Inference and other aspects for $ q- $Weibull distribution via generalized order statistics with applications to medical datasets

M. Nagy; H. M. Barakat; M. A. Alawady; I. A. Husseiny; A. F. Alrasheedi; T. S. Taher; A. H. Mansi; M. O. Mohamed; M. Nagy; H. M. Barakat; M. A. Alawady; I. A. Husseiny; A. F. Alrasheedi; T. S. Taher; A. H. Mansi; M. O. Mohamed

doi:10.3934/math.2024404

AIMS Mathematics

2024, Volume 9, Issue 4: 8311-8338. doi: 10.3934/math.2024404

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Research article Special Issues

Inference and other aspects for $q-$ Weibull distribution via generalized order statistics with applications to medical datasets

1.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O.Box 2455, Riyadh 11451, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
3.
DICA, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, MI, Italy

Received: 03 January 2024 Revised: 27 January 2024 Accepted: 02 February 2024 Published: 27 February 2024
MSC : 62B10, 62G30

This work utilizes generalized order statistics (GOSs) to study the $q$ -Weibull distribution from several statistical perspectives. First, we explain how to obtain the maximum likelihood estimates (MLEs) and utilize Bayesian techniques to estimate the parameters of the model. The Fisher information matrix (FIM) required for asymptotic confidence intervals (CIs) is generated by obtaining explicit expressions. A Monte Carlo simulation study is conducted to compare the performances of these estimates based on type Ⅱ censored samples. Two well-established measures of information are presented, namely extropy and weighted extropy. In this context, the order statistics (OSs) and sequential OSs (SOSs) for these two measures are studied based on this distribution. A bivariate $q$ -Weibull distribution based on the Farlie-Gumbel-Morgenstern (FGM) family and its relevant concomitants are studied. Finally, two concrete instances of medical real data are ultimately provided.

Keywords:

Citation: M. Nagy, H. M. Barakat, M. A. Alawady, I. A. Husseiny, A. F. Alrasheedi, T. S. Taher, A. H. Mansi, M. O. Mohamed. Inference and other aspects for $q-$ Weibull distribution via generalized order statistics with applications to medical datasets[J]. AIMS Mathematics, 2024, 9(4): 8311-8338. doi: 10.3934/math.2024404

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Abstract

1. Introduction

Let $q$ be a positive integer. For each integer $a$ with $1 \leqslant a < q, (a, q) = 1$ , we know that there exists one and only one $\bar{a}$ with $1 \leqslant\bar{a} < q$ such that $a \bar{a} \equiv 1(q)$ . Let $r(q)$ be the number of integers $a$ with $1 \leqslant a < q$ for which $a$ and $\bar{a}$ are of opposite parity.

D. H. Lehmer (see ^[1]) posed the problem to investigate a nontrivial estimation for $r(q)$ when $q$ is an odd prime. Zhang ^[2,3] gave some asymptotic formulas for $r(q)$ , one of which reads as follows:

$r(q) = \frac{1}{2} \phi(q)+O\left(q^{\frac{1}{2}} d^{2}(q) \log ^{2} q\right).$

Zhang ^[4] generalized the problem over short intervals and proved that

$\sum\limits_{\substack{a \leq N \\ a \in R(q)}} 1 = \frac{1}{2} N \phi(q) q^{-1}+O\left(q^{\frac{1}{2}} d^{2}(q) \log ^{2} q\right),$

where

$R(q): = \{a: 1 \leqslant a \leqslant q,(a, q) = 1,2 \nmid a+\bar{a}\} .$

Let $n \geqslant 2$ be a fixed positive integer, $q \geqslant 3$ and $c$ be two integers with $(n, q) = (c, q) = 1$ . Let $0 < \delta_{1}, \delta_{2} \leq 1.$ Lu and Yi ^[5] studied the Lehmer problem in the sense of short intervals as

$r_{n}\left(\delta_{1}, \delta_{2}, c ; q\right): = \mathop {\sum\limits_{{a \leqslant \delta_{1} q}} {\sum\limits_{{\bar{a} \leqslant \delta_{2} q}} {} } }\limits_{\scriptstyle{a \bar{a}\equiv c\bmod q}\atop \scriptstyle {n \nmid a+\bar{a} }} 1 ,$

and obtained an interesting asymptotic formula,

$r_{n}\left(\delta_{1}, \delta_{2}, c ; q\right) = \left(1-n^{-1}\right) \delta_{1} \delta_{2} \phi(q)+O\left(q^{\frac{1}{2}} d^{6}(q) \log ^{2} q\right).$

Liu and Zhang ^[6] $r$ -th residues and roots, and obtained two interesting mean value formulas. Guo and Yi ^[7] found the Lehmer problem also has good distribution properties on Beatty sequences. For fixed real numbers $\alpha$ and $\beta$ , the associated non-homogeneous Beatty sequence is the sequence of integers defined by

$\mathcal{B}_{\alpha, \beta}: = (\lfloor\alpha n+\beta\rfloor)_{n = 1}^{\infty},$

where $\lfloor t\rfloor$ denotes the integer part of any $t \in \mathbb{R}$ . Such sequences are also called generalized arithmetic progressions. If $\alpha$ is irrational, it follows from a classical exponential sum estimate of Vinogradov ^[8] that $\mathcal{B}_{\alpha, \beta}$ contains infinitely many prime numbers; in fact, one has the asymptotic estimate

$\#\left\{\text { prime } p \leqslant x: p \in \mathcal{B}_{\alpha, \beta}\right\} \sim \alpha^{-1} \pi(x) \quad \text { as } \quad x \rightarrow \infty$

where $\pi(x)$ is the prime counting function.

