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On weighted cumulative past (residual) inaccuracy for record values
Journal of Inequalities and Applications volume 2019, Article number: 134 (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.
Introduction
Let X and Y be two nonnegative random variables with distribution functions \(F(x)\), \(G(x)\) and survival functions \(\bar{F}(x)\), \(\bar{G}(x)\), respectively. If \(f (x)\) is the actual probability density function (pdf) corresponding to the observations and \(g(x)\) is the density assigned by the experimenter, then the inaccuracy measure of X and Y is defined by Kerridge [9] as
Recently, Kundu [10] considered a weighted measure of inaccuracy as
Analogous to the Kerridge measure of inaccuracy (1.1), Thapliyal and Taneja [17] proposed a cumulative past inaccuracy (CPI) measure as
When \(G(x)= F(x)\), Eq. (1.3) becomes cumulative entropy studied by Di Crescenzo and Longobardi [4]. Kundu et al. [11] studied some properties of CPI for truncated random variables. In analogy with (1.2), we define the weighted cumulative past inaccuracy (WCPI) as
Similarly, Kundu et al. [11] introduced the concept of cumulative residual inaccuracy (CRI) which is defined as
In analogy with (1.4), we define the weighted cumulative residual inaccuracy (WCRI) as
Let \(X_{1},X_{2},\ldots \) be a sequence of iid random variables having an absolutely continuous cdf \(F(x)\) and pdf \(f(x)\). An observation \(X_{j}\) is called a lower record (upper record) value if its value is less (greater) than that of all previous observations. Thus, \(X_{j}\) is a lower (upper) record if \(X_{j}<(>)X_{i}\) for every \(i< j\). Further, assume that \(T_{1}=1\) and \(T_{n}=\min \{j: j>T_{n1}, X_{j}< X _{T_{n1}}\}\) are known as lower record time sequence. Then, the lower record value sequence can be defined by \(L_{n}=X_{T_{n}}\), \(n\geq 1\). The density function and cumulative distribution function (cdf) of \(L_{n}\), which are denoted by \(f_{L_{n}}\) and \(F_{L_{n}}\), respectively, are given by
Similarly, assume that \(Z_{1}=1\) and \(Z_{n}=\min \{j^{*}: j^{*}>Z_{n1}, X_{j^{*}}>X_{Z_{n1}}\}\) are known as upper record time sequence. Then, \(R_{n}=X_{Z_{n}}\), \(n\geq 1\) are said to be upper record values. The pdf of \(R_{n}\) is given by
Also, the survival function of \(R_{n}\) can be obtained as
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 nonstrict sense.
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
Let X be a nonnegative absolutely continuous random variable with cdf F. Then, we define the WCPI between \(F_{L_{n}}\) (distribution function of the nth lower record value \(L_{n}\)) and F as
where \(\tilde{\lambda }(x)=\frac{f(x)}{F(x)}\) is the reversed hazard rate function and \(L_{j+2}\) is a random variable with density function \(f_{L_{j+2}}(x)=\frac{[\log F(x)]^{j+1}f(x)}{(j+1)!}\).
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
$$ I^{w}(F_{L_{n}},F)=\frac{\alpha ^{2}}{\beta }\sum _{j=0}^{n1}\frac{ \varGamma (\frac{(j+1)\beta 2}{\beta } )}{j!}. $$ 
(ii)
If X is uniformly distributed on \([0,\theta ]\), then we obtain
$$ I^{w}(F_{L_{n}},F)=\theta ^{2}\sum _{j=0}^{n1}(j+1) \biggl(\frac{1}{3} \biggr) ^{j+2}. $$ 
(iii)
If X has a power distribution with cdf \(F(x)=[\frac{x}{\alpha }]^{\beta }\), \(0< x<\alpha \), \(\beta >0\), then we obtain
$$ I^{w}(F_{L_{n}},F)=\alpha ^{2}\sum _{j=0}^{n1}(j+1)\frac{\beta ^{j+1}}{(2+ \beta )^{j+2}}. $$
Proposition 2.2
Let X be a nonnegative random variable with cdf F, then we have
Proof
By (2.1) and using the relation \(\log F(x)=\int _{x}^{\infty } \tilde{\lambda }(z)\,dz\), we have
So, the proof is completed. □
The weighted mean inactivity time (WMIT) function of a nonnegative random variable X is given by
Now, the WMIT of \(L_{n}\) is given by
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
Let X be a nonnegative random variable with cdf F. Then, we have
Proof
From (2.2) and (2.3), we obtain
yielding the claim. □
Proposition 2.4
Let X be an absolutely continuous nonnegative random variable with \(I^{w}(F_{L_{n}},F)<\infty \), for all \(n\geq 1\). Then we have
where \(\tilde{h}^{w}_{j+1}(t)=\int _{t}^{\infty }x [\log F(x) ] ^{j+1}\,dx\).
