TX_1~ABS:AT/ADD:TX_2~ABS:AT 18 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 ReseaRch aRticle Estimating Hazard Function through Reliability Function and Empirical Methods Azhin M. Khudhur, Shvan A. Hama Noory, Bestun M. Abdulkareem Department of Statistics and Informatics, College of Administration and Economics, Salahaddin University-Erbil, Kurdistan Region - F.R. Iraq ABSTRACT In this research, the reliability functions are applied to estimate the hazard function of four used car components such as (tires, brakes, lights, and engine), which are inspected by aperiodic vehicle inspection (PVI) established in Erbil city, a specialized company that conducts the annual technical inspection of vehicles to detect the failure component, that either require repair or replace it with a new one. For our purpose, the data about the failure components of a sample of size (50,000) cars are obtained from the Erbil traffic directorate, which are annually inspected for 11 years (2010–2020) by a (PVI) company. From the available data, the reliability function, hazard function, and probability density function of the failure time of each component are found by the non-parametric method and the estimated Rayleigh distribution since the failure rates of the components are the linear functions of time, also the comparison between their reliability values have made by the mean absolute error method. Keywords: Basics of reliability function, empirical reliability, Rayleigh distribution, hazard function, mean absolute error INTRODUCTION Reliability theory played a great role in various areas of life such as medicine, mechanical, electrical, and electronic engineering. It has been increasingly applied after a wide expansion of industry and become an independent entity since the fast technological developments and the increasing complexity of the equipment parts in the last century. The reliability is mainly conserved with the determination of the probability that a system consisting possible of several components will operate adequately for a given period of time, thus the reliability of the system is through attention to the internal relations of the system components and the impact of these relations in the system reliability, so it is necessary to know the pattern of behavior of these components and then impact on the behavior of the system. In our cities, the vehicles are inspected annually by a (PVI) company, to reduce road accidents and preserve the safety of the passengers, which began to record high numbers of deaths and injuries due to the increase in the number of vehicles and the lack of control systems. Smith[1] the scale is said to be highly reliable if it produces similar results under constant conditions. It is the characteristics of a set of test scores that relate to the amount of random error of the measurement process that may be included in the scores. Highly reliable scores are accurate, reproduced, and consistent when the test is repeated. That is, if you repeat the test process with a group of test takers, we will get essentially the same results, usually using different types of reliability scores, ranging from 0.00 (big error) to 1.00 (without error), indicating the amount of error in the scores. Tanner and Wong[2] presented the smoothing of the empirical hazards, a kernel estimate of the hazard function from censored data is obtained and criteria for asymptotic normality are considered. The mean and variance of the estimator are presented in small and large sample expressions. Hubbard[3] proposed an empirical examination of moral hazard in the vehicle inspection market, when vendors have an incentive to misrepresent a buyer’s condition in “diagnosis- cure” sellers such as auto repair and health care, moral hazard occurs. This article looks at the California automobile emission inspection market to determine if there are any incentives for inspectors to assist vehicles pass. Consumers, in my experience, Corresponding