Quantile-based cumulative inaccuracy measures

Accepted Manuscript Quantile-based cumulative inaccuracy measures Suchandan Kayal

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S0378-4371(18)30854-9 https://doi.org/10.1016/j.physa.2018.06.130 PHYSA 19824

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Physica A

Received date : 24 January 2018 Revised date : 28 June 2018 Please cite this article as: S. Kayal, Quantile-based cumulative inaccuracy measures, Physica A (2018), https://doi.org/10.1016/j.physa.2018.06.130 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Highlights 1. We have introduced quantile version of the cumulative inaccuracy measures. 2. The proposed measures are useful for the models having no tractable distribution functions. 3. Examples are provided in support of the proposed quantile-based measures over the conventional distribution function-based approach. 4. Various properties, bounds, effect of transformations and characterizations are obtained.

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Quantile-based Cumulative Inaccuracy Measures Suchandan Kayal Department of Mathematics, National Institute of Technology Rourkela, Rourkela-769008, India

Abstract Numerous statistical models do not have explicitly known distribution functions. Because of this, the study of properties of the cumulative residual (past) inaccuracy measures using distribution function-based approach are difficult. Further, it is well-known that in modeling and analyzing statistical data, an equivalent alternative to distribution function is quantile function. The objective of this paper is to introduce and study quantile versions of the cumulative residual (past) inaccuracy measures and their dynamic forms. We obtain some bounds, relations with other quantile-based reliability measures, monotonicity results and characterizations. Various examples are provided to show the importance of the proposed quantile-based measures and the associated results. Keywords: Quantile function, cumulative residual (past) inaccuracy, reliability measures, proportional (reversed) hazards model, equilibrium distributions. Mathematics Subject Classification: 94A17, 60E15, 62B10, 62N05.

1

Introduction

Various efforts have been made by numerous researchers to generalize the well-known concept of Shannon’s entropy since 1948. In this direction, an important development is the Kerridge inaccuracy measure. We often undergo two types of errors when expressing some statements on probabilities of various experimental events. One of them is due to the lack of sufficient information in experimental results and other is due to wrong specification of the model. To take these errors into account, Kerridge (1961) proposed inaccuracy measure. Let X1 and X2 be two absolutely continuous nonnegative random variables with probability density functions f1 (.), f2 (.); cumulative distribution functions F1 (.), F2 (.) and survival functions F¯1 (.), F¯2 (.), respectively. Assume that X1 and X2 describe lifetimes of the components or living organisms. Consider f1 (.) as the actual probability density function corresponding to some observations and f2 (.) the assigned density function. Then, the inaccuracy measure of X1 and X2 is given by (see Kerridge, 1961 and Nath, 1968) ∫ ∞ I(X1 , X2 ) = − f1 (x) ln f2 (x)dx = Ef1 (− ln f2 (X)), (1.1) 0

1

where expectation is evaluated with respect to the density function f1 (.). Equation (1.1) plays significant role in reliability and survival analysis in modeling survival data. It has been extensively used as a tool for the measurement of error (inaccuracy) in experimental results. Further, it has various applications in statistical inference, estimation and coding theory. The inaccuracy measure given by (1.1) reduces to the Shannon∫entropy (see Shan∞ non, 1948) for f1 (.) = f2 (.). The Shannon entropy of X1 is H(X1 ) = − 0 f1 (x) ln f1 (x)dx. Kundu and Nanda (2015) obtained characterizations for truncated distributions based on (1.1). Rajesh et al. (2017) proposed nonparametric estimators of the inaccuracy measure for lifetime distributions based on censored data. Very recently, Kayal et al. (2017) and Kayal and Sunoj (2017) proposed a generalization of (1.1) and developed various theoretical merits. The concept of entropy plays an important role in various real-life problems (see Cover and Thomas, 2006). In spite of its great success as an uncertainty measure, it has some demerits. For example, it is defined for distributions having density functions and takes any value in the extended real line. For more on this, we refer to Rao et al. (2004). To overcome such drawbacks, the authors proposed an alternative measure based on the distribution function. This is dubbed as the cumulative residual entropy (CRE). For a ∫∞ ¯ nonnegative random variable X1 , it is given by E(X1 ) = − 0 F1 (x) ln F¯1 (x)dx. The idea is to substitute reliability function in place of density function to the expression of H(X1 ). An analogous measure based on the distribution function was introduced by Di Crescenzo and Longobardi (2009).∫ It is known as the cumulative past entropy (CPE), and for X1 , it ∞ is given by E ∗ (X1 ) = − 0 F1 (x) ln F1 (x)dx. Motivated by CRE, Taneja and Kumar (2012) proposed cumulative residual inaccuracy (CRI) measure of X1 and X2 as ∫ ∞ CI(X1 , X2 ) = − F¯1 (x) ln F¯2 (x)dx. (1.2) 0

When the distributions of X1 and X2 coincide, (1.2) becomes CRE introduced by Rao et al. (2004). Analogue to the inaccuracy measure in (1.2), F¯1 (x) can be treated as the actual survival function and F¯2 (x) the approximate survival function assigned by an experimenter. Similar to the CPE, Kumar and Taneja (2015) proposed cumulative past inaccuracy (CPI) measure of X1 and X2 as ∫ ∞ ∗ CI (X1 , X2 ) = − F1 (x) ln F2 (x)dx. (1.3) 0

The function in (1.3) reduces to the CPE due to Di Crescenzo and Longobardi (2009) when F1 (.) = F2 (.). The measures in (1.2) and (1.3) are defined for the cases when X1 and X2 do not posses density functions. Kundu et al. (2016) considered the measures given by (1.2) and (1.3), and obtained several properties when random variables are left, right as well as both-side truncated. Very recently, bivariate extension of CRI and CPI has been addressed by Ghosh and Kundu (2017).

2

It is seen that all the theoretical developments on the measures given by (1.2) and (1.3) are carried out based on the distribution function approach. But, there are numerous statistical models (for instance, various forms of lambda distributions, power-Pareto distribution, Govindarajulu distribution etc.) for which distribution functions are not analytically tractable. In this case, an alternative approach should be used to study CRI and CPI. Various authors have shown that quantile function-based investigation can be treated as an efficient and parallel approach to distribution function-based study in analyzing statistical data. See the volumes by Gilchrist (2000) and Nair et al. (2013) for some relevant details in this direction. It can be noted that both distribution and quantile functions convey the same information about the distribution of a random variable. However, there are some properties of a quantile function which the distribution function does not have. For example, (i) the sum of two quantile functions is a quantile function, (ii) the product of two positive quantile functions is also a quantile function, (iii) the convex combination αQ1 (p) + (1 − α)Q2 (p), where 0 < α < 1 is a quantile function which lies between Q1 (p) and Q2 (p), and (iv) ξ(Q1 (p)) is a quantile function when ξ(x) is a nondecreasing function of x. For more features of quantile function, one may refer to Nair et al. (2013). The quantile function of X1 is given by Q1 (p) = F1−1 (p) = inf (x|F1 (x) ≥ p) , 0 ≤ p ≤ 1.

