:A moment inequality is derived for the system whose life distribution is in an overall decreasing life (ODL) class of life distributions. A new nonparametric test statistic for testing exponentiality against ODL is investigated based on this inequality. The asymptotic normality of the proposed statistic is presented. Pitman's asymptotic efficiency, power and critical values of this test are calculated to assess the performance of the test. Real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. Wealso proposed a test for testing exponentiality versus ODL for right censored data and the power estimates of this test are also simulated for censored data for some commonly used distributions in reliability. Finally, real data are used as an example for practical problems.
A Study of Some Tests of Uniformity and Their PerformancesIOSRJM
The uniform distribution appears due to natural random events or to the application of methods for transforming samples from any other distribution to samples with uniformly distributed values in the interval (0,1). Thus, in order to test whether a samplecomes from a given distribution, one can test whether its transformed sample is distributed according to the uniform distribution or not. Several test procedures are developed to test the goodness of fit for uniformity. In this paper, we want to study the performance of eleven different tests for uniformity by considering different sample sizes as well as different alternatives. The results so obtained are displayed in various tables and graphs. Finally, conclusions are made on the basis of the results.
Solution to the practice test ch 10 correlation reg ch 11 gof ch12 anovaLong Beach City College
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Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
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Chapter 11: Goodness-of-Fit and Contingency Tables
11.1: Goodness of Fit Notation
A Study of Some Tests of Uniformity and Their PerformancesIOSRJM
The uniform distribution appears due to natural random events or to the application of methods for transforming samples from any other distribution to samples with uniformly distributed values in the interval (0,1). Thus, in order to test whether a samplecomes from a given distribution, one can test whether its transformed sample is distributed according to the uniform distribution or not. Several test procedures are developed to test the goodness of fit for uniformity. In this paper, we want to study the performance of eleven different tests for uniformity by considering different sample sizes as well as different alternatives. The results so obtained are displayed in various tables and graphs. Finally, conclusions are made on the basis of the results.
Solution to the practice test ch 10 correlation reg ch 11 gof ch12 anovaLong Beach City College
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Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
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Chapter 11: Goodness-of-Fit and Contingency Tables
11.1: Goodness of Fit Notation
Design of experiments - Dr. Manu Melwin Joy - School of Management Studies, C...manumelwin
Planning an experiment to obtain appropriate data and drawing inference out of the data with respect to any problem under investigation is known as design and analysis of experiments.
This might range anywhere from the formulations of the objectives of the experiment in clear terms to the final stage of the drafting reports incorporating the important findings of the enquiry
Stochastic Model to Find the Diagnostic Reliability of Gallbladder Ejection F...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
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Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
BioSHaRE: Analysis of mixed effects models using federated data analysis appr...Lisette Giepmans
BioSHaRE conference July 28th, 2015, Milan - Latest tools and services for data sharing
Stream 3: Study application and results
Contact info:
Prof. Edwin van den Heuvel
University of Eindhoven
e.r.v.d.heuvel@tue.nl
key words: biobank, bioshare, cohort, data sharing, epidemiology, harmonisation, statistics
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Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
This paper tries to compare more accurate and efficient L1 norm regression algorithms. Other comparative studies are mentioned, and their conclusions are discussed. Many experiments have been performed to evaluate the comparative efficiency and accuracy of the selected algorithms.
2016 Sonoma County Chinese Waldorf Community GatheringLeslie Wiser, PMP
A gathering of Sonoma County Chinese Waldorf Families and Supporters. This event was open to all families enrolled in the numerous Waldorf Schools in Sonoma County - Charter and Private.
Design of experiments - Dr. Manu Melwin Joy - School of Management Studies, C...manumelwin
Planning an experiment to obtain appropriate data and drawing inference out of the data with respect to any problem under investigation is known as design and analysis of experiments.
This might range anywhere from the formulations of the objectives of the experiment in clear terms to the final stage of the drafting reports incorporating the important findings of the enquiry
Stochastic Model to Find the Diagnostic Reliability of Gallbladder Ejection F...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
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Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
BioSHaRE: Analysis of mixed effects models using federated data analysis appr...Lisette Giepmans
BioSHaRE conference July 28th, 2015, Milan - Latest tools and services for data sharing
Stream 3: Study application and results
Contact info:
Prof. Edwin van den Heuvel
University of Eindhoven
e.r.v.d.heuvel@tue.nl
key words: biobank, bioshare, cohort, data sharing, epidemiology, harmonisation, statistics
Please Subscribe to this Channel for more solutions and lectures
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Chapter 11: Goodness-of-Fit and Contingency Tables
11.2: Contingency Tables
This paper tries to compare more accurate and efficient L1 norm regression algorithms. Other comparative studies are mentioned, and their conclusions are discussed. Many experiments have been performed to evaluate the comparative efficiency and accuracy of the selected algorithms.
2016 Sonoma County Chinese Waldorf Community GatheringLeslie Wiser, PMP
A gathering of Sonoma County Chinese Waldorf Families and Supporters. This event was open to all families enrolled in the numerous Waldorf Schools in Sonoma County - Charter and Private.