Author: Azhin M. Khudhur, Department of Statistics and Informatics, College of Administration and Economics, Salahaddin University-Erbil, Kurdistan Region - F.R. Iraq E-mail: azhin.khudhur@su.edu.krd Received: January 27, 2024 Accepted: February 28, 2024 Published: March 20, 2024 DOI: 10.24086/cuesj.v8n1y2024.pp18-28 Copyright © 2024 Azhin M. Khudhur, Shvan A. Hama Noory, Bestun M. Abdulkareem. This is an open-access article distributed under the Creative Commons Attribution License. Cihan University-Erbil Scientific Journal (CUESJ) Khudhur, et al.: Estimating hazard function through reliability function 19 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 can typically give companies and inspectors incentives to assist them pass. Thomas et al.[4] predicted the reliability of automotive components by the study of fatigue. Although calculations yield a wealth of data, an experimental investigation is always required to establish this reliability. The “Stress-Strength interference analysis” approach is used for results exploitation. The “Stresses” describe the severity of the distribution of the automobile owner’s stresses, whereas the “Strength” depicts the strength of the components’ dispersed fatigue resistance. Klyatis[5] studied why existing accelerated reliability testing results for passenger automobiles frequently fail to provide appropriate information for field evaluation and prediction of reliability, fatigue, and durability. The fundamental ideas of strategy that can aid in the elimination of these factors will be discussed, when a simultaneous combination of fundamental environmental elements (temperature, humidity, pollution, radiation, etc.) is employed, how may expedited environmental testing be improved, how can accelerated corrosion testing of automotive components be improved when chemical, mechanical, motion, and other variables are taken into account. Przybysz[6] studied the reliability and tests of a sample (37) in the operation military vehicles, during 2-year observation period to determine their reliability of them, using operational data, an empirical reliability function. The reliability function of mileage to damage of military units has a logarithmic distribution. METHODOLOGY This paper includes the basic concepts of reliability function, empirical reliability functions, and estimated Rayleigh distribution where its scale parameter is estimated by the maximum likelihood estimator, also it includes the mean and variance time to component failure and the mean absolute error (MAE) for different reliability values of components. Basics of Reliability Function The reliability function is defined as the probability that a system or device will operate for a given period of time (t) under given operating conditions, denoted by R(t),[7] Let T be the lifetime of an item, then: R t Pr T t� � � �� � R t f t dt t � � � � �� � (1) Where R (0) = 1, R(∞) = 0 and f(t) is the failure p. d. f in a time interval (t, t + ∆t) ( ) ( ), 0= < < + ∆ ≥f t Pr t T t t when t With a cumulative distribution function (CDF) F t Pr T t� � � �� � � � �� f t dt t 0 � �1 R t( ) f t dR t dt � � � � ( ) (2) Reliability declines over time, suggesting that the likelihood of failure will rise as the entire system or component matures as shown in Figure 1. Failure (hazard) rate function Is the conditional probability of a component or an item to fail in the interval (t, t + ∆t) given that is operated until time (t),[8] h t Pr t T t t|T t� � � � � � � �� � � � � � � �� � �� �lim Pr t t T t t|T t t 0 t 0 Pr(t T t t)lim t Pr(T t)∆ → < < + ∆ = ∆ > ( ) t 0 F t t F(t)1 lim R(t) t ∆ → + ∆ − = ∆ The failure (hazard) rate h(t) can be written as: h t f t R t � � � ( ) ( ) (3) h t R t dR t dt � � � � � � � �1 � � � �dlnR t dt � � � � � �� h t dt lnR t t 0 Then the reliability function can be written in terms of failure rate as: R t e h t t dt � � � � � ��0 � �e H t( ) (4) The probability density function (PDF) of failure can be written as f t h t e H