(1.4)

In a similar manner, the quantile function of X2 denoted by Q2 (p) can be defined. A wide variety of research has been done based on the quantile function approach. However, the study of uncertainty measures, divergence measures, Chernoff distance and inaccuracy measures based on quantile function is of very recent interest. For the development on various entropy measures based on quantile function, we refer to Sunoj and Sankaran (2012), Sunoj et al. (2013), Nanda et al. (2014), Sankaran and Sunoj (2017), Baratpour and Khammar (2018) and Khammar and Jahanshahi (2018). For the study of quantile-based divergence measures, one may refer to Sankaran et al. (2016), Sunoj et al. (2017, 2018) and Kayal and Tripathy (2018). We refer to Sunoj et al. (2013) and Kayal (2017) for the investigation to the quantile-based inaccuracy measure and Chernoff distance. As discussed above, there are various distributions which do not have closed-form (manageable) distribution functions, even though they have tractable quantile functions. Thus, the study of the CRI and CPI proposed by Taneja and Kumar (2012) and Kumar and Taneja (2015) is difficult for the random variables associated with these distributions. A quantile-based approach is useful in this direction since it has various advantages. Firstly, quantile-based CRI and CPI, and their dynamic versions are easy to evaluate for the distributions having closed-form quantile functions but no tractable distribution functions. Further, our proposed approach provides a parallel methodology in this direction. Because of this, in this paper, we consider and study quantile-based CRI and CPI, and their dynamic versions. In the rest of the paper, random variables are assumed to be nonnegative and absolutely continuous. The terms increasing and decreasing are used in wide sense. The derivatives are 3

assumed to exist. Further, we presume that F1 (x) and F2 (x) are absolutely continuous and strictly increasing so that F1 (x) = p is equivalent to x = Q1 (p). Similarly, for the random variable X2 . The remainder of the paper proceeds as follows. In the next section, we propose quantilebased CRI and CPI, and study some of its important theoretical properties. Section 3 explores the quantile-based CRI for residual lifetime distributions and its various properties. Bounds are obtained. Characterization results for exponential distribution are provided. Further, we introduce quantile-based CPI for past lifetime distributions and study some of its properties. Finally, concluding remarks are added in Section 4.

2

Quantile-based CRI and CPI

This section investigates various properties of quantile-based CRI and CPI. Denote Q1 (.) and Q2 (.) the quantile functions for the nonnegative absolutely continuous random variables X1 and X2 , respectively. Moreover, we have F1 Q1 (p) = p, 0 < p < 1, where F1 Q1 (p) is the composite function. On differentiating F1 Q1 (p) = p, we obtain q1 (p)f1 (Q1 (p)) = 1, 0 < p < 1,

(2.1)

where f1 (Q1 (p)) = f1 Q1 (p) represents the density quantile function (see Parzen, 1979) and d Q1 (p) denotes the quantile density function corresponding to the distribution q1 (p) = dp function F1 (.). Next, we consider some quantile-based reliability measures which are useful to obtain main results. For X1 , let h1 (x) = f1 (x)/F¯1 (x) and h∗1 (x) = f1 (x)/F1 (x) be the hazard and reversed hazard functions, respectively. Then, the hazard quantile and the reversed hazard quantile functions of X1 are given by h1Q (p) = [(1 − p)q1 (p)]−1 and h∗1Q (p) = [pq1 (p)]−1 ,

(2.2)

respectively, where 0 < p < 1. It is known that h1Q (p) and h∗1Q (p) determine the distribution of X1 uniquely. Another two important reliability measures are mean residual and mean past lifetimes. Denote m1 (x) = E(X1 − x|X1 > x) and m∗1 (x) = E(x − X1 |X1 < x) the mean residual lifetime and the mean past lifetime of X1 , respectively. In analogy to (2.2), the mean residual quantile and mean past quantile functions for X1 are defined as ∫ v ∫ 1 −1 ∗ −1 pq1 (p)dp, (2.3) (1 − p)q1 (p)dp and m1Q (v) = v m1Q (u) = (1 − u) 0

u

respectively, where 0 < u < 1 and 0 < v < 1. The following relations can be easily obtained: (h1Q (u))−1 = m1Q (u) − (1 − u)m1Q ′ (u) and (h∗1Q (v))−1 = m∗1Q (v) + vm∗1Q ′ (v),

(2.4)

where ′ denotes derivative. We note that based on some relationships between hazard rate and mean residual life function, Sankaran and Sunoj (2004) obtained characterizations for various lifetime distributions. 4

2.1

Results on quantile-based CRI

In this subsection, we consider quantile version of the CRI of X1 and X2 . Denote Q3 (.) = d the quantile function of F1 (F2−1 (.)). Further, let q3 (p) = dp Q3 (p) be the quantile density function of Q3 (p). First, let us state the following definition.

Q−1 2 (Q1 (.))

Definition 2.1 Let Q1 (.) and Q2 (.) be the quantile functions of two absolutely continuous and nonnegative random variables X1 and X2 , respectively. Then, the quantile-based CRI of X1 and X2 is ∫ 1 ( ) CIQ (X1 , X2 ) = − F¯1 (Q1 (p)) ln F¯2 (Q1 (p)) dQ1 (p) ∫0 1 = − (1 − p) ln (1 − Q3 (p))) q1 (p)dp. (2.5) 0

When the distributions of X1 and X2 coincide, that is, Q1 (.) = Q2 (.), we have Q3 (p) = p. So, Equation (2.5) reduces to the quantile-based CRE given by (see Sankaran and Sunoj, 2017) ∫ 1 EQ (X1 ) = − (1 − p) ln(1 − p)q1 (p)dp. (2.5′ ) 0

Note that the measures CIQ (X1 , X2 ) and CIQ (X2 , X1 ) are not symmetric. Thus, in addition to CIQ (X1 , X2 ), it is also reasonable to study CIQ (X2 , X1 ) for the comparison of two independent lifetime distributions in reliability modeling. Also, it is worth pointing that CIQ (X1 , X2 ) quantifies information contained when the actual distribution is F1 (.) and is compared with F2 (.). For the case of CIQ (X2 , X1 ), the role gets reversed. Below, we show that (2.5) can be expressed in terms of the hazard quantile function of X1 which is given by ∫ 1 CIQ (X1 , X2 ) = − (2.6) ln (1 − Q3 (p)) [h1Q (p)]−1 dp. 0

To illustrate the importance of the proposed quantile-based measure given by (2.5), we consider the following examples. Example 2.1 Let X1 and X2 represent two random variables with corresponding quantile functions Q1 (p) = 2p−p2 and Q2 (p) = p, 0 < p < 1. Note that Q1 (p) is the quantile function of a special case of Govindarajulu distribution ( see Govindarajulu, 1977) and Q2 (p) is the 2 quantile function of uniform distribution. Here, Q3 (p) = Q−1 2 (Q1 (p)) = 2p − p . Thus, from Equation (2.5), the quantile-based CRI of X1 and X2 is obtained as ∫ 1 4 CIQ (X1 , X2 ) = −4 (1 − p)2 ln (1 − p) dp = . 9 0 5

Example 2.2 Let X1[ and X2 be two variables] with respective quantile ] rescaled-beta random [ 1 1 functions Q1 (p) = r1 1 − (1 − p) c1 and Q2 (p) = r2 1 − (1 − p) c2 , where r1 , r2 , c1 , c2 > 0 and 0 < p < 1. For simplicity in calculation, we assume that r1 = r2 = 1. Here, Q3 (p) = c2 1 −1 1 − (1 − p) c1 and q1 (p) = c11 (1 − p) c1 . From (2.5), we obtain CIQ (X1 , X2 ) = −

(

c2 c21

)∫

0

1

1

(1 − p) c1 ln(1 − p)dp =

c2 . (1 + c1 )2

Remark 2.1 Note that the special case of Govindarajulu distribution considered in Example 2.1 does not have tractable distribution function though it has closed-form quantile function. Thus, the distribution function-based tool in (1.2) proposed by Taneja and Kumar (2012) is useless in computing CRI of Govindarajulu and uniform distributions. However, for this case, the quantile-based tool given in (2.5) plays useful role to compute the same as seen in Example 2.1. Further, note that the models considered in Example 2.2 have closed form distribution functions and quantile functions. Thus, it is not necessary to apply the tool (2.5) to compute CRI of two rescaled-beta distributions. In this case, one could use (1.2). Now, we consider example which deals with the Cox proportional hazards (PH) model and proportional reversed hazards (PRH) model. In survival studies, these models are widely used. Let X1 and X2 have hazard rates h1 (x) and h2 (x), and reversed hazard rates h∗1 (x) and h∗2 (x), respectively. Suppose X1 and X2 satisfy PH model, that is, they satisfy h2 (x) = θh1 (x), where θ > 0. Analogously, if X1 and X2 satisfy PRH model, then h∗2 (x) = θh∗1 (x), where θ is positive integer. We refer to Nair et al. (2013) for discussion on PH and PRH models. Below, we consider equivalent relations to PH and PRH models in terms of quantile functions and obtain quantile-based CRI using (2.5). For results concerning proportional quantile functions, one may refer to Di Crescenzo et al. (2016). Example 2.3 (i) Let X1 and X2 be two nonnegative and absolutely continuous random variables with quantile functions Q1 (.) and Q2 (.), respectively, which satisfy PH model, that is, ( ) 1 θ Q2 (p) = Q1 1 − (1 − p) , (2.7) θ where ∫ 1 θ > 0. In this case, Q3 (p) = 1 − (1 − p) . Thus, from (2.5) we get CIQ (X1 , X2 ) = −θ 0 (1 − p) ln(1 − p)q1 (p)dp.

