 and 1:

$lim_{n \to \infty} \frac{{log}^{5} (n)}{n h_{H}^{j} ϕ_{r} (h_{k})} = 0 .$

4. Remarks Concerning the Hypotheses

The attribute of concentration of the explanatory variable is denoted by the assumption (P1) in the context of tiny balls. The condition (P2) is applied to our model in order to regulate the regularity of the functional space. They are essential for calculating the bias component of the convergence rates and are needed to be present. Both the (P3) and (P4) hypotheses center their attention on the cumulative function H as well as the kernels

K^{^{'}} H^{^{'}}

, and

K^{^{'}}

, respectively. In particular, this assumption is utilized to get rid of the word "bias" in the conclusion of asymptotic normalcy. The fifth assumption, denoted as P5, is widely regarded as a classical assumption in functional estimation within spaces of both finite and infinite dimensions. The hypothesis (P6) is a notation that refers to a structural condition that is applied to the quasi-associated data. In order for us to demonstrate the asymptotic normality of our model when it is exposed to quasi-association, it is required for us to have the assumption (P7). Following is a definition of the asymptotic behavior of the joint distribution of the pair (

R_{i}, R_{j}

5. Main Results

Theorem 1.

Under hypotheses

(P 1) - (P 7)

, we have:

{\hat{F}}^{r} (s) - F^{r} (s) = O (h_{K}^{b 1} + h_{K}^{b 2}) + O ({(\frac{L o g (n)}{n ϕ_{R} (h_{K})})}^{\frac{1}{2}})

(1)

The following decomposition, together with the lemmas that are listed below, provide the foundation for the proof of Theorem 1:

\begin{matrix} {\hat{F}}^{r} (s) - F^{r} (s) & = \frac{1}{{\hat{F}}_{D}^{r}} (({\hat{F}}_{N}^{r} (s) - E {\hat{F}}_{N}^{r} (s)) - (F^{r} (s) - E {\hat{F}}_{N}^{r} (s))) \\ + \frac{F^{r} (s)}{{\hat{F}}_{D}^{r}} ({\hat{F}}_{D}^{r} - E {\hat{F}}_{D}^{r}) \end{matrix}

(2)

We can write:

{\hat{F}}^{r} (s) = \frac{{\hat{F}}_{N}^{r} (s)}{{\hat{F}}_{D}^{r}}

Where:

{\hat{F}}_{N}^{r} (s) = \frac{1}{n E [K_{1} (r)]} \sum_{i = 1}^{n} K_{i} (r) H_{i} (s)

and:

{\hat{F}}_{D} (r) = \frac{1}{n E [K_{1} (r)]} \sum_{i = 1}^{n} K_{i} (r)

with :

K_{i} (r) = K (h_{K}^{- 1} d (r, R_{i})) a n d H_{i} (s) = H (h_{H}^{- 1} (s - S_{i}))

Lemma 1.

Under hypotheses

(P 1)

(P 4)

and

(P 6)

\frac{1}{{\hat{F}}_{D}^{r} (s)} ({\hat{F}}_{N}^{r} (s) - E {\hat{F}}_{N}^{r} (s)) = O_{a . c o} ({(\frac{L o g (n)}{n ϕ_{R} (h_{K})})}^{\frac{1}{2}})

(3)

Corollary 1.

Under hypotheses

(P 1)

(P 4)

and

(P 6)

, we have:

\sum_{i = 1}^{\infty} P (| {\hat{F}}_{D}^{r} (s) | < 1 / 2) < \infty

(4)

Lemma 2.

Under hypotheses

(P 1)

(P 6)

, we have:

\frac{1}{{\hat{F}}_{D}^{r} (s)} (F^{r} (s) - E {\hat{F}}_{N}^{r} (s)) = O (h_{K}^{b 1} + h_{K}^{b 2})

(5)

Lemma 3.

Under hypotheses

(P 1)

(P 3)

and

(P 6)

, we have:

{\hat{F}}_{D}^{r} (s) - E {\hat{F}}_{D}^{r} (s) = O_{a . c o} ({(\frac{l o g (n)}{n ϕ_{r} (h_{K})})}^{\frac{1}{2}})

(6)

6. Application on the Conditional Hazard Function Estimate

Using our most important result, The estimator of the conditional hazard function

{\hat{Z}}^{r} (s)

exhibits convergence behavior , it was examined in our analysis. This estimator is defined as follows:

{\hat{Z}}^{r} (s) = \frac{{\hat{f}}^{r} (s)}{1 - {\hat{F}}^{r} (s)} \forall s \in R,

(7)

where

{\hat{f}}^{r} (s)

is an estimate of the conditional density given by

{\hat{f}}^{r} (s) = \frac{h_{H}^{- 1} \sum_{i = 1}^{n} K (h_{K}^{- 1} d (r, R_{i})) H^{'} (h_{H}^{- 1} (s - S_{i}))}{\sum_{i = 1}^{n} K (h_{K}^{- 1} d (r, R_{i}))} \forall s \in R

(8)

This is the derivative of H expressed as

H^{'}

6.1. Almost Complete Convergence with the Rate

Theorem 2.

Assuming (P1)-(P7) and Theorem 1. of [23], we obtain:

| {\hat{Z}}^{r} (s) - Z^{r} (s) | = O (h_{K}^{b_{1}} + h_{K}^{b_{2}}) + O_{a . c o} ({(\frac{log n}{n h_{H} ϕ_{r} (h_{K})})}^{\frac{1}{2}}) .

(9)

The proof of Theorem 2 is the sum of this paper’s proof and theorem’s proof in [23]

6.2. Asymptotic Normality

Theorem 3.

Assuming (P1)-(P7) and Theorem 1. of [10] and [11], we obtain: for any

r \in A

\sqrt{n h_{H} ϕ_{} (r, h_{K})} ({\hat{Z}}^{r} (s) - Z^{r} (s)) \overset{D}{⟶} N (0, σ_{h}^{2} (r)) a s n \to \infty

where

A = {r \in H, f^{r} (s) (1 - F^{r} (s)) \neq 0}

σ_{h}^{2} (r) = \frac{C_{2} Z^{r} (s)}{C_{1}^{2} (1 - F (s, r))} \int H^{' 2} (t) d t,

C_{j} = K (1) - \int_{0}^{1} {(K^{j})}^{'} (s) β (r, s) d s, f o r j = 1, 2 .

The symbol

\overset{D}{⟶}

denotes convergence in distribution.

7. Confidence Bands

One important aspect in statistical analysis is the establishment of confidence bands for estimations. These confidence bands provide a range of values within which we can be confident that the true value lies. By calculating and interpreting these confidence bands, we can gain a better understanding of the uncertainty associated with our estimations. The objective of this section is to utilize our asymptotic normality result (Theorem 3) to create confidence intervals for the actual value of

Z^{r} (s)

for a specified curve with the format

< X, X^{^{'}} > = < x, x^{^{'}} >

. Estimation using nonparametric methods relies on the asymptotic variance, which is determined by a number of unknown functions. Regarding our situation, we have

σ_{Z}^{2} (r) = \frac{C_{2} Z^{r} (s)}{C_{1}^{2} (1 - F^{r} (s))}

The variables

Z^{r} (s), F^{r} (s), C_{1}

, and

C_{2}

are not known in advance and must be approximated during the practical implementation. It is possible to derive confidence bands even when

σ_{Z}^{2} (r)

is functionally given. An estimate for the asymptotic standard deviation

{\hat{σ}}_{Z}^{2} (r)

may be produced using the estimators

{\hat{Z}}^{(r)} (s)

{\hat{F}}^{(r)} (s)

{\hat{C}}_{1}

, and

{\hat{C}}_{2}

for

Z^{(r)} (s)

F^{(r)} (s)

C_{1}

, and

C_{2}

, respectively.

{\hat{σ}}_{Z}^{2} (r) = \frac{{\hat{C}}_{2} {\hat{Z}}^{r} (s)}{{\hat{C}}_{1}^{2} (1 - {\hat{F}}^{r} (s))} .