t� � � � � �. ( ) Figure l: Reliability function declines over time Khudhur, et al.: Estimating hazard function through reliability function 20 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 Such that, h(t) ≥0 limt h t dt� � � � �� � � 0 The relationship between failure functions is in the following planned (Hashimoto, 1998) Mean and variance time to failure Mean E(t) Time to Failure: E t tf t dt� � � � ��0 � � � �� R t dt0 � (5) Variance σ t 2 Time to Failure: � � t t f t dt E t2 2 0 2 � � � � � �� ( ) (6) Estimate Empirical Reliability The reliability and hazard function can be estimated from the failure times t1 =    t t e tf t o w (14) Where; t: Variable of time β: Scale parameter With mean E t� � � �� 4 (15) And variance � � � T 2 1 4 � �� � � � � � (16) Where the failure rate (hazard function) is a linear function of time (t): h t t� � � 2 2� (17) Moreover, the reliability function is: R t e t � � � � � � � � � � � 2 (18) Khudhur, et al.: Estimating hazard function through reliability function 21 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 Figure 2 displays reliability over time. In the Rayleigh distribution when time increases, reliability decreases. Figure 3 illustrates the hazard function over time. In the Rayleigh distribution when time increases, also hazard function increases. Figure 4 shows how the distribution grows more dispersed and the peak of the Rayleigh function moves to the right as β grows. Accordingly, the probability mass is dispersed across a wider range of values for bigger values of β, suggesting a higher degree of variability in the underlying randomized process. The function’s representation of a time-dependent process with a Rayleigh distribution, maybe in engineering or physics, is indicated by the time axis. Estimate the scale parameter of Rayleigh distribution This is one important way of method to estimate the parameters of any distribution, and this method relies on the use of possible functions. The following steps represent the estimation of the shape and the scale parameters of the Rayleigh distribution, as the maximum likelihood function will be as follows: [11,12] ( ) 2 1 2 3 21 2, , , , 1ββ β   −    = … = >∏ tin n i i L t t t t t e i ( ) 2 2 1 2 3 2 1 2, , , , ββ β − =   … =     ∏ tn n i n i i L t t t t t e Taking ln to both sides ( ) 2 2 1 2 3 2 1 2, , , , ln ββ β − =    … =      ∏ tn in n i i lnL t t t t t e ( ) ( ) 2 2 1 2 3 2 1 2, , , , ln ln ββ β − =      … = +        ∏ tin n n n i i lnL t t t t t e ( ) ( ) 2 1 1 2 3 21 , , , , 2 2β β β = = … = − + −∑∑ n n i i n i i t lnL t t t t nln nln lnt Taking partial derivative for scale parameter β is going to be: ( ) ( ) 2 1 21 1 2 3 2 2 , , , , β β β β β = =    ∂ − + −   ∂ …  = ∂ ∂ ∑∑ n n i i i i n t nln nln lnt lnL t t t t ( ) ( ) 2 1 2 3 21 , , , , 1 2 β β β β β β= ∂ … ∂ ∂ = − − ∂ ∂ ∂∑ n n i i lnL t t t t n ln t ( ) 2 1 2 3 1 3 2, , , , 2 β β β β =∂ … = − + ∂ ∑ n in i tlnL t t t t n (19) Then, equaling to zero, we get � � ���2 2 01 2 3 n t i n i � � n t i n i � � � �� 1 2 3 Figure 2: Reliability function Figure 4: Rayleigh function over timeFigure 3: Hazard function Khudhur, et al.: Estimating hazard function through reliability function 22 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 n t i n i � 2 1 2 � �� � 2 1 2 � �� i n it n Taking the second partial derivative of equation (19) for scale parameter β is going to be: ( ) 22 2 2 1 2 3 1 2 2 2 3 2, , , , 2β β β β β β =   ∂ …  ∂ ∂  = − +   ∂ ∂ ∂      ∑ n in i tlnL t t t t n ( ) 22 1 2 3 1 2 2 4 6, , , , 2β β β β =∂ … = − ∂ ∑ n in i tlnL t t t t n ( ) 2 22 1 2 3 1 2 4 2 6, , , , ββ β β = −∂ … = ∂ ∑ n in i n tlnL t t t t Since n t i n i� 2 1 2� � � So ( ) 2 22 1 2 3 1 1 2 4 2 6, , , ,β β β = = −∂ … = ∂ ∑ ∑ n n i in i i t tlnL t t t t ( ) 22 1 2 3 1 2 4 4, , , , 0 β β β = −∂ … = < ∂ ∑ n in i tlnL t t t t Hence, the maximum likelihood estimation for β2 is 2 2 1β̂ == ∑ n i i t n The maximum likelihood estimation for β will be obtained as 1 22 2 1 1β̂ = =    = =     ∑ ∑ n