(ii) Assume that X1 and X2 satisfy PRH model, that is, 1

Q2 (p) = Q1 (p θ ), where θ is a positive integer. Here, Q3 (p) = pθ . So, CIQ (X1 , X2 ) = − pθ )q1 (p)dp. 6

(2.8) ∫1 0

(1 − p) ln(1 −

Given a set of observations, it is usual practice to apply transformation of data for finding the best statistical model. A parallel approach to this is to keep the actual data fixed and transform the quantile function in searching the best model. The following result provides a tool for the evaluation of the CRI when transformation is applied to the quantile functions of X1 and X2 . Theorem 2.1 Consider two continuous, nondecreasing and invertible transformations τ1 (.) and τ2 (.). Then, ∫ 1 ) ( CIQ (τ1 (X1 ), τ2 (X2 )) = − (1 − p) ln 1 − Q2−1 (τ2−1 (τ1 (Q1 (p)))) dτ1 (Q1 (p)). (2.9) 0

Proof. Let F¯τ1 (X1 ) (x) and F¯τ2 (X2 ) (x) denote the survival functions of τ1 (X1 ) and τ2 (X2 ), respectively. Then, from (1.2), we have ∫ ∞ CI (τ1 (X1 ), τ2 (X2 )) = − F¯τ1 (X1 ) (x) ln F¯τ2 (X2 ) (x)dx. (2.10) 0

Now, using analogous idea of Sunoj et al. (2018), it can be shown that the quantile functions −1 of Fτ1 (X1 ) (x) and Fτ2 (X2 ) (x) are τ1 (Q1 (p)) and Q−1 2 (τ2 (τ1 (Q1 (p)))), respectively. Substituting these in Equation (2.10), the desired result follows. This completes the proof.  From Theorem 2.1, we observe that CI (X1 , X2 ) is not closed with respect to nondecreasing and invertible transformations. Now, as an application of Theorem 2.1, we consider the following examples. Example 2.4 We know that when a random variable X1 follows exponential distribution 1/α with quantile function Q1 (p) = − λ11 ln(1 − p), λ1 > 0, 0 < p < 1, then X1 follows Weibull distribution with quantile function Q1 (p) = (− λ11 ln(1 − p))1/α , α > 0. Now, let X1 and X2 be two independent exponential random variables with respective quantile functions Q1 (p) = − λ11 ln(1 − p) and Q2 (p) = − λ12 ln(1 − p), respectively, where λ1 , λ2 > 0 and 0 < p < 1. Thus, as an application of Theorem 2.1, the quantile-based CRI of two Weibull distributions can 1/α 1/α be obtained, which is described as follows. Here, τ1 (X1 ) = X1 and τ2 (X2 ) = X2 . Thus, −1 λ2 /λ1 τ1 (Q1 (p)) = (− λ11 ln(1 − p))1/α and Q−1 . Using these in 2 (τ2 (τ1 (Q1 (p)))) = 1 − (1 − p) (2.9), we get CIQ

(

1/α 1/α X1 , X2

)

λ2 Γ( α1 + 1) = . αλ21

Example 2.5 Suppose that X1 and X2 follow two independent Pareto distributions with respective quantile functions Q1 (p) = (1 − p)−1/a1 and Q2 (p) = (1 − p)−1/a2 , where a1 , a2 > 0 and 0 < p < 1. Further, it is known that if X1 follows Pareto distribution with quantile function Q1 (p) = (1 − p)−1/c , c > 0, then ln X1 follows exponential distribution with quantile 7

function Q1 (p) = − 1c ln(1−p). Thus, using the result of Theorem 2.1, the quantile-based CRI for two exponential distributions can be computed. Let τ1 (X1 ) = ln X1 and τ2 (X2 ) = ln X2 . a2 −1 a1 Then, τ1 (Q1 (p)) = − a11 ln(1 − p) and Q−1 2 (τ2 (τ1 (Q1 (p)))) = 1 − (1 − p) . Thus, from (2.9), we obtain ∫ 1 2 CIQ (ln X1 , ln X2 ) = −(a2 /a1 ) ln(1 − p)dp = a2 /a21 . 0

In this part of the subsection, we discuss the quantile-based CRI for equilibrium distributions. Assume that a nonnegative random variable X1 has finite mean, that is, E(X1 ) < ∞. Then, the density function of the equilibrium random variable X1e of X1 is given by (see Gupta, 2007) f1e (x) =

F¯1 (x) , x > 0. E(X1 )

(2.11)

It is known that in renewal processes, the equilibrium distribution arises as a limiting distribution of the forward recurrence times. Because of this, the equilibrium distribution is of interest in various applications in queuing and reliability theory. Below, we show a relation between the quantile-based CRI and the quantile-based inaccuracy measure of the equilibrium distributions. To be specific, the quantile-based CRI of X1 and X2 can be expressed in terms of the quantile-based inaccuracy measure of the equilibrium distributions of X1 and X2 , and quantile-based means of X1 and X2 . We ∫1 ∫ 1 denote the quantile-based means of X1 and X2 by µ1Q (= 0 (1 − p)q1 (p)dp) and µ2Q (= 0 Q1 (p)q3 (p)dp), respectively. Theorem 2.2 Let X1 and X2 be two nonnegative random variables with E(X1 ) < ∞ and E(X2 ) < ∞. Then, CIQ (X1 , X2 ) = µ1Q [IQ (X1e , X2e ) − ln µ2Q ] ,

(2.12)

where IQ (X1e , X2e ) denotes ∫the quantile-based inaccuracy measure of X1e and X2e , µ1Q = ∫1 1 (1 − p)q1 (p)dp and µ2Q = 0 Q1 (p)q3 (p)dp. 0 Proof. The quantile-based inaccuracy measure of X1e and X2e is ) ( ) ∫ 1( 1−p 1 − Q3 (p) IQ (X1e , X2e ) = − ln q1 (p)dp. µ1Q µ2Q 0

Now, rest of the proof follows after straightforward calculations.

(2.13) 

From (2.12), we observe that CIQ (X1 , X2 ) is expressed in terms of the quantile-based inaccuracy measure of X1e and X2e in the unity measure of µ1Q apart from a constant term. Further, for µ1Q = 1 and µ2Q = 1, we obtain CIQ (X1 , X2 ) = IQ (X1e , X2e ). Let X1 and X2 be two nonnegative random variables with quantile functions Q1 (p) = 23 (2p − p2 ) 8

and Q2 (p) = 98 p, respectively, where 0 < p < 1. Thus, q3 (p) = 83 (1 − p), 0 < p < 1. ∫1 ∫1 In this case, it can be obtained µ1Q = 3 0 (1 − p)2 dp = 1 and µ2Q = 2 0 (2p − p2 )(2 − 2p)dp = 1. Another measure of interest in information theory is Kullback-Leibler (KL) divergence. For two random variables X1 and X2 , the KL divergence is given ∫ ∞nonnegative f1 (x) by KL(X1 , X2 ) = 0 f1 (x) ln( f2 (x) )dx. Sankaran et al. (2016) studied various properties ∫1 of quantile-based KL divergence. It is given by KLQ (X1 , X2 ) = − 0 ln (q1 (p)f2 (Q1 (p))) dp. Next, we express quantile-based KL divergence of X1e and X2e in terms of the quantile-based CRE and quantile-based CRI. Theorem 2.3 Let X1 and X2 be two nonnegative absolutely continuous random variables with finite means. Then, ( ) ] µ2Q 1 [ KLQ (X1e , X2e ) = ln + CIQ (X1 , X2 ) − EQ (X1 ) , (2.14) µ1Q µ1Q where µ1Q and µ2Q are defined in Theorem 2.2.