We define type $\tau = \tau(\alpha)$ for any irrational number $\alpha$ by the following definition:

$\tau: = \sup \left\{t \in \mathbb{R}: \liminf \limits_{n \rightarrow \infty} n^t\|\alpha n\| = 0\right\}.$

Based on the results obtained, we consider the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals in this paper. That is,

$r_{n}\left(\delta_{1}, \delta_{2}, \cdots ,\delta_{k}, c, \alpha, \beta ; q\right): = \mathop {\sum\limits_{{x_{1} \leqslant \delta_{1} q}} { \cdots \sum\limits_{{x_{k} \leqslant \delta_{k} q}} {} } }\limits_{\scriptstyle {x_{1} \cdots x_{k} \equiv c\bmod q } \atop {\scriptstyle {x_{1}, \cdots x_{k-1} \in B_{\alpha,\beta}} \atop \scriptstyle {n \nmid x_{1}+\cdots+x_{k}}}} 1,(0 < \delta_{1}, \delta_{2},\cdots, \delta_{k} \leq 1),$

and where k = 2, we get the result of ^[7].

By using the properties of Beatty sequences and the estimates for hyper Kloosterman sums, we obtain the following result.

Theorem 1.1. Let $k \geq 2$ be a fixed positive integer, $q\geq n^{3}$ and $c$ be two integers with $(n, q) = (c, q) = 1$ , and $\delta_{1}, \delta_{2}, \cdots, \delta_{k}$ be real numbers satisfying $0 < \delta_{1}, \delta_{2}, \cdots, \delta_{k} \leq 1$ . Let $\alpha > 1$ be an irrational number of finite type. Then, we have the following asymptotic formula:

$r_{n}\left(\delta_{1}, \delta_{2}, \cdots ,\delta_{k}, c, \alpha, \beta ; q\right) = \left(1-n^{-1}\right) \alpha^{-(k-1)} \delta_{1} \delta_{2} \cdots \delta_{k}\phi^{k-1}(q)+O(q^{k-1-\frac{1}{\tau+1}+\varepsilon} ),$

where $\phi(\cdot)$ is the Euler function, $\varepsilon$ is a sufficiently small positive number, and the implied constant only depends on $n$ .

Notation. In this paper, we denote by $\lfloor t\rfloor$ and $\{t\}$ the integral part and the fractional part of $t$ , respectively. As is customary, we put

$\mathbf{e}(t): = e^{2 \pi i t} \quad \text { and } \quad\{t\}: = t-\lfloor t\rfloor .$

The notation $\|t\|$ is used to denote the distance from the real number $t$ to the nearest integer; that is,

$\|t\|: = \min \limits_{n \in \mathbb{Z}}|t-n| .$

Let $\chi^{0}$ be the principal character modulo $q$ . The letter $p$ always denotes a prime. Throughout the paper, $\varepsilon$ always denotes an arbitrarily small positive constant, which may not be the same at different occurrences; the implied constants in symbols $O, \ll$ and $\gg$ may depend (where obvious) on the parameters $\alpha, n, \varepsilon$ but are absolute otherwise. For given functions $F$ and $G$ , the notations $F \ll G$ , $G \gg F$ and $F = O(G)$ are all equivalent to the statement that the inequality $|F| \leqslant \mathcal{C}|G|$ holds with some constant $\mathcal{C} > 0$ .

2. Preliminary lemmas

To complete the proof of the theorem, we need the following several definitions and lemmas.

Definition 2.1. For an arbitrary set $\mathcal{S}$ , we use $\mathbf{1}_{\mathcal{S}}$ to denote its indicator function:

$\mathbf{1}_{\mathcal{S}}(n): = \begin{cases}1 & { if } \;n \in \mathcal{S}, \\ 0 & { if }\; n \notin \mathcal{S} .\end{cases}$

We use $\mathbf{1}_{\alpha, \beta}$ to denote the characteristic function of numbers in a Beatty sequence:

$\mathbf{1}_{\alpha, \beta}(n): = \begin{cases}1 & { if } \;n \in \mathcal{B}_{\alpha, \beta}, \\ 0 & { if }\; n \notin \mathcal{B}_{\alpha, \beta}.\end{cases}$

Lemma 2.2. Let $a, q$ be integers, $\delta \in(0, 1)$ be a real number, $\theta$ be a rational number. Let $\alpha$ be an irrational number of finite type $\tau$ and $H = q^{\varepsilon} > 0$ . We have

$\sum\limits_{\scriptstyle {a \le \delta q} \atop \scriptstyle{a \in {{\cal B}_{\alpha ,\beta }}}} ' 1 = \alpha^{-1} \delta \phi(q)+O\left((\phi(q))^{\frac{\tau}{\tau+1}+\varepsilon}\right),$

and

$\sum\limits_{\substack{a \leqslant \delta q \\ a \in \mathcal{B}_{\alpha, \beta}}} \mathbf{e}(\theta a) = \alpha^{-1} \sum\limits_{a \leqslant \delta_1 q} \mathbf{e}(\theta a)+O\left(\|\theta\|^{-1} q^{-\varepsilon}+q^{\varepsilon}\right).$

Taking

$H = \|\theta\|^{-\frac{1}{\tau+1}+\varepsilon},$

we have

$\sum\limits_{\substack{a \leqslant \delta q \\ a \in B_{\alpha, \beta}}} \mathbf{e}(\theta a) = \alpha^{-1} \sum\limits_{a \leqslant \delta_1 q} \mathbf{e}(\theta a)+O\left(\|\theta\|^{-\left(\frac{\tau}{\tau+1}+\varepsilon\right)}\right) .$

Proof. This is Lemma 2.4 and Lemma 2.5 of ^[7].

Lemma 2.3. Let

$\mathbf{Kl}(r_{1},r_{2},\cdots,r_{k};q) = \sum\limits_{x_{1} \leqslant q-1} \cdots \sum\limits_{x_{k-1} \leqslant q-1} \mathbf{e}\left(\frac {r_{1}x_{1}+\cdots+r_{k-1}x_{k-1}+ r_{k}\overline{x_{1} \cdots x_{k-1}}}{p}\right).$

Then

$\mathbf{Kl}(r_{1},r_{2},\cdots,r_{k};q) \ll q^{\frac{k-1}{2}} k^{\omega(q)}\left(r_{1}, r_{k}, q\right)^{\frac{1}{2}} \cdots\left(r_{k-1}, r_{k}, q\right)^{\frac{1}{2}}$

where $(a, b, c)$ is the greatest common divisor of $a, b$ and $c$ .