The constants

C_{1}

and

C_{2}

are estimated empirically in the following manner:

{\hat{C}}_{1} = \frac{1}{n ϕ (r, h_{K})} \sum_{i = 1}^{n} K (h_{K}^{- 1} d (r, R_{i})), {\hat{C}}_{2} = \frac{1}{n ϕ (r, h_{K})} \sum_{i = 1}^{n} K^{2} (h_{K}^{- 1} d (r, R_{i}))

where

ϕ (r, h_{K}) = \frac{1}{n} \sum_{i = 1}^{n} I_{{| < r - R_{i} > | < h_{K}}} .

Now the asymptotic confidence band at asymptotic level

1 - ζ

for

Z^{r} (s)

is given by

[{\hat{Z}}^{r} (s) - u_{1 - \frac{ζ}{2}} {(\frac{{\hat{σ}}_{Z}^{2} (r)}{n ϕ (r, h_{K})})}^{1 / 2}, {\hat{Z}}^{r} (s) + u_{1 - \frac{ζ}{2}} {(\frac{{\hat{σ}}_{Z}^{2} (r)}{n ϕ (r, h_{K})})}^{1 / 2}]

where

u_{1 - \frac{ζ}{2}}

denotes the

1 - \frac{ζ}{2}

quantile of the standard normal distribution.

8. A simulation Study

In this part, we examine how our asymptotic normality conclusions behave across data from a limited sample.

Our primary purpose is to demonstrate how simple the conditional hazard function is to develop and to study the effect of dependency on this asymptotic characteristic.

We produce functional observations for this purpose by examining the following functional nonparametric model:

Z_{i} = r (W_{i}) + ϵ_{i} for i = 1, \dots, n

(10)

Where

ϵ_{i} ⇝ N (0, . 5)

. The linear process with quasi-associated variables is well known to satisfy the requirement (H7).

As a result, the functional regressor with quasi-associates shown below is constructed.

Z_{i} (t) = \sum_{j = i + 1}^{i + m} Γ_{j} (t)

Where

Γ_{j} (t) = s_{j} t^{2} + h_{j} * t + g_{j} t \in [0, 1]

{(s_{j})}_{j} ⇝ N (0, \frac{1}{2})

(resp.

{(h_{j})}_{j} ⇝ N (- 1, \frac{1}{2})

and

{(g_{j})}_{j} ⇝ N (1, \frac{1}{2})

The Figure 1 shows the

Z_{i}

’s curves discretized in the same 100-point grid in [0, 1] for three separate m=1 (independent case), and 10 values

Furthermore, the regression operator computes the scalar variable

Z_{i}

r (w) = 5 \int_{0}^{1} exp \{W (t)\} d t .

Given

W = w

, we may derive the theoretical conditional from the error distribution (

ϵ_{i}

). The distribution function of Z is supplied by

e_{i}

’s distribution shifted by

r (w)

. As a result, the theoretical conditional hazard function is simple to compute. To demonstrate this function’s asymptotic normality, we fix one curve,

w = W_{0}

, and

z = Z_{0}

, from the created data, then collect m independent n-samples of the same data and compute the quantity:

\sqrt{n h_{H} ϕ_{(} r, h_{K})} {\hat{σ}}_{h_{K}}^{- 1} (r) ({\hat{Z}}^{r} (s) - Z^{r} (s))

where

{\hat{σ}}_{h_{K}}

is the previous section’s standard deviation estimation.

The collected sample is next tested for normality. While the cross-validation approach determines , We utilized a quadratic kernel K represented as: the bandwidth parameters

h_{H}

and

h_{K}

dependent on the number of nearest neighbors and we have used quadratic kernel K expressed by

K (t) = \frac{3}{2} (1 - t^{2}) I_{[0, 1]} .

The Figure 2 depicts a Graph of the collected sample vs a typical normal distribution at varied m=1 and 10 values.

It is evident that our estimator is simple to use and behaves well in practice. Furthermore, the pace of asymptotic normality convergence is substantially related to data correlation. It diminishes in particular with respect to m values.

9. Auxiliary Results and Proofs

Corollary 2.

19 (See, [13]). Let

{(R_{n})}_{n \in N}

be a quasi-associated sequence of random variables with values in

H

. Let

h \in B L (H^{∣ I ∣}) \cap L^{\infty}

and

M \in B L (H^{∣ J ∣}) \cap L^{\infty}

, for some finite disjoint subsets

I, J \subset N

. Then

C o v (h (R_{i}, i \in I), M (R_{j}, j \in J)) \leq L i p (h) L i p (M) \sum_{i \in I} \sum_{j \in J} \sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} |C o v (R_{i}^{k}, R_{j}^{l})|

where

(B L (H^{u}; u > 0)

is the set of bounded Lipschitz functions

h : H^{u} \to R

and

L^{\infty}

is the set of bounded functions.

Corollary 3.(See, [25]). Let

R_{1}, . . ., R_{n}

the real random variables such that

E (R_{j}) = 0

and

P (∣ R_{j} ∣ \leq M) = 1

for all

j = 1, . . ., n

and some

M < \infty

, Let

σ_{n}^{2} = V a r (\sum_{i = 1}^{n} Δ_{i})

Assume, furthermore, that there exist

K < \infty

and

β > 0

such that, for all u-uplets

(s_{1}, . . ., s_{u}) \in N^{u}

(t_{1}, . . ., t_{v}) \in N^{v}

with

1 \leq s_{1} \leq \dots \leq s_{u} \leq t_{1} \leq \dots \leq t_{v} \leq n .

The following inequality is fulfilled:

∣ c o v (R_{s_{1}} . . . R_{s_{u}}, R_{t_{1}} . . . R_{t_{v}}) ∣ \leq K^{2} M^{u + v - 2} v e^{- β (t_{1} - s_{u})} .

Then,

P (∣ \sum_{j = 1}^{n} R_{j} ∣ > t) \leq exp {- \frac{t^{2} / 2}{A_{n} + B_{n}^{1 / 3} t^{5 / 2}}}

for some

A_{n} \leq σ_{n}^{2}

and

B_{n} = (\frac{16 n K^{2}}{9 A n (1 - e^{-} β) \lor 1}) \frac{2 (K \lor M)}{1 - e^{-} β} .

The proof of Lemma 1

We put:

Δ_{i} = \frac{1}{n h_{H} E [K_{1}]} χ (R_{i}, S_{i}), 1 \leq i \leq n .

(11)

where :

ψ (R_{i}, S_{i}) = K (h_{K}^{- 1} d (r, R_{i})) H (s - S_{i}) - E [K_{1} H_{1}] 1 \leq i \leq n,

(12)

R_{i} \in H

S_{i} \in R

E (Δ_{i}) = 0

and,

| {\hat{F}}_{N}^{r} (s) - E {\hat{F}}_{N}^{r} (s) | = \sum_{i = 1}^{n} Δ_{i}

(13)

Moreover, we can write :

{| | ψ | |}_{\infty} \leq {2 C | | K | |}_{\infty} {| | H | |}_{\infty}

(14)

and :

L i p (ψ) \leq C (h_{K}^{- 1} {| | H | |}_{\infty} L i p (K) + h_{H}^{- 1} {| | K | |}_{\infty} L i p (H)) .