Proof. Proof is straightforward, and hence omitted for brevity.



If µ1Q = 1 and µ2Q = 1, then KLQ (X1e , X2e ) = CIQ (X1 , X2 ) − EQ (X1 ). Now, in the following theorem we obtain a bound of CIQ (X1 , X2 ). Note that the study of bounds are useful when the closed-form expression of CIQ (X1 , X2 ) are difficult to obtain. Theorem 2.4 Let X1 and X2 be two nonnegative and absolutely continuous random variables with finite CIQ (X1 , X2 ). Then, CIQ (X1 , X2 ) ≥ EQ (X1 ) + µ1Q − µ2Q ,

(2.15)

provided EQ (X1 ) given in (2.5′ ), µ1Q and µ2Q are finite. Proof. Using nonnegativity property of KL divergence measure, we get from (2.14) as ( ) µ1Q CIQ (X1 , X2 ) ≥ EQ (X1 ) + µ1Q ln . (2.16) µ2Q ( ) Thus, the desired result follows using inequality u ln uv ≥ u − v for nonnegative u and v in (2.16).  Kundu et al. (2016) introduced the concept of cumulative residual inaccuracy ratio (CRIR) as CIR(X1 , X2 ) =

CI(X1 , X2 ) , E(X1 )

(2.17)

where CI(X1 , X2 ) and E(X1 ) are known as CRI and CRE, respectively. In the following definition we introduce the notion of quantile-based CRIR. 9

Definition 2.2 Let X1 and X2 be two nonnegative and absolutely continuous random variables with respective quantile functions Q1 (.) and Q2 (.). Then, the quantile-based CRIR is given by CIQ (X1 , X2 ) CIRQ (X1 , X2 ) = . (2.18) EQ (X1 )

For two identical distributions, CIRQ (X1 , X2 ) = 1. Similar to CIQ (X1 , X2 ), CIRQ (X1 , X2 ) given by (2.18) is also not symmetric. A casual statement on the interpretation of CIRQ (X1 , X2 ) may be like that it measures the discrepancy in the amount of information carried by the CRE when the true distribution F1 (.) is replaced by another distribution F2 (.). Further, CIRQ (X1 , X2 ) > 1 implies that using F2 (.) instead of F1 (.) provides larger information in the sense of quantile-based CRI rather than that carried by quantile-based CRE of F1 (.). Further, using the nonnegativity property of KL divergence measure in (2.14), we get the following bound of CIRQ (X1 , X2 ). The proof is simple and hence omitted.

Theorem 2.5 For two random variables X1 and X2 , we have ( ) µ1Q µ1Q CIRQ (X1 , X2 ) ≥ 1 + ln . EQ (X1 ) µ2Q

(2.19)

To illustrate Theorem 2.5, we consider the following example.

Example 2.6 Consider two exponential random variables X1 and X2 with quantile functions Q1 (p) = − λ11 ln(1 − p) and Q2 (p) = − λ12 ln(1 − p), respectively, where λ1 , λ2 > 0 and 0 < p < 1. Then, a lower bound of CIRQ (X1 , X2 ) given by (2.19) can be obtained as 1 + ln( λλ12 ). It is worth pointing that in this case, the exact value of CIRQ (X1 , X2 ) can be obtained as λλ21 . Below, we present a table (Table 1) to compare the exact value of CIRQ (X1 , X2 ) with the lower bound for various values of λ1 and λ2 . Table 1: Comparison of exact value of CIRQ (X1 , X2 ) and its lower bound (λ1 , λ2 ) (0.5,0.5) Exact value 1 Lower bound 1

(0.5,1) 2 1.69

(0.5,2) 4 2.38

(1,1.5) 1.5 1.41

(1,2) 2 1.69

(1,2.5) 2.5 1.92

(2,3) 1.5 1.41

(3,4) 1.33 1.29

(4,5) 1.25 1.22

Example 2.7 Consider X1 and X2 with respective quantile functions Q1 (p) = 2p − p2 and − 1 Q2 (p) = −1/ ln p, where 0 < p < 1. In this case, Q3 (p) = e 2p−p2 , and hence q3 (p) = − 1 1−p e 2p−p2 . After simplification, CIRQ (X1 , X2 ) can be obtained as (2p−p2 )2 ∫ 1 ( ) − 1 CIRQ (X1 , X2 ) = −9 (1 − p)2 ln 1 − e 2p−p2 dp, 0

which is complicated to evaluate. For such case, Theorem 2.5 is useful it provides a ∫ ∞since −u lower bound of CIRQ (X1 , X2 ). It is obtained as 1 + 3 ln(4/3) − 3 ln 1 e u du≈ 6.413841. This is useful to assess the proper information. 10

2.2

Results on quantile-based CPI

Here, we will discuss various properties of the quantile-based CPI in analogy with the quantile-based CRI. Consider the following definition. Definition 2.3 Let X1 and X2 be two nonnegative absolutely continuous random variables with quantile functions Q1 (.) and Q2 (.), respectively. Then, the quantile-based CPI of X1 and X2 is given by ∫ 1 ∗ F1 (Q1 (p)) ln(F2 (Q1 (p)))dQ1 (p) CIQ (X1 , X2 ) = − 0 ∫ 1 p ln(Q3 (p))q1 (p)dp. = − (2.20) 0

If two quantile functions∫ Q1 (.) and Q2 (.) coincide, CIQ ∗ (X1 , X2 ) reduces to the quantile1 based CPE EQ ∗ (X1 ) = − 0 p ln(p)q1 (p)dp (see Sankaran and Sunoj, 2017). Analogous to CIQ (X1 , X2 ), the quantile-based CPI given by (2.20) is also not symmetric. Further, (2.20) can be written in terms of the reversed hazard quantile function of X1 , which is defined in (2.2). It is given by ∫ 1 ∗ CIQ (X1 , X2 ) = − (2.21) ln(Q3 (p))[h∗1Q (p)]−1 dp. 0

Next, we present examples to describe the usefulness of (2.20). It specifically shows that even though the distribution function-based CPI due to Kumar and Taneja (2015) fails to compute cumulative past inaccuracy measure for some distributions, the tool proposed in this subsection (see (2.20)) is able to compute the same. Example 2.8 Let Q1 (p) = 2p − p2 and Q2 (p) = −γ(ln p)−1 , γ > 0, where 0 < p < 1 be the quantile functions of X1 and X2 , respectively. Note that the distribution of X1 is a special case of Govindarajulu distribution (see Govindarajulu, 1977) which has no explicit form of the distribution function. The random γvariable X2 has reciprocal exponential distribution. − Here, q1 (p) = 2(1−p) and Q3 (p) = e 2p−p2 . Thus, from (2.20), CIQ ∗ (X1 , X2 ) = 2(1−ln 2)γ. Let γ = 2. Then, CIQ ∗ (X1 , X2 ) ≈ 1.22741. Example 2.9 Consider the quantile functions of a generalized lambda distribution and a uniform distribution as Q1 (p) = λ11 + λ12 (pλ3 −(1−p)λ4 ) and Q2 (p) = p, respectively, where p ∈ (0, 1), λ1 , λ2 , λ4 > 0 and λ3 is a positive integer. Further, it is known that Q1 (p) represents a life distribution if λ1 − λ−1 2 ≥ 0. For simplicity of our calculation, we consider λ1 = λ2 = 1. In this case, Q3 (p) = 1 + pλ3 − (1 − p)λ4 and q1 (p) = λ3 pλ3 −1 + λ4 (1 − p)λ4 −1 . Substituting these in (2.20), we obtain ∫ 1 ( )( ) ∗ CIQ (X1 , X2 ) = − p ln 1 + pλ3 − (1 − p)λ4 λ3 pλ3 −1 + λ4 (1 − p)λ4 −1 dp, 0

which can be computed numerically for specific values of the parameters. Let λ3 = 1 and λ4 = 0.2. Then, CIQ ∗ (X1 , X2 ) ≈ 0.062943. 11

Let us now discuss the effect of transformations on (2.20). It shows that CIQ ∗ (X1 , X2 ) is not closed under nondecreasing and invertible transformations. Theorem 2.6 Consider two continuous, nondecreasing and invertible transformations τ1 (.) and τ2 (.). Then, ∫ 1 ( ) ∗ −1 CIQ (τ1 (X1 ), τ2 (X2 )) = − p ln Q−1 (2.22) 2 (τ2 (τ1 (Q1 (p)))) dτ1 (Q1 (p)). 0

Proof. The proof is similar to that of Theorem 2.1. Thus, it is omitted.