This paper proposes development of an android app to improve the transportation services for
bus rental companies that lift Taibah University students. It intends to reduce the waiting time
for bus students, thereby to stimulate sharing of updated information between the bus drivers
and students. The application can run only on android devices. It would inform the students
about the exact time of arrival and departure of buses on route. This proposed app would
specifically be used by students and drivers of Taibah University. Any change in the scheduled
movement of the buses would be updated in the software. Regular alerts would be sent in case of
delays or cancelation of buses. Bus locations and routes are shown on dynamic maps using
Google maps. The application is designed and tested where the users assured that the
application gives the real time service and it is very helpful for them.
Research from leading IT analyst firm Enterprise Management Associates (EMA) has evidenced strong and growing interest in cloud deployment models. While public cloud has gotten the earliest attention, stronger adoption is happening within private and hybrid models. In the EMA April 2014 report “Managing Networks in the Age of Cloud, SDN, and Big Data Network Management Megatrends 2014,” over 50% of respondents reported public/hybrid cloud initiatives were driving network management priorities. Since 2012, cloud projects have moved from early adopter status to mainstream business initiatives, and their impact on network management grew from 36% in 2012 to over 50% in 2014. These slides cover some of the insights from this new research.
Computational Pool-Testing with Retesting StrategyWaqas Tariq
Pool testing is a cost effective procedure for identifying defective items in a large population. It also improves the efficiency of the testing procedure when imperfect tests are employed. This study develops computational pool-testing strategy based on a proposed pool testing with re-testing strategy. Statistical moments based on this applied design have been generated. With advent of computers in 1980‘s, pool-testing with re-testing strategy under discussion is handled in the context of computational statistics. From this study, it has been established that re-testing reduces misclassifications significantly as compared to Dorfman procedure although re-testing comes with a cost i.e. increase in the number of tests. Re-testing considered improves the sensitivity and specificity of the testing scheme.
A GENERALIZED SAMPLING THEOREM OVER GALOIS FIELD DOMAINS FOR EXPERIMENTAL DESIGNcscpconf
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
A Generalized Sampling Theorem Over Galois Field Domains for Experimental Des...csandit
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
KEY
In this paper we focus on mixed model analysis for regression model to take account of over dispersion in random effects. Moreover, we present the Data Exploration, Box plot, QQ plot, Analysis of variance, linear models, linear mixed –effects model for testing the over dispersion parameter in the mixed model. A mixed model is similar in many ways to a linear model. It estimates the effects of one or more explanatory variables on a response variable. In this article, the mixed model analysis was analyzed with the R-Language. The output of a mixed model will give you a list of explanatory values, estimates and confidence intervals of their effect sizes, P-values for each effect, and at least one measure of how well the model fits. The application of the model was tested using open-source dataset such as using numerical illustration and real datasets
Inferential statistics takes data from a sample and makes inferences about the larger population from which the sample was drawn.
Make use of the PPT to have a better understanding of Inferential statistics.
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Top of Form1. Stream quality is based on the levels of many .docxedwardmarivel
Top of Form
1.
Stream quality is based on the levels of many variables, including the following. Which of these variables is quantitative?
The amount of dissolved oxygen
The number of distinct species present
The amount of phosphorus
All of the above
2.
Which of the following is a discrete variable?
Weight of a fish
Length of a fish
None of the above
Number of toxins present in a fish
3.
During winter, red foxes hunt small rodents by jumping into thick snow cover. Researchers report that a hunting trip lasts on average 19 minutes and involves on average 7 jumps. They also report that, surprisingly, 79% of all successful jumps are made in the northeast direction. Three variables are mentioned in this report. The first variable mentioned is
ordinal.
quantitative and discrete.
quantitative and continuous.
categorical.
4.
A sample of 55 streams in severe distress was obtained during 2007. The following is a bar graph of the number of streams that are from the Northeast, Northwest, Southeast, or Southwest. In the bar graph, the bar for the Northeast has been omitted.
The number of streams from the Northeast is
35.
25.
15.
45.
5.
Here is a stemplot (with split stems) of body temperatures (in degrees Fahrenheit) for 65 healthy adult women.
The first quartile for this data set is
97.6.
97.5.
98.0.
97.9.
6.
Researchers measured the length of the central retrix (R1), a flight-involved tail feather, in 21 female long-tailed finches. Here is a boxplot of the length, in millimeters (mm).
Based on this boxplot, which of the following statements is TRUE?
The distribution of R1 lengths is bimodal.
The distribution of R1 lengths is mildly right-skewed with a high outlier.
75% of the birds in this study had an R1 length above 70 mm.
All of the above
7.
Geckos are lizards with specialized toe pads that enable them to easily climb all sorts of surfaces. A research team examined the adhesive properties of 7 Tokay geckos. Below are their toe-pad areas (in square centimeters, cm2).
5.6
4.9
6.0
5.1
5.5
5.1
7.5
To be an outlier, an observation must fall outside the range
4.9 to 7.5.
4.2 to 6.9.
3.75 to 7.35.
5.1 to 6.0.
8.
The median age of five people on a committee is 30 years. One of the members, whose age is 50 years, resigns. The median age of the remaining four people in the committee is
not able to be determined from the information given.
25 years.
30 years.
40 years.
9.
By inspection, determine which of the following sets of numbers has the smallest standard deviation.
7, 8, 9, 10
0, 0, 10, 10
0, 1, 2, 3
5, 5, 5, 5
10.
The volume of oxygen consumed (in liters per minute) while a person is at rest and while he or she is exercising (running on a treadmill) was measured for each of 50 subjects. The goal is to determine if the volume of oxygen consumed during aerobic exercise can be estimated from the amount consumed at rest. The results are plotted below.
