Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms

Mohamed S. Eliwa; Essam A. Ahmed; Mohamed S. Eliwa; Essam A. Ahmed

doi:10.3934/math.2023002

AIMS Mathematics

2023, Volume 8, Issue 1: 29-60. doi: 10.3934/math.2023002

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Research article

Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms

Mohamed S. Eliwa ^{1,2,3
,
, ,},
Essam A. Ahmed ^4,5

1.
Department of Statistics and Operation Research, College of Science, Qassim University, Buraydah 51482, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
3.
Department of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy
4.
Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
5.
Faculty of Business Administration, Taibah University, Saudi Arabia

Received: 19 July 2022 Revised: 25 August 2022 Accepted: 01 September 2022 Published: 27 September 2022
MSC : 60E05, 62F10, 62N05, 62P10

Examining life-testing experiments on a product or material usually requires a long time of monitoring. To reduce the testing period, units can be tested under more severe than normal conditions, which are called accelerated life tests (ALTs). The objective of this study is to investigate the problem of point and interval estimations of the Lomax distribution under constant stress partially ALTs based on progressive first failure type-II censored samples. The point estimates of unknown parameters and the acceleration factor are obtained by using maximum likelihood and Bayesian approaches. Since reliability data are censored, the maximum likelihood estimates (MLEs) are derived utilizing the general expectation-maximization (EM) algorithm. In the process of Bayesian inference, the Bayes point estimates as well as the highest posterior density credible intervals of the model parameters and acceleration factor, are reported. This is done by using the Markov Chain Monte Carlo (MCMC) technique concerning both symmetric (squared error) and asymmetric (linear-exponential and general entropy) loss functions. Monte Carlo simulation studies are performed under different sizes of samples for comparison purposes. Finally, the proposed methods are applied to oil breakdown times of insulating fluid under two high-test voltage stress level data.

Keywords:

Citation: Mohamed S. Eliwa, Essam A. Ahmed. Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms[J]. AIMS Mathematics, 2023, 8(1): 29-60. doi: 10.3934/math.2023002

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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.

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AIMS Mathematics

Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms

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Abstract

1. Introduction

2. Preliminary lemmas

3. Proof of theorem

3.1. Estimation of $S_{11}$

3.2. Estimation of $S_{21}$

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

4. Conclusions

Acknowledgment

Conflict of interest

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AIMS Mathematics

Reliability analysis of constant partially accelerated life tests under progressive first failure type-II censored data from Lomax model: EM and MCMC algorithms

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Abstract

1. Introduction

2. Preliminary lemmas

3. Proof of theorem

3.1. Estimation of S11 S_{11}

3.2. Estimation of S21 S_{21}

3.3. Estimation of S22 S_{22} and S12 S_{12}

4. Conclusions

Acknowledgment

Conflict of interest

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3.1. Estimation of $S_{11}$

3.2. Estimation of $S_{21}$

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