As an application of Theorem 2.6, we consider the following example. Example 2.10 Let X1 and X2 be two independent reciprocal exponential random variables with respective quantile functions Q1 (p) = −λ1 / ln p and Q2 (p) = −λ2 / ln p, where λ1 , λ2 > 0 and 0 < p < 1. Consider the transformations τ1 (X1 ) = e−λ1 /X1 and τ2 (X2 ) = e−λ2 /X2 . Then, −1 Q−1 2 (τ2 (τ1 (Q1 (p)))) = p. Thus, from (2.22), we obtain ∫ 1 ( −λ1 /X1 −λ2 /X2 ) 1 ∗ CIQ e ,e =− p ln pdp = , 4 0 which is equal to the CPI for two standard uniform distributions.

Similar to Theorem 2.4, we get the following result which gives a lower bound of(CI)Q ∗ (X1 , X2 ). The proof readily follows on using the log-sum inequality and the inequality u ln uv ≥ u − v for all u, v > 0. Thus, it is omitted for the sake of brevity. Theorem 2.7 Consider two random variables with a common support [0, r], where r is finite. Further, assume Q1 (.) and Q2 (.) the quantile functions of X1 and X2 , respectively. Then, CIQ ∗ (X1 , X2 ) ≥ EQ ∗ (X1 ) + µ2Q − µ1Q .

(2.23)

Similarly to the quantile-based CRIR, here, we define cumulative past inaccuracy ratio (CPIR) based on the quantile function. Definition 2.4 Let X1 and X2 be two nonnegative random variables with quantile functions Q1 (.) and Q2 (.), respectively. Then, the quantile-based CPIR of X1 and X2 can be defined as CIQ ∗ (X1 , X2 ) CIRQ ∗ (X1 , X2 ) = . (2.24) EQ ∗ (X1 )

When two distributions coincide, CIRQ ∗ (X1 , X2 ) = 1. The measure CIRQ (X1 , X2 ) may be interpreted that it measures the discrepancy in the amount of information carried by the CPE when the true distribution F1 (.) is replaced by another distribution F2 (.). If CIRQ ∗ (X1 , X2 ) < 1, it implies that using F2 (.) instead of F1 (.) provides less information in the sense of quantile-based CPI rather than that carried by quantile-based CPE of F1 (.). Below, we obtain a lower bound of CIRQ ∗ (X1 , X2 ) in terms of the quantile-based CPE and means of X1 and X2 . 12

Theorem 2.8 Consider two random variables X1 and X2 with a common support [0, r], where r is finite. Then, CIRQ ∗ (X1 , X2 ) ≥ 1 +

1 (µ2Q − µ1Q ) . EQ (X1 ) ∗

Proof. The proof follows using (2.23) and (2.24).

3

(2.25) 

Quantile-based dynamic CRI and CPI

In the context of reliability and survival analysis, when the current ages of two components need to be taken into the account, the measures given by (1.2) and (1.3) are not appropriate. In this case, dynamic measures are useful to describe the information carried out by the random lifetimes when age changes. This led various authors to consider dynamic information measures. Taneja and Kumar (2012) considered the dynamic version of CRI which is analog to the dynamic CRE due to Asadi and Zohrevand (2007). The dynamic CRI of X1 and X2 is defined as (¯ ) ∫ ∞ ¯ F2 (x) F1 (x) CI(X1 , X2 ; t) = − ln ¯ dx, (3.1) F¯1 (t) F2 (t) t for t > 0 such that F¯1 (t) > 0. The function given in (3.1) can be considered as the CRI for the residual random lifetimes X1t = [X1 |X1 ≥ t] and X2t = [X2 |X2 ≥ t]. Moreover, CI(X1 , X2 ; t) reduces to CRI given by (1.2) when t → 0. Further, there are some situations, where it is reasonable to study inaccuracy measure based on past random lifetimes X1(t) = [X1 |X1 ≤ t] and X2(t) = [X2 |X2 ≤ t]. For example, this random variable is used in economics to study the income distribution of the poor for a poverty line t. Because of this, in analogy to Di Crescenzo and Longobardi (2009), Kumar and Taneja (2015) proposed dynamic version of the CPI. The dynamic CPI of X1 and X2 is ( ) ∫ t F2 (x) F1 (x) ∗ ln dx, (3.2) CI (X1 , X2 ; t) = − F2 (t) 0 F1 (t) for t > 0 such that F1 (t) > 0. As discussed earlier, (3.1) and (3.2) can not be used for the cases where distribution functions are not available in tractable form. In the subsequent subsections, we consider quantile-based dynamic CRI and CPI of X1 and X2 .

3.1

Results on quantile-based dynamic CRI

We start this subsection with the following definition.

13

Definition 3.1 Let X1 and X2 be two nonnegative and absolutely continuous random variables with respective quantile functions Q1 (.) and Q2 (.). Then, the quantile-based dynamic CRI of X1 and X2 is defined as )∫ 1 ( ) ( 1 1 − Q3 (p) (1 − p) ln (3.3) CIQ (X1 , X2 ; u) = − q1 (p)dp, 1−u 1 − Q3 (u) u where 0 < u < 1 and Q3 (p) = Q−1 2 (Q1 (p)). Note that the functional (3.3) reduces to the quantile version of the dynamic CRE given by EQ (X1 ; u) = −

(

1 1−u

)∫

u

1

(1 − p) ln

(

1−p 1−u

)

q1 (p)dp

(3.3′ )

when two distributions coincide. Further, when u tends to 0, (3.3) becomes quantile-based CRI given by (2.5). Below, we consider examples to evaluate quantile-based dynamic CRI using (3.3). Example 3.1 Consider two exponential random variables X1 and X2 with quantile functions Q1 (p) = − θ11 ln(1−p) and Q2 (p) = − θ12 ln(1−p), respectively, where θ1 , θ2 > 0 and 0 < p < 1. Using these, it can be obtained Q3 (p) = 1 − (1 − p)θ2 /θ1 and q1 (p) = (θ1 (1 − p))−1 . Thus, from (3.3), we obtain CIQ (X1 , X2 ; u) = θ2 /θ12 . Note that the family of van Staden-Loots distributions (see van Staden and Loots, 2009) does not have explicitly known distribution function. Hence, the measure (3.1) proposed by Taneja and Kumar (2012) is not capable to quantify the dynamic CRI. In the following example, we show that the mathematical tool (3.3) can be used in this direction. Example 3.2 Consider two random variables X1 and X2 which follow van Staden-Loots 3 and uniform distributions with respective quantile functions Q1 (p) = θ1 + θ2 [( 1−θ )(pθ4 − 1) − θ4 ( θθ43 )((1 − p)θ4 − 1)] and Q2 (p) = p, where θ1 , θ2 , θ3 , θ4 > 0 and 0 < p < 1. Here, it is easy to 3 obtain Q3 (p) = θ1 + θ2 [( 1−θ )(pθ4 − 1) − ( θθ34 )((1 − p)θ4 − 1)]. Thus, from (3.3), we get θ4 )[ ∫ 1 ( [( 1 − θ ) 1 3 (pθ4 − 1) CIQ (X1 , X2 ; u) = − θ2 (1 − p) ln 1 − θ1 − θ2 1−u θ 4 u ] ( θ )( )])[ 3 − (1 − p)θ4 − 1 (1 − θ3 )pθ4 −1 + θ3 (1 − p)θ4 −1 dp θ4 ( [( 1 − θ ) ( θ )( )]) 3 3 −θ2 ln 1 − θ1 − θ2 (uθ4 − 1) − (1 − u)θ4 − 1 θ4 θ4 ] ∫ 1 ] [ × (1 − p) (1 − θ3 )pθ4 −1 + θ3 (1 − p)θ4 −1 dp , 0 < u < 1, (