Proof. See ^[9].

Lemma 2.4. Assume that $U$ is a positive real number, $K$ is a positive integer and that $a$ and $b$ are two real numbers. If

$a = \frac{s}{r}+\frac{\theta}{r^{2}}, \quad(r, s) = 1, r \geq 1,|\theta| \leq 1,$

then

$\sum\limits_{k \leqslant K} \min (U, \frac{1}{\|a k+b\|}) \ll (\frac{K}{r}+1 )(U+r \log r).$

Proof. The proof is given in ^[10].

3. Proof of theorem

We begin by the definition

$r_{n}\left(\delta_{1}, \delta_{2}, \cdots ,\delta_{k}, c, \alpha, \beta ; q\right) = S_{1}-S_{2},$

where

$S_{1}: = \mathop {\sum\limits_{{x_{1} \leqslant \delta_{1} q}} { \cdots \sum\limits_{{x_{k} \leqslant \delta_{k} q}} {} } }\limits_{\scriptstyle {x_{1} \cdots x_{k} \equiv c\bmod q } \atop {\scriptstyle {x_{1}, \cdots x_{k-1} \in \mathcal{B}_{\alpha,\beta}} }} 1,$

and

$S_{2}: = \mathop {\sum\limits_{{x_{1} \leqslant \delta_{1} q}} { \cdots \sum\limits_{{x_{k} \leqslant \delta_{k} q}} {} } }\limits_{\scriptstyle{x_{1} \cdots x_{k} \equiv c\bmod q }\atop {\scriptstyle{x_{1}, \cdots x_{k-1} \in \mathcal{B}_{\alpha,\beta}}\atop \scriptstyle{n \mid x_{1}+\cdots+x_{k}}}} 1.$

By the Definition 2.1, Lemma 2.2 and congruence properties, we have

$\begin{aligned} S_{1}& = \mathop{\sum\limits_{x_{1} \leqslant \delta_{1} q} \cdots \sum\limits_{x_{k} \leqslant \delta_{k} q}}_{x_{1} \cdots x_{k} \equiv c\bmod q }\mathbf{1}_{\alpha,\beta}\left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right)\\ & = \frac{1}{\phi(q)} \sum\limits_{x_{1} \leqslant \delta_{1} q} \cdots \sum\limits_{x_{k} \leqslant \delta_{k} q} \sum\limits_{\chi \bmod q}\chi(x_{1}) \cdots \chi(x_{k}) \chi(\overline c)\mathbf{1}_{\alpha,\beta} \left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right)\\ & = S_{11}+S_{12}, \end{aligned}$

where

$\begin{align*} S_{11}: = \frac{1}{\phi(q)}\mathop{ {\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \cdots \mathop{ {\sum}^{\prime}}_{x_{k} \leqslant \delta_{k} q} \mathbf{1}_{\alpha,\beta}\left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right), \end{align*}$

and

$S_{12}: = \frac{1}{\phi(q)} \mathop{\sum\limits_{\chi \bmod q}}_{\chi \neq \chi_{0}} \chi(\overline c) \left(\sum\limits_{x_{1} \leqslant \delta_{1} q} \cdots \sum\limits_{x_{k} \leqslant \delta_{k} q} \chi(x_{1}) \cdots \chi(x_{k}) \mathbf{1}_{\alpha,\beta} ( x_{1}) \cdots \mathbf{1}_{\alpha,\beta}( x_{k-1} )\right).$

For $S_{2}$ , it follows that

$\begin{aligned} S_{2}& = \frac{1}{\phi(q)} \mathop{\sum\limits_{x_{1} \leqslant \delta_{1} q} \cdots \sum\limits_{x_{k} \leqslant \delta_{k} q}}_{n \mid x_{1}+\cdots+x_{k}} \sum\limits_{\chi \bmod q}\chi(x_{1}) \cdots \chi(x_{k}) \chi(\overline c)\mathbf{1}_{\alpha,\beta} \left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right)\\ & = S_{21}+S_{22}, \end{aligned}$

where

$\begin{align*} S_{21}: = \frac{1}{\phi(q)} \mathop{\mathop{ {\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \cdots \mathop{ {\sum}^{\prime}}_{x_{k} \leqslant \delta_{k} q} }_{n \mid x_{1}+\cdots+x_{k}} \mathbf{1}_{\alpha,\beta}\left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right), \end{align*}$

and

$\begin{align*} S_{22}: = \frac{1}{\phi(q)} \mathop{\sum\limits_{\chi \bmod q}}_{\chi \neq \chi_{0}} \chi(\overline c) \mathop{{\sum\limits_{x_{1} \leqslant \delta_{1} q}} \cdots {\sum\limits_{x_{k} \leqslant \delta_{k} q}}}_{n \mid x_{1}+\cdots+x_{k}} \chi(x_{1}) \cdots \chi(x_{k-1}) \mathbf{1}_{\alpha,\beta} \left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1}\right) . \end{align*}$

3.1. Estimation of $S_{11}$

From the classical bound

$\sum \limits_{a \le \delta q}' 1 = \delta \phi(q)+O\left(d(q)\right)$

and Lemma 2.2, we have

$\begin{align} S_{11}& = \frac{1}{\phi(q)} \left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right) \left(\mathop{{\sum}^{\prime}}_{x_{k} \leqslant \delta_{k} q}1\right) \\ & = \left(\delta_{k}+O\left(\frac{d(q)}{\phi(q)}\right)\right)\prod \limits_{i = 1}^{k-1}\left( \alpha^{-1} \delta_{i} \phi(q)+O\left((\phi(q))^{\frac{\tau}{\tau+1}+\varepsilon}\right)\right) \\ & = \alpha^{-(k-1)}\phi^{k-1}(q)\prod \limits_{i = 1}^{k-1} \delta_{i}+O\left(q^{k-1-\frac{1}{\tau+1}+\varepsilon}\right). \end{align}$