n i i i i t t n n (20) With it, a quadratic mean and β are one-to-one relationships of β2, and it has an invariant property of maximum likelihood estimation. Then ( ) ˆ 2β   −    = t R t e (21) APPLICATION This section includes the computations of the reliability function R(t), hazard function h(t), and the PDF of failure time f(t) of four failure car components (tires, lights, brakes, and engine) by estimated Rayleigh distribution and empirical method, with mean and variance time to failure components Table 1: Number of failures and survivals of tire Tire Year ti No. of failure No. of survival Total test 2010 1 132 49868 50000 2011 2 307 49562 49868 2012 3 512 49050 49562 2013 4 748 48302 49050 2014 5 1110 47192 48302 2015 6 1329 45863 47192 2016 7 1639 44223 45863 2017 8 1674 42549 44223 2018 9 1924 40625 42549 2019 10 2665 37960 40625 2020 11 2503 35457 37960 and the comparison between the different reliability values have made by the MAE. Data Collection The data about the failure car components for a sample of size (50,000) cars, which are annually inspected for 11 years (2010–2020) by one of the PVI centers obtained from the Erbil traffic directorate, as defined in the following [Tables 1-4]. Table 1 shows from 2010 to 2019, there was an increase in a number of failures of the tire from 132 to 2503, respectively, whereas the quantity of surviving tires decreased over time, with the number declining from 49,868 in 2010 to 35,457 in 2020. The statistics in Table 2 shows that brake failures vary throughout the years, with a high in 2015 (2193 failures) and then following a decreasing trend. The survival rate, which indicates the lack of brake failures, follows the opposite trend, peaking in 2015 and then declining. The “Light” demonstrated in Table 2 showed system data show a distinct trend from 2010 to 2020. The quantity of system failures continues to climb yearly, achieving a high of Table 2: Number of failures and survivals of brake Brake Year ti No. of failure No. of survival Total test 2010 1 217 49783 50000 2011 2 506 49277 49783 2012 3 844 48433 49277 2013 4 1235 47198 48433 2014 5 1831 45367 47198 2015 6 2193 43174 45367 2016 7 2705 40469 43174 2017 8 2762 37707 40469 2018 9 3175 34531 37707 2019 10 4397 30134 34531 2020 11 3509 26625 30134 Khudhur, et al.: Estimating hazard function through reliability function 23 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 Table 4: Number of failures and survivals of engine Engine Year ti No. of failure No. of survival Total test 2010 1 243 49757 50000 2011 2 567 49189 49757 2012 3 947 48243 49189 2013 4 1384 46858 48243 2014 5 2053 44805 46858 2015 6 2459 42346 44805 2016 7 3033 39313 42346 2017 8 3097 36216 39313 2018 9 3560 32656 36216 2019 10 4930 27727 32656 2020 11 4301 23426 27727 Table 3: Number of failures and survivals of light Light Year ti No. of failure No. of survival Total test 2010 1 230 49770 50000 2011 2 537 49233 49770 2012 3 895 48338 49233 2013 4 1309 47028 48338 2014 5 1942 45086 47028 2015 6 2326 42760 45086 2016 7 2869 39891 42760 2017 8 2930 36961 39891 2018 9 3368 33594 36961 2019 10 4664 28930 33594 2020 11 4131 24799 28930 4,664 in 2019. In contrast, the number of survivors declined, reaching its lowest level in 2020 with 24,799 survivors. Despite such changes, the overall number of tests performed has remained largely steady, with a minor decline over time. Table 4 indicates that the engine test data throughout time. In 2019 (ti = 10), there were approximately 4930 failures among a total of 32,656 tests performed, suggesting a 15% failure rate. There were 27,727 surviving engines. There were 4301 failures out of 27,727 tests in 2020 (ti = 11), with 23,426 engines surviving. In 2020, the total failure rate was about 15.5%. The Estimated Mean and Variance Time to Failure Components Empirically The mean and the variance time to failure of components are estimated, [Table 5] by: t f mt f i i i � � � � � 2 2 ( ) 1 − = − ∑ ∑ i i i f mt t S f 2( ) 1 − = − ∑ ∑ i i i f mt t S f Where: t : Meantime S2: Variance S: Standard deviation mti: Midpoint of time fi: Number failure Table 5 shows the statistical overview for car parts, the tire component, for example, has an average measurement (mean) of 8.42, a variation from the mean (variance) of 6.48, and an indicator of data dispersion (standard deviation) of 2.55. Estimating the Scale Parameter of Rayleigh Distribution for Components The scale parameter (β) values of Rayleigh distribution for the components (tire, light, break, and engine) are estimated as defined in the following [Tables 6-9] 11 2 1 11 1 β̂ = = = ∑ ∑ i i i i i f mt f Where Table 6 displayed an estimated scale parameter of tires for the years 2010–2020, with tire sizes mti ranging from 1.5 to 11.5”, is ˆ 8.795β =Tire . This numerical data, as shown in Table 7, emphasizes the critical importance of braking-related parameters determining the analyzed outcomes. The predicted brake parameter ( β̂Brake ) is computed at 8.1207, offering quantitative insights into the significant influence of braking coefficients. Table 8 shows a perceptible numerical rise in the meantime to failure from 1.5 to 11.5. Concurrently, the failure rate increases from 230 to 4131. The calculated parameter β̂Light , fixed at 8.7647. Table 9 illustrates engine failure numbers over a decade, with (mti) reflecting the average time between failures. has an estimated parameter coefficient of the engine counted to 8.121. Table 5: Mean and the variance time to failure of components Mean Variance Standard deviation Tire 8.42 6.48 2.55 Brake 8.34 6.40 2.53 Light 8.39 6.45 2.54 Engine 8.38 6.44 2.54 Khudhur, et al.: Estimating hazard function through reliability function 24 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 Table 9: Estimated scale parameter of engine Engine Year mti No. of failure f mti i 2 2010 1.5 243 517.5 2011 2.5 567 3356.25 2012 3.5 947 10963.75 2013 4.5 1384 26507.25 2014 5.5 2053 58745.5 2015 6.5 2459 98273.5 2016 7.5 3033 161381.25 2017 8.5 3097 211692.5 2018 9.5 3560 303962 2019 10.5 4930 514206 2020 11.5 4301 546324.75 Table 6: Estimated scale parameter of tire Tire Year mti No. of failure f mti i 2 2010 1.5 132 297 2011 2.5 307 1918.75 2012 3.5 512 6272 2013 4.5 748 15147 2014 5.5 1110 33577.5 2015 6.5 1329 56150.25 2016 7.5 1639 92193.75 2017 8.5 1674 120946.5 2018 9.5 1924 173641 2019 10.5 2665 293816.25 2020 11.5 2503 331021.75 Table 7: Estimated scale parameter of brake Brake Year mti No. of failure f mti i 2 2010 1.5 217 488.25 2011 2.5 506 3162.5 2012 3.5 844 10339 2013 4.5 1235 25008.75 2014 5.5 1831 55387.75 2015 6.5 2193 92654.25 2016 7.5 2705 152156.25 2017 8.5 2762 199554.5 2018 9.5 3175 286543.75 2019 10.5 4397 484769.25 2020 11.5 3509 464065.25 Table 8: Estimated scale parameter of light Light Year mti No. of failure f mti i 2 2010 1.5 230 517.5 2011 2.5 537 3356.25 2012 3.5 895 10963.75 2013 4.5 1309 26507.25 2014 5.5 1942 58745.5 2015 6.5 2326 98273.5 2016 7.5 2869 161381.25 2017 8.5 2930 211692.5 2018 9.5 3368 303962 2019 10.5 4664 514206 2020 11.5 4131 546324.75 Determine the PDF, Failure Rates, and Reliability of Components by Estimated Rayleigh Distribution Since the failure rates h(t) of the car components are linear functions of time (t), [Figures 5-8], thus the PDF of failure f(t), hazard function h(t), and reliability R(t) by estimated Rayleigh distribution of (tire, brake, light, and engine) components are determined by equations (14), (17), (18), as defined in the following [Tables 10-13]. The hazard function regarding the failure rate of the components over time is estimated in Figures 5-8. “Estimate hazard function of components” the chance of failing rises with time. This could be used in reliability engineering when modeling a tire component’s life expectancy and determining when to do maintenance or replacements. The hazard function regarding the failure rate of the tire component over time is estimated in Figure 5. Table 10 illustrates a