u

14

which can be evaluated numerically for specific values of θ1 , θ2 , θ3 , θ4 and u. Note that for Q1 (p) to be a life distribution, the condition θ1 − θ2 (1 − θ3 )θ4−1 ≥ 0 must be satisfied ( see Nair et al., 2013). Consider θ1 = 1, θ2 = 3, θ3 = 0.2 and θ4 = 3. For these values, CIQ (X1 , X2 ; 0.1) ≈ 0.168331. The following result provides a tool for the evaluation of the quantile-based dynamic CRI given by (3.3). Theorem 3.1 Consider two continuous, nondecreasing and invertible transformations τ1 (.) and τ2 (.). Then )) ) (( ∫ 1( −1 −1 1 − Q (τ (τ (Q (p)))) 1−p 1 1 2 2 ) dτ1 (Q1 (p)).(3.4) CIQ (τ1 (X1 ), τ2 (X2 ); u) = − ln ( −1 −1 1−u 1 − Q2 (τ2 (τ1 (Q1 (u)))) u Proof. The proof is similar to Theorem 2.1, and hence omitted.



The following example illustrates the result in Theorem 3.1. Example 3.3 Consider the random variables as in Example 2.5. Let τ1 (X1 ) = X1 − 1 −1 and τ2 (X2 ) = X2 − 1. Then, τ1 (Q1 (p)) = (1 − p)−1/a1 − 1 and Q−1 2 (τ2 (τ1 (Q1 (p)))) = 1 − (1 − p)a2 /a1 . Thus, as an application of (3.4), we obtain the quantile-based dynamic CRI of two Lomax distributions with a common location parameter 1 and different shape parameters a1 and a2 as ( ) ∫ 1 1−u a2 − a1 (1 − p) 1 ln dp CIQ (X1 − 1, X2 − 1; u) = (1 − u)a21 u 1−p a2 = 1 , a1 < 1, 0 < u < 1. (a1 − 1)2 (1 − u) a1 In the following, we obtain a relation between the quantile-based dynamic CRI and the mean residual quantile function, which is useful to obtain further results. Theorem 3.2 The quantile-based dynamic CRI given by (3.3) satisfies the following ordinary differential equation: ( ) ( ) 1 q3 (u) ′ CIQ (X1 , X2 ; u) = CIQ (X1 , X2 ; u) − m1Q (u), (3.5) 1−u 1 − Q3 (u) where 0 < u < 1 and CIQ ′ (X1 , X2 ; u) stands for the derivative with respect to u. Proof. On differentiating (3.3) with respect to u, we obtain ( )2 ∫ 1 ( ) 1 1 − Q3 (p) ′ CIQ (X1 , X2 ; u) = − ln (1 − p)q1 (p)dp 1−u 1 − Q3 (u) u )( )∫ 1 ( 1 q3 (u) (1 − p)q1 (p)dp. − (1 − Q3 (u)) 1−u u 15

(3.6)

Thus, from (3.3) and (3.6), the desired result follows.



Note that q1 (u) = [(1 − u)h1Q (u)]−1 , where h1Q (u) denotes the hazard quantile function of X1 given by (2.2). Using this, we get q3 (u) h2Q (Q1 (u)) = , 1 − Q3 (u) (1 − u)h1Q (u)

(3.7)

where h2Q (Q1 (u)) = f2 (Q1 (u))/F¯2 (Q1 (u)). Using (3.7), Equation (3.5) can be further written as ( ( ) ) 1 h2Q (Q1 (u)) ′ CIQ (X1 , X2 ; u) = CIQ (X1 , X2 ; u) − m1Q (u). (3.8) 1−u (1 − u)h1Q (u) Note that there exist distributions which are not monotone in terms of quantile-based dynamic CRI. We consider the following example in this purpose. Example 3.4 Let X1 and X2 follow reciprocal exponential distributions with quantile functions Q1 (p) = −λ1 / ln p and Q2 (p) = −λ2 / ln p, where λ1 , λ2 > 0 and 0 < p < 1. Then, the quantile-based dynamic CRI is λ2 ) )∫ 1( ) ( ( 1−p 1 − p λ1 λ1 ln (3.9) dp. CIQ (X1 , X2 ; u) = − λ2 1−u p(ln p)2 λ1 u 1−u Now, Figure 1 shows that CIQ (X1 , X2 ; u) given by (3.9) is not monotone with respect to u. The following result provides conditions under which CIQ (X1 , X2 ; u) is increasing and decreasing. Theorem 3.3 The dynamic measure CIQ (X1 , X2 ; u) is increasing (decreasing) in u if and only if CIQ (X1 , X2 ; u) ≥ (≤)

m1Q (u) , h3Q (u)(1 − Q3 (u))

(3.10)

where h3Q (u) = 1/((1 − u)q3 (u)) denotes the hazard quantile function of Q−1 1 (Q2 (p)). Proof. We know that CIQ (X1 , X2 ; u) is increasing (decreasing) if and only if CIQ ′ (X1 , X2 ; u) ≥ (≤)0. Hence, the result follows from (3.8) and (3.7). 

3.1.1

Bounds and characterizations

In the following, we obtain bounds for quantile-based dynamic CRI given by (3.3).

16

CIQ H X1 , X2 ;uL 50

40

30

20

10

u 0.2

0.4

0.6

0.8

1.0

(a)

Figure 1: Plot of CIQ (X1 , X2 ; u) for u ∈ (0, 1) as in Example 3.4 when λ1 = 2 and λ2 = 3. Theorem 3.4 Consider two random variables X1 and X2 with quantile functions Q1 (.) and Q2 (.), respectively. Then, for 0 < u < 1, ) ( m1Q (u) CIQ (X1 , X2 ; u) ≥ EQ (X1 ; u) + m1Q (u) ln . m2Q (u) Proof. The result can be obtained upon using the log-sum inequality. (

m1Q (u) m2Q (u)



)

Remark 3.1 Note that m1Q (u) ln ≥ m1Q (u) − m2Q (u). Using this, it is easy to obtain another lower bound of CIQ (X1 , X2 ; u) given by EQ (X1 ; u)+m1Q (u)−m2Q (u). However, lower bound obtained in Theorem 3.4 is greater than or equal to EQ (X ( 1 ; u)+m ) 1Q (u)−m2Q (u). Thus, in terms of the sharpness of bounds, EQ (X1 ; u) + m1Q (u) ln than EQ (X1 ; u) + m1Q (u) − m2Q (u).

m1Q (u) m2Q (u)

is always better

Theorem 3.5 Let X1 and X2 be two nonnegative and absolutely continuous random variables with quantile functions Q1 (.) and Q2 (.), respectively. Further, assume that they satisfy PH model given by (2.7). Then, we have CIQ (X1 , X2 ; u) = θEQ (X1 ; u), θ > 0, where EQ (X1 ; u) is given in (3.3′ ). 17

(3.11)

Proof. Proof follows from (2.7) and (3.3). Thus, it is omitted.



