Lot acceptance criteria for products following attribute single sampling plan and its impact towards discretization of some continuous lifetime distributions

Introduction

The process of discretizing continuous models, functions, or attributes is known as discretization. In order to provide a better alternative model for describing count data sets with complex behaviour, a number of discrete distributions have been created in recent decades as analogues of continuous distributions. In actual life, it can be challenging or inconvenient to collect samples from a continuous distribution. Almost always, observed values are discrete as they are measured to a finite number of decimal places and cannot truly represent all points on a continuum. Although the measurements are conducted on a continuous scale, the observations may be documented in a fashion that lends itself to a discrete model. In some cases, due to measuring device precision or space constraints, continuous variables are measured using frequencies of non-overlapping class intervals, the union of which constitutes the entire range of random variables, and multinomial law is employed to model the situation. In categorical data analysis using the econometric technique, the existence of a continuous unobserved or latent variable under an observed categorical variable is assumed. When an unobserved continuous variable passes a threshold value, it is observed as a categorical variable. As a result, the conclusion is based on observed discrete values, which merely represent the intervals to which the unobserved continuous variable belongs, not its real values. As a result, this is an example of using a discretization of an underlying continuous variable. According to the preceding discussion, the original variables in the real world are frequently continuous in nature but discrete by observation, so it is reasonable and easy to model the situation using an appropriate discrete distribution produced from the underlying continuous models while preserving one or more important characteristics of the continuous distribution. Researchers have focused their emphasis on obtaining discrete counterparts (discretization) of continuous distributions. There are various approaches to deriving discrete distributions from continuous ones. In survival analysis, the survival function may be defined as a function of count random variables, which are discrete versions of the underlying continuous random variable.

• Discretizing the continuous distribution's cdf function • Discretizing its hazard rate function • Discretizing using the probability density function or any other function of it.

The analogue form of Weibull-Geometric (WG) distribution had been introduced fromthe survival function of this distribution by Jayakumar and Babu (2018) by followingthe steps given below, 1. A continuous random variable X is considered. 2. The discrete form of this variable can be determined from the following as, 𝑃(𝑌 = 𝑦) = 𝑃(𝑋 ≥ 𝑦) -𝑃(𝑋 ≥ 𝑦 + 1) = 𝑆 𝑋 (𝑦) -𝑆 𝑋 (𝑦 + 1); 𝑦 = 0,1,2, ….

In similar manner, by using the survival function of exponential-logarithmic distribution, exponential-geometric distribution, Weibull-logarithmic distribution and Weibull-geometric distribution their discretized versions are derived in this section.

Discrete Exponential-Logarithmic distribution

Exponential-Logarithmic (EL) distribution was proposed by Tahmasbi and Rezaei (2008) [46] , a compound distribution formed from the exponential a continuous distribution and logarithmic, a discrete distribution. It is characterized with many salient features and its discretized version is derived here from its survival function which is given as follows,

The cumulative distribution function of discrete form of Exponential-Logarithmic distribution is derived as follows,

Where 𝑦 = 0,1,2, … … ; 0 < 𝛾 < 1; 0 < 𝜌 < 1

Discrete Exponential-Geometric distribution

Discrete Exponential-geometric distribution is derivedfrom the survival function of exponential-geometric distribution introduced by Adamidis and Loukas (1998) [1] and is given as follows,

Thus, Re-parameterizing we get,

The cumulative distribution function of discrete form of exponential-geometricdistribution is derived below,

Cancelling like terms we get,

Where 𝑦 = 0,1,2, … … ; 0 < 𝜅 < 1; 0 < 𝜌 < 1

Discrete Weibull-Logarithmic distribution

Discrete form of Weibull-Logarithmic distribution is obtained from the survival function of Weibull-logarithmic distribution propounded by Roxana Ciumara and Vasile preda (2009) [13] and the survival function of Weibull-Logarithmic distribution is given as,

By re-parameterizing 𝜌 = 𝑒 -𝜃 , the survival function can be re-written as follows,