(3.1)

3.2. Estimation of $S_{21}$

From Lemma 2.2, we obtain

$\begin{align} S_{21}& = \frac{1}{\phi(q)}\left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right)\left(\mathop{\mathop{{\sum}^{\prime}}_{x_{k} \leqslant \delta_{k} q}}_{n \mid x_{k}+(x_{1}+ \cdots +x_{k-1})}1\right) \\ & = \frac{1}{\phi(q)}\left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right) \left(\mathop{\sum\limits_{x_{k} \leqslant \delta_{k} q }}_{x_{k} \equiv-(x_{1}+ \cdots +x_{k-1}) \bmod n} \sum\limits_{\substack{d \mid(x_{k}, q)}} \mu(d)\right)\\ & = \frac{1}{\phi(q)}\left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right)\left( \sum\limits_{\substack{d \mid q}} \mu(d) \mathop{\mathop{\sum\limits_{x_{k} \leqslant \delta_{k}q}}_{d \mid x_{k}}}_{x_{k} \equiv-(x_{1}+ \cdots +x_{k-1})\bmod n} 1 \right) \\ & = \frac{1}{\phi(q)} \left(\mathop{{\sum}^{\prime}}_{x_{1} \leqslant \delta_{1} q} \mathbf{1}_{\alpha,\beta}( x_{1} )\right) \cdots \left(\mathop{{\sum}^{\prime}}_{x_{k-1} \leqslant \delta_{k-1} q}\mathbf{1}_{\alpha,\beta}( x_{k-1} )\right)\left( \sum\limits_{\substack{d \mid q}} \mu(d) \left( \frac{\delta_{k}q}{nd}+O(1)\right)\right) \\ & = \frac{1}{\phi(q)}\left(\frac{\delta_{k}\phi(q)}{n}+O\left(d(q)\right) \right)\prod \limits_{i = 1}^{k-1}\left( \alpha^{-1} \delta_{i} \phi(q)+O\left((\phi(q))^{\frac{\tau}{\tau+1}+\varepsilon}\right)\right)\\ & = \alpha^{-(k-1)}n^{-1}\phi^{k-1}(q)\prod \limits_{i = 1}^{k-1} \delta_{i}+O (q^{k-1-\frac{1}{\tau+1}+\varepsilon} ). \end{align}$

(3.2)

3.3. Estimation of $S_{22}$ and $S_{12}$

By the properties of exponential sums,

$\begin{align} S_{22} = &\frac{1}{n \phi(q)} \mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}} \chi(\overline c) \left({\sum\limits_{x_{1} \leqslant \delta_{1} q}}\cdots {\sum\limits_{x_{k} \leqslant \delta_{k-1} q}}\chi(x_{1}) \cdots \chi(x_{k}) \mathbf{1}_{\alpha,\beta} \left( x_{1} \right) \cdots \mathbf{1}_{\alpha,\beta}\left( x_{k-1} \right)\right) \\ &\times \left(\sum \limits_{l = 1}^{n}\mathbf{e}(\frac{x_{1}+\cdots+x_{k}}{n}l) \right)\\ = &\frac{1}{n \phi(q)} \mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}} \chi(\overline c)\sum \limits_{l = 1}^{n} \prod \limits_{i = 1}^{k-1}\left( \sum\limits_{x_{i} \leqslant \delta_{i} q}\mathbf{1}_{\alpha, \beta}(x_{i}) \chi(x_{i}) \mathbf{e}(\frac{x_{i}}{n} l)\right)\left( \sum\limits_{x_{k} \leqslant \delta_{k} q} \chi(x_{k}) \mathbf{e} (\frac{x_{k}}{n} l)\right). \end{align}$

(3.3)

Let

$G(r, \chi): = \sum\limits_{h = 1}^{q} \chi(h) \mathbf{e} (\frac{r h}{q} )$

be the Gauss sum, and we know that for $\chi \neq \chi_{0}$ ,

$\chi(x_{i}) = \frac{1}{q} \sum\limits_{r = 1}^{q} G(r, \chi) \mathbf{e} (-\frac{x_{i} r}{q} ) = \frac{1}{q} \sum\limits_{r = 1}^{q-1} G(r, \chi) \mathbf{e} (-\frac{x_{i} r}{q} ),$

and

$\frac{l}{n}-\frac{r}{q} \neq 0$

for $1 \leqslant l \leqslant n, 1 \leqslant r \leqslant q-1$ and $(n, q) = 1$ .

Therefore,

$\begin{equation} \sum\limits_{x_{k} \leqslant \delta_{k} q} \chi(x_{k}) \mathbf{e} (\frac{x_{k}}{n} l ) = \frac{1}{q} \sum\limits_{r_{k} = 1}^{q-1} G\left(r_{k}, \chi\right) \frac{f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\mathbf{e} (\frac{r_{k}}{q}-\frac{l}{h})-1}, \end{equation}$

(3.4)

where

$f(\delta, l, r ; n, p): = 1-\mathbf{e}\left( (\frac{l}{n}-\frac{r}{q} )\lfloor\delta q\rfloor\right)$