Corollary 3.1 Let X1 and X2 satisfy PH model given by (2.7). Assume that θ ≥ 1. Then, CIQ (X1 , X2 ; u) ≥ EQ (X1 ; u). Proof. Under the condition made, the proof follows from (3.11).



Next, we compare two statistical distributions using quantile-based dynamic CRI. In this direction, we use the notion of quantile-based hazard rate order. For some definitions of useful ageing concepts involving quantile functions, one may see the paper by Nair and Vineshkumar (2011). Definition 3.2 Let X1 and X2 be two nonnegative absolutely continuous random variables with quantile-based hazard rate functions h1Q (p) and h2Q (p), respectively. Then, X1 is said to be smaller than X2 in quantile-based hazard rate order denoted by X1 ≤qh X2 if h1Q (p) ≥ h2Q (p), where 0 < p < 1. Theorem 3.6 Suppose that X1 ≤qh X2 . Then, CIQ (X1 , X2 ; u) ≤ EQ (X1 ; u), where EQ (X1 ; u) is given in (3.3′ ). Proof. From (3.3), we get )[ ( ) ( ) ( )] ∫ 1( 1 − Q3 (p) 1 − Q3 (u) 1−p 1−p CIQ (X1 , X2 ; u) = − ln − ln + ln q1 (p)dp 1−u 1−p 1−u 1−u u )[ ( ) ( )] ∫ 1( 1 − Q3 (p) 1 − Q3 (u) 1−p = − ln − ln q1 (p)dp 1−u 1−p 1−u u ) ( ) ∫ 1( 1−p 1−p − ln q1 (p)dp. 1−u 1−u u Further, under the condition made, Q3 (1 − p)/p is decreasing in p, which implies (1 − Q3 (p))/(1 − p) is increasing in p (see p.-304, Nair et al. 2013). Thus, the first expression is non-positive. Hence, the desired result follows.  Various authors have studied characterizations of PH models for several information measures. In this direction, we refer to Sankaran et al. (2016), Kayal (2017), Sunoj et al. (2017) and Kayal and Tripathy (2018). Besides these, we also refer to Belzunce and MartinezRiquelme (2017) on characterizations and conditions for the comparison of various quantilebased measures. Thus, naturally, it is of interest to study such property for the quantile-based dynamic CRI. Note that when X1 and X2 satisfy (2.7), the similar characterization result for CIQ (X1 , X2 ; u) depends on the distribution of X1 . Indeed, CIQ (X1 , X2 ; u) is constant and equal to a nonnegative real number if m1Q (u) = c, 18

(3.12)

where c is a nonnegative real number. Equation (3.5) can be used to characterize lifetime distributions based on CIQ (X1 , X2 ; u). Assume that F1 (.) is the distribution function of a true model and F2 (.) is that of the chosen model. Then, a useful problem in modeling is to identify the true model when the chosen model is known. In the following theorem, we show that along with known F2 (.), the functional CIQ (X1 , X2 ; u) uniquely determines F1 (.). Theorem 3.7 Let X2 follow exponential distribution with quantile function Q2 (p) = − λ12 ln(1− p), λ2 > 0, 0 < p < 1. Then, X1 is exponential if and only if CIQ (X1 , X2 ; u) given by (3.3) is a positive constant. Proof. Let CIQ (X1 , X2 ; u) = A, where A > 0. Thus, from (3.8), noting that h2Q is constant, we obtain m1Q (u) A = . h1Q (u) θ2

(3.13)

Using (2.4) to (3.13), we get m1Q (u) − (1 − u)m1Q ′ (u) =

A . θ2 m1Q (u)

(3.14)

Denote η(u) = (1 − u)m1Q (u), which implies η ′ (u) = (1 − u)m1Q ′ (u) − m1Q (u). Thus, (3.14) reduces to a first order ordinary differential equation in η(u), which is given by ( ) A 1−u ′ η (u) = − . (3.15) θ2 η(u) Solving this, we get η(u) = (A/θ2 )1/2 (1 − u) + C, where C is an arbitrary constant. Moreover, when u → 1, we have C = η(1) = 0. Thus, m1Q (u) = (A/θ2 )1/2 , a constant. Hence, X1 is exponentially distributed. To prove ‘reverse’ part, we assume that X1 has exponential distribution with quantile function Q1 (p) = − λ11 ln(1 − p), λ1 > 0, 0 < p < 1. Thus, the rest of the proof follows from Example 3.1. This completes the proof of theorem.  Next, we characterize a lifetime model when quantile-based dynamic CRI is expressed in terms of mean residual quantile function. Theorem 3.8 Let X1 and X2 be two random variables with quantile functions Q1 (.) and Q2 (.), respectively. Further, assume that they satisfy PH model given by (2.7). Let m1Q (u) < ∞. Then, CIQ (X1 , X2 ; u) = cm1Q (u) if and only if X1 follows exponential distribution for c = θ.

19

Proof. First, we prove ‘if’ part. Let X1 follow exponential distribution with quantile function Q1 (p) = − λ11 ln(1 − p), λ1 > 0. For this distribution, the mean residual quantile function can be obtained as m1Q (u) = 1/λ1 . Further, the quantile-based dynamic CRI is CIQ (X1 , X2 ; u) = θ/λ1 = cm1Q (u) implies c = θ. Hence, the proof of ‘if’ part follows. To prove ‘only if’ part, assume that the given relation holds, that is, CIQ (X1 , X2 ; u) = cm1Q (u). Since X1 and X2 have PH model, thus we have from Equation (3.16) as ∫ 1 1 c ln(1 − u)m1Q (u) − (1 − p) ln(1 − p)q1 (p)dp = m1Q (u). 1−u u θ On differentiating (3.17) with respect to u, we get ] c m1Q (u) [ c ln(1 − u) [m1Q ′ (u) + q1 (u)] + − ln(1 − u) − 1 = m1Q ′ (u). 1−u θ θ m

(3.16)

(3.17)

(3.18)

(u)

1Q Furthermore, it can be shown that m1Q ′ (u) = 1−u − q1 (u). Thus, from (3.18) we obtain after some simplification as ) (c − 1 q1 (u). (3.19) m1Q ′ (u) = θ Solving the above ordinary differential equation, we get (c ) m1Q (u) = − 1 Q1 (u) + m1Q (0). (3.20) θ Thus, the result follows. Hence, the proof is complete. 

3.2

Quantile-based dynamic CPI

In reliability and life testing studies, statistical observations are of truncated nature. The truncation may occur from right side. Further, to fit the available data, sometime, we get a model having no closed-form distribution function, but closed-form quantile function. In such case, CI ∗ (X1 , X2 ; t) given in (3.2) is not an appropriate tool to study dynamic CPI. In this subsection, we propose quantile-based dynamic CPI and study its various properties. First, consider the following definition. Definition 3.3 Let X1 and X2 be two nonnegative and absolutely continuous random variables with respective quantile functions Q1 (.) and Q2 (.). Then, the quantile-based dynamic CPI of X1 and X2 is given by ( )∫ v ( ) 1 Q3 (p) ∗ CIQ (X1 , X2 ; v) = − p ln q1 (p)dp, (3.21) v Q3 (v) 0 where 0 < v < 1 and Q3 (p) = Q−1 2 (Q1 (p)). 20

Clearly, CIQ ∗ (X1 , X2 ; v) measures the cumulative inaccuracy between two past random variables based on their quantile functions. Note that ∫the functional given by (3.21) reduces (p) ∗ 1 v to the quantile-based dynamic CPE EQ (X1 ; v) = − v 0 p ln v q1 (p)dp when two distributions coincide. Further, when X1 and X2 have PRH model given by (2.8), (3.21) reduces to a constant multiple of EQ ∗ (X1 ; v), that is, CIQ ∗ (X1 , X2 ; v) = θEQ ∗ (X1 ; v), where θ is a positive integer. Now, consider the following examples to compute CIQ ∗ (X1 , X2 ; v) using (3.21). Example 3.5 Consider two random variables X1 and X2 which satisfy PRH model given by (2.8). In this case, using ∫ v (3.21), the quantile-based dynamic CPI can be obtained as ∗ CIQ (X1 , X2 ; v) = −(θ/v) 0 pq1 (p) ln(p/v)dp, 0 < v < 1.