substantial decrease in performance from 2010 to 2020 using the Rayleigh distribution for tire dependability. The increasing failure rates from 0.026 in 2010 to 0.284 in 2020, along with a reduction in dependability from 0.987 to 0.209, suggest a concerning pattern of decreasing tire lifespan during the indicated time. Over the 2010–2020 timeframe, the brake reliability data, as provided in Table 11 and examined with the Rayleigh distribution, demonstrates a disturbing pattern of increasing failure rates from 0.030 to 0.334, coupled with a reduction in dependability from 0.985 to 0.160. The PDF likewise rises, showing an increased possibility of brake failure at key time intervals. Table 12 displayed the Rayleigh distribution’s PDF, failure rate, and reliability for each year from 2010 to 2020. Notably, the PDF increases from 0.026 to 0.286, demonstrating an increasing chance of failure, yet the failure rate decreases from 0.987 to 0.207, demonstrating improved system robustness during the same time. Table 13 summarizes the Rayleigh distribution- based reliability study of an engine from 2010 to 2020, Khudhur, et al.: Estimating hazard function through reliability function 25 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 Figure 6: Estimate hazard function of brake component Figure 5: Estimate hazard function of tire component Figure 8: Estimate hazard function of engine component Figure 7: Estimate hazard function of light component including characteristics such as periods (ti), hazard rates h(t), reliability rates R(t), and PDFs f(t). For example, in 2010, the reliability was 0.985, showing a high likelihood of failure-free operation, however in 2020, the reliability dropped to 0.160, indicating a Table 10: Probability density function, failure rate, and reliability by Rayleigh distribution of tire Tire Year ti h(t) R(t) f(t) 2010 1 0.026 0.987 0.026 2011 2 0.052 0.950 0.049 2012 3 0.078 0.890 0.069 2013 4 0.103 0.813 0.084 2014 5 0.129 0.724 0.094 2015 6 0.155 0.628 0.097 2016 7 0.181 0.531 0.096 2017 8 0.207 0.437 0.090 2018 9 0.233 0.351 0.082 2019 10 0.259 0.275 0.071 2020 11 0.284 0.209 0.060 significant fall in the engine’s dependability all over the investigated time. Table 11: Probability density function, failure rate, and reliability by Rayleigh distribution of brake Brake Year ti h(t) R(t) f(t) 2010 1 0.030 0.985 0.030 2011 2 0.061 0.941 0.057 2012 3 0.091 0.872 0.079 2013 4 0.121 0.785 0.095 2014 5 0.152 0.684 0.104 2015 6 0.182 0.579 0.105 2016 7 0.212 0.476 0.101 2017 8 0.243 0.379 0.092 2018 9 0.273 0.293 0.080 2019 10 0.303 0.220 0.067 2020 11 0.334 0.160 0.053 Khudhur, et al.: Estimating hazard function through reliability function 26 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 Table 12: Probability density function, failure rate, and reliability by Rayleigh distribution of light Light Year ti h(t) R(t) f(t) 2010 1 0.026 0.987 0.026 2011 2 0.052 0.949 0.049 2012 3 0.078 0.889 0.069 2013 4 0.104 0.812 0.085 2014 5 0.130 0.722 0.094 2015 6 0.156 0.626 0.098 2016 7 0.182 0.528 0.096 2017 8 0.208 0.435 0.091 2018 9 0.234 0.348 0.082 2019 10 0.260 0.272 0.071 2020 11 0.286 0.207 0.059 Table 13: Probability density function, failure rate, and reliability by Rayleigh distribution of engine Engine Year ti h(t) R(t) f(t) 2010 1 0.030 0.985 0.030 2011 2 0.061 0.941 0.057 2012 3 0.091 0.872 0.079 2013 4 0.121 0.785 0.095 2014 5 0.152 0.685 0.104 2015 6 0.182 0.579 0.105 2016 7 0.212 0.476 0.101 2017 8 0.243 0.379 0.092 2018 9 0.273 0.293 0.080 2019 10 0.303 0.220 0.067 2020 11 0.334 0.160 0.053 Mean and Variance Time to Failure by Estimated Rayleigh Distribution for Components The mean E(t) and the variance ( )σT 2 of failure components by estimated Rayleigh distribution are calculated by equation (15) and equation (16), as shown in the following in [Table 14]. According to Table 14, the car’s component with the greatest mean is “Tire” (mean = 2.6276), whereas the component with the smallest variance is “Brake” (variance = 4.9807), indicating a more closely packed distribution around its mean. Estimate PDF, Failure Rate, and Reliability by Empirical Method for Components The PDF of failure ˆ ( )f t , hazard function ˆ( )h t , and reliability ˆ ( )R t by empirical method for the components (tire, brake, light, and engine) are estimated by equation (7), Table 14: Mean and the variance time to failure of components Components Mean Variance Tire 2.6276 5.6552 Brake 2.5248 4.9807 Light 2.6230 5.6247 Engine 2.5249 4.9811 Table 15: Estimate probability density function, failure rate, and reliability by empirical method for tire Tire Year ti ˆ ( )R t ˆ ( )f t ˆ( )h t 2010 1 0.997 0.006 0.006 2011 2 0.994 0.010 0.010 2012 3 0.990 0.015 0.015 2013 4 0.985 0.023 0.023 2014 5 0.977 0.028 0.028 2015 6 0.972 0.035 0.036 2016 7 0.964 0.037 0.038 2017 8 0.962 0.044 0.045 2018 9 0.955 0.063 0.066 2019 10 0.934 0.062 0.066 2020 11 0.934 -- -- Table 16: Estimate probability density function, failure rate, and reliability by empirical method for brake Brake Year ti ˆ ( )R t ˆ ( )f t ˆ( )h t 2010 1 0.996 0.010 0.010 2011 2 0.990 0.017 0.017 2012 3 0.983 0.025 0.025 2013 4 0.975 0.038 0.039 2014 5 0.961 0.046 0.048 2015 6 0.952 0.060 0.063 2016 7 0.937 0.064 0.068 2017 8 0.932 0.078 0.084 2018 9 0.916 0.117 0.127 2019 10 0.873 0.102 0.116 2020 11 0.884 -- -- equation (9), and equations (11), likewise demonstrated in [Tables 15-18]. Based on Table 15 supplied, the tire component’s dependability shows a distinct pattern over time. Estimated reliability was high in 2010, at 0.997, indicating great effectiveness of operations. However, dependability has since declined, with scores falling to 0.972 in 2015, 0.955 in 2018, and even lower to 0.934 in both 2019 and 2020. Khudhur, et al.: Estimating hazard function through reliability function 27 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 Table 17: Estimate probability density function, failure rate, and reliability by empirical method for light Light Year ti ˆ ( )R t ˆ ( )f t ˆ( )h t 2010 1 0.995 0.011 0.011 2011 2 0.989 0.018 0.018 2012 3 0.982 0.027 0.027 2013 4 0.973 0.040 0.041 2014 5 0.959 0.049 0.052 2015 6 0.948 0.064 0.067 2016 7 0.933 0.069 0.073 2017 8 0.927 0.084 0.091 2018 9 0.909 0.126 0.139 2019 10 0.861 0.123 0.143 2020 11 0.857 -- -- Table 18: Estimate probability density function, failure rate, and reliability by empirical method for engine Engine Year ti ˆ ( )R t ˆ ( )f t ˆ( )h t 2010 1 0.995 0.011 0.011 2011 2 0.989 0.019 0.019 2012 3 0.981 0.028 0.029 2013 4 0.971 0.043 0.044 2014 5 0.956 0.052 0.055 2015 6 0.945 0.068 0.072 2016 7 0.928 0.073 0.079 2017 8 0.921 0.091 0.098 2018 9 0.902 0.136 0.151 2019 10 0.849 0.132 0.155 2020 11 0.845 -- -- Table 16 indicates the dependability data for the braking system over time. The dependability has steadily reduced from 0.996 in 2010 to 0.884 in 2020, indicating a possible performance degradation. It is worth noting, however, that particular information on the failure rate and hazard rate for 2020 is missing from Table 16, preventing an exhaustive evaluation for that year. The reliability calculation of the light system from 2010 to 2020 is shown in Table 17, with estimated reliability (R [t]) dropping from 0.995 in 2010 to 0.857 in 2020. However, the absence of missing data in the PDF and the rate of failure columns for 2020 (NaN) emphasizes the requirement for additional data. Table 18 includes empirical estimates for an engine’s PDF, failure rate, and reliability from 2010 to 2020. The estimations, which are based on observable data, are critical in reliability engineering for analyzing the engine’s efficiency and probable failure features. MAE for Components The comparison between the reliability values which are found by the estimated Rayleigh distribution R(t) and empirical method ˆ ( )R t of (tire, brake, light, and engine) components has made by MAE method, as illustrated in [Tables 19-22] where: ˆ− = ∑ R R MAE n Between the years 