and

$\left|f\left(\delta_{k}, l, r_{k} ; n, q\right)\right| \leqslant 2.$

For $x_{i}(1\leqslant i \leqslant k-1)$ , using Lemma 2.2, we also have

$\begin{align} & \sum\limits_{x_{i} \leqslant \delta_{i} q} \mathbf{1}_{\alpha, \beta}(x_{i}) \chi(x_{i}) \mathbf{e} (\frac{x_{i}}{n} l ) \\ = & \frac{1}{q} \sum\limits_{x_{i} \leqslant \delta_{i} q} \mathbf{1}_{\alpha, \beta}(x_{i}) \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \mathbf{e}\left( (\frac{l}{n}-\frac{r_{i}}{q} ) x_{i}\right) \\ = & \frac{1}{q} \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \sum\limits_{x_{i} \leqslant \delta_{i} q} \mathbf{1}_{\alpha, \beta}(x_{i}) \mathbf{e}\left( (\frac{l}{n}-\frac{r_{i}}{q} ) x_{i}\right) \\ = & \frac{1}{q} \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \left(\alpha^{-1}\sum\limits_{a \leqslant \delta_{i} q} \mathbf{e}\left( (\frac{l}{n}-\frac{r_{i}}{q} ) x_{i}\right)+O\left(\frac{q^{-\varepsilon}}{\|\frac{l}{n}-\frac{r_{i}}{q}\|} +q^{\varepsilon}\right)\right) \\ = &\frac{1}{q \alpha} \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \left(\frac{f\left(\delta_{i} , l, r_{i} ; n, q\right)}{\mathbf{e} (\frac{r_{i}}{q}-\frac{l}{n} )-1}+O\left(\frac{q^{-\varepsilon}}{\|\frac{l}{n}-\frac{r_{i}}{q}\|} +q^{\varepsilon}\right)\right) . \end{align}$

(3.5)

Let

$\begin{align} S_{23}& = \frac{1}{n \phi(q)} \mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}} \chi(\overline c)\sum \limits_{l = 1}^{n}\prod\limits_{i = 1}^{k-1} \left(\frac{1}{q \alpha} \sum\limits_{r_{i} = 1}^{q-1} G\left(r_{i}, \chi\right) \frac{f\left(\delta_{i}, l, r_{i} ; n, q\right)}{\mathbf{e} (\frac{r_{i}}{q}-\frac{l}{n})-1}\right)\left(\frac{1}{q} \sum\limits_{r_{k} = 1}^{q-1} G\left(r_{k}, \chi\right) \frac{f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\mathbf{e}(\frac{r_{k}}{q}-\frac{l}{n})-1}\right) \\ & = \frac{1}{n \phi(q) q^{k} \alpha^{k-1}} \sum\limits_{l = 1}^{n}\sum\limits_{r_{1} = 1}^{q-1}\cdots \sum\limits_{r_{k} = 1}^{q-1} \frac{f\left(\delta_{1} , l, r_{1} ; n, q\right)\cdots f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\left(\mathbf{e} (\frac{r_{1}}{q}-\frac{l}{n} )-1\right)\cdots \left(\mathbf{e} (\frac{r_{k}}{q}-\frac{l}{n} )-1\right)} \\ &\times \mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}} \chi(\overline c)G\left(r_{1}, \chi\right)\cdots G\left(r_{k}, \chi\right). \end{align}$

(3.6)

From the definition of Gauss sum and Lemma 2.3, we know that

$\begin{align} &\sum\limits_{\chi \mathrm{mod} q}\chi(\overline c)G\left(r_{1}, \chi\right)\cdots G\left(r_{k}, \chi\right)\\ = &\sum\limits_{h_{1} = 1}^{q-1}\cdots \sum\limits_{h_{k} = 1}^{q-1}\sum\limits_{\chi \mathrm{mod}q}\chi(\overline c)\chi(h_{1})\cdots \chi(h_{k})\mathbf{e} ( \frac{r_{1}h_{1}+\cdots +r_{k}h_{k}}{q} )\\ = &\phi(q)\mathop{\sum\limits_{h_{1} = 1}^{q-1}\cdots \sum\limits_{h_{k} = 1}^{q-1}}_{h_{1} \cdots h_{k} \equiv c \bmod q}\mathbf{e} ( \frac{r_{1}h_{1}+ \cdots +r_{k}h_{k}}{q} )\\ = &\phi(q)\sum\limits_{h_{1} = 1}^{q-1}\cdots \sum\limits_{h_{k} = 1}^{q-1}\mathbf{e} ( \frac{r_{1}h_{1}+ \cdots r_{k-1}h_{k-1}+r_{k}c\overline{h_{1} \cdots h_{k-1}}}{q} )\\ = &\phi(q) \mathbf{Kl}(r_{1},r_{2},\cdots,r_{k}c;q) \\ \ll& \phi(q) q^{\frac{k-1}{2}} k^{\omega(q)}\left(r_{1}, r_{k}c, q\right)^{\frac{1}{2}} \cdots\left(r_{k-1}, r_{k}c, q\right)^{\frac{1}{2}} \\ \ll&\phi(q) q^{\frac{k-1}{2}} k^{\omega(q)}\left(r_{1}, q\right) \cdots\left(r_{k}, q\right). \end{align}$

(3.7)

By Mobius inversion, we get

$G(r, \chi_{0}) = \sum\limits_{h = 1}^{q}' \mathbf{e} (\frac{r h}{q} ) = \mu\left(\frac{q}{(r, q)}\right) \frac{\varphi(q)}{\varphi(q /(r, q))} \ll(r, q),$

and

$\chi_{0}(\overline c)G\left(r_{1}, \chi_{0}\right)\cdots G\left(r_{k}, \chi_{0}\right) \ll\left(r_{1}, q\right) \cdots\left(r_{k}, q\right).$

Hence,

$\begin{align} &\mathop{\sum\limits_{\chi \mathrm{mod} q}}_{\chi \neq \chi_{0}}\chi(\overline c)G\left(r_{1}, \chi\right)\cdots G\left(r_{k}, \chi\right)\\ = &\sum\limits_{\chi \mathrm{mod} q}\chi(\overline c)G\left(r_{1}, \chi\right)\cdots G\left(r_{k}, \chi\right)-\chi_{0}(\overline c)G\left(r_{1}, \chi_{0}\right)\cdots G\left(r_{k}, \chi_{0}\right)\\ \ll&\phi(q) q^{\frac{k-1}{2}} k^{\omega(q)}\left(r_{1}, q\right) \cdots\left(r_{k}, q\right). \end{align}$

(3.8)

From (3.8) we may deduce the following result:

