- Research
- Open Access
On weighted cumulative past (residual) inaccuracy for record values
- Safieh Daneshi^{1},
- Ahmad Nezakati^{1} and
- Saeid Tahmasebi^{2}Email author
https://doi.org/10.1186/s13660-019-2082-y
© The Author(s) 2019
- Received: 21 December 2018
- Accepted: 26 April 2019
- Published: 14 May 2019
Abstract
In this paper, we propose the weighted cumulative past (residual) inaccuracy for record values. For this concept, we obtain some properties and characterization results such as relationships with other reliability functions, bounds, stochastic ordering and effect of linear transformation. Dynamic versions of this weighted measure are considered. We also study a problem of estimating the weighted cumulative past inaccuracy by means of the empirical cumulative inaccuracy for lower record values.
Keywords
- Cumulative past (residual) inaccuracy
- Cumulative entropy
- Lower (upper) record values
- Weighted inaccuracy
MSC
- 62N05
- 62B10
- 94A17
1 Introduction
Record values are applied in problems such as industrial stress testing, meteorological analysis, hydrology, sporting and economics. In reliability theory, record values are used to study, for example, technical systems which are subject to shocks, e.g., peaks of voltages. For more details about records and their applications, one may refer to Arnold et al. [1]. Several authors have worked on measures of inaccuracy for ordered random variables. Thapliyal and Taneja [16] proposed a measure of inaccuracy between the ith order statistic and the parent random variable. Thapliyal and Taneja [17] developed measures of dynamic cumulative residual and past inaccuracy. They studied characterization results of these dynamic measures under proportional hazard model and proportional reversed hazard model. Thapliyal and Taneja [18] have introduced the measure of residual inaccuracy of order statistics and proved a characterization result for it. Tahmasebi and Daneshi [14] and Tahmasebi et al. [15] obtained some results for inaccuracy measures of record values. In this paper, we propose a weighted cumulative past (residual) inaccuracy of record values and study its characterization results. The paper is organized as follows. In Sect. 2, we consider a weighted measure of inaccuracy associated with \(F_{L_{n}}\) and F and obtain some results of its properties. In Sect. 3, we study a dynamic version of WCPI between \(F_{L_{n}}\) and F. In Sect. 4, we propose empirical WCPI of lower record values. In Sect. 5, we study WCRI and its dynamic version between \(\bar{F}_{R_{n}}\) and F̄, and obtain some results about their properties. Throughout the paper we assume that the terms increasing and decreasing are used in non-strict sense.
2 Weighted cumulative past inaccuracy for \(L_{n}\)
In this section, we propose a weighted measure of CPI between \(F_{L_{n}}\) and F. For this concept, we study some properties and characterization results under some assumptions.
Definition 2.1
In the following, we present some examples and properties of \(I^{w}(F_{L_{n}},F)\).
Example 2.1
- (i)If X has an inverse Weibull distribution with the cdf \(F(x)=\exp (-(\frac{\alpha }{x})^{\beta })\), \(x>0\), then we have$$ I^{w}(F_{L_{n}},F)=\frac{\alpha ^{2}}{\beta }\sum _{j=0}^{n-1}\frac{ \varGamma (\frac{(j+1)\beta -2}{\beta } )}{j!}. $$
- (ii)If X is uniformly distributed on \([0,\theta ]\), then we obtain$$ I^{w}(F_{L_{n}},F)=\theta ^{2}\sum _{j=0}^{n-1}(j+1) \biggl(\frac{1}{3} \biggr) ^{j+2}. $$
- (iii)If X has a power distribution with cdf \(F(x)=[\frac{x}{\alpha }]^{\beta }\), \(0< x<\alpha \), \(\beta >0\), then we obtain$$ I^{w}(F_{L_{n}},F)=\alpha ^{2}\sum _{j=0}^{n-1}(j+1)\frac{\beta ^{j+1}}{(2+ \beta )^{j+2}}. $$
Proposition 2.2
Proof
Note that \(\mu ^{w}_{n}(t)\) is analogous to the mean residual waiting time used in reliability and survival analysis (for more details, see Bdair and Raqab [2]).
Proposition 2.3
Proof
Proposition 2.4
Proof
Remark 2.1
- 1.
A random variable X is said to be smaller than Y according to stochastic ordering (denoted by \(X\leq ^{st}Y\)) if \(P(X\geq x)\leq P(Y \geq x)\) for all x. It is known that \(X \leq ^{st}Y \Leftrightarrow \mathbb{E}(\phi (X))\leq \mathbb{E}(\phi (Y))\) for all increasing functions (equivalency (1.A.7) in Shaked and Shanthikumar [13]).
- 2.
A random variable X is said to be smaller than Y in likelihood ratio ordering (denoted by \(X\leq ^{lr}Y\)) if \(\frac{g(x)}{f(x)}\) is increasing in x.