The scatterplot sugges ...
Estimating Reconstruction Error due to Jitter of Gaussian Markov ProcessesMudassir Javed
This paper presents estimation of reconstruction error due to jitter of Gaussian Markov Processes. Two samples are considered for the analysis in two different situations. In one situation, the first sample does not have jitter while the other one is effected by jitter. In the second situation, both the samples are effected by jitter. The probability density functions of the jitter are given by Uniform Distribution and Erlang Distribution. Statistical averaging is applied to conditional expectation of random variable of jitter. From that, conditional variance is obtained which is defined as reconstruction error function and by knowing that, the reconstruction error of a Gaussian Markov Process is determined.
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Austin Statistics is an open access, peer reviewed, scholarly journal dedicated to publish articles in all areas of statistics.
The aim of the journal is to provide a forum for scientists, academicians and researchers to find most recent advances in the field statistics.
Austin Statistics accepts original research articles, review articles, case reports and rapid communication on all the aspects of statistics.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
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Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
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Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Cosmetic shop management system project report.pdf
A Moment Inequality for Overall Decreasing Life Class of Life Distributions with Hypotheses Testing Applications
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 5 Issue 1 || January. 2017 || PP-62-71
www.ijmsi.org 62 | Page
A Moment Inequality for Overall Decreasing Life Class of Life
Distributions with Hypotheses Testing Applications
L.S. Diab and E.S. El-Atfy
College of Science for (girls), Dept. of Mathematics, Al-Azhar University, Nasr City,11884, Egypt
ABSTRACT:A moment inequality is derived for the system whose life distribution is in an overall decreasing life
(ODL) class of life distributions. A new nonparametric test statistic for testing exponentiality against ODL is
investigated based on this inequality. The asymptotic normality of the proposed statistic is presented. Pitman's
asymptotic efficiency, power and critical values of this test are calculated to assess the performance of the test.
Real examples are given to elucidate the use of the proposed test statistic in the reliability analysis. Wealso
proposed a test for testing exponentiality versus ODL for right censored data and the power estimates of this test
are also simulated for censored data for some commonly used distributions in reliability. Finally, real data are
used as an example for practical problems.
KEYWORDS:life distributions, ODL, Moment inequalities,exponentiality U-statistic, asymptotic normality,
efficiency, Monte Carlo method, power and censored data.
I. INTRODUCTION AND MOTIVATION
The mathematical theory of reliability has developed in two main branches. The first branch is mostly constructed
on the mathematical aspects of practical problems that face reliability and design engineers and reliability analysis
specialists. The second branch concentrates on the mathematical and statistical aspects of reliability such as life
testing and estimation.
Note that the assumption of exponentiality is the back bone in life testing, the theory of reliability, maintenance
modeling, biometrics and biological science. It is very important because of its implications concerning the
random mechanism operating in the experiment under consideration. In reliability theory, the exponential
assumption may apply when one is dealing with failure times of items or equipment without any moving parts
such as, for example, fuses, transistors, air monitors and bulbs. In these examples the failure is caused due to
sudden shocks rather than wear and tear. The exponential assumption corresponds to assuming that this shock
follows a Poisson process distribution. Testing for exponentiality of the failure time is, in effect, the same as
testing the Poisson assumption about the process producing the shock that causes failure. An assumption of
exponentially distributed life time's indicate that a used item is stochastically as good as new, so there is no reason
to replace a functioning unit. This distribution is the most commonly used life distribution in applied probability
analysis primarily. See Barlow and Proschan[10] and Zachks[33].
For more than four decades, investigates have led to declaring many families of life distributions to characterize
aging. There are a several number of classes that have been presented in reliability, see for example Siddiqui and
Bryson [31] Ahmad and abouammoh[7], Abu-Youssef [3], Mahmoud et al. [27] and Diab ([13],[14]). To
categorize distributions based on their aging properties or their dual. Among the most practical aging classes of
life distributions, the increasing failure rate (IFR). Many other have been extensively studied in the literature
Barlow et al. [9],Proschan and Pyke[28]. For testing against increasing failure rate in expectation (IFRA), see
Aly[8] and Ahmad [6]. For testing versus new better than used (NBU), see koul[24], and Kumazawa[25].
Ahmad [4], one of the authors presented moments inequalities for classes of life distributions IFR, NBU, NBUE
and harmonic new better than used in expectation (HNBUE). For testing exponentiallity against an alternative
among the classes NBUE, HNBUE, NBUFR, NBAFR, NBURFR, HNBUE, NRBU, DMRL, NBUC, NBUL and
DVRL classes, we refer to Klefsjo ([22],[23]), Deshpaned et al. [11] Ahmad and Abuammoh[5], EL-Arishy et al.
[16], Abdul Alim and Mahmoud [1], Mahmoud and Diab[26] and Diab et al. [12] respectively. Finally, for
exponentiallity testing versus an alternative RNBU and HNRBUE, see El -Arishy, et al. [17] and Diab and
Mahmoud [15].
LetX be a nonnegative continuous random variable with distribution function F , survival function 𝐹 = 1 − 𝐹 ,
with finite mean, with survival of a device in operation at any time𝑡 ≥ 0 is given by,
𝑊 𝑡 =
1
𝜇
𝐹 𝑢 𝑑𝑢
∞
𝑡
𝑡 ≥ 0 .