$\begin{align} S_{23}&\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{r = 1}^{q-1} \frac{(r,q)}{\left|\mathbf{e} (\frac{r}{q}-\frac{l}{n} )-1\right|}\right)^{k}\\ &\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{r = 1}^{q-1} \frac{(r,q)}{\left|\sin \pi (\frac{r}{q}-\frac{l}{n} )\right|}\right)^{k}\\ &\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{r = 1}^{q-1} \frac{(r,q)}{\|\frac{r}{q}-\frac{l}{n}\|}\right)^{k}\\ & = \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\mathop{\sum\limits_{d \mid q}}_{d < q }\mathop{\sum\limits_{r \leq q-1}}_{(r,q) = d }\frac{d}{\|\frac{r}{q}-\frac{l}{n}\|}\right)^{k }\\ & = \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\mathop{\sum\limits_{d \mid q}}_{d < q }d\mathop{\sum\limits_{m \leq\frac{q-1}{d} }}_{(m,q) = 1}\frac{1}{\|\frac{md}{q}-\frac{l}{n}\|}\right)^{k }\\ & = \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\mathop{\sum\limits_{d \mid q}}_{d < q }d\sum\limits_{k \mid q}\mu(k)\sum\limits_{m \leq\frac{q-1}{kd} }\frac{1}{\|\frac{mkd}{q}-\frac{l}{n}\|}\right)^{k }. \end{align}$

It is easy to see

$\|\frac{mkd}{q}-\frac{l}{n}\| = \|\frac{mkn-l(q/d)}{(q/d)n}\| \geq \frac{1}{(q/d)n},$

and we obtain

$S_{23}\ll\frac{k^{\omega(q)}}{n \phi(q) q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\mathop{\sum\limits_{d \mid q\\{d < q }}}d\sum\limits_{k \mid q}\sum\limits_{m \leq\frac{q-1}{kd} }\min (\frac{qn}{d},\frac{1}{\|\frac{mkd}{q}-\frac{l}{n}\|} )\right)^{k }.$

Let $k d / q = h_{0} / q_{0}$ , where $q_{0} \geq 1, \left(h_{0}, q_{0}\right) = 1$ , and we will easily obtain $q /(k d) \leq q_{0} \leq q / d$ . By using Lemma 2.4, we have

$\begin{align} S_{23}&\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{\substack{d \mid q \\ d < q}} d \sum\limits_{k \mid q}\left(\frac{(q-1) /(k d)}{q_{0}}+1\right) (\frac{q n}{d}+q_{0} \log q_{0} )\right)^{k}\\ &\ll \frac{k^{\omega(q)}}{n q^{\frac{k+1}{2}} \alpha^{k-1}}\sum\limits_{l = 1}^{n}\left(\sum\limits_{\substack{d \mid q \\ d < q}} d \sum\limits_{k \mid q}\left(\frac{(q-1) /(k d)}{q/(kd)}+1\right) (\frac{q n}{d}+\frac{q}{d} \log \frac{q}{d} )\right)^{k}\\ &\ll \frac{k^{\omega(q)}q^{\frac{k-1}{2}}}{ \alpha^{k-1}}\left(\sum\limits_{\substack{d \mid q \\ d < q}} \sum\limits_{k \mid q}n+\log q\right)^{k}\\ &\ll q^{\frac{k-1}{2}}d^{2k}(q)(\log q+n)^{k}. \end{align}$

Let

$S_{24}: = \frac{q^{(k-1)(-\varepsilon)}}{n \phi(q)}\mathop{\sum\limits_{\chi \mathrm{mod} q\\{\chi \neq \chi_{0}}} }\chi(\overline c)\sum \limits_{l = 1}^{n}\prod\limits_{i = 1}^{k-1}\left(\frac{1}{q \alpha} \sum\limits_{r_{i} = 1}^{q-1} G(r_{i}, \chi)\frac{1}{\|\frac{l}{n}-\frac{r_{i}}{q}\|} \right) \left(\frac{1}{q} \sum\limits_{r_{k} = 1}^{q-1} G\left(r_{k}, \chi\right) \frac{f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\mathbf{e}(\frac{r_{k}}{q}-\frac{l}{n})-1}\right)$

and

$S_{25}: = \frac{q^{(k-1)(\varepsilon)}}{n \phi(q)}\mathop{\sum\limits_{\chi \mathrm{mod} q\\{\chi \neq \chi_{0}}}} \chi(\overline c)\sum \limits_{l = 1}^{n}\prod\limits_{i = 1}^{k-1}\left(\frac{1}{q \alpha} \sum\limits_{r_{i} = 1}^{q-1} G(r_{i}, \chi) \right) \left(\frac{1}{q} \sum\limits_{r_{k} = 1}^{q-1} G\left(r_{k}, \chi\right) \frac{f\left(\delta_{k}, l, r_{k} ; n, q\right)}{\mathbf{e}(\frac{r_{k}}{q}-\frac{l}{n})-1}\right).$

By the same argument of $S_{23}$ , it follows that

$S_{24} \ll q^{\frac{k-1}{2}-\varepsilon}d^{2k}(q)(\log q+n)^{k},$

$S_{25} \ll q^{\frac{k-3}{2}+\varepsilon}(\log q+n).$

Since $n\ll q^{\frac{1}{3}}$ , we have

$\begin{equation} S_{25} \ll S_{24} \ll S_{23} \ll q^{\frac{k-1}{2}+\varepsilon}n^{k}\ll q^{k-2+\varepsilon}. \end{equation}$

(3.9)

Taking $n = 1$ , we get

$\begin{equation} S_{12}\ll q^{\frac{k-1}{2}+\varepsilon}. \end{equation}$

(3.10)

With (3.1), (3.2), (3.9) and (3.10), the proof is complete.

4. Conclusions

This paper considers the high-dimensional Lehmer problem related to Beatty sequences over incomplete intervals. And we give an asymptotic formula by the properties of Beatty sequences and the estimates for hyper Kloosterman sums.