- 3.
A random variable X is said to be smaller than a random variable Y in the increasing convex order, denoted by \(X \leq ^{icx}Y\), if \(\mathbb{E}(\phi (X))\leq \mathbb{E}(\phi (Y))\) for all increasing convex functions ϕ such that the expectations exist.
- 4.
A non-negative random variable X is said to have a decreasing reversed hazard rate on average (DRHRA) if \(\frac{\tilde{\lambda }(x)}{x}\) is decreasing in x.
- 5.
A non-negative random variable X is said to have a decreasing hazard rate on average (DHRA) if \(\frac{\lambda (x)}{x}\) is decreasing in x.
Theorem 2.5
Proof
Thus the proof is completed. □
Proposition 2.6
Proof
Since \(-\log F(x)\geq 1-F(x)\), the proof follows by recalling (2.1). □
Proposition 2.7
Proposition 2.8
Proof
Proposition 2.9
Proposition 2.10
Proof
Remark 2.2
3 Dynamic weighted cumulative past inaccuracy
In the following theorem, we prove that \(I^{w}(F_{L_{n}},F;t)\) uniquely determines the distribution function.
Theorem 3.1
Let X be a non-negative continuous random variable with distribution function \(F(\cdot )\). Let the weighted dynamic cumulative inaccuracy of the nth lower record value be finite, that is, \(I^{w}(F_{L_{n}},F;t)< \infty \), \(t\geq 0\). Then \(I^{w}(F_{L_{n}},F;t)\) characterizes the distribution function.
Proof
4 Empirical weighted cumulative past inaccuracy
Example 4.1
Numerical values of \(\mathbb{E}[\hat{I}^{w}(F_{L_{n}},F)]\) and \(\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]\) for Weibull distribution
m | n = 2 | n = 3 | n = 4 | n = 5 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
λ = 0.5 | λ = 1 | λ = 2 | λ = 0.5 | λ = 1 | λ = 2 | λ = 0.5 | λ = 1 | λ = 2 | λ = 0.5 | λ = 1 | λ = 2 | |
\(\mathbb{E}[\hat{I}^{w}(F_{L_{n}},F)] \) | ||||||||||||
10 | 0.196 | 0.098 | 0.049 | 0.239 | 0.120 | 0.060 | 0.261 | 0.131 | 0.065 | 0.271 | 0.135 | 0.068 |
15 | 0.134 | 0.067 | 0.033 | 0.165 | 0.082 | 0.041 | 0.181 | 0.090 | 0.045 | 0.189 | 0.094 | 0.047 |
20 | 0.102 | 0.051 | 0.025 | 0.125 | 0.063 | 0.031 | 0.138 | 0.069 | 0.034 | 0.145 | 0.072 | 0.036 |
\(\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]\) | ||||||||||||
10 | 0.063 | 0.016 | 0.004 | 0.072 | 0.018 | 0.004 | 0.076 | 0.019 | 0.005 | 0.077 | 0.019 | 0.005 |
15 | 0.043 | 0.011 | 0.002 | 0.050 | 0.012 | 0.003 | 0.053 | 0.013 | 0.003 | 0.054 | 0.014 | 0.003 |
20 | 0.033 | 0.008 | 0.002 | 0.038 | 0.009 | 0.002 | 0.041 | 0.010 | 0.002 | 0.042 | 0.010 | 0.003 |
Example 4.2
Numerical values of \(\mathbb{E}[\hat{I}^{w}(F_{L_{n}},F)]\) and \(\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]\) for beta distribution
m | \(\mathbb{E}[\hat{I}^{w}(F_{L_{n}},F)]\) | \(\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]\) | ||||||
---|---|---|---|---|---|---|---|---|
n = 2 | n = 3 | n = 4 | n = 5 | n = 2 | n = 3 | n = 4 | n = 5 | |
10 | 0.219 | 0.291 | 0.330 | 0.349 | 0.003 | 0.004 | 0.004 | 0.005 |
15 | 0.230 | 0.309 | 0.356 | 0.380 | 0.002 | 0.003 | 0.004 | 0.004 |
20 | 0.235 | 0.319 | 0.370 | 0.397 | 0.001 | 0.002 | 0.003 | 0.003 |
Theorem 4.1
Proof
5 Weighted cumulative residual inaccuracy for \(R_{n}\)
In this section, we propose WCRI between \(\bar{F}_{R_{n}}\) and F̄. We discuss some properties of WCRI such as the effect of a linear transformation, relationships with other reliability functions, bounds and stochastic ordering.
Definition 5.1
In the following example, we calculate \(\bar{I}^{w}(\bar{F}_{R_{n}}, \bar{F})\) for some specific lifetime distributions which are widely used in reliability theory and life testing.