Proof
By using (2.1) and Fubini’s theorem, we obtain
□
Remark 2.1
Let X be a symmetric random variable with respect to the finite mean \(\mu =E(X)\), i.e., \(F(x+\mu )=1F(\mu x)\) for all \(x\in \mathbb{R}\). Then
where \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})\) is the weighted cumulative residual measure of inaccuracy between \(\bar{F}_{R_{n}}\) (survival function of the nth upper record value \(R_{n}\)) and F̄.
Now we can prove an important property of the inaccuracy measure using some properties of stochastic ordering. For that we present the following definitions:

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 nonnegative 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 nonnegative 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
Suppose that a nonnegative random variable X is DRHRA, then
Proof
Let \(f_{L_{n}}(x)\) be the pdf of the nth lower record value \(X_{L_{n}}\). Then, the ratio \(\frac{f_{L_{n}}(x)}{f _{L_{n+1}}(x)}=\frac{n}{\log F(x)}\) is increasing in x. Therefore, \(X_{n+1}\leq ^{lr}X_{n}\), and this implies that \(X_{n+1}\leq ^{st}X _{n}\), i.e., \(\bar{F}_{n+1}(x)\leq \bar{F}_{n}(x)\) (for more details, see Shaked and Shanthikumar ([13], Chap. 1)). This is equivalent (see Shaked and Shanthikumar ([13], p. 4)) to having
for all increasing functions ϕ such that these expectations exist. Thus, if X is DRHRA and \(\tilde{\lambda }(x)\) is its reversed hazard rate, then \(\frac{x}{\tilde{\lambda }(x)}\) is increasing in x. From (2.1), we have that
Thus the proof is completed. □
Proposition 2.6
Let X be a nonnegative random variable with absolutely continuous cumulative distribution function \(F(x)\). Then for \(n=1,2,\dots \), we have
Proof
Since \(\log F(x)\geq 1F(x)\), the proof follows by recalling (2.1). □
Proposition 2.7
Let X be a nonnegative random variable with absolutely continuous cumulative distribution function \(F(x)\). Then for \(n=1,2,\dots \), we have
Assume that \(\tilde{X}_{\theta }\) denotes a nonnegative absolutely continuous random variable with the distribution function \(H_{\theta }(x)=[F(x)]^{\theta }\), \(x\geq 0\). We now obtain the cumulative measure of inaccuracy between \(H_{L_{n}}\) and H as follows:
Proposition 2.8
If \(\theta \geq 1\), then for any \(n=1,2,\dots \), we have
Proof
Suppose that \(\theta \geq 1\), then it is clear that \([F(x)]^{\theta }\leq F(x)\), and hence we have
□
Proposition 2.9
Let X be a nonnegative random variable with cdf F, then an analytical expression for \(I^{w}(F_{L_{n}},F)\) is given by
where
is a weighted generalized cumulative entropy (WGCE) which was introduced by Kayal and Moharana [7].
Proposition 2.10
Let \(a,b>0\). Then for \(n=1,2,\ldots \) , it holds that
Proof
From (2.8), we have
The proof is completed. □
Recently, Cali et al. [3] introduced a generalized CPI of order m defined as
In analogy with the measure defined in Eq. (2.11), we now introduce a weighted generalized CPI (WGCPI) of order m defined as
Remark 2.2
Let X be a nonnegative absolutely continuous random variable with cdf F. Then, the WGCPI of order m between \(F_{L_{n}}\) and F is
Dynamic weighted cumulative past inaccuracy
In this section, we study a dynamic version of \(I^{w}( F_{L_{n}},F )\). If a system that begins to work at time 0 is observed only at deterministic inspection times, and is found to be ‘down’ at time t, then we consider a dynamic cumulative measure of inaccuracy as
Note that \(\lim_{t\rightarrow \infty }I^{w}(F_{L_{n}},F;t)=I^{w}(F _{L_{n}},F)\). Since \(\log F(t)\leq 0\) for \(t\geq 0\), we have
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 nonnegative 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
From (3.1) we have
Differentiating both sides of (3.2) with respect to t, we obtain
Taking derivative with respect to t again, we get
Suppose that there are two functions F and \(F^{*}\) such that
Then for all t, from (3.3) we get
where
and \(s(t)=\mu _{n}^{w}(t)\). By using Theorem 2.1 and Lemma 2.2 of Gupta and Kirmani [5], we have \(\tilde{\lambda }_{F}(t)=\tilde{\lambda } _{F^{*}}(t)\), for all t. Since the reversed hazard rate function characterizes the distribution function uniquely, we complete the proof. □
Empirical weighted cumulative past inaccuracy
In this section we address the problem of estimating the weighted cumulative measure of inaccuracy by means of the empirical weighted cumulative inaccuracy of lower record values. Let \(X_{1},X_{2}, \dots ,X_{m}\) be a random sample of size m from an absolutely continuous cumulative distribution function \(F(x)\). Then according to (2.8), the empirical cumulative measure of inaccuracy is
where
is the empirical distribution of the sample and I is the indicator function. If we denote \(X_{(1)} \leq X_{(2)}\leq \cdots \leq X_{(m)}\) as the order statistics of the sample, then (4.1) can be written as
Moreover,
Hence, (4.2) can be written as
where \(U_{k}=\frac{X^{2}_{(k+1)}X^{2}_{(k)}}{2}\), \(k=1,2,\dots ,m1\) are the sample spacings.