2010 and 2020, Table 19 showed both predicted and real tire ratings decreased, with a notable rise in absolute errors, notably in 2019. The MAE MAETire = 0.3517 is an average measure of the model’s precision, showing the total level of departure from expected and real scores. Table 20 shows a significant reduction in brake ratings from 2010 to 2020, shown in both projected ˆ ( )R t and real R(t) values. The absolute errors, including the MAE MAEBrake = 0.3659, showed a significant difference Table 19: Mean absolute error for tire Tire Year ˆ ( )R t R(t) ˆ−R R 2010 0.997 0.987 0.01024 2011 0.994 0.950 0.04429 2012 0.990 0.890 0.09953 2013 0.985 0.813 0.17154 2014 0.977 0.724 0.25314 2015 0.972 0.628 0.34389 2016 0.964 0.531 0.43351 2017 0.962 0.437 0.52487 2018 0.955 0.351 0.60383 2019 0.934 0.275 0.65986 2020 0.9341 0.209 0.72482 Table 20: Mean absolute error for bake Brake Year ˆ ( )R t R(t) ˆ−R R 2010 0.996 0.985 0.00875 2011 0.990 0.941 0.04117 2012 0.983 0.872 0.09468 2013 0.975 0.785 0.16457 2014 0.961 0.684 0.24182 2015 0.952 0.579 0.32934 2016 0.937 0.476 0.41298 2017 0.932 0.379 0.50141 2018 0.916 0.293 0.57181 2019 0.873 0.220 0.60485 2020 0.884 0.160 0.68047 Khudhur, et al.: Estimating hazard function through reliability function 28 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2024, 8 (1): 18-28 Table 21: Mean absolute error for light Light Year ˆ ( )R t R(t) ˆ−R R 2010 0.995 0.987 0.00833 2011 0.989 0.949 0.03994 2012 0.982 0.889 0.09236 2013 0.973 0.812 0.16093 2014 0.959 0.722 0.23650 2015 0.948 0.626 0.32255 2016 0.933 0.528 0.40449 2017 0.927 0.435 0.49192 2018 0.909 0.348 0.56052 2019 0.861 0.272 0.58916 2020 0.857 0.207 0.65023 Table 22: Mean absolute error for engine Engine Year ˆ ( )R t R(t) ˆ−R R 2010 0.995 0.985 0.00806 2011 0.989 0.941 0.03943 2012 0.981 0.872 0.09155 2013 0.971 0.785 0.15963 2014 0.956 0.685 0.23442 2015 0.945 0.579 0.31978 2016 0.928 0.476 0.40060 2017 0.921 0.379 0.48718 2018 0.902 0.293 0.55398 2019 0.849 0.220 0.57760 2020 0.845 0.160 0.63852 CONCLUSION According to the results of the application, the following conclusions are found: The number of failure components increased annually, except in 2020, since the number of arrived customers to the system had lessened, because of COVID-19. The reliability values of components defined by estimated Rayleigh distribution are more decreasing with time than reliability values defined by empirical method. The mean and variance times of failure components which are determined by estimated Rayleigh distribution and empirical method prepared in the same ascending order as follows brakes, engine, lights, and tires successively. MAE between R(t) by Rayleigh distribution and ˆ ( )R t values by Empirical method of car components sorting from minimum to maximum value is (light < tire < engine < brake). REFERENCES 1. D. J. Smith. Reliability Engineering. Pitman, London, 1972. 2. M. A. Tanner and W. H. Wong. The estimation of the hazard function from randomly censored data by the kernel method. The Annals of Statistics, vol. 11, no. 3, pp. 989-993, 1983. 3. T. N. Hubbard. An empirical examination of moral hazard in the vehicle inspection market. The RAND Journal of Economics, vol. 29, pp. 406-426, 1998. 4. J. Thomas, G. Perroud, A. 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John Wiley and Sons, Inc., Hoboken, 2000. 12. A. J. Gross and V. Clark. Survival Distributions: Reliability Applications in the Biomedical Sciences. Wiley, New York, c1975. between predicted and real brake ratings during the selected years. In addition, Tables 21 and 22 displayed the predicted ˆ ( )R t and real R(t) rating for light and engine, respectively, from 2010 to 2020, with appropriate absolute errors. The MAEs MAEEngine and MAEEngine are 0.3235 and 0.355, indicating the average absolute errors in predicted and real ratings throughout the years of interest. Both tables show an ongoing decrease in ratings and a rise in absolute errors throughout the duration.