Thus probability mass function of discrete form of Weibull-logarithmic distribution is given below as,

Cumulative distribution function of discrete Weibull-Logarithmic distribution is derived as follows,

Cancelling like terms we get,

Where 𝑦 = 0,1,2, … … ; 0 < 𝜅 < 1; 0 < 𝜌 < 1; 𝛾 > 0

Discrete Weibull-Geometric Distribution

The pdf and cdf of the Weibull-Geometric (WG) distribution can be obtained, respectively, as follow

for the parameters ∝ and 𝛽 is positive, and 0 < 𝑝 < 1. 𝛼,𝛽 and 𝑝 are the shape, scale and mixing parameters, respectively. The survival and hazard functions of the WG distribution are given:

Formation of Discrete Weibull-Geometric distribution

Discrete form of Weibull-Geometric distribution is derived by Jayakumar and Babu (2018) [19] from the survival function of Weibull-Geometric distributionwhich was studied by Barreto-Souza, De Morais and Cordeiro (2011) [6] and is obtained as follows,

Discrete version of Weibull-Geometric distribution is obtained below from the following formula, 𝑝 𝑦 = 𝑆 𝑋 (𝑦) -𝑆 𝑋 (𝑦 + 1)

The cumulative distribution function of DWG (𝜅, 𝜌, 𝛾) is obtained as,

Where 𝑦 = 0,1,2, … … ; 0 < 𝜅 < 1; 0 < 𝜌 < 1; 𝛾 > 0 Survival function of DWG(𝜅, 𝜌, 𝛾) is given below as,

Taking 𝜌 (𝑦+1) 𝛾 as a common term in the numerator, we get the survival function of DWG(𝜅, 𝜌, 𝛾) as follows,

Hazard function of DWG(𝜅, 𝜌, 𝛾) is given as follows,

By cancelling the like terms reduced form of Hazard function of DWG(𝜅, 𝜌, 𝛾) is given as follows,

Properties of Discrete Weibull-Geometric distribution ➢ When 𝛾=1, this distribution pmf reduces to pmf of discrete exponential-geometric distribution as,

➢ As 𝜅 → 0, 𝑝 𝑦 (𝑦; 𝜅, 𝛼) = (𝜌 𝑦 𝛾 -𝜌 (𝑦+1) 𝛾 ) is obtained which is the probability mass function of discrete Weibull distribution (Nakagawa and Osaki 1975) [31] ➢ When → 0𝑎𝑛𝑑𝛾 → 2, 𝑤𝑒𝑔𝑒𝑡𝑝 𝑦 (𝑦; 𝜌) = (𝜌 𝑦 2 -𝜌 (𝑦+1) 2 ) the probability mass function of discrete Rayleigh distribution. ➢ The geometric distribution is obtained when, 𝜅 → 0𝑎𝑛𝑑𝛾 → 1 as 𝑝 𝑦 (𝑦; 𝜌) = (𝜌 𝑦 -𝜌 (𝑦+1) ) with the parameter 𝜌.

Reliability Acceptance Sampling Plans for Discrete Weibull-Geometric distribution

In this section, reliability acceptance sampling plan has been drafted for the discrete Weibull-geometric (WG) distribution by equating its cdf top, the proportion ofproduct failing before time t and for sketching these plans, pivotal parameters are theassumed distribution's scale and shape parameter. Let us take t to be a lifetime random variable that is used to model certain electronic components which is assumed to follow the discrete Weibull-geometric distribution. In certain cases, the life time random variable t has to be represented in the form of discrete variable and so in such cases discrete distributions takes the main role. In order to design the reliability acceptance sampling plans for this discrete distribution, we equate the cumulative distribution function of the distribution to p, proportion of product failing before time t

𝑝 = 𝑃(𝑇 ≤ 𝑡) = 𝐹(𝑡)

Here we take the lifetime random variable t as y as it is represented in the discrete form and so p is equated as follows,

𝑝 =

(1 -𝜌 (𝑦+1) 𝛾 ) (1 -𝜅𝜌 (𝑦+1) 𝛾 )

Single Sampling Plan

Acceptance Sampling plans have the following plans, but not limited to that alone; some of them are single sampling plans (SSP), double sampling plans, sequential sampling plans, etc. In connection with the title ASP, it is clear that its main objective is to check the quality of products with some criteria and decide whether to accept or reject.