Example 4.1
Consider a random sample \(X_{1},X_{2}, \dots ,X_{m}\) from the Weibull distribution with density function
Then \(Y_{k}=X_{k}^{2}\) has an exponential distribution with mean \(\frac{1}{\lambda }\). In this case, the sample spacings \(2U_{k}=X^{2} _{(k+1)}X^{2}_{(k)}\) are independent and exponentially distributed with mean \(\frac{1}{\lambda (mk)}\) (for more details, see Pyke [12]). Now from (4.3) we obtain
and
We have computed the values of \(\mathbb{E}[\hat{I}^{w}(F_{L_{n}},F)]\) and \(\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]\) for sample sizes \(m=10, 15,20\), \(\lambda =0.5,1,2\) and \(n=2,3,4,5\) in Table 1. We can easily see that \(\mathbb{E}[\hat{I}^{w}(F_{L_{n}},F)]\) and \(\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]\) are decreasing in m. Also, we consider that \(\lim_{m\rightarrow \infty }\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]=0\).
Example 4.2
Let \(X_{1},X_{2}, \dots ,X_{m}\) be a random sample from a population with pdf \(f(x)=2x\), \(0< x<1\). Then the sample spacings \(2U_{k}\) are independent and beta distributed with parameters 1 and m (for more details, see Pyke [12]). Now from (4.3) we obtain
and
We have computed the values of \(\mathbb{E}[\hat{I}^{w}(F_{L_{n}},F)]\) and \(\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]\) for sample sizes \(m=10, 15,20\) and \(n=2,3,4,5\) in Table 2. We can easily see that \(\operatorname{Var}[\hat{I}^{w}(F_{L_{n}},F)]\) is decreasing in m and \(\lim_{m\rightarrow \infty }\operatorname{Var}[\hat{I}^{w}(F _{L_{n}},F)]=0\).
Theorem 4.1
Let X be an absolutely continuous nonnegative random variable such that \(I^{w}(F_{L_{n}},F)<\infty \), for all \(n\geq 1\). Then we have
Proof
From (2.8) we have
where
Now we can obtain
where
Using dominated convergence (DCT) and Glivenko–Cantelli theorems, we have
It follows that
Moreover, by using SLLN, \(\frac{1}{m}\sum_{i=1}^{m} X_{i}^{p} \longrightarrow \mathbb{E}(X^{p})\) and \(\sup_{m}(\frac{1}{m}\sum_{i=1}^{m} X_{i}^{p})< \infty \), so \(\hat{F}_{m} (x)\leq x^{p} (\sup_{m}(\frac{1}{m} \sum_{i=1}^{m} X_{i}^{p}) )=Cx^{p}\). Now applying DCT, we have
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
Let X be a nonnegative absolutely continuous random variable with survival function F̄. Then, we define the WCRI between \(\bar{F}_{R_{n}}\) and F̄ as follows:
where \(\lambda (x)=\frac{f(x)}{\bar{F}(x)}\) is the hazard rate function and \(R_{j+2}\) is a random variable with reliability \(\bar{F}_{R_{j+2}}\).
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 \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})=\theta ^{2}\sum_{j=0} ^{n1}\frac{3^{j+2}2^{j+2}}{6^{j+2}}(j+1)\), 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 \(\bar{I}^{w}(\bar{F}_{R _{n}},\bar{F})=\frac{1}{\beta }\sum_{j=0}^{n1}\frac{ \alpha ^{2(1+j\frac{1}{\beta })}(j+\frac{2}{\beta })!}{j!}\).