(15)

In order to use the lemma that was developed by [25], we need to do an evaluation of the variance term

V a r (\sum_{i = 1}^{n} Δ_{i})

as well as the covariance term

C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}})

. Both of these terms are denoted by the following for all

(s_{1}, . . ., s_{u}) \in N_{u}, (t_{1}, . . ., t_{v}) \in N_{v}

with

1 \leq s_{1} \leq . . . \leq s_{u} \leq t_{1} \leq . . . \leq t_{v} \leq n

We explore the following cases with respect to the covariance term: if

t_{1} = s_{u}

. Utilizing the Reality that

E [| K_{1} H_{1} |] = O_{a . c o} (ϕ_{r} (h_{K}))

and

E [| K_{1} |] = O_{a . c o} (ϕ_{r} (h_{K}))

we have:

\begin{matrix} | C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | & \leq {(\frac{C}{n E [K_{1}]})}^{u + v} E_{ψ} {| R_{1}, S_{1} |}^{u + v} \\ \leq {(\frac{{C | | K | |}_{\infty} {| | H | |}_{\infty}}{n E [K_{1}]})}^{u + v} E | K_{1} H_{1} | \\ \leq ϕ_{r} (h_{K}) {(\frac{C}{n ϕ_{r} (h_{K})})}^{u + v} \end{matrix}

(16)

If it

t_{1} > s_{u}

, Under

(P 5)

, the quasi-association yields:

\begin{matrix} | C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | & \leq {(\frac{h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H)}{n E [K_{1}]})}^{2} \times {(\frac{C}{n E [K_{1}]})}^{u + v - 2} \sum_{i = 1}^{u} \sum_{j = 1}^{v} λ_{s_{i}, t_{j}} \\ \leq {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} {(\frac{C}{n E [K_{1}]})}^{u + v} v λ_{t_{1} - s_{u}} \\ \leq {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} {(\frac{C}{ϕ_{r} (h_{K})})}^{u + v} v e^{- α (t_{1} - s_{u})} \end{matrix}

(17)

On the other hand, if we take into account

(P 6)

, we have:

\begin{matrix} | C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | & \leq {(\frac{{C | | K | |}_{\infty} {| | H | |}_{\infty}}{n E [K_{1}]})}^{u + v - 2} (E | Δ_{s_{u}}, Δ_{t_{1}} | + E | Δ_{s_{u}} | E | Δ_{t_{1}} |) \\ \leq {(\frac{{C | | K | |}_{\infty} {| | H | |}_{\infty}}{n E [K_{1}]})}^{u + v - 2} (\frac{C}{n E [K_{1}]}) \\ \times h_{H} (s u p_{i \neq j} P ((R_{i}, R_{j}) \in B (r, h_{K}) \times B_{θ} (r, h_{K}) \\ + P {(R_{1} \in B (r, h_{K}))}^{2})) \\ \leq {(\frac{C}{h_{H} ϕ_{r} (h_{K})})}^{u + v} {(ϕ_{r} (h_{K}))}^{2} \end{matrix}

(18)

In addition, by taking a

γ - p o w e r

of (10), and a

(1 - γ) - p o w e r

of (11), we are able to derive an upper-bound of the tree terms as follows, for:

1 \leq s_{1} \leq . . . s_{u} \leq t_{1} \leq . . . . \leq t_{v} \leq n,

| C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | \leq h_{H} ϕ_{r} (h_{K}) {(\frac{C}{n h_{H} ϕ_{r} (h_{K})})}^{u + v}

(19)

Second, we entered the following information for the

V a r (\sum_{i = 1}^{n} Δ_{i})

variance term for all

1 \leq i \leq n

\begin{matrix} | V a r (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | & = {(\frac{1}{n E [K_{1}]})}^{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} C o v (K_{i} H_{i}, K_{j} H_{j}) \\ = \underset{T_{1}}{\underset{︸}{{(\frac{1}{n E [K_{1}]})}^{2} V a r (K_{1} H_{1})}} \\ + \underset{T_{2}}{\underset{︸}{{(\frac{1}{n E [K_{1}]})}^{2} \sum_{i = 1}^{n} \sum_{j = 1, i \neq j}^{n} C o v (K_{i} H_{i}, K_{j} H_{j})}} \end{matrix}

(20)

For the first term

T_{1}

, we have :

V a r (K_{1} H_{1}) = E (K_{1}^{2} H_{1}^{2}) - {(E (K_{1} H_{1}))}^{2}

(21)

Then,

E [K_{1}^{2} H_{1}^{2}] = E [K_{1}^{2} E [H_{1}^{2} / X_{1}]]

(22)

As a result, considering (P2) and (P3) and integrating over the real component y gives us the following:

E [H_{1}^{2} ∣ R_{1}] = O (h_{H})

(23)

As for all

j \geq 1

I E [K_{1}^{j}] = O (ϕ_{x} (h_{K})) .

Then,

I E [K_{1}^{2} H_{1}^{2}] = O (ϕ_{x} (h_{K})) .

It follows that:

{(\frac{1}{n {(I E [K_{1}])}^{2}})}^{2} V a r (K_{1} H_{1}) = O (n ϕ_{x} (h_{K})) .

(24)

Regarding the covariance term in

(r e f 26)

, the following decomposition will be utilized.

\begin{matrix} \sum_{i = 1}^{n} \sum_{j = 1, i \neq j}^{n} C o v (K_{i} H_{i}, K_{j} H_{j}) & = & \underset{I}{\underset{︸}{\sum_{i = 1}^{n} \sum_{j = 1, 0 < ∣ i - j ∣ \leq m_{n}}^{n} C o v (K_{i} H_{i}, K_{j} H_{j})}} \\ + & \underset{I I}{\underset{︸}{\sum_{i = 1}^{n} \sum_{j = 1, ∣ i - j ∣ > m_{n}}^{n} C o v (K_{i} H_{i}, K_{j} H_{j})}} \end{matrix}

where

(m_{n})

is an infinite series of positive integers as n tends to infinity. The

(P 1)

(P 3)

and

(P 6)

Presuppositions

we obtain, if

i \neq j

\begin{matrix} I & \leq n m_{n} (max_{i \neq j} |E (K_{i} H_{i} K_{j} H_{j})| + {(E (K_{1} H_{1}))}^{2}) \\ \leq C n m_{n} (ϕ_{r}^{2} (h_{K}) + {(ϕ_{r} (h_{K}))}^{2}) \\ \leq error (ϕ_{r}^{2} (h_{K})) . \end{matrix}

(25)

H and K are both bounded and Lipschitz kernels, we obtain:

\begin{matrix} I I & \leq & {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} \sum_{i = 1}^{u} \sum_{j = 1_{∣ i - j ∣ > m_{n}}}^{v} λ_{i, j} \\ \leq & C {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} \sum_{i = 1}^{u} \sum_{j = 1_{∣ i - j ∣ > m_{n}}}^{v} λ_{i, j} \\ \leq & C n {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} λ_{m_{n}} \\ \leq & C n {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} e^{- α m_{n}} . \end{matrix}

(26)

Then, by (20) and (21), we get

\sum_{j = 1, i \neq j}^{n} C o v (K_{i} H_{i}, K_{j} H_{j}) \leq C (n m_{n} (h_{H}^{2} ϕ_{r}^{2} (h_{K})) + n {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} e^{- α m_{n}})

By choosing :

m_{n} = log (\frac{{(h_{K}^{- 1} Lip (K) + h_{H}^{- 1} Lip (H))}^{2}}{α ϕ_{r}^{2} (h_{K})})

We get :

\frac{1}{ϕ_{r} (h_{K})} \sum_{j = 1, i \neq j}^{n} C o v (K_{i} H_{i}, K_{j} H_{j}) \underset{n \to + \infty}{\to} 0 .

(27)

Finally, we get this result by integrating the previous three results (13), (18), and (22).

V a r (\sum_{i = 1}^{n} Δ_{i}) = O (\frac{1}{n ϕ_{r} (h_{K})})

Therefore, the criteria of the lemma are satisfied by the variables

Δ_{i}

, where

i = 1, . . ., n

K_{n} = \frac{C}{n \sqrt{ϕ_{r} (h_{K})}}, M_{n} = \frac{C}{n ϕ_{r} (h_{K})}, a n d A_{n} = V a r (\sum_{i = 1}^{n} Δ_{i})

Thus,

\begin{matrix} P (|{\hat{F}}_{N}^{r} (s) - E {\hat{F}}_{N}^{r} (s)| > η \frac{log n}{n ϕ_{r} (h_{K})}) & = P (|\sum_{i = 1}^{n} Δ_{i}| > η \frac{log n}{n ϕ_{r} (h_{K})}) \\ \leq exp \{- \frac{η^{2} log n}{n ϕ_{r} (h_{K}) Var (\sum_{i = 1}^{n} Δ_{i}) + \frac{{log}^{5 / 6} n}{{(n ϕ_{r} (h_{K}))}^{(7 / 6)}}}\} \\ \leq exp \{- \frac{η^{2} log n}{C + \frac{{(log n)}^{5 / 6}}{n ϕ_{r} {(h_{K})}^{(7 / 6)}}}\} \\ \leq C^{'} exp \{- C η^{2} log n\} \end{matrix}

(28)

(P 7)

. Last but not least, with an appropriate selection of

η

, Borel-Cantelli’s lemma makes it possible to conclude the proof of Lemma 1.