Sepehrifar et al. [30] defined the ODL .And investigated the probabilistic characteristics of this class of life
distribution.
2. A Moment Inequality For Overall Decreasing Life Class Of Life Distributions With Hypotheses…
www.ijmsi.org 63 | Page
Definition 1.1: A life distribution Fon 0, ∞ , with𝐹 0−
= 0 is called overall decreasing life (ODL) if,
𝑊 𝑥 𝑑𝑥
∞
𝑡
≤ 𝜇𝑊 𝑡 , 𝑡 ≥ 0, (1.1)
Where𝜇 = 𝐹 𝑢 𝑑𝑢
∞
0
.
Remark 1.1:Life distribution F belongs to ODL if and onlyif
𝑓(𝑥)
𝐹(𝑥)
<
1
𝜇
.
The rest of the article is structured as follows. The moment inequality developed in section 2, for ODL class of life
distribution based on these inequalities test statistics for testing 𝐻0 ∶is exponential against 𝐻1 ∶ is ODLand not
exponential. In section 3, a new test statistic based on U-statistic is established and have exceptionally Pitman
asymptotic efficiency of some of well-known alternatives, Monte Carlo null distribution critical points are
simulated for sample sizes𝑛 = 5 1 30 5 50and the power estimates of this test are also calculated at the
significant level 𝛼 = 0.05 for some common alternatives distribution followed by some numerical example. In
section 4, we dealing with right-censored data and selected critical values are tabulated. Finally, the power
estimates for censor data of this test are tabulated and numerical example are calculated.
II. MOMENT INEQUALITY
In this section we derive our main results. The following theorem gives the moment inequality for the ODLclass of
life distributions.
Theorem 2.1:IfF is ODL ,then for all integer 𝑟 ≥ 0,
1
(𝑟 + 3)
𝜇 𝑟+3 ≤ 𝜇𝜇 𝑟+2 , 𝑟 ≥ 0. (2.1)
Where
𝜇(𝑟) = 𝐸 𝑋 𝑟
= 𝑟 𝑋 𝑟−1
∞
0
𝐹 𝑢 𝑑𝑢.
Proof: Since Fis said to be overall decreasing life ODL class, then from (1.1) we multiplying both sides by
𝑡 𝑟
, 𝑟 ≥ 0 and integrating over (0, ∞)with respect tot,we get
𝑡 𝑟
∞
0
𝑊 𝑥 𝑑𝑥𝑑𝑡
∞
𝑡
≤ 𝜇 𝑡 𝑟
𝑊 𝑡 𝑑𝑡
∞
0
, (2.2)
Now,
𝑅. 𝐻. 𝑆 = 𝑡 𝑟
𝑊 𝑡 𝑑𝑡
∞
0
,
=
1
𝑟 + 1
𝑥 𝑟+1
∞
0
𝐹 𝑥 𝑑𝑥,
=
1
𝑟 + 1
𝐸 𝑥 𝑟+1
𝑋
0
𝑑𝑥 ,
=
1
𝑟 + 1 𝑟 + 2
𝜇 𝑟+2 . (2.3)
Also,
𝐿. 𝐻. 𝑆 = 𝑡 𝑟
∞
0
𝑊 𝑥 𝑑𝑥𝑑𝑡
∞
𝑡
,
=
1
𝜇 𝑟 + 1 𝑟 + 2
𝑥 𝑟+2
𝐹 𝑥 𝑑𝑥
∞
0
,
=
1
𝜇 𝑟 + 1 𝑟 + 2 𝑟 + 3
𝜇 𝑟+3. (2.4)
Using (2.3),(2.4) in (2.2) the result follows.
2.1 Testing against ODL class for non-censored data
Let 𝑋1, 𝑋2, … , 𝑋 𝑛 be a random sample from a population with distribution function F. We test the null hypothesis
𝐻0 ∶ 𝐹 is exponential with mean 𝜇 against 𝐻1 ∶ 𝐹 is ODL and not exponential. Using Theorem (2.1), we can
use the following quantity as a measure of departure from 𝐻0 in favor of 𝐻1
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𝛿 𝑂𝐷𝐿 𝑟 = 𝜇𝜇(𝑟+2) −
1
𝑟 + 3
𝜇 𝑟+3 (3.1)
Note that under 𝐻0 ∶ 𝛿 𝑂𝐷𝐿 𝑟 = 0,and it is positive under 𝐻1.To make the test scale invariant under 𝐻0 we use
∆ 𝑂𝐷𝐿 𝑟 =
𝛿 𝑂𝐷𝐿 (𝑟)
𝜇 𝑟+3
It could be estimated based on a random sample 𝑋1, 𝑋2, … , 𝑋 𝑛 from F by
∆ 𝑂𝐷𝐿 𝑟 =
𝛿 𝑂𝐷𝐿 𝑟
𝑋 𝑟+3
=
1
𝑋 𝑟+3
1
𝑛 𝑛 − 1
𝑋𝑖 𝑋𝑗
𝑟+2
−
1
𝑟 + 3
𝑋𝑖
𝑟+3
𝑛
𝑗 =1
𝑛
𝑖=1
3.2
Where𝑋 =
1
𝑛
𝑋𝑖
𝑛
𝑖=1 is the sample mean, and 𝜇 is estimated by 𝑋
Setting,
𝜙 𝑋1, 𝑋2 = 𝑋1 𝑋2
𝑟+2
−
1
𝑟 + 3
𝑋1
𝑟+3
Again Δ 𝑂𝐷𝐿 𝑟 and
𝛿 𝑂𝐷𝐿 𝑟
𝑋 𝑟+3 have the same limiting distribution. But since Δ 𝑂𝐷𝐿 𝑟 is the usual U-statistics
theory,it is asymptotically normal and all we need to evaluate𝑉𝑎𝑟
𝛿 𝑂𝐷𝐿 𝑟
𝜇 𝑟+3 . The following theorem summarized
the large sample properties of Δ 𝑂𝐷𝐿 𝑟 or U-statistic.