(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
Let X be an absolutely continuous nonnegative random variable with survival function F̄. Then, we have
where \(\mu _{n}=\int _{0}^{+\infty }\bar{F}_{R_{n}}(x)\,dx\).
Proof
□
Proposition 5.3
Let \(a,b > 0\). For \(n =1,2,\dots \), it holds that
Proof
From (5.1) and noting that \(\bar{F}_{aX+b}(x)= \bar{F}(\frac{xb}{a})\), we have
□
Kayid et al. [8] proposed the combination mean residual life (CMRL) function of X as the reciprocal hazard rate of the lengthbiased equilibrium distribution given by
Now, the CMRL of \(R_{n}\) is given by
Proposition 5.4
Let X be an absolutely continuous nonnegative random variable with survival function F̄. Then, we have
Proof
By (5.1) and the fact that \(\log \bar{F}(x)=\int _{0}^{x}\lambda (z)\,dz\), we have
□
Proposition 5.5
Let X be a nonnegative random variable with survival function F̄. Then, we have
Proof
From (5.3) and using Proposition 5.4, we obtain
This completes the proof. □
Proposition 5.6
Let X be an absolutely continuous nonnegative random variable such that \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})<\infty \), for \(n\geq 1\). Then, we have
where
Proof
From (5.1) and using Fubini’s theorem, we obtain
□
Proposition 5.7
Let X be a nonnegative and absolutely continuous random variable with cdf F. Then
where \(\mathcal{E}^{w_{j+1}}(X)=\int _{0}^{\infty }x^{ (\frac{1}{j+1} )} \bar{F}(x)\log {\bar{F}(x)}\,dx\).
Proof
From (5.1), we have
This completes the proof. □
The next propositions give some lower and upper bounds for \(\bar{I} ^{w}(\bar{F}_{R_{n}},\bar{F})\).
Proposition 5.8
Let X be a nonnegative random variable with absolutely continuous cumulative distribution function \(F(x)\). Then for \(n=1,2,\dots \), we have
Proof
By using (5.1), the proof is easy. □
Proposition 5.9
Let X be a nonnegative random variable with survival function \(\bar{F}(x)\). Then for \(n=1,2,\dots \), we have
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
If X is DFRA, then for \(n=1,2,\dots \), we have
Proof
Suppose that \(f_{R_{n}}\) is the pdf of of the nth record value \(R_{n}\). Then, the ratio \(\frac{f_{R_{n+1}}(t)}{f _{R_{n}}(t)}=\frac{\log \bar{F}(t)}{n}\) is increasing in t. Therefore, \(R_{n}\leq ^{lr}R_{n+1}\), and this implies that \(R_{n}\leq ^{st}R_{n+1}\), i.e., \(\bar{F}_{R_{n}}\leq \bar{F}_{R_{n+1}}\) (for more details, see Shaked and Shanthikumar ([13], Chap. 1)). Hence, if X is DFRA and \(\lambda (x)\) is its hazard rate, then \(\frac{x}{\lambda (x)}\) is incre asing function of x. So, from (5.1) we have
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
Let X and Y be two nonnegative random variables with reliability functions \(\bar{F}(x)\) and \(\bar{G}(x)\), respectively. If \(X\leq ^{hr}Y\) and X is DFRA, then
for \(n=1,2,\dots \).