The proof of Corollary 1

We have:

\{| F_{D}^{R} | \leq \frac{1}{2}\} \subseteq \{| F_{D}^{R} - 1 | > \frac{1}{2}\}

(29)

Therefor,

\begin{matrix} P \{| F_{D}^{R} | \leq \frac{1}{2}\} & \leq P \{| F_{D}^{R} - 1 | > \frac{1}{2}\} \\ \leq P \{| F_{D}^{X} - E F_{D}^{X} | > \frac{1}{2}\} \end{matrix}

(30)

For

E (F_{D}^{R}) = 1

, we apply the result of Lemma 1 we show that :

P \{| F_{D}^{R} | \leq \frac{1}{2}\} < \infty

(31)

The proof of the Corollary 1 has finished.

The proof of Lemma 2

We have :

\begin{matrix} E {\hat{F}}_{N}^{R} (S) - F^{R} (S) & = \frac{1}{n E K_{1}} \sum_{i = 1}^{n} E K_{i} H_{i} (y) - F^{R} (S) \\ = \frac{1}{E K_{1}} [E K_{1} H_{1} (\frac{s - S_{i}}{h_{H}}) - F^{R} (S)] \\ = \frac{1}{E K_{1}} E (K_{1} [E (H_{1} (h_{H}^{- 1} (s - S_{i})) / R) - F^{R} (S)]) \end{matrix}

(32)

We derive the following by making use of the stationarity of the data, the conditioning by the variable that is doing the explaining, and the customary change in the variable

t = \frac{s - u}{h_{H}}

\begin{matrix} E (H_{1} (h_{H}^{- 1} (s - S_{i})) / R) & = \int_{R} H (\frac{s - u}{h_{H}}) f^{R} (u) d u \\ = \int_{R} H^{(1)} (t) F^{R} (s - h_{H} t) d t \end{matrix}

(33)

and we deduce ,

\begin{matrix} |E (H_{1} (h_{H}^{- 1} (s - S_{i})) / R) - F^{R} (s)| & = |\int_{R} H^{(1)} (t) F^{R} (s - h_{H} t) d t - F^{R} (s)| \\ \leq \int_{R} H^{(1)} (t) |F^{R} (s - h_{H} t) - F^{R} (s)| d t \end{matrix}

So, by (H2) we get

|E (H_{1} (h_{H}^{- 1} (s - S_{i})) / R) - F^{R} (s)| \leq A_{R} \int_{R} H^{(1)} (h_{K}^{b_{1}} + {| t |}^{b_{2}} h_{H}^{b_{2}}) d t

This inequality holds true everywhere in y and after substituting in (29) and simplifying the expression

(E K_{1})

we get that:

E {\hat{F}}_{N}^{R} (s) - F^{R} (s) \leq A_{R} (h_{K}^{b_{1}} \int_{R} H^{(1)} (t) d t + h_{H}^{b_{2}} \int_{R} {| t |}^{b_{2}} H^{(1)} (t) d t)

In conclusion, the evidence of Lemma 2 is provided by Hypothesis (P4) and Corollary 1.

Substituting for

ψ (., .)

in Lemma 1’s proof of Lemma 3 yields a simple application of Lemma 3.

ψ (R_{i}) = K (h_{K}^{- 1} d (r, R_{i})) - E [K_{1}], \forall R_{i} \in H .

(34)

The proof of Theorem 1 has finished. □

Author Contributions

Author Contributions Conceptualization, Z.C.M. and D.H.; methodology, Z.C.M. and D.H.; software, D.H.; validation, Z.C.M. and D.H.; formal analysis, Z.C.M., F. A. and D.H.; investigation, Z.C.M., F. A. and D.H.; resources, Z.C.M., F. A. and D.H.; data curation, D.H.; writing—original draft preparation, D.H.; writing—review and editing, Z.C.M., F. A. and D.H.; visualization, D.H.; supervision, Z.C.M. and F. A.; project administration, Z.C.M.; funding acquisition, Z.C.M. and F.A.; All authors have read and agreed to the published version of the manuscript.

Funding

This research project was funded by (1) Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2023R358), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia, and (2) The Deanship of Scientific Research at King Khalid University this work through Small group Research Project under grant number R.G.P. 1/366/44.

Informed Consent Statement

Not applicable

Data Availability Statement

Not applicable

Data Availability Statement

Not applicable

Acknowledgments

The authors express their gratitude to the Editors, the Associate Editor, and the anonymous reviewers for their insightful comments and suggestions, which significantly enhanced the overall quality of a previous iteration of this manuscript. The authors acknowledge the project funding: Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2023R358), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia; The Deanship of Scientific Research at King Khalid University through the Research Groups Program under grant number R.G.P. 1/366/44.

Conflicts of Interest

No conflict of interest.

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Figure 1. A sample of 200 curves.

Figure 2. The QQ-plot of the obtained sample.

Table 1. Kolmogorov-Smirnov analysis.

m	1	10
P-value	0.88	0.43

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Submitted:

02 August 2023

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03 August 2023

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Abstract

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

2. Model and Estimator

To commence, we provide a precise delineation of quasi-association pertaining to random variables that possess values within a separable Hilbert space.

Consider a separable Hilbert space

(H, < ., . >)

furnished with an orthonormal basis. The sequence of elements is represented by

e_{k}, k \geq 1

. Let

{(R_{n})}_{n \in N}

be a sequence of real random variables with values in

H

. The statement is made that this sequence exhibits quasi-association in relation to the basis. The user defines

e_{k}

as a variable. The term "quasi-associated" is used to describe the sequence

e_{k}

if, for any positive integer d, the d-dimensional sequence

{(< R_{i}, e_{j 1} >, . . ., < R_{i}, e_{j d} >), i \in N}

is also quasi-associated.

In this analysis, we will examine a group of n quasi-associated random variables, which we will represent as

W_{i} = {(R_{i}, S_{i})}_{1 \leq i \leq n}

. The random variables mentioned exhibit the same distribution as the random variable

W = (R, S)

, which represents values in a separable real Hilbert space denoted as

H \times R

. This Hilbert space is equipped with an inner product denoted as

< ., . >

, which generates the norm. The user’s text does not contain any meaningful information. The semi-metric d is considered, as given by.

\forall r, r^{'} \in H / d (x, x^{'}) = ∥ r - r^{'} ∥

. A point r in the

H

space, its fixed neighborhood

N_{r}

, and a compact subset of

R

all have the notation

S

For each

r \in N_{r}

, there is a S such that

R = r

. Using a sample of n dependent observations from random variables with the same distribution as

W : = (R, S)

, the conditional distribution function

F^{r} (s)

is estimated.

We present the

{\hat{F}}^{r}

estimator of

F^{r}

, a kernel type estimator, defined as:

{\hat{F}}^{r} (s) = \frac{\sum_{i = 1}^{n} K (h_{H}^{- 1} d (r, R_{i})) H (h_{H}^{- 1} (s - S_{i}))}{K (h_{H}^{- 1} d (r, R_{i}))} \forall s \in R

As n increases to infinity, the sequence of positive real integers

h_{K} = h_{K, n}

(resp.

h_{H} = h_{H, n}

) converges to zero.

Let K denote the kernel and H represent a particular distribution function.