Acknowledgment

This work is supported by Natural Science Foundation No. 12271422 of China. The authors would like to express their gratitude to the referee for very helpful and detailed comments.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

[1]	U. Kamps, A concept of generalized order statistics, J. Statist. Plan. Inf., 48 (1995), 1–23. https://doi.org/10.1016/0378-3758(94)00147-N doi: 10.1016/0378-3758(94)00147-N
[2]	C. A. Charalambides, Discrete q-distributions on Bernoulli trials with a geometrically varying success probability, J. Statist. Plann. Inf., 140 (2010), 2355–2383. https://doi.org/10.1016/j.jspi.2010.03.024 doi: 10.1016/j.jspi.2010.03.024
[3]	R. Diaz, E. Pariguan, On the Gaussian q-distribution, J. Math. Anal. Appl., 358 (2009), 1–9. https://doi.org/10.1016/j.jmaa.2009.04.046 doi: 10.1016/j.jmaa.2009.04.046
[4]	R. Diaz, C. Ortiz, E. Pariguan, On the k-gamma q-distribution, Cent. Eur. J. Math., 8 (2010), 448–458. https://doi.org/10.2478/s11533-010-0029-0 doi: 10.2478/s11533-010-0029-0
[5]	X. Jia, S. Nadarajah, B. Guo, Inference on q-Weibull parameters, Statist. Papers, 61 (2020), 575–593. http://doi.org/10.1007/s00362-017-0951-3 doi: 10.1007/s00362-017-0951-3
[6]	B. Singh, R. U. Khan, M. A. Khan, Characterizations of q-Weibull distribution based on generalized order statistics, J. Statist. Manag. Sys., 22 (2019), 1573–1595. http://doi.org/10.1080/09720510.2019.1643554 doi: 10.1080/09720510.2019.1643554
[7]	X. Jia, Reliability analysis for q-Weibull distribution with multiply type-Ⅰ censored data, Qual. Reliab. Eng. Int., 37 (2021), 2790–2817. http://doi.org/10.1002/qre.2890 doi: 10.1002/qre.2890
[8]	F. Lad, G. Sanflippo, G. Agro, Extropy: complementary dual of entropy, Statist. Sci., 30 (2015), 40–58. http://doi.org/10.1214/14-sts430 doi: 10.1214/14-sts430
[9]	I. A. Husseiny, A. H. Syam, The extropy of concomitants of generalized order statistics from Huang-Kotz-Morgenstern bivariate distribution, J. Math., 2022 (2022), 6385998. http://dx.doi.org/10.1155/2022/6385998 doi: 10.1155/2022/6385998
[10]	S. Bansal, N. Gupta, Weighted extropies and past extropy of order statistics and k-record values, Commun. Statist. Theory Meth., 51 (2022), 6091–6108. http://dx.doi.org/10.1080/03610926.2020.1853773 doi: 10.1080/03610926.2020.1853773
[11]	S. Picoli, R. S. Mendes, L. C. Malacarne, q-exponential, Weibull, and q-Weibull distributions: an empirical analysis, Phys. A, 324 (2003), 678–688. http://doi.org/10.1016/s0378-4371(03)00071-2 doi: 10.1016/s0378-4371(03)00071-2
[12]	Z. A. Aboeleneen, Inference for Weibull distribution under generalized order statistics, Math. Comput. Sim., 81 (2010), 26–36. http://dx.doi.org/10.1016/j.matcom.2010.06.013 doi: 10.1016/j.matcom.2010.06.013
[13]	A. A. Jafari, H. Zakerzadeh, Inference on the parameters of the Weibull distribution using records, SORT, 39 (2015), 3–18.
[14]	P. H. Garthwaite, J. B. Kadane, A. OHagan, Statistical methods for eliciting probability distributions, J. Am. Stat. Assoc., 100 (2005), 680–701. http://dx.doi.org/10.1198/016214505000000105 doi: 10.1198/016214505000000105
[15]	K. M. Hamdia, X. Zhuang, P. He, T. Rabczuk, Fracture toughness of polymeric particle nanocomposites: Evaluation of models performance using Bayesian method, Composites Sci. Tech., 126 (2016), 122–129. http://dx.doi.org/10.1016/j.compscitech.2016.02.012 doi: 10.1016/j.compscitech.2016.02.012
[16]	X. Jia, D. Wang, P. Jiang, B. Guo, Inference on the reliability of Weibull distribution with multiply type-Ⅰ censored data, Reliab. Eng. Syst. Saf., 150 (2016), 171–181. http://doi.org/10.1016/j.ress.2016.01.025 doi: 10.1016/j.ress.2016.01.025
[17]	M. Xu, E. L. Droguett, I. D. Lins, M. das Chagas Moura, On the q-Weibull distribution for reliability applications: An adaptive hybrid artificial bee colony algorithm for parameter estimation, Reliab. Eng. Syst. Saf., 158 (2017), 93–105. http://doi.org/10.1016/j.ress.2016.10.012 doi: 10.1016/j.ress.2016.10.012
[18]	R. B. Nelsen, An Introduction to Copulas, New York: Springer-Verlag, 2006.
[19]	A. Sklar, Random variables, joint distribution functions, and copulas, Kybernetika, 9 (1973), 449–460.
[20]	E. J. Gumbel, Bivariate exponential distributions, J. Am. Stat. Assoc., 55 (1960), 698–707.
[21]	M. A. Alawady, H. M. Barakat, M. A. Abd Elgawad, Concomitants of generalized order statistics from bivariate Cambanis family of distributions under a general setting, Bull. Malays. Math. Sci. Soc., 44 (2021), 3129–3159. http://dx.doi.org/10.1007/s40840-021-01102-1 doi: 10.1007/s40840-021-01102-1
[22]	M. A. Alawady, H. M. Barakat, S. Xiong, M. A. Abd Elgawad, Concomitants of generalized order statistics from iterated Farlie-Gumbel-Morgenstern type bivariate distribution, Commun. Statist. Theory Meth., 51 (2022), 5488–5504. http://dx.doi.org/10.1080/03610926.2020.1842452 doi: 10.1080/03610926.2020.1842452