Theorem 3.1:As 𝑛 → ∞, 𝑛 Δ 𝑂𝐷𝐿 𝑟 − 𝛿 𝑂𝐷𝐿 𝑟 is asymptotically normal with mean zero and variance
𝜎(𝑟)
2
= 𝑉𝑎𝑟 −
1
𝑟 + 3
𝑋 𝑟+3
+ 𝑋 𝑟+2
+ 𝑟 + 2 ! 𝑋 − 𝑟 + 2 ! (3.3)
Under𝐻0 this value reduced to
𝜎0
2
= 2𝑟 + 4 ! − 𝑟 + 2 !
2
(3.4)
Proof: Since Δ 𝑂𝐷𝐿 𝑟 and
𝛿 𝑂𝐷𝐿 𝑟
𝑋 𝑟+3 have the same limiting distribution, weconcentrate on 𝑛 Δ 𝑂𝐷𝐿 𝑟 −
𝛿𝑂𝐷𝐿𝑟. Now this is asymptotic normal with mean zero and variance 𝜎2=𝑉𝑎𝑟𝜙𝑋1, where
𝜙 𝑋1 = 𝐸 𝜙 𝑋1, 𝑋2 |𝑋1 + 𝐸 𝜙 𝑋2, 𝑋1 |𝑋1 (3.5)
But
𝜙 𝑋1 = 𝜇(𝑟+2) 𝑋1 −
1
𝑟 + 3
𝑋1
𝑟+3
+ 𝑋1
𝑟+2
− 𝜇 𝑟+2 (3.6)
Hence (3.3) follows.Under 𝐻0
𝜙 𝑋1 = 𝑟 + 2 ! 𝑋1 −
1
𝑟 + 3
𝑋1
𝑟+3
+ 𝑋1
𝑟+2
− 𝑟 + 2 ! (3.7)
Thus it is easy to get 𝜎0
2
as it is defined in (3.4). When 𝑟 = 0,
𝛿 𝑂𝐷𝐿 = 𝜇𝜇 2 −
1
3
𝜇 3 (3.8)
In this case𝜎0 = 4.47214 and the test statistic is
𝛿 𝑂𝐷𝐿 =
1
𝑛 𝑛 − 1
𝑋𝑖 𝑋𝑗
2
−
1
3
𝑋𝑖
3
𝑛
𝑗 =1
𝑛
𝑖=1
. (3.9)
And
∆ 𝑂𝐷𝐿 =
𝛿 𝑂𝐷𝐿
𝑋3
, (3.10)
which is quite simple statistics. One can use the proposed test to calculate
𝑛Δ 𝑂𝐷𝐿
𝜎0
and reject 𝐻0 if
𝑛Δ 𝑂𝐷𝐿
𝜎0
≥
𝑍 𝛼 ,where𝑍 𝛼 is the 𝛼 −quintile of the standard normal distribution.
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2.2 The Pitman asymptotic efficiency
This subsection includes the calculations of the asymptotic efficiencies of the ODL test statistic. We use the
concept of Pitman's asymptotic efficiency (PAE) which is defined as 𝑃𝐴𝐸 ∆ 𝑟 𝜃 =
𝑑
𝑑𝜃
∆ 𝑟 𝜃
𝜃→𝜃0
𝜎0
. These
calculations are done by using the following common alternative families:
(i) Linear failure rate family (LFR)
𝐹1 𝑥 = 𝑒−𝑥−
𝜃
2
𝑥2
, 𝑥 ≥ 0, 𝜃 ≥ 0,
(ii) Makeham family
𝐹2 𝑥 = 𝑒−𝑥−𝜃 𝑥+𝑒−𝑥−1
, 𝑥 ≥ 0, 𝜃 ≥ 0,
(iii) Weibull family
𝐹3 𝑥 = 𝑒−𝑥 𝜃
, 𝑥 > 0, 𝜃 ≥ 0,
(iv) Gamma family
𝐹4 𝑥 = 𝑒−𝑢
∞
𝑥
𝑢 𝜃−1
𝑑𝑢/Γ 𝜃 , 𝑥 > 0, 𝜃 ≥ 0.
Since,
𝛿 𝑂𝐷𝐿 𝑟 = 𝜇 𝜃 𝜇(𝑟+2) 𝜃 −
1
𝑟 + 3
𝜇 𝑟+3 𝜃 .
Then we define
𝛿 𝑂𝐷𝐿 𝑟 𝜃→𝜃0
= 𝜇 𝜃0 𝜇(𝑟+2) 𝜃0 + 𝜇 𝜃0 𝜇(𝑟+2) 𝜃0 −
1
𝑟 + 3
𝜇 𝑟+3 𝜃0 .