3. Hypotheses and Notations

When there is no chance of misunderstanding, we shall designate any strictly positive generic constants along the actual paper by the notation l or/and

l^{^{'}}

. This will only happen when there is no risk of mistake. w in the process signifies a fixed point in

H

, and

N_{w}

stands for a fixed neighborhood of w. Taking into account the fact that the random couple

{(W_{i}, T_{i}), i \in N}

is a process that is stationary.

Let

λ_{k}

denote the covariance coefficient, which may be found by using the equation:

λ_{k} = s u p_{s \geq k} \sum_{| i - j | \geq s} λ_{i, j} .

Where

λ_{i, j} = \sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} | C o v (W_{i}^{k}, W_{j}^{l}) | + \sum_{k = 1}^{\infty} | C o v (W_{i}^{k}, T_{j}) | + \sum_{l = 1}^{\infty} | C o v (W_{i}, X_{j}^{l}) | + | C o v (T_{i}, T_{j}) | .

The

k^{t h}

component of

W_{i}

, as represented by

R_{i}^{k}

, where

R_{i}^{k} : = < W_{i}, e^{k} >

is the definition of

R k_{i}

. Let’s denote the ball as

B (w, h) : = {r^{'} \in H / d (w^{'}, w) < h}

. This represents the ball with a center of w and a radius

h,

when

h > 0

For the purpose of establishing the virtually full convergence of the estimator of the

{\hat{F}}^{r} (s)

, the following assumptions are required, and in order to have a functional explanation, we need to incorporate he following assumptions:

(P1): $P (R \in B (r, h)) = ϕ_{r} (h) > 0$ and the function $ϕ_{r} (h)$ is a differentiable at 0.
(P2): The conditional distribution function $F^{r} (s)$ satisfies the Holder condition, that is: $\forall (r_{1}, r_{2}) \in N_{r} \times N_{x}, \forall (s_{1}, s_{2}) \in S^{2}$

$| F^{r_{1}} (s_{1}) - F^{r_{2}} (s_{2}) | \leq l (d^{z_{1}} (r_{1}, r_{2}) + {| s_{1} - s_{2} |}^{z_{2}}), z_{1} > 0, z_{2} > 0 .$

$S$ is a fixed compact subset of $R$ .
(P3): The kernel, denoted by H, is a differentiable function, and its inverse, denoted by $H^{'}$ , is a positive, bounded, and Lipschitzian continuous function:

${\int | t |}^{z_{2}} H^{'} (t) d t < \infty a n d \int H^{' 2} (t) d t < \infty .$
(P4): For K, we have the necessary conditions for it to be a bounded continuous Lipschitz function, which are:

${l ‖}_{[0, 1]} (.) < K (.) < l^{^{'}} ‖_{[0, 1]} (.)$

$‖_{[0, 1]}$ is referred to as an indicator function.
(P5): A quasi-association exists between the sequence of random pairings $(R_{i}, S_{i})$ , where $i \in N$ , and the covariance coefficient $λ_{k}, k \in N$ , as long as the conditions are met.

$\exists a > 0, \exists l > 0, t h e n λ_{k} \leq l e^{- a k}$
(P6): The joint distribution functions are defined so that they hold for each and every pair $(i, j)$ .

$Ψ_{i, j} (h) = P [(R_{i}, R_{j}) \in B (r, h) \times B (r, h))]$

satisfy :

$sup_{i \neq j} Ψ_{i, j} (h) = O (ϕ_{r}^{2} (h_{k})) > 0 .$
(P7): The sequences of positive values that make up the bandwidths $h_{K}$ and $h_{H}$ fulfill the following equations when j is equal to 0 and 1:

$lim_{n \to \infty} \frac{{log}^{5} (n)}{n h_{H}^{j} ϕ_{r} (h_{k})} = 0 .$

4. Remarks Concerning the Hypotheses

K^{^{'}} H^{^{'}}

, and

K^{^{'}}

R_{i}, R_{j}

5. Main Results

Theorem 1.

Under hypotheses

(P 1) - (P 7)

, we have:

{\hat{F}}^{r} (s) - F^{r} (s) = O (h_{K}^{b 1} + h_{K}^{b 2}) + O ({(\frac{L o g (n)}{n ϕ_{R} (h_{K})})}^{\frac{1}{2}})

(1)

The following decomposition, together with the lemmas that are listed below, provide the foundation for the proof of Theorem 1:

\begin{matrix} {\hat{F}}^{r} (s) - F^{r} (s) & = \frac{1}{{\hat{F}}_{D}^{r}} (({\hat{F}}_{N}^{r} (s) - E {\hat{F}}_{N}^{r} (s)) - (F^{r} (s) - E {\hat{F}}_{N}^{r} (s))) \\ + \frac{F^{r} (s)}{{\hat{F}}_{D}^{r}} ({\hat{F}}_{D}^{r} - E {\hat{F}}_{D}^{r}) \end{matrix}

(2)

We can write:

{\hat{F}}^{r} (s) = \frac{{\hat{F}}_{N}^{r} (s)}{{\hat{F}}_{D}^{r}}

Where:

{\hat{F}}_{N}^{r} (s) = \frac{1}{n E [K_{1} (r)]} \sum_{i = 1}^{n} K_{i} (r) H_{i} (s)

and:

{\hat{F}}_{D} (r) = \frac{1}{n E [K_{1} (r)]} \sum_{i = 1}^{n} K_{i} (r)

with :

K_{i} (r) = K (h_{K}^{- 1} d (r, R_{i})) a n d H_{i} (s) = H (h_{H}^{- 1} (s - S_{i}))

Lemma 1.

Under hypotheses

(P 1)

(P 4)

and

(P 6)

\frac{1}{{\hat{F}}_{D}^{r} (s)} ({\hat{F}}_{N}^{r} (s) - E {\hat{F}}_{N}^{r} (s)) = O_{a . c o} ({(\frac{L o g (n)}{n ϕ_{R} (h_{K})})}^{\frac{1}{2}})

(3)

Corollary 1.

Under hypotheses

(P 1)

(P 4)

and

(P 6)

, we have:

\sum_{i = 1}^{\infty} P (| {\hat{F}}_{D}^{r} (s) | < 1 / 2) < \infty

(4)

Lemma 2.

Under hypotheses

(P 1)

(P 6)

, we have:

\frac{1}{{\hat{F}}_{D}^{r} (s)} (F^{r} (s) - E {\hat{F}}_{N}^{r} (s)) = O (h_{K}^{b 1} + h_{K}^{b 2})

(5)

Lemma 3.

Under hypotheses

(P 1)

(P 3)

and

(P 6)

, we have:

{\hat{F}}_{D}^{r} (s) - E {\hat{F}}_{D}^{r} (s) = O_{a . c o} ({(\frac{l o g (n)}{n ϕ_{r} (h_{K})})}^{\frac{1}{2}})

(6)

6. Application on the Conditional Hazard Function Estimate

Using our most important result, The estimator of the conditional hazard function

{\hat{Z}}^{r} (s)

exhibits convergence behavior , it was examined in our analysis. This estimator is defined as follows:

{\hat{Z}}^{r} (s) = \frac{{\hat{f}}^{r} (s)}{1 - {\hat{F}}^{r} (s)} \forall s \in R,

(7)

where

{\hat{f}}^{r} (s)

is an estimate of the conditional density given by

{\hat{f}}^{r} (s) = \frac{h_{H}^{- 1} \sum_{i = 1}^{n} K (h_{K}^{- 1} d (r, R_{i})) H^{'} (h_{H}^{- 1} (s - S_{i}))}{\sum_{i = 1}^{n} K (h_{K}^{- 1} d (r, R_{i}))} \forall s \in R

(8)

This is the derivative of H expressed as

H^{'}

6.1. Almost Complete Convergence with the Rate

Theorem 2.

Assuming (P1)-(P7) and Theorem 1. of [23], we obtain:

| {\hat{Z}}^{r} (s) - Z^{r} (s) | = O (h_{K}^{b_{1}} + h_{K}^{b_{2}}) + O_{a . c o} ({(\frac{log n}{n h_{H} ϕ_{r} (h_{K})})}^{\frac{1}{2}}) .