The procedure for the SSP is as follows:

• At the initial stage, choose a random sample n from the lot and subject it to inspection.

• Note down the number of defectives dfrom the sample of size n, where c is the acceptance number.

• If d≤ c, then accept the lot, otherwise reject it.

Minimum Sample Size

Designing an acceptance sampling plan's foremost requirement is finding the appropriate sample size n for conducting life tests, which is obtained with optimality considering two different kind of situations such as finite or infinite lot sizes modelled with the Hypergeometric distribution in the first case and Binomial or Poisson distribution in the latter case. In this contributory work, prime importance has been given for designing sampling plans with optimum sample size n and acceptance number c which are obtained by verifying the conditions given below in equations ( 23) and (24).

Producer and Consumer risk

The probability of rejecting a good lot is a risk for the producer, so it is named as Producer risk (α) and here the producer risk is fixed as α=0.05. Probability of accepting a bad lot is termed as Consumer risk in terms of acceptance sampling plans and it is taken as 𝛽 = 0.10 for which the sampling plans in this contributory work are drafted by satisfying both the producer and consumer's risks for the probability of acceptance while the Binomial distribution is used for modelling as given in equations( 19) and (20).

Operating Characteristic function for designing the reliability acceptance sampling plan for Discrete Weibull-Geometric distribution

The operating characteristic curve plays a significant role for assessing the strength of a sampling plan which is mainly obtained from the operating characteristic function evaluated using the discrete models such as the Hypergeometric, Binomial and the Poisson distributions where the first one is used when the lot is of finite size and the other two models are used when the lot is of infinite sizes or formed from the process. For designing the reliability acceptance sampling plan, Binomial or Poisson distributions are used and their probability mass functions are as follows,

Where, n is the sample size, p is the proportion defective,0 ≤ 𝑝 ≤ 1, q is the proportion effective, 𝑞 = 1 -𝑝 and x is the number of occurrences. The procedure for designing the reliability acceptance sampling plan for the discrete Weibull-geometric distribution involves the distribution parameters and the sampling plan parameters such as sample size n and the acceptance number c. In order to design a single sampling plan, the OC curve should pass through two vital points namely (𝑝 1 , 𝛼) and (𝑝 2 , 𝛽)satisfying the following conditions in terms of probability of acceptance modelled using the discrete distributions such as the Binomial or Poisson distribution equations given above. The above operating characteristic curve is a depiction of different OC curves with varying sample size n and a constant acceptance number c obtained from the sampling plan table given below. OC curve 1 is graphed for sample size 101 and acceptance number 1, while the OC curve 2 represents the sample size 88 and acceptance number 1 and OC curve 3is graphed for a sample size of 78 and acceptance number 1. This picture is a visual representation that as the sample size decreases keeping the acceptance number as constant value, the OC curve bends towards the left.

Numerical Illustration

An experimenter conducts a life test experiment for his manufactured products and their lifetime are represented in terms of number of days and the products are assumed to follow the discrete Weibull-Geometric (WG) distribution. The proportion defective is got from the table 1 as while the sample size and acceptance number for this kind of experiment are obtained as n=160 and c=2 from the table 1 when p 1 =0.0032 and p 2 =0.0330 values are taken as 0.0060 and 0.0364 • For a lot inspection, the sample size of n=145 is taken and the life test is performed for a certain number of days.

• The number of failures observed are compared to the acceptance number c=2 that occurs within the specified number of days or months or years according to the product taken for testing. • It the number of failures is less than or equal the acceptance number c, then the respective lot is accepted, otherwise the lot is rejected.

