Mathematically 1 and 2 are equivalent respective to the dispersive order and excess wealth order, respectively, although the two have different interpretations and reliability prop- erties. For details see Vineshkumar et al. (2015). It may be noted that the hazard (**mean** **residual**) quantile orders mentioned above are different from the usual hazard rate (**mean** **residual** **life**) orders. For detailed study of the concepts and definitions given in this sec- tion we refer to Nair et al. (2013) and Vineshkumar et al. (2015).

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The concept of **residual** **life** is of special interest in reliability theory and survival anal- ysis as it measures the **life** remaining to a device or an individual after it has attained a specific age. Various characteristics of **residual** **life** such as its **mean**, variance, coefficient of variation, higher moments and percentiles have been extensively studied in literature. Among these, the variance **residual** **life** has attracted many researchers including Dallas (1981), Karlin (1982), Chen et al. (1983), Gupta (1987), Gupta et al. (1987), Abouammoh et al. (1990), Adatia et al. (1991), Stein and Dattero (1999), Gupta and Kirmani (2000, 2004), Stoyanov and Al-Sadi (2004), Gupta (2006) and Nair and Sudheesh (2010) when lifetime is treated as a continuous random variable. These works emphasize the im- portance of variance **residual** **life** as (i) a reliability function useful in modelling lifetime data with special reference to inference procedures and characterizations (ii) a means to classify lifetime distributions through the monotonicity properties and (iii) through its relationship with the **mean** **residual** **life** in the same way as the **mean** to the variance; see Hall and Wellner (1981). In the discrete case also the topic in the univariate case has been well studied by several authors that includes Hitha and Nair (1989), Roy (2005), El- Arishy (2005), Sudheesh and Nair (2010), Khorashadizadeh et al. (2010) and Al-Zahrani et al. (2013). The only study that appears to be made in higher dimensional discrete case is that of Roy (2005) who characterized some **bivariate** discrete distributions by cer- tain simple properties of the variance **residual** **life**. There are several multicomponent devices and systems in which the lifetimes of the components are measured as the num- ber of time units completed, or the number of cycles in operation before failure. Also,

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Gupta (2016) has analysed the FGM from a reliability point of view. We extend some of these results to the Cambanis family and provide some new applications. Two basic concepts required for our discussions are the **bivariate** hazard rate and the **mean** **residual** **life**. There are several deﬁnitions for the hazard rate in the multivariate case, of which we ﬁrst consider the **bivariate** scalar hazard rate of Basu (1971), deﬁned as

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Pending a future clarification of the physics of line for- mation, we have attempted to stabilize the algorithm by per- forming simple “corrections” to the forward model to re- move, as far as possible, the spectral bias. In the first in- stance, we have performed retrievals on the Lamont spec- tra discussed in Sect. 4.2, accounting for systematic spectral residuals as follows. We have modified the forward model to include addition of a spectral basis vector, multiplied by a scale factor, to the modeled spectrum. We calculate the **mean** **residual** spectrum from a large set of the Lamont retrievals (the set to be defined shortly). We then use those **mean** resid- uals as the basis vector to be added to the modeled spectrum. The scale factor which multiplies the basis vector is incor- porated in the state vector, to be retrieved for each measured spectrum. It is typically ∼ 1.

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In **life** testing situations, the **mean** additional **life** time given that a component has survived until time t is a function of t, called the **mean** **residual** **life**. We consider the special case of **mean** **residual** **life** in terms of fuzzy random variables. More specifically, if the fuzzy random variables X represent the **life** of a component, then the **mean** **residual** **life** is given by 𝑚 𝛼 (t) = E[ 𝑋 𝛼 − 𝑡/𝑋 𝛼 > 𝑡 ]

Background: Multiple imputation is a commonly used method for handling incomplete covariates as it can provide valid inference when data are missing at random. This depends on being able to correctly specify the parametric model used to impute missing values, which may be difficult in many realistic settings. Imputation by predictive **mean** matching (PMM) borrows an observed value from a donor with a similar predictive **mean**; imputation by local **residual** draws (LRD) instead borrows the donor’s **residual**. Both methods relax some assumptions of parametric imputation, promising greater robustness when the imputation model is misspecified.

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With the delay time concept, see chapter 14, system **life** is assumed to be classified into two stages. The first is the normal working stage where no abnormal condition parameters are to be expected. The second starts when a hidden defect is first initiated with possible abnormal signals. The identification of the initial point in the evolution of such a defect is important and has a direct impact on the subsequent prediction model. Most research on fault diagnosis focuses on the location of the fault, the possible cause of the fault, and of course, the type of fault. This serves for the engineering purpose of deciding what to repair, but does not aid the decision of when to do the task. This initial point defect identification has received very little attention in prognosis literature. Wang (2006b) addressed this problem to some extent using a combination of the delay time concept and the HMM. Much work still remains. It is possible that a multi-stage (>2) failure process could be used, which might be more appropriate to some cases.

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Abstract. Several inequalities involving **bivariate** means introduced by Z.-H. Yang in [15] are established. Also, lower and upper bounds for the means under discussion are obtained. Bounding quantities are expressed in terms of the geometric and quadratic means. Results presented in this paper are obtained with the aid of the Schwab- Borchardt **mean**.