(9)

The proof of Theorem 2 is the sum of this paper’s proof and theorem’s proof in [23]

6.2. Asymptotic Normality

Theorem 3.

Assuming (P1)-(P7) and Theorem 1. of [10] and [11], we obtain: for any

r \in A

\sqrt{n h_{H} ϕ_{} (r, h_{K})} ({\hat{Z}}^{r} (s) - Z^{r} (s)) \overset{D}{⟶} N (0, σ_{h}^{2} (r)) a s n \to \infty

where

A = {r \in H, f^{r} (s) (1 - F^{r} (s)) \neq 0}

σ_{h}^{2} (r) = \frac{C_{2} Z^{r} (s)}{C_{1}^{2} (1 - F (s, r))} \int H^{' 2} (t) d t,

C_{j} = K (1) - \int_{0}^{1} {(K^{j})}^{'} (s) β (r, s) d s, f o r j = 1, 2 .

The symbol

\overset{D}{⟶}

denotes convergence in distribution.

7. Confidence Bands

Z^{r} (s)

for a specified curve with the format

< X, X^{^{'}} > = < x, x^{^{'}} >

. Estimation using nonparametric methods relies on the asymptotic variance, which is determined by a number of unknown functions. Regarding our situation, we have

σ_{Z}^{2} (r) = \frac{C_{2} Z^{r} (s)}{C_{1}^{2} (1 - F^{r} (s))}

The variables

Z^{r} (s), F^{r} (s), C_{1}

, and

C_{2}

are not known in advance and must be approximated during the practical implementation. It is possible to derive confidence bands even when

σ_{Z}^{2} (r)

is functionally given. An estimate for the asymptotic standard deviation

{\hat{σ}}_{Z}^{2} (r)

may be produced using the estimators

{\hat{Z}}^{(r)} (s)

{\hat{F}}^{(r)} (s)

{\hat{C}}_{1}

, and

{\hat{C}}_{2}

for

Z^{(r)} (s)

F^{(r)} (s)

C_{1}

, and

C_{2}

, respectively.

{\hat{σ}}_{Z}^{2} (r) = \frac{{\hat{C}}_{2} {\hat{Z}}^{r} (s)}{{\hat{C}}_{1}^{2} (1 - {\hat{F}}^{r} (s))} .

The constants

C_{1}

and

C_{2}

are estimated empirically in the following manner:

{\hat{C}}_{1} = \frac{1}{n ϕ (r, h_{K})} \sum_{i = 1}^{n} K (h_{K}^{- 1} d (r, R_{i})), {\hat{C}}_{2} = \frac{1}{n ϕ (r, h_{K})} \sum_{i = 1}^{n} K^{2} (h_{K}^{- 1} d (r, R_{i}))

where

ϕ (r, h_{K}) = \frac{1}{n} \sum_{i = 1}^{n} I_{{| < r - R_{i} > | < h_{K}}} .

Now the asymptotic confidence band at asymptotic level

1 - ζ

for

Z^{r} (s)

is given by

[{\hat{Z}}^{r} (s) - u_{1 - \frac{ζ}{2}} {(\frac{{\hat{σ}}_{Z}^{2} (r)}{n ϕ (r, h_{K})})}^{1 / 2}, {\hat{Z}}^{r} (s) + u_{1 - \frac{ζ}{2}} {(\frac{{\hat{σ}}_{Z}^{2} (r)}{n ϕ (r, h_{K})})}^{1 / 2}]

where

u_{1 - \frac{ζ}{2}}

denotes the

1 - \frac{ζ}{2}

quantile of the standard normal distribution.

8. A simulation Study

In this part, we examine how our asymptotic normality conclusions behave across data from a limited sample.

Our primary purpose is to demonstrate how simple the conditional hazard function is to develop and to study the effect of dependency on this asymptotic characteristic.

We produce functional observations for this purpose by examining the following functional nonparametric model:

Z_{i} = r (W_{i}) + ϵ_{i} for i = 1, \dots, n

(10)

Where

ϵ_{i} ⇝ N (0, . 5)

. The linear process with quasi-associated variables is well known to satisfy the requirement (H7).

As a result, the functional regressor with quasi-associates shown below is constructed.

Z_{i} (t) = \sum_{j = i + 1}^{i + m} Γ_{j} (t)

Where

Γ_{j} (t) = s_{j} t^{2} + h_{j} * t + g_{j} t \in [0, 1]

{(s_{j})}_{j} ⇝ N (0, \frac{1}{2})

(resp.

{(h_{j})}_{j} ⇝ N (- 1, \frac{1}{2})

and

{(g_{j})}_{j} ⇝ N (1, \frac{1}{2})

The Figure 1 shows the

Z_{i}

’s curves discretized in the same 100-point grid in [0, 1] for three separate m=1 (independent case), and 10 values

Furthermore, the regression operator computes the scalar variable

Z_{i}

r (w) = 5 \int_{0}^{1} exp \{W (t)\} d t .

Given

W = w

, we may derive the theoretical conditional from the error distribution (

ϵ_{i}

). The distribution function of Z is supplied by

e_{i}

’s distribution shifted by

r (w)

. As a result, the theoretical conditional hazard function is simple to compute. To demonstrate this function’s asymptotic normality, we fix one curve,

w = W_{0}

, and

z = Z_{0}

, from the created data, then collect m independent n-samples of the same data and compute the quantity:

\sqrt{n h_{H} ϕ_{(} r, h_{K})} {\hat{σ}}_{h_{K}}^{- 1} (r) ({\hat{Z}}^{r} (s) - Z^{r} (s))

where

{\hat{σ}}_{h_{K}}

is the previous section’s standard deviation estimation.

The collected sample is next tested for normality. While the cross-validation approach determines , We utilized a quadratic kernel K represented as: the bandwidth parameters

h_{H}

and

h_{K}

dependent on the number of nearest neighbors and we have used quadratic kernel K expressed by

K (t) = \frac{3}{2} (1 - t^{2}) I_{[0, 1]} .

The Figure 2 depicts a Graph of the collected sample vs a typical normal distribution at varied m=1 and 10 values.

9. Auxiliary Results and Proofs

Corollary 2.

19 (See, [13]). Let

{(R_{n})}_{n \in N}

be a quasi-associated sequence of random variables with values in

H

. Let

h \in B L (H^{∣ I ∣}) \cap L^{\infty}

and

M \in B L (H^{∣ J ∣}) \cap L^{\infty}

, for some finite disjoint subsets

I, J \subset N

. Then

C o v (h (R_{i}, i \in I), M (R_{j}, j \in J)) \leq L i p (h) L i p (M) \sum_{i \in I} \sum_{j \in J} \sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} |C o v (R_{i}^{k}, R_{j}^{l})|

where

(B L (H^{u}; u > 0)

is the set of bounded Lipschitz functions

h : H^{u} \to R

and

L^{\infty}

is the set of bounded functions.

Corollary 3.(See, [25]). Let

R_{1}, . . ., R_{n}

the real random variables such that

E (R_{j}) = 0

and

P (∣ R_{j} ∣ \leq M) = 1

for all

j = 1, . . ., n

and some

M < \infty

, Let

σ_{n}^{2} = V a r (\sum_{i = 1}^{n} Δ_{i})

Assume, furthermore, that there exist

K < \infty

and

β > 0

such that, for all u-uplets

(s_{1}, . . ., s_{u}) \in N^{u}

(t_{1}, . . ., t_{v}) \in N^{v}

with

1 \leq s_{1} \leq \dots \leq s_{u} \leq t_{1} \leq \dots \leq t_{v} \leq n .

The following inequality is fulfilled:

∣ c o v (R_{s_{1}} . . . R_{s_{u}}, R_{t_{1}} . . . R_{t_{v}}) ∣ \leq K^{2} M^{u + v - 2} v e^{- β (t_{1} - s_{u})} .