In particular, we can relate these kinds of discrete distributions with several applications in real life where the lifetime is taken as a discrete case such as for instance, the medical equipment that are used in the hospitals specifically mentioning thermometer used for measuring the body temperature of the patient can be tested for its life and come to a conclusion about its functioning strength for a specified number of years which is a discrete case. This is a valid example of where the discrete form distributions take a major role in real time. Not only limited to this, but the expiry date of tablets is mentioned either by number of days or years which is also a best example of discrete case that can be modelled using this type of distributions and also certain kind of devices has its lifetime recorded in number of weeks, a discrete case.

Conclusion

In this contributory work, the probability mass function and the cumulative distribution functions are derived for the discrete versions of the Exponential-Logarithmic distribution, Exponential-Geometric distribution and Weibull-Logarithmic distributions. Reliability acceptance sampling plans has been designed considering the assumption that the products follow discrete Weibull-Geometric distribution. As a future scope, the derivations related to mean, skewness, kurtosis, and other measures for the above said discrete distributions are to be derived, and the plans related to them are extended.

References

1. Adamidis K, Loukas S. A lifetime distribution with decreasing failure rate. Stat Probab Lett. 1998;39(1):35-42.
2. Afify AZ, Ahsan-ul-Haq M, Aljohani HM, Alghamdi AS, Babar A, Gómez HW. A new one-parameter discrete exponential distribution: properties, inference, and applications to COVID-19 data. J King Saud Univ Sci. 2022;34(6):102199.
3. Afify AZ, Elmorshedy M, Eliwa MS. A new skewed discrete model: properties, inference, and applications. Pak J Stat Oper Res. 2021;799-816.
4. Alghamdi AS, Ahsan-ul-Haq M, Babar A, Aljohani HM, Afify AZ, Cell QE. The discrete power-Ailamujia distribution: properties, inference, and applications. AIMS Math. 2022;7(5):8344-8360.
5. Al-Babtain AA, Ahmed AHN, Afify AZ. A new discrete analog of the continuous Lindley distribution, with reliability applications. Entropy. 2020;22(6):603.
6. Barreto-Souza W, de Morais AL, Cordeiro GM. The Weibull-geometric distribution. J Stat Comput Simul. 2011;81(5):645657.
7. Bracquemond C, Gaudoin O. A survey on discrete lifetime distributions. Int J Reliab Qual Saf Eng. 2003;10(1):69-98.
8. Chakraborty S. A new discrete distribution related to generalized gamma distribution and its properties. Commun Stat Theory Methods. 2015;44(8):1691-1705.
9. Chakraborty S. Generating discrete analogues of continuous probability distributions-A survey of methods and constructions. J Stat Distribut Appl. 2015;2:1-30.
10. Chakraborty S, Chakravarty D. Discrete gamma distribution: properties and parameter estimation. Commun Stat Theory Methods. 2012;41:3301-3324.
11. Chakraborty S, Chakravarty D. A discrete Gumbel distribution. arXiv preprint. 2014. Available from: https://arxiv.org/abs/1410.7568
12. Eldeeb AS, Ahsan-Ul-Haq M, Babar A. A discrete analog of inverted Topp-Leone distribution: properties, estimation and applications. Int J Anal Appl. 2021;19(5):695-708.
13. Gómez-Déniz E, Calderín-Ojeda E. The discrete Lindley distribution: properties and applications. J Stat Comput Simul. 2011;81(11):1405-1416.