𝛿 𝑂𝐷𝐿 𝜃→𝜃0
= 𝐹𝜃0
𝑥
∞
0
𝑑𝑥 2 𝑥𝐹𝜃0
𝑥
∞
0
𝑑𝑥 +
2 𝑥𝐹𝜃0
𝑥
∞
0
𝑑𝑥 𝐹𝜃0
𝑥
∞
0
𝑑𝑥 − 𝑥2
𝐹𝜃0
𝑥
∞
0
𝑑𝑥
Note that 𝐻0(the exponential distribution) is attained at 𝜃0 = 0 in (i), (ii) and at 𝜃0 = 1 in (iii), (iv)
Direct calculations for the families in (i) and (ii) are given at 𝑟 = 0.
(i) Linear failure rate
𝑃𝐴𝐸 ∆ 𝑟 𝜃 =
1
𝜎0
𝑟 + 2 𝑟 + 2 ! .
(ii) Makeham family
𝑃𝐴𝐸 ∆ 𝑟 𝜃 =
1
𝜎0
𝑟 + 2 !
1
2
− 2− 3+𝑟
.
Using mathematica program to calculate PAE for Weibull family and Gamma family at 𝑟 = 0.
(iii) Gamma family
𝑃𝐴𝐸 ∆ 𝑟 𝜃 = 0.298142.
(iv) Weibull family
𝑃𝐴𝐸 ∆ 𝑟 𝜃 = 0.67082.
Direct calculations of the asymptotic efficiencies of ODL test are given in Table 1. At𝑟 = 0.
Table 1. Pitman asymptotic efficiencies for ODL test
Efficiency Linear failure rate Makeham family Weibull family Gamma family
𝑃𝐴𝐸 ∆ 𝑟 𝜃 0.894427 0.167705 0.67082 0.298142
It is clear from Table 1,we can see that the new ODL test statistic is more efficiency.
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III. MONTE CARLO NULL DISTRIBUTION CRITICAL POINTS
In practice, simulated percentiles for small samples are commonly used by applied statisticians and reliability
analyst. We have simulated the upper percentile values for 90%, 95%, 98% and 99%. Table 2 presented these
percentile values of the statistics ∆ 𝑂𝐷𝐿 in (3.10) and the calculations are based on 5000 simulated samples of sizes
𝑛 = 5(1)30(5)50.
Table 2. Critical values of statistic ∆ 𝑂𝐷𝐿 .
n 90% 95% 98% 99%
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
0.6803 0.69300.7044 0.7100
0.6726 0.6844 0.6957 0.7032
0.6675 0.6799 0.6902 0.6961
0.6611 0.6741 0.6860 0.6928
0.6540 0.6691 0.6820 0.6873
0.6468 0.6645 0.6773 0.6852
0.6405 0.6597 0.6752 0.6828
0.6381 0.6563 0.6708 0.6791
0.6275 0.65240.6709 0.6780
0.6180 0.6413 0.6624 0.6719
0.6141 0.6406 0.6601 0.6706
0.6066 0.6354 0.6547 0.6660
0.6050 0.6319 0.6547 0.6646
0.5948 0.6232 0.6465 0.6630
0.5921 0.6214 0.6438 0.6579
0.5885 0.6198 0.6424 0.6517
0.5834 0.6128 0.6387 0.6519
0.5760 0.6083 0.6334 0.6449
0.5766 0.6066 0.6334 0.6443
0.5640 0.5982 0.6257 0.6411
0.5603 0.5944 0.6265 0.6423
0.5572 0.5915 0.6234 0.6369
0.5527 0.5888 0.6170 0.6322
0.5501 0.5855 0.6127 0.6301
0.5437 0.5776 0.6092 0.6256
0.5385 0.5770 0.6071 0.6250
0.5259 0.56500.5974 0.6109
0.5055 0.5488 0.5843 0.6016
0.4825 0.5289 0.5675 0.5888
0.4778 0.5182 0.5583 0.5764
In view of Table 2, and Fig. 1, it is noticed that the critical values are increasing as the confidence level increasing
and is almost decreasing as the sample size increasing.
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3.1 The Power Estimates
Now, we present an estimation of the power estimate of the test statistic∆ 𝑂𝐷𝐿 at the significance level 𝛼 = 0.05
using LFR, Gamma and Weibull distribution. The estimates are based on 5000 simulated samples for sizes
𝑛 = 10, 20 and 30 with parameter 𝜃 = 2,3 𝑎𝑛𝑑 4.
Table 3. Power estimates using𝛼 = 0.05.
Distribution
Parameter
𝜃
Sample Size
n=10 n=20n=30
LFR
2
3
4
0.9988 0.9998 1.0000
0.99981.0000 1.0000
0.99921.0000 1.0000
Weibull
2
3
4
1.0000 1.0000 1.0000
1.0000 1.0000 1.0000
1.0000 1.0000 1.0000
Gamma
2
3
4
0.99780.99860.9988
1.0000 1.0000 1.0000
1.0000 1.0000 1.0000
It is clear from the Table 3 that our test has good powers for all alternatives and the power increases as the sample
size increases. The power is getting as smaller as the ODL approaches the exponential distribution.
3.2 Applications Using Complete (Uncensored) Data
Here, we present some of good real examples to illustrate the use of our test statistics ∆ 𝑶𝑫𝑳 in the case of
non-censored data at 95% confidence level.