Example 5.1
- (a)
If X is uniformly distributed on \([0,\theta ]\), then it is easy to see that \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})=\theta ^{2}\sum_{j=0} ^{n-1}\frac{3^{j+2}-2^{j+2}}{6^{j+2}}(j+1)\), for all integers \(n\geq 1\).
- (b)
If X has a Weibull distribution with survival function \(\bar{F}(x)=e^{-\alpha x^{\beta }}\), \(x\geq 0\), \(\alpha ,\beta >0\), then for all integers \(n\geq 1\), we obtain \(\bar{I}^{w}(\bar{F}_{R _{n}},\bar{F})=\frac{1}{\beta }\sum_{j=0}^{n-1}\frac{ \alpha ^{2(1+j-\frac{1}{\beta })}(j+\frac{2}{\beta })!}{j!}\).
- (c)
Let X be an exponential distribution with mean \(\frac{1}{ \lambda }\), then \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})=\frac{n(n+1)(n+2)}{3 \lambda ^{2}}\).
Proposition 5.2
Proof
Proposition 5.3
Proof
Proposition 5.4
Proof
Proposition 5.5
Proof
Proposition 5.6
Proof
Proposition 5.7
Proof
The next propositions give some lower and upper bounds for \(\bar{I} ^{w}(\bar{F}_{R_{n}},\bar{F})\).
Proposition 5.8
Proof
By using (5.1), the proof is easy. □
Proposition 5.9
Proof
Since \(-\log \bar{F}(x)\geq 1-\bar{F}(x)\), the proof follows by recalling (5.1). □
In the following, we obtain some results on \(I^{w}(\bar{F}_{R_{n}}, \bar{F})\) and its connection with notions of reliability theory.
Proposition 5.10
Proof
The proof is completed. □
Proposition 5.11
If X has the exponential distribution with mean \(\mu =\frac{1}{ \theta }\), then as the hazard rate is constant, we obtain that \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})=\frac{n(n+1)(n+2)}{3}\mu ^{2}\), which is an increasing function of n.
Proposition 5.12
Proof
Proposition 5.13
Proof
Since \(h_{j+1}^{w}(\cdot )\) is an increasing convex function for \(j\geq 0\), it follows by Shaked and Shanthikumar [13] that \(X\leq ^{icx}Y\) implies \(h_{j+1}^{w}(X)\leq ^{icx}h_{j+1} ^{w}(Y)\). By recalling the definition of increasing convex order and Proposition 5.6, the proof is complete. □
Proposition 5.14
Proof
Proposition 5.15
Proof
Proposition 5.16
- (i)Let X be a continuous random variable with survival function \(\bar{F}(\cdot )\) that takes values in \([0, b]\), with finite b. Then,$$ \bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})\leq b\bar{I}( \bar{F}_{R_{n}}, \bar{F}). $$
- (ii)Let X be a non-negative continuous random variable with survival function \(\bar{F}(\cdot )\) that takes values in \([a, \infty )\), with finite \(a>0\). Then,$$ \bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})\geq a\bar{I}( \bar{F}_{R_{n}}, \bar{F}). $$
Proposition 5.17
Proof
Proposition 5.18
Theorem 5.19
\(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})=0\) if and only if X is degenerate.
Proof
Suppose X is degenerate at point a. Then, obviously, by definition of degenerate function and definition of \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})\), we have \(\bar{I}^{w}(\bar{F} _{R_{n}},\bar{F})=0\).
Remark 5.1
Theorem 5.20
Let X be a non-negative continuous random variable with distribution function \(F(\cdot )\). Let the weighted dynamic cumulative inaccuracy of the nth record value satisfy \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F};t)< \infty \), \(t\geq 0\). Then \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F};t)\) characterizes the distribution function.
Proof
6 Conclusions
In this paper, we discussed the concept of a weighted past inaccuracy measure between \(F_{L_{n}}\) and F. We proposed a dynamic version of WCPI and studied its characterization results. We have also proved that \(I^{w}(F_{L_{n}},F;t)\) uniquely determines the parent distribution F. Moreover, we studied some new basic properties of \(I^{w}(F_{L _{n}},F)\) such as the effect of a linear transformation, relationships with other reliability functions, bounds and stochastic order properties. We estimated the WCPI by means of the empirical cumulative inaccuracy of lower record values. Finally, we proposed the WCRI measure between the survival function \(\bar{F}_{R_{n}}\) and F̄. We also studied some properties of \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})\) such as the connections with other reliability functions, several useful bounds and stochastic orderings. These concepts can be applied in measuring the weighted inaccuracy contained in the associated past (residual) lifetime.
Declarations
Acknowledgements
The authors would like to thank the editors and reviewers for their valuable contributions, which greatly improved the readability of this paper.
Availability of data and materials
Not applicable.
Funding
The authors state that they have received no funding for this paper.
Authors’ contributions
The authors have made equal contributions. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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