14. Gómez-Déniz E, Vázquez-Polo FJ, García-García V. A discrete version of the half-normal distribution and its generalization with applications. Stat Papers. 2014;55(2):497-511.
15. Hussain T, Aslam M, Ahmad M. A two-parameter discrete Lindley distribution. Rev Colomb Estadística. 2016;39(1):45-61.
16. Inusah S, Kozubowski TJ. A discrete analogue of the Laplace distribution. J Stat Plan Inference. 2006;136:1090-1102.
17. Jayakumar K, Babu MG. Discrete Weibull geometric distribution and its properties. Commun Stat Theory Methods. 2018;47(7):1767-1783.
18. Jazi MA, Lai CD, Alamatsaz MH. A discrete inverse Weibull distribution and estimation of its parameters. Stat Methodol. 2010;7(2):121-132.
19. Jia JM, Yan ZZ, Peng XY. A new discrete extended Weibull distribution. IEEE Access. 2019;7:175474-175486.
20. Kemp AW. Characterization of a discrete normal distribution. J Stat Plan Inference. 1997;63:223-229.
21. Krishna H, Pundir PS. Discrete Maxwell distribution. InterStat. 2007;3.
22. Krishna H, Pundir PS. Discrete Burr and discrete Pareto distributions. Stat Methodol. 2009;6:177-188.
23. Krishna H, Singh P. A bivariate geometric distribution with applications to reliability. Commun Stat Theory Methods. 2009;38:1079-1093.
24. Kozubowski TJ, Inusah S. A skew Laplace distribution on integers. Ann Inst Stat Math. 2006;58:555-571.
25. Lai CD. Issues concerning constructions of discrete lifetime models. Qual Technol Quant Manag. 2013;10(2):251-262.
26. Lisman JHC, Van Zuylen MCA. Note on the generation of the most probable frequency distribution. Statistica Neerlandica. 1972;26:19-23.
27. Mabel YO, Novita M, Nurrohmah S. Discrete Weibull-geometric distribution. J Phys Conf Ser. 2021;1725(1):012033.
28. Nakagawa T, Osaki S. The discrete Weibull distribution. IEEE Trans Reliab. 1975;24:300-301.
29. Nekoukhou V, Alamatsaz MH, Bidram H. A discrete analog of the generalized exponential distribution. Commun Stat Theory Methods. 2012;41(11):2000-2013.
30. Nekoukhou V, Alamatsaz MH, Bidram H. Discrete generalized exponential distribution of a second type. Statistics. 2013;47(4):876-887.
31. Opone FC, Izekor EA, Akata IU, Osagiede FE. A discrete analogue of the continuous Marshall-Olkin Weibull distribution with application to count data. Earthline J Math Sci. 2021;5(2):415-428.
32. Ord JK. The discrete Student's t distribution. Ann Math Stat. 1968;1513-1516.
33. Padgett WJ, Spurrier JD. Discrete failure models. IEEE Trans Reliab. 1985;34:253-256.
34. Para BA, Jan TR. Discretization of Burr-Type III distribution. J Reliab Stat Stud. 2014;87-94.
35. Ramos PL, Louzada F. A distribution for instantaneous failures. Stats. 2019;2(2):247-258.
36. Roknabadi AR, Borzadaran GM, Khorashadizadeh M. Some aspects of discrete hazard rate function in telescopic families. 2009.
37. Roy D. The discrete normal distribution. Commun Stat Theory Methods. 2003;32:1871-1883.
38. Roy D. Discrete Rayleigh distribution. IEEE Trans Reliab. 2004;53(2):255-260.
39. Salvia AA, Bollinger RC. On discrete hazard functions. IEEE Trans Reliab. 1982;31:458-459.
40. Sato H, Ikota M, Aritoshi S, Masuda H. A new defect distribution meteorology with a consistent discrete exponential formula and its applications. IEEE Trans Semicond Manuf. 1999;12:409-418.
41. Shafaei Nooghabi M, Mohtashami Borzadaran GR, Rezaei Roknabadi H. Discrete modified Weibull distribution. Metron. 2011;69:207-222.
42. Stein WE, Dattero R. On proportional odds models. IEEE Trans Reliab. 1984;33:196-197.
43. Tahmasbi R, Rezaei S. A two-parameter lifetime distribution with decreasing failure rate. Comput Stat Data Anal. 2008;52(8):3889-3901.