Example 3.6 Let Q1 (p) = c1 p/(1 − p) and Q2 (p) = c2 p1/c3 , where c1 , c2 , c3 > 0 and 0 < p < p c3 1. In this case, Q3 (p) = ( cc12 )c3 ( 1−p ) . Thus, as an application of (3.21), we obtain CIQ

∗

) ( ) (c c ) ∫ v ( p p 1−v 1 3 (X1 , X2 ; v) = − ln × dp, 0 < v < 1, v (1 − p)2 1−p v 0

which can be computed numerically for some values of c1 , c3 and v. Let c1 = 1 and c3 = 2. Then, CIQ ∗ (X1 , X2 ; 0.7) ≈ 2.05479. Next, we consider the effect of transformations to the quantile-based dynamic CPI. Theorem 3.9 Consider two continuous, nondecreasing and invertible transformations τ1 (.) and τ2 (.). Then, )) ∫ v ( ) ( ( −1 −1 Q (τ (τ (Q (p)))) p 1 1 2 2 ) dτ1 (Q1 (p)). (3.22) CIQ ∗ (τ1 (X1 ), τ2 (X2 ); v) = − ln ( −1 −1 v Q2 (τ2 (τ1 (Q1 (v)))) 0 Proof. The proof is similar to Theorem 3.1. Hence, it is omitted for the sake of brevity.  To illustrate Theorem 3.9, we consider the following example. Example 3.7 Consider the random variables and the transformations as in Example 2.4. Then, using (3.22), the quantile-based dynamic CPI measure for two Weibull distributions with scale parameters λ1 , λ2 > 0 and a common shape parameter α > 0 can be obtained as ) ( )1/α ∫ v( ) ( ( ) 1 − (1 − p)λ2 /λ1 1 p 1/α 1/α ∗ , 0 < v < 1. ln d − ln(1 − p) CIQ X1 , X2 ; v = − v 1 − (1 − v)λ2 /λ1 λ1 0 This integration can be evaluated numerically for particular values of the parameters and v. Consider α = 1, λ1 = 2 and λ2 = 4. Then, CIQ ∗ (X1 , X2 ; 0.9) ≈ 0.105546. Similar to Theorem 3.2, the following theorem provides a connection between CIQ ∗ (X1 , X2 ; v) and m1Q ∗ (v). 21

Theorem 3.10 The quantile-based dynamic CPI measure CIQ ∗ (X1 , X2 ; v) satisfies the differential equation given by ( ) ( ) q3 (v) 1 ∗′ ∗ CIQ (X1 , X2 ; v) = m1Q (v) − CIQ ∗ (X1 , X2 ; v), 0 < v < 1, (3.23) Q3 (v) v where CIQ ∗ ′ (X1 , X2 ; v) denotes the derivative with respect to v. Proof. The proof of the theorem is straightforward, and hence it is omitted.



Again, we have q3 (v) h2Q ∗ (Q1 (v)) = , Q3 (v) vh1Q ∗ (v)

(3.24)

where h2Q ∗ (Q1 (v)) = f2 (Q1 (v))/F2 (Q1 (v)). Now, making use of (3.24) in (3.23), we get ) ( ) ( 1 h2Q ∗ (Q1 (v)) ∗′ ∗ m1Q (v) − CIQ ∗ (X1 , X2 ; v). (3.25) CIQ (X1 , X2 ; v) = ∗ vh1Q (v) v Next, we deal with an example to show that the quantile-based dynamic CPI is not monotone for some cases. Example 3.8 Consider two random variables X1 and X2 as in Example 2.2. In this case, we obtain ( )∫ v ( ) ( ) c2 /c1 1 1 p 1 − (1 − p) −1 ∗ CIQ (X1 , X2 ; v) = − (1 − p) c1 ln dp, 0 < v < 1. (3.26) c1 v 1 − (1 − v)c2 /c1 0 We plot (3.26) in Figure 2 for c1 = 0.02 and c2 = 0.05 which shows that CIQ ∗ (X1 , X2 ; v) is not monotone with respect to v.

The following result provides condition under which CIQ ∗ (X1 , X2 ; v) is decreasing and increasing. Theorem 3.11 Let CIQ ∗ (X1 , X2 ; v) be increasing (decreasing) in v. Then, ) ( h2Q ∗ (Q1 (v)) ∗ m1Q ∗ (v). CIQ (X1 , X2 ; v) ≤ (≥) h1Q ∗ (v) Proof. The proof follows from (3.25), and hence omitted.

22

(3.27) 

CIQ * H X1 , X2 ;vL

0.20

0.15

0.10

0.05

0.2

0.4

0.6

0.8

1.0

v

(a)

Figure 2: Plot of CIQ ∗ (X1 , X2 ; v) for v ∈ (0, 1) as in Example 3.8 when c1 = 0.02 and c2 = 0.05.

3.2.1

Bounds and characterization

Analogous to Theorem 3.4, we obtain the following bounds for the quantile-based dynamic CPI. Theorem 3.12 Let X1 and X2 have quantile functions Q1 (.) and Q2 (.), respectively. Then, for 0 < v < 1, ( ) m1Q ∗ (v) ∗ ∗ ∗ CIQ (X1 , X2 ; v) ≥ EQ (X1 ; v) + m1Q (v) ln . m2Q ∗ (v) 

Proof. Proof is simple, and hence omitted.

Theorem 3.13 Let X1 and X2 satisfy PRH model given by (2.8). Further, let θ ≥ 1. Then, CIQ ∗ (X1 , X2 ; v) ≥ EQ ∗ (X1 ; v). Proof. Under the condition made, we obtain CIQ ∗ (X1 , X2 ; v) = θEQ ∗ (X1 ; v) ≥ EQ ∗ (X1 ; v).



Hence, the proof is complete. 23

Definition 3.4 Let X1 and X2 be two nonnegative absolutely continuous random variables with reversed hazard quantile functions h1Q ∗ (p) and h2Q ∗ (p), respectively. Then, X1 is said to be smaller than X2 in reversed hazard quantile function order denoted by X1 ≤rq X2 if h1Q ∗ (p) ≤ h2Q ∗ (p) for all 0 < p < 1. Theorem 3.14 Suppose that X1 ≤rq X2 . Then, CIQ ∗ (X1 , X2 ; v) ≥ EQ ∗ (X1 ; v). Proof. Under the assumption, Q3 (p)/p is increasing in p (see p-307, Nair et al. 2013). Using this, the proof follows similarly to Theorem 3.6.  Various authors have studied characterization results based on the constancy of the quantile-based divergence and Chernoff distance measures when X1 and X2 have PRH model. Let two nonnegative and absolutely continuous random variables X1 and X2 satisfy (2.8). For this case, it is of interest to check whether CIQ ∗ (X1 , X2 ; v) is free from v or viceversa. Here, it can be shown that characterization result for the constancy of CIQ ∗ (X1 , X2 ; v) depends on the distribution of X1 . More specifically, CIQ ∗ (X1 , X2 ; v) is equal to a nonnegative real number if m1Q ∗ (v) = c,

(3.28)

where c is a nonnegative real number.

4

Concluding remarks

We have proposed quantile-based CRI, CPI and their dynamic versions. It can be noticed that the proposed measures are not replacements to the existing measures. They are alternative tools to obtain inaccuracy measure when lifetime distributions do not have tractable distribution functions but have tractable quantile functions. We have provided some justifications for the need of the proposed quantile-based study of the existing measures. The effect of transformations have been discussed. We obtained bounds, characterizations to the proposed measures. The results presented in this paper generalize some of the existing results in the context with quantile-based CRE and CPE. Acknowledgements The author sincerely wishes to thank the reviewer for the suggestions which have considerably improved the content and the presentation of the paper.

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