Example 1: The data set of 40 patients suffering from blood cancer (Leukemia) from one of ministry of health
hospitals in Saudi Arabia sees Abouammoh et al. [2]. The ordered life times (in years)
0.315 0.496 0.616 1.145 1.208 1.263 1.414 2.025 2.036 2.162
2.211 2.370 2.532 2.693 2.805 2.910 2.912 3.192 3.263 3.348
3.348 3.427 3.499 3.534 3.767 3.751 3.858 3.986 4.049 4.244
4.323 4.381 4.392 4.397 4.647 4.753 4.929 4.973 5.074 4.381
It was found that ∆ 𝑂𝐷𝐿 = 0.678917 and this value exceeds the tabulated critical value in Table 2. It is evident that
at the significant level %95 this data set has ODL property.
Example 2: The following data in keating et al.[21] set on the time, in operating days, between successive failures
of air conditioning equipment in an aircraft. These data are recorded
3.750 0.417 2.500 7.750 2.542 2.042 0.583 1.000 2.333 0.833
3.292 3.500 1.833 2.458 1.208 4.917 1.042 6.500 12.917 3.167
1.083 1.833 0.958 2.583 5.417 8.667 2.917 4.208 8.667
by using (3.3) it is found that ∆ 𝑂𝐷𝐿 = 0.373289 which is less than the critical value of Table 2, then we accept the
null hypothesis. This means that the data set has the exponential property.
Example 3:Using the data set given in Grubbs [19], this data set gives the times between arrivals of 25 customers
at a facility
1.80 2.89 2.93 3.03 3.15 3.43 3.48 3.57 3.85 3.92
3.98 4.06 4.11 4.13 4.16 4.23 4.34 4.73 4.53 4.62
4.65 4.84 4.91 4.99 5.17
It is easily to show that∆ 𝑂𝐷𝐿 = 0.668617 which is greater than the critical value ofTable 2. Thenwe accept 𝐻1
which states that the data set have ODL property and not exponential.
Example 4: Consider the well-known Darwin data (Fisher [18]) that represent the differences in heights between
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cross- and self-fertilized plants of the same pair grown together in one pot
4.25 8.708 0.583 2.375 2.25 1.333 2.792 2.458
5.583 6.333 1.125 0.583 9.583 2.75 2.542 1.417
We can see that the value of test statistic for the data set by formula (3.10) is given by ∆ 𝑂𝐷𝐿 = 2.41461 and this
value greater than the tabulated critical value in Table 2. This means that the set of data have ODLproperty and not
exponential.
IV. TESTING AGAINST ODL CLASS FOR CENSORED DATA
In this section, a test statistic is proposed to test 𝐻0versus 𝐻1with randomly right-censored data. Such a censored
data is usually the only information available in a life-testing model or in a clinical study where patients may be
lost (censored) before the completion of a study. This experimental situation can formally be modeled as follows.
Suppose n objects are put on test, and 𝑋1, 𝑋2, … , 𝑋 𝑛 denote their true life time. We assume that 𝑋1, 𝑋2, … , 𝑋 𝑛 be
independent, identically distributed (i.i.d.) according to a continuous life distribution F. Let 𝑌1, 𝑌2, … , 𝑌𝑛 be
(i.i.d.) according to a continuous life distribution G. Also we assume that 𝑋,
𝑠 and 𝑌,
𝑠 are independent.
In the randomly right-censored model, we observe the pairs 𝑍𝑗 , 𝛿𝑗 , 𝑗 = 1, … , 𝑛 where𝑍𝑗 = 𝑚𝑖𝑛 𝑋𝑗 , 𝑌𝑗 and
𝛿𝑗 =
1 𝑖𝑓𝑍𝑗 = 𝑋𝑗 𝑗 − 𝑡ℎ𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑖𝑠𝑢𝑛𝑐𝑒𝑛𝑠𝑜𝑟𝑒𝑑 .
0 𝑖𝑓𝑍𝑗 = 𝑌𝑗 𝑗 − 𝑡ℎ𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑖𝑠𝑐𝑒𝑛𝑠𝑜𝑟𝑒𝑑 .
Let 𝑍 0 = 0 < 𝑍 1 < 𝑍 2 < ⋯ < 𝑍 𝑛 denote the ordered 𝑍,
𝑠 and 𝛿 𝑗 is the 𝛿𝑗 corresponding to 𝑍 𝑗
respectively.
Using the censored data 𝑍𝑗 , 𝛿𝑗 , 𝑗 = 1, … , 𝑛 Kaplan and Meier [20] proposed the product limit estimator.
𝐹𝑛 𝑋 = 1 − 𝐹𝑛 𝑋 =
𝑛 − 𝑗
𝑛 − 𝑗 + 1
𝛿 𝑗
𝑗 :𝑍 𝑗 ≤𝑋
, 𝑋 ∈ 0, 𝑍 𝑛
Now, for testing 𝐻0 ∶ ∆ 𝑂𝐷𝐿 = 0, against 𝐻1 ∶ ∆ 𝑂𝐷𝐿 > 0,using the randomly right censored data, we propose the
following test statistic:
∆ 𝑂𝐷𝐿
𝑐
=
1
𝜇3
𝜇𝜇2 −
1
3
𝜇3 .