Then,

P (∣ \sum_{j = 1}^{n} R_{j} ∣ > t) \leq exp {- \frac{t^{2} / 2}{A_{n} + B_{n}^{1 / 3} t^{5 / 2}}}

for some

A_{n} \leq σ_{n}^{2}

and

B_{n} = (\frac{16 n K^{2}}{9 A n (1 - e^{-} β) \lor 1}) \frac{2 (K \lor M)}{1 - e^{-} β} .

The proof of Lemma 1

We put:

Δ_{i} = \frac{1}{n h_{H} E [K_{1}]} χ (R_{i}, S_{i}), 1 \leq i \leq n .

(11)

where :

ψ (R_{i}, S_{i}) = K (h_{K}^{- 1} d (r, R_{i})) H (s - S_{i}) - E [K_{1} H_{1}] 1 \leq i \leq n,

(12)

R_{i} \in H

S_{i} \in R

E (Δ_{i}) = 0

and,

| {\hat{F}}_{N}^{r} (s) - E {\hat{F}}_{N}^{r} (s) | = \sum_{i = 1}^{n} Δ_{i}

(13)

Moreover, we can write :

{| | ψ | |}_{\infty} \leq {2 C | | K | |}_{\infty} {| | H | |}_{\infty}

(14)

and :

L i p (ψ) \leq C (h_{K}^{- 1} {| | H | |}_{\infty} L i p (K) + h_{H}^{- 1} {| | K | |}_{\infty} L i p (H)) .

(15)

In order to use the lemma that was developed by [25], we need to do an evaluation of the variance term

V a r (\sum_{i = 1}^{n} Δ_{i})

as well as the covariance term

C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}})

. Both of these terms are denoted by the following for all

(s_{1}, . . ., s_{u}) \in N_{u}, (t_{1}, . . ., t_{v}) \in N_{v}

with

1 \leq s_{1} \leq . . . \leq s_{u} \leq t_{1} \leq . . . \leq t_{v} \leq n

We explore the following cases with respect to the covariance term: if

t_{1} = s_{u}

. Utilizing the Reality that

E [| K_{1} H_{1} |] = O_{a . c o} (ϕ_{r} (h_{K}))

and

E [| K_{1} |] = O_{a . c o} (ϕ_{r} (h_{K}))

we have:

\begin{matrix} | C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | & \leq {(\frac{C}{n E [K_{1}]})}^{u + v} E_{ψ} {| R_{1}, S_{1} |}^{u + v} \\ \leq {(\frac{{C | | K | |}_{\infty} {| | H | |}_{\infty}}{n E [K_{1}]})}^{u + v} E | K_{1} H_{1} | \\ \leq ϕ_{r} (h_{K}) {(\frac{C}{n ϕ_{r} (h_{K})})}^{u + v} \end{matrix}

(16)

If it

t_{1} > s_{u}

, Under

(P 5)

, the quasi-association yields:

\begin{matrix} | C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | & \leq {(\frac{h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H)}{n E [K_{1}]})}^{2} \times {(\frac{C}{n E [K_{1}]})}^{u + v - 2} \sum_{i = 1}^{u} \sum_{j = 1}^{v} λ_{s_{i}, t_{j}} \\ \leq {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} {(\frac{C}{n E [K_{1}]})}^{u + v} v λ_{t_{1} - s_{u}} \\ \leq {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} {(\frac{C}{ϕ_{r} (h_{K})})}^{u + v} v e^{- α (t_{1} - s_{u})} \end{matrix}

(17)

On the other hand, if we take into account

(P 6)

, we have:

\begin{matrix} | C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | & \leq {(\frac{{C | | K | |}_{\infty} {| | H | |}_{\infty}}{n E [K_{1}]})}^{u + v - 2} (E | Δ_{s_{u}}, Δ_{t_{1}} | + E | Δ_{s_{u}} | E | Δ_{t_{1}} |) \\ \leq {(\frac{{C | | K | |}_{\infty} {| | H | |}_{\infty}}{n E [K_{1}]})}^{u + v - 2} (\frac{C}{n E [K_{1}]}) \\ \times h_{H} (s u p_{i \neq j} P ((R_{i}, R_{j}) \in B (r, h_{K}) \times B_{θ} (r, h_{K}) \\ + P {(R_{1} \in B (r, h_{K}))}^{2})) \\ \leq {(\frac{C}{h_{H} ϕ_{r} (h_{K})})}^{u + v} {(ϕ_{r} (h_{K}))}^{2} \end{matrix}

(18)

In addition, by taking a

γ - p o w e r

of (10), and a

(1 - γ) - p o w e r

of (11), we are able to derive an upper-bound of the tree terms as follows, for:

1 \leq s_{1} \leq . . . s_{u} \leq t_{1} \leq . . . . \leq t_{v} \leq n,

| C o v (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | \leq h_{H} ϕ_{r} (h_{K}) {(\frac{C}{n h_{H} ϕ_{r} (h_{K})})}^{u + v}

(19)

Second, we entered the following information for the

V a r (\sum_{i = 1}^{n} Δ_{i})

variance term for all

1 \leq i \leq n

\begin{matrix} | V a r (\prod_{i = 1}^{u} Δ_{s_{i}}, \prod_{j = 1}^{v} Δ_{t_{j}}) | & = {(\frac{1}{n E [K_{1}]})}^{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} C o v (K_{i} H_{i}, K_{j} H_{j}) \\ = \underset{T_{1}}{\underset{︸}{{(\frac{1}{n E [K_{1}]})}^{2} V a r (K_{1} H_{1})}} \\ + \underset{T_{2}}{\underset{︸}{{(\frac{1}{n E [K_{1}]})}^{2} \sum_{i = 1}^{n} \sum_{j = 1, i \neq j}^{n} C o v (K_{i} H_{i}, K_{j} H_{j})}} \end{matrix}

(20)

For the first term

T_{1}

, we have :

V a r (K_{1} H_{1}) = E (K_{1}^{2} H_{1}^{2}) - {(E (K_{1} H_{1}))}^{2}

(21)

Then,

E [K_{1}^{2} H_{1}^{2}] = E [K_{1}^{2} E [H_{1}^{2} / X_{1}]]

(22)

As a result, considering (P2) and (P3) and integrating over the real component y gives us the following:

E [H_{1}^{2} ∣ R_{1}] = O (h_{H})

(23)

As for all

j \geq 1

I E [K_{1}^{j}] = O (ϕ_{x} (h_{K})) .

Then,

I E [K_{1}^{2} H_{1}^{2}] = O (ϕ_{x} (h_{K})) .

It follows that:

{(\frac{1}{n {(I E [K_{1}])}^{2}})}^{2} V a r (K_{1} H_{1}) = O (n ϕ_{x} (h_{K})) .

(24)

Regarding the covariance term in

(r e f 26)

, the following decomposition will be utilized.

\begin{matrix} \sum_{i = 1}^{n} \sum_{j = 1, i \neq j}^{n} C o v (K_{i} H_{i}, K_{j} H_{j}) & = & \underset{I}{\underset{︸}{\sum_{i = 1}^{n} \sum_{j = 1, 0 < ∣ i - j ∣ \leq m_{n}}^{n} C o v (K_{i} H_{i}, K_{j} H_{j})}} \\ + & \underset{I I}{\underset{︸}{\sum_{i = 1}^{n} \sum_{j = 1, ∣ i - j ∣ > m_{n}}^{n} C o v (K_{i} H_{i}, K_{j} H_{j})}} \end{matrix}

where

(m_{n})

is an infinite series of positive integers as n tends to infinity. The

(P 1)

(P 3)

and

(P 6)

Presuppositions

we obtain, if

i \neq j

\begin{matrix} I & \leq n m_{n} (max_{i \neq j} |E (K_{i} H_{i} K_{j} H_{j})| + {(E (K_{1} H_{1}))}^{2}) \\ \leq C n m_{n} (ϕ_{r}^{2} (h_{K}) + {(ϕ_{r} (h_{K}))}^{2}) \\ \leq error (ϕ_{r}^{2} (h_{K})) . \end{matrix}