[23]	S. P. Arun, C. Chesneau, R. Maya, M. R. Irshad, Farlie-Gumbel-Morgenstern bivariate moment exponential distribution and its inferences based on concomitants of order statistics, Stats, 6 (2023), 253–267. http://dx.doi.org/10.3390/stats6010015 doi: 10.3390/stats6010015
[24]	H. M. Barakat, M. A. Alawady, I. A. Husseiny, G. M. Mansour, Sarmanov family of bivariate distributions: statistical properties, concomitants of order statistics, and information measures, Bull. Malays. Math. Sci. Soc., 45 (2022), 49–83. http://dx.doi.org/10.1007/s40840-022-01241-z doi: 10.1007/s40840-022-01241-z
[25]	H. M. Barakat, E. M. Nigm, M. A. Alawady, I. A. Husseiny, Concomitants of order statistics and record values from the generalization of FGM bivariate-generalized exponential distribution, J. Statist. Theory Appl., 18 (2019), 309–322. http://dx.doi.org/10.2991/jsta.d.190822.001 doi: 10.2991/jsta.d.190822.001
[26]	H. M. Barakat, E. M. Nigm, M. A. Alawady, I. A. Husseiny, Concomitants of order statistics and record values from the iterated FGM type bivariate-generalized exponential distribution, REVSTAT Statist. J., 19 (2021), 291–307. http://dx.doi.org/10.2298/fil1809313b doi: 10.2298/fil1809313b
[27]	S. Cambanis, Some properties and generalizations of multivariate Eyraud-Gumbel-Morgenstern distributions, J. Mul. Anal., 7 (1977), 551–559. http://dx.doi.org/10.1016/0047-259x(77)90066-5 doi: 10.1016/0047-259x(77)90066-5
[28]	I. A. Husseiny, H. M. Barakat, G. M. Mansour, M. A. Alawady, Information measures in record and their concomitants arising from Sarmanov family of bivariate distributions, J. Comput. Appl. Math., 408 (2022), 114120. http://doi.org/10.1016/j.cam.2022.114120 doi: 10.1016/j.cam.2022.114120
[29]	I. A. Husseiny, M. A. Alawady, H. M. Barakat, M. A. Abd Elgawad, Information measures for order statistics and their concomitants from Cambanis bivariate family, Commun. Statist. Theory Meth., 53 (2024), 865–881. http://doi.org/10.1080/03610926.2022.2093909 doi: 10.1080/03610926.2022.2093909
[30]	J. Scaria, B. Thomas, Second order concomitants from the Morgenstern family of distributions, J. Appl. Statist. Sci., 21 (2014), 63–76.
[31]	W. Schucany, W. C. Parr, J. E. Boyer, Correlation structure in Farlie-Gumbel-Morgenstern distributions, Biometrika, 65 (1978), 650–653. http://doi.org/10.1093/biomet/65.3.650 doi: 10.1093/biomet/65.3.650
[32]	R. A. Attwa, T. Radwan, E. O. Abo Zaid, Bivariate q-extended Weibull Morgenstern family and correlation coefficient formulas for some of its sub-models, AIMS Math., 8 (2023), 25325–25342. http://dx.doi.org/10.3934/math.20231292 doi: 10.3934/math.20231292
[33]	M. Gurvich, A. Dibenedetto, S. Ranade, A new statistical distribution for characterizing the random strength of brittle materials, J. Mater. Sci., 32 (1997), 2559–2564.
[34]	H. A. David, Concomitants of order statistics, Bull. Int. Statist. Inst., 45 (1973), 295–300.
[35]	I. Bairamov, S. Kotz, M. Becki, New generalized Farlie-Gumbel-Morgenstern distributions and concomitants of order statistics, J. Appl. Statist., 28 (2001), 521–536. http://dx.doi.org/10.1080/02664760120047861 doi: 10.1080/02664760120047861
[36]	M. I. Beg, M. Ahsanullah, Concomitants of generalized order statistics from Farlie-Gumbel-Morgenstern distributions, Statist. Methodol., 5 (2008), 1–20. http://dx.doi.org/10.1016/j.stamet.2007.04.001 doi: 10.1016/j.stamet.2007.04.001
[37]	F. Domma, S. Giordano, Concomitants of m-generalized order statistics from generalized Farlie-Gumbel-Morgenstern distribution family, J. Comput. Appl. Math., 294 (2016), 413–435. http://dx.doi.org/10.1016/j.cam.2015.08.022 doi: 10.1016/j.cam.2015.08.022
[38]	S. Eryilmaz, On an application of concomitants of order statistics, Commun. Statist. Theory Meth., 45 (2016), 5628–5636. http://dx.doi.org/10.1080/03610926.2014.948201 doi: 10.1080/03610926.2014.948201
[39]	J. Scaria, N. U. Nair, Distribution of extremes of rth concomitant from the Morgenstern family, Statist. Papers, 49 (2008), 109–119. http://doi.org/10.1007/s00362-006-0365-0 doi: 10.1007/s00362-006-0365-0
[40]	S. Tahmasebi, A. A. Jafari, Concomitants of order statistics and record values from Morgenstern type bivariate-generalized exponential distribution, Bull. Malays. Math. Sci. Soc., 38 (2015), 1411–1423. http://doi.org/10.1007/s40840-014-0087-8 doi: 10.1007/s40840-014-0087-8
[41]	H. S. Klakattawi, W. H. Aljuhani, L. A. Baharith, Alpha power exponentiated new Weibull-Pareto distribution: Its properties and applications, Pakistan J. Statist. Oper. Res., 18 (2022), 703–720. http://doi.org/10.18187/pjsor.v18i3.3937 doi: 10.18187/pjsor.v18i3.3937
[42]	E. T. Lee, J. Wang, Statistical Methods for Survival Data Analysis, New york: John Wiley & Sons, 2003.

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