For computational purposes, ∆ 𝑂𝐷𝐿
𝑐
may be rewritten as
∆ 𝑂𝐷𝐿
𝑐
=
1
𝜓3
𝜓Ω −
1
3
Φ . (4.1)
Where
𝜓 = 𝐶 𝑚
𝛿 𝑚
𝑖−1
𝑚=1
𝑛
𝑖=1
𝑍 𝑖 − 𝑍 𝑖−1 , Ω = 2 𝑍 𝑗
𝑛
𝑗 =1
𝐶𝑝
𝛿 𝑝
𝑗 −1
𝑝=1
𝑍 𝑗 − 𝑍 𝑗−1 ,
and,
Φ = 3 𝑍 𝑖
2
𝑛
𝑖=1
𝐶𝑞
𝛿 𝑞
𝑖−1
𝑞=1
𝑍 𝑖 − 𝑍 𝑖−1 .
Where 𝐶𝑘 =
𝑛−𝑘
𝑛−𝑘+1
.
Table 4.gives the critical values percentiles of ∆ 𝑂𝐷𝐿
𝑐
test for sample sizes 𝑛 = 5 5 30 10 70,81,86.based on
5000 replications.
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Table 4. Critical values for percentiles of ∆ 𝑂𝐷𝐿
𝑐
test
n 90% 95% 98% 99%
5
10
15
20
25
30
40
50
60
70
81
86
0.09651 0.13859 0.20003 0.25226
0.08166 0.12182 0.17443 0.21400
0.06461 0.09050 0.13029 0.15792
0.05159 0.07348 0.10401 0.12619
0.04419 0.06126 0.08593 0.10513
0.03738 0.05214 0.07011 0.08734
0.03067 0.04256 0.05678 0.06852
0.02538 0.03482 0.04918 0.05767
0.02244 0.03059 0.04002 0.04884
0.01968 0.02658 0.03639 0.04561
0.01735 0.02344 0.03169 0.03797
0.01646 0.02233 0.02917 0.03653
We noticed that from Table 4 and Fig. 2, the critical values are increasing as the confidence level increasing and is
almost decreasing as the sample size increasing.
4.1 The power estimates for ∆ 𝑶𝑫𝑳
𝒄
Her, we present an estimation of the power for testing exponentiality Versus ODL. Using significance level
𝛼 = 0.05with suitable parameter values of 𝜃at 𝑛 = 10,20 𝑎𝑛𝑑 30 , and for commonly used distributions in
reliability such as LFR family, Gamma family and Weibull family alternatives which include in Table 5.
Table 5. Power estimates
Distribution
Parameter
𝜃
Sample Size
n=10 n=20n=30
LFR
2
3
4
0.9974 0.9992 0.9982
0.99961.0000 1.0000
0.9998 1.0000 1.0000
Weibull
2
3
4
0.9968 0.9990 0.9994
0.9990 1.0000 1.0000
0.9954 1.0000 1.0000
Gamma
2
3
4
0.92940.94800.9548
0.9334 0.9682 0.9662
0.93100.9768 0.9784
We notice from Table 5. We can show that our test has a good power, and the power increases as the sample size
increases.
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4.2 Applications Using Incomplete (Censored) Data
Example 1.
Consider the data of Susarla and Vanryzin [32], which represent 81 survival times (in weeks) of patients
melanoma. Out of these 46 represents non-censored data and the ordered values are:
13 14 19 19 20 21 23 23 25 26
26 27 27 31 32 34 34 37 38 38
40 46 50 53 54 57 58 59 60 65
65 66 70 85 90 98 102 103 110 118
124 130 136 138 141 234
The ordered censored observations are:
16 21 44 50 55 67 73 76 80 81
86 93 100 108 114 120 124 125 129 130
132 134 140 147 148 151 152 152 158 181
190 193 194 213 215
Now, taking into account the whole set of survival data (both censored and uncensored). It was found that the
value of test statistic for the data set by formula (4.1) is given by∆ 𝑂𝐷𝐿
𝑐
= 0.0492696 and this value greater than the
tabulated critical value in Table 5. This means that the data set hasODL property.
Example 2.
On the basis of right censored data for lung cancer patients from Pena [29]. These data consists of 86 survival
times (in month) with 22 right censored:
The whole life times (non-censored data):
0.99 1.28 1.77 1.97 2.17 2.63 2.66 2.76 2.79 2.86
2.99 3.06 3.15 3.45 3.71 3.75 3.81 4.11 4.27 4.34
4.4 4.63 4.73 4.93 4.93 5.03 5.16 5.17 5.49 5.68
5.72 5.85 5.98 8.15 8.62 8.48 8.61 9.46 9.53 10.05
10.15 10.94 10.94 11.24 11.63 12.26 12.65 12.78 13.18 13.47
13.96 14.88 15.05 15.31 16.13 16.46 17.45 17.61 18.2 18.37
19.06 20.7 22.54 23.36
The ordered censored observations are:
11.04 13.53 14.23 14.65 14.91 15.47 16.49 17.05 17.28 17.88
17.97 18.83 19.55 19.58 19.75 19.78 19.95 20.04 20.24 20.73
21.55 21.98
We now account the whole set of survival data (both censored and uncensored), and computing the test statistic
given by formula (4.1). It was found that ∆ 𝑂𝐷𝐿
𝑐
= 0.433052 which is exceeds the tabulated value in Table 5. It is
evident that at the significant level 0.95 .Then this data set has ODL property.
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