The Supreme Court of Virginia revisited the role of residual doubt at sentencing in Stockton v. is Stockton contended that the trial court erred by refusing to allow argument [r]

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Generally, the **mean** of the entire population ranges from 1.0 to 2.8 ø, with an average of 1.92 ø, indicating the sediments are medium sands. The median is 1.92 ø. The Sorting Co-efficient (So) or Standard Deviation (GSD) range from 0.26 to 0.74, with an average of 0.44, which means the sediments are well sorted. The measure of the symmetry of the grain size distribution: skewness (GSK), range from -0.007 to 0.117, with an average of -0.05. The sediments are therefore symmetrical. The kurtosis (K) range from 0.41 to 1.84, with an average of 1.18, indicating the sediment population is leptokurtic. Most of the samples are unimodal, some are bimodal and a few are trimodal. The samples in the sand dunes and backshore are 54.5% unimodal, 36.4% bimodal and 9.1% trimodal. While those of the beach face are 72.7% unimodal, 18.2% bimodal and 9.1% trimodal. The modal class which is the commonest grain size in the distribution falls within the 0.25 to 0.125 mm size grade - which is fine sand. There are two other size grades that are significant in the population; they are the medium sand – 0.5 – 0.25 mm and the very fine sand -0.125 – 0.063 mm. The separation between the modal class and these other subordinate classes are close that one mode of transport can be inferred for them. The transport mode inferred for the sediment is by saltation. The coarser sand fragments which are greater than 0.5 mm are not common; they are transported by traction mode.

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We study the high order equilibrium distributions of a counting random variable. Properties such as moments, the probability generating function, the stop—loss transform and the **mean** **residual** lifetime, are derived. Ex- pressions are obtained for higher order equilibrium distribution functions under mixtures and convolutions of a counting distribution. Recursive formulas for higher order equilibrium distribution functions of the

ing with respect to p ∈ R for ﬁxed a, b > with a = b, the Schwab-Borchardt **mean** SB(a, b) is strictly increasing in both a and b, nonsymmetric and homogeneous of de- gree with respect to a and b. Many symmetric **bivariate** means are special cases of the Schwab-Borchardt **mean**. For example, P(a, b) = (a – b)/[ arcsin((a – b)/(a + b))] =

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II. S TATISTICAL F EATURES -ANN R ECOGNIZER Statistical features-ANN recognizer comprises two main stages, i.e., (i) process data streams and input representation, and (ii) process monitoring and diagnosis, as shown in Figure 1. In first stage, data streams of two dependent process variables were plotted on a scatter diagram to yield **bivariate** shift patterns. Based on the scatter diagram, data streams were then transformed into statistical features input representation into an ANN. This will be described further in Section IV.

tribution (LD), of which the one-parameter LD is a par- ticular case, for modeling waiting and survival times data. Several properties of the two-parameter LD such as mo- ments, failure rate function, **mean** **residual** **life** function, stochastic orderings, estimation of parameters by the method of maximum likelihood and the method of mo- ments have been discussed. Finally, the proposed distri- bution has been fitted to a number of data sets relating to waiting and survival times to test its goodness of fit to which earlier the one-parameter LD has been fitted and it is found that two-parameter LD provides better fits than those by the one-parameter LD.

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, ( , ) D R x y is a specified point in D , y is the unknown function. By the solution of the IVP on an interval I R , we **mean** a function ( ) x such that (i) is differentiable on I , (ii) the growth of lies in D , (iii) ( ) = x 0 y 0 and (iv) ' ( ) = ( , ( )), x f x x 0 for all x I . The

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The remainder of our paper is organized as follows: In Section 2, we define the new 𝐸𝑃𝑃𝑆 class of distributions. Some statistical properties of the 𝐸𝑃𝑃𝑆 distribution including the ordinary and conditional, **mean** deviations, **residual** and reversed **residual** lifes and order statistics are obtained In Section 3. Maximum likelihood estimates of the unknown parameters are presented in Section 4. In Section 5, we investigate three special cases of the 𝐸𝑃𝑃𝑆 class of distributions. In Section 6, we demonstrate the flexibility and applicability of one special model using a real data set. We provide some concluding remarks in Section 7.

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Fatigue test were carried out in air at room temperature and in 0.842% 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 corrosive aqueous solution. Both fatigue test in air and in corrosive solution were performed using 6 mm diameter cylindrical specimens, as shown in Figure 1. The specimen geometry was designed according to ASTM E466-07 [9]. The roughness of the gauge length surface was evaluated for each specimen through the arithmetic **mean** of four different measurements. Values were between Ra=0.095 micron and Ra=0.130 micron.

The Intervention – Anxiety Relationship Over Time ............................................................. 109 #-).$+')""$!('%)$#222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222287@ #-).$+')""$!'(*!)(222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222287@ $!-)(2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222287@ $!+'#$"%$##)(222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222287@ $!$$#((3$3)22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222887 The Intervention – Quality of **Life** Relationship Over Time .................................................. 112 *!).$!$+')""$!('%)$#2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222889 *!).$!$+')""$!'(*!)(2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222889 $!-)(22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222889 $!+'#$"%$##)(2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222889 $!$$#((3$3)2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222288: The Moderating Effect of Out-of-Class Engagement on the Relationship Between the Intervention with Depression Over Time ................................................................................ 115

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• Overall, there is no evidence to suggest the pipes will have their service **life** limited by erosion of the wall or changes to the strength or stiffness of the material. Moreover, the joints continue to not only function, but also meet the requirements applied to new pipes. There is nothing in the test results to suggest the **life** of the pipes will be limited to 50 years. Given the pipes have been in service for 25 years and are in such good condition, there is no reason to suppose they will not achieve upwards of 100 years service.

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All equipment, engineering parts, communication systems and civil engineering structures of the whole power plant unit have to be liable to a life-time management system. I[r]