(25)

H and K are both bounded and Lipschitz kernels, we obtain:

\begin{matrix} I I & \leq & {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} \sum_{i = 1}^{u} \sum_{j = 1_{∣ i - j ∣ > m_{n}}}^{v} λ_{i, j} \\ \leq & C {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} \sum_{i = 1}^{u} \sum_{j = 1_{∣ i - j ∣ > m_{n}}}^{v} λ_{i, j} \\ \leq & C n {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} λ_{m_{n}} \\ \leq & C n {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} e^{- α m_{n}} . \end{matrix}

(26)

Then, by (20) and (21), we get

\sum_{j = 1, i \neq j}^{n} C o v (K_{i} H_{i}, K_{j} H_{j}) \leq C (n m_{n} (h_{H}^{2} ϕ_{r}^{2} (h_{K})) + n {(h_{K}^{- 1} L i p (K) + h_{H}^{- 1} L i p (H))}^{2} e^{- α m_{n}})

By choosing :

m_{n} = log (\frac{{(h_{K}^{- 1} Lip (K) + h_{H}^{- 1} Lip (H))}^{2}}{α ϕ_{r}^{2} (h_{K})})

We get :

\frac{1}{ϕ_{r} (h_{K})} \sum_{j = 1, i \neq j}^{n} C o v (K_{i} H_{i}, K_{j} H_{j}) \underset{n \to + \infty}{\to} 0 .

(27)

Finally, we get this result by integrating the previous three results (13), (18), and (22).

V a r (\sum_{i = 1}^{n} Δ_{i}) = O (\frac{1}{n ϕ_{r} (h_{K})})

Therefore, the criteria of the lemma are satisfied by the variables

Δ_{i}

, where

i = 1, . . ., n

K_{n} = \frac{C}{n \sqrt{ϕ_{r} (h_{K})}}, M_{n} = \frac{C}{n ϕ_{r} (h_{K})}, a n d A_{n} = V a r (\sum_{i = 1}^{n} Δ_{i})

Thus,

\begin{matrix} P (|{\hat{F}}_{N}^{r} (s) - E {\hat{F}}_{N}^{r} (s)| > η \frac{log n}{n ϕ_{r} (h_{K})}) & = P (|\sum_{i = 1}^{n} Δ_{i}| > η \frac{log n}{n ϕ_{r} (h_{K})}) \\ \leq exp \{- \frac{η^{2} log n}{n ϕ_{r} (h_{K}) Var (\sum_{i = 1}^{n} Δ_{i}) + \frac{{log}^{5 / 6} n}{{(n ϕ_{r} (h_{K}))}^{(7 / 6)}}}\} \\ \leq exp \{- \frac{η^{2} log n}{C + \frac{{(log n)}^{5 / 6}}{n ϕ_{r} {(h_{K})}^{(7 / 6)}}}\} \\ \leq C^{'} exp \{- C η^{2} log n\} \end{matrix}

(28)

(P 7)

. Last but not least, with an appropriate selection of

η

, Borel-Cantelli’s lemma makes it possible to conclude the proof of Lemma 1.

The proof of Corollary 1

We have:

\{| F_{D}^{R} | \leq \frac{1}{2}\} \subseteq \{| F_{D}^{R} - 1 | > \frac{1}{2}\}

(29)

Therefor,

\begin{matrix} P \{| F_{D}^{R} | \leq \frac{1}{2}\} & \leq P \{| F_{D}^{R} - 1 | > \frac{1}{2}\} \\ \leq P \{| F_{D}^{X} - E F_{D}^{X} | > \frac{1}{2}\} \end{matrix}

(30)

For

E (F_{D}^{R}) = 1

, we apply the result of Lemma 1 we show that :

P \{| F_{D}^{R} | \leq \frac{1}{2}\} < \infty

(31)

The proof of the Corollary 1 has finished.

The proof of Lemma 2

We have :

\begin{matrix} E {\hat{F}}_{N}^{R} (S) - F^{R} (S) & = \frac{1}{n E K_{1}} \sum_{i = 1}^{n} E K_{i} H_{i} (y) - F^{R} (S) \\ = \frac{1}{E K_{1}} [E K_{1} H_{1} (\frac{s - S_{i}}{h_{H}}) - F^{R} (S)] \\ = \frac{1}{E K_{1}} E (K_{1} [E (H_{1} (h_{H}^{- 1} (s - S_{i})) / R) - F^{R} (S)]) \end{matrix}

(32)

We derive the following by making use of the stationarity of the data, the conditioning by the variable that is doing the explaining, and the customary change in the variable

t = \frac{s - u}{h_{H}}

\begin{matrix} E (H_{1} (h_{H}^{- 1} (s - S_{i})) / R) & = \int_{R} H (\frac{s - u}{h_{H}}) f^{R} (u) d u \\ = \int_{R} H^{(1)} (t) F^{R} (s - h_{H} t) d t \end{matrix}

(33)

and we deduce ,

\begin{matrix} |E (H_{1} (h_{H}^{- 1} (s - S_{i})) / R) - F^{R} (s)| & = |\int_{R} H^{(1)} (t) F^{R} (s - h_{H} t) d t - F^{R} (s)| \\ \leq \int_{R} H^{(1)} (t) |F^{R} (s - h_{H} t) - F^{R} (s)| d t \end{matrix}

So, by (H2) we get

|E (H_{1} (h_{H}^{- 1} (s - S_{i})) / R) - F^{R} (s)| \leq A_{R} \int_{R} H^{(1)} (h_{K}^{b_{1}} + {| t |}^{b_{2}} h_{H}^{b_{2}}) d t

This inequality holds true everywhere in y and after substituting in (29) and simplifying the expression

(E K_{1})

we get that:

E {\hat{F}}_{N}^{R} (s) - F^{R} (s) \leq A_{R} (h_{K}^{b_{1}} \int_{R} H^{(1)} (t) d t + h_{H}^{b_{2}} \int_{R} {| t |}^{b_{2}} H^{(1)} (t) d t)

In conclusion, the evidence of Lemma 2 is provided by Hypothesis (P4) and Corollary 1.

Substituting for

ψ (., .)

in Lemma 1’s proof of Lemma 3 yields a simple application of Lemma 3.

ψ (R_{i}) = K (h_{K}^{- 1} d (r, R_{i})) - E [K_{1}], \forall R_{i} \in H .

(34)

The proof of Theorem 1 has finished. □

Author Contributions

Funding

Informed Consent Statement

Not applicable

Data Availability Statement

Not applicable

Data Availability Statement

Not applicable

Acknowledgments

Conflicts of Interest

No conflict of interest.

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Figure 1. A sample of 200 curves.

Figure 2. The QQ-plot of the obtained sample.

Table 1. Kolmogorov-Smirnov analysis.

m	1	10
P-value	0.88	0.43

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The Consistency of the Distribution Function Conditional Estimate and Application on the Consistency and Asymptotic Normality of the Conditional Hazard Function Estimate for High Dimensional Quasi-Associated Data

Abstract

1. Introduction

2. Model and Estimator

3. Hypotheses and Notations

4. Remarks Concerning the Hypotheses

5. Main Results

6. Application on the Conditional Hazard Function Estimate

6.1. Almost Complete Convergence with the Rate

6.2. Asymptotic Normality

7. Confidence Bands

8. A simulation Study

9. Auxiliary Results and Proofs

Author Contributions

Funding

Informed Consent Statement

Data Availability Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Abstract

1. Introduction

2. Model and Estimator

3. Hypotheses and Notations

4. Remarks Concerning the Hypotheses

5. Main Results

6. Application on the Conditional Hazard Function Estimate

6.1. Almost Complete Convergence with the Rate

6.2. Asymptotic Normality

7. Confidence Bands

8. A simulation Study

9. Auxiliary Results and Proofs

Author Contributions

Funding

Informed Consent Statement

Data Availability Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

MDPI Initiatives

Important Links

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