Proof
It is well known that \(X\leq ^{hr} Y\) implies \(X\leq ^{st} Y\) (see Shaked and Shanthikumar [13]). Hence, we have
where \(\bar{G}_{\tilde{R}_{j+2}}\) is the survival function of \(\tilde{R}_{j+2}\). That is, \(R_{j+2}\leq ^{st}\tilde{R}_{j+2}\) holds. This is equivalent (see Shaked and Shanthikumar [13], p. 4) to having
for all increasing functions ϕ such that these expectations exist. Thus, if we assume that X is DFRA and \(\lambda (x)\) is its failure rate, then \(\frac{x}{\lambda (x)}\) is increasing and we have
On the other hand, \(X\leq ^{hr} Y\) implies that the respective failure rate functions satisfy \(\lambda _{F}(x)\geq \lambda _{G}(y)\). Hence, we have
Therefore, using both expressions, we obtain \(\bar{I}^{w}(\bar{F}_{R _{n}},\bar{F})\leq \bar{I}^{w}(\bar{G}_{\tilde{R}_{n}},\bar{G})\). □
Proposition 5.13
Let X and Y be two nonnegative random variables with reliability functions \(\bar{F}(x)\) and \(\bar{G}(x)\), respectively. If \(X\leq ^{icx}Y\), then
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
If X is IFRA (DFRA), then for \(n=1,2,\dots \), we have
Proof
From (5.1), we have
Now, since X is IFRA (DFRA), \(\frac{\log \bar{F}(x)}{x}\) is increasing (decreasing) with respect to \(x>0\), which implies that
Hence, the proof is completed by noting (5.9) and (5.10). □
Proposition 5.15
Let X and Y be two nonnegative random variables with survival functions \(\bar{F}(x)\) and \(\bar{G}(x)\), respectively. If \(X\leq ^{hr}Y\), then for \(n=1,2,\dots \), it holds that
Proof
By noting that the function \(h_{j+1}^{w}(x)= \int _{0}^{x}z[\log \bar{F}(z)]^{j+1}\,dz\) is an increasing convex function, under the assumption \(X\leq ^{hr}Y\), it follows, by Shaked and Shanthikumar [13], that
Hence, the proof is completed by recalling (5.1). □
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 nonnegative continuous random variable with survival function \(\bar{F}(\cdot )\) that takes values in \([a, \infty )\), with finite \(a>0\). Then,
$$ \bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})\geq a\bar{I}( \bar{F}_{R_{n}}, \bar{F}). $$
Assume that \(X^{*}_{\theta }\) denotes a nonnegative absolutely continuous random variable with the survival function \(\bar{H}_{ \theta }(x)=[\bar{F}(x)]^{\theta }\), \(x\geq 0\). This model is known as a proportional hazards rate model. We now obtain the weighted cumulative residual measure of inaccuracy between \(\bar{H}_{R_{n}}\) and H̄ as follows:
Proposition 5.17
If \(\theta \geq (\leq )1\), then for any \(n\geq 1\), we have
where \({\mathcal{E}}^{w}_{j+1}(X)\) is the weighted generalized cumulative residual entropy of X, defined by Kayal [6] as
Proof
Suppose that \(\theta \geq (\leq )1\), then it is clear that \([\bar{F}(x)]^{\theta }\leq (\geq )\bar{F}(x)\), and hence (5.11) yields
□
Proposition 5.18
Let X be a nonnegative random variable with survival function \(\bar{F}(\cdot )\), then an analytical expression for \(I^{w}(\bar{F} _{R_{n}},\bar{F})\) is given by
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 \(\bar{I}^{w}(\bar{F} _{R_{n}},\bar{F})=0\).
Now, suppose that \(I^{w}(\bar{F}_{R_{n}},\bar{F})=0\), i.e.,
Then, by noting that the integrand of (5.13) is nonnegative, we conclude that
for almost all \(x\in \mathbb{R}^{+}\). Thus, \(\bar{F}(x)=0 \text{ or } 1\), for almost all \(x\in \mathbb{R}^{+}\). □
Remark 5.1
Let X be a nonnegative absolutely continuous random variable with survival function \(\bar{F}(\cdot )\). Then in analogy with the measure defined in (2.13), the WGCRI of order m between \(\bar{F} _{R_{n}}\) and F̄ is given by
In the remainder of this section, we study a dynamic version of \(\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})\). Let X be the lifetime of a system under the condition that the system has survived up to age t. Analogously, we can also consider a dynamic version of \(\bar{I} ^{w}(\bar{F}_{R_{n}},\bar{F})\) as
Note that \(\lim_{t\rightarrow 0}\bar{I}^{w}(\bar{F}_{R_{n}},\bar{F};t)= \bar{I}^{w}(\bar{F}_{R_{n}},\bar{F})\). Since \(\log \bar{F}(t)\leq 0\) for \(t\geq 0\), we have
Theorem 5.20
Let X be a nonnegative 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
From (5.14) we have
Differentiating both sides of (5.15) with respect to t, we obtain
Taking derivative with respect to t again, we get
Suppose that there are two functions F and \(F^{*}\) such that
Then for all t, from (5.16) we get
where
and \(\tilde{s}(t)=m_{n}^{c}(t)\). By using Theorem 3.2 and Lemma 3.3 of Gupta and Kirmani [5], we have \(\lambda _{F}(t)=\lambda _{F^{*}}(t)\), for all t. Since the hazard rate function characterizes the distribution function uniquely, we complete the proof. □
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.
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Daneshi, S., Nezakati, A. & Tahmasebi, S. On weighted cumulative past (residual) inaccuracy for record values. J Inequal Appl 2019, 134 (2019). https://doi.org/10.1186/s136600192082y
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MSC
 62N05
 62B10
 94A17
Keywords
 Cumulative past (residual) inaccuracy
 Cumulative entropy
 Lower (upper) record values
 Weighted inaccuracy