1. An Empirical Likelihood Ratio Based Goodness-of-Fit Test for Two-parameter Weibull Distributions Presented by: Ms. Ratchadaporn Meksena Student ID: 555020227-5 Advisor: Assoc. Prof. Dr. Supunnee Ungpansattawong Date: 29 th November 2013 Department of Statistics, Faculty of Science, Khon Kaen University

2. OUTLINE 1.Introduction  Rationale and Background  Objective of Study  Scope and Limitation of Study  Anticipated Outcomes 2.Literature Review 3.Research Methodology  Empirical Likelihood Method  Goodness-of-Fit Test Based on Empirical Likelihood Ratio  Calculation of Critical Values and Evaluation of Type I Error Control  Evaluation of the Power of the Proposed Test

3. 1. Introduction  Rationale and Background Weibull distribution is commonly used in many fields such as Survival Analysis Reliability Engineering & Failure Analysis Extreme Value Theory Weather Forecasting General Insurance etc. The two-parameter Weibull distribution is the most widely used distribution for life data analysis.

4. 1. Introduction  Rationale and Background (cont.) The important part of data analysis is ensuring that the data come from a particular family of distributions. The goodness-of-fit tests for Weibull distribution are generally based on the empirical distribution function (EDF), such as the Kolmogorov-Smirnov (KS) test, Cramer-von Mises (CvM) test, or the Anderson-Darling (AD) test. Recently, there are some literature about a goodness-of-fit test based on empirical likelihood ratio which the study results showed the goodness-of-fit tests based on empirical likelihood ratio is competitive when compared with other available tests. Therefore, in this study, we will propose an empirical likelihood ratio based goodness of fit test for two-parameter Weibull distributions.

5. 1. Introduction  Objective of Study The objective of this study is to propose a new goodness-of-fit statistic based on empirical likelihood ratio for two-parameter Weibull distributions.

6. 1. Introduction  Scope and Limitation of Study In this study, we will derive an empirical likelihood ratio based goodness-of-fit test for two-parameter Weibull distributions and its asymptotic properties, calculate the critical values for fixed sample sizes using Monte Carlo simulations, and evaluate the performance of the proposed test in controlling the Type I error. Finally, we will compare the power of the test between the proposed test statistic and Kolmogorov-Smirnov, Cramér-von Mises, and Anderson-Darling statistic.

7. 1. Introduction  Anticipated Outcomes We expect that we will get a new goodness-of-fit test based on empirical likelihood ratio for two- parameter Weibull distributions.

8. 2. Literature Review  Examples of Goodness-of-Fit Tests for Two-Parameter Weibull Distributions: Shapiro and Brain (1987) proposed the test statistic is based on similar principles used in the derivation of the well known W-test for normality. Coles (1989) proposed a test via the stabilized probability plot, which involves estimating scale and shape parameters. Khamis (1997) proposed the δ -corrected Kolmogorov-Smirnov test, where the MLE for scale and shape parameters was employed.

9. 2. Literature Review  Examples of Goodness-of-Fit Tests for Two-Parameter Weibull Distributions (cont.): Cabana and Quiroz (2005) proposed to employ the empirical moment generating function and a ffi ne invariant estimators for estimating scale and shape parameters such as moment estimators.

10. 2. Literature Review  Examples of Goodness-of-Fit Tests Based on Empirical Likelihood Ratio: Vexler and Gurevich (2010) constructed an empirical likelihood ratio based goodness of fit test to approximate the optimal Neyman–Pearson ratio test with an unknown alternative density function. Vexler et al. (2011) proposed a similar goodness of fit test based on the empirical likelihood method to test the null hypothesis of an inverse Gaussian distribution.

11. 2. Literature Review  Examples of Goodness-of-Fit Tests Based on Empirical Likelihood Ratio (cont.): Ning and Ngunkeng (2013) proposed a similar goodness of fit test based on the empirical likelihood method to test the null hypothesis of a skew normality.

12. 3. Research Methodology Consider the two-parameter Weibull distribution which has the cumulative distribution function and the probability density function defined as and respectively, where x > 0, β > 0 is the scale parameter and α > 0 is the shape parameter.

13. 3. Research Methodology  Empirical Likelihood Method Let X 1, X 2, …, X n be independently and identically distributed observations, which follow an unknown population distribution F. The empirical likelihood function of F be defined as where the component p i, i =1, 2, …, n, maximize the likelihood L p (F) and satisfy empirical constraints corresponding to hypotheses of interest. For example, when a population parameter θ identified by E(X) = θ is of interest, and the true value of θ is θ 0. The null hypothesis is H o ∶ E(X) = θ 0. To maximize L p (F), the values of p i in L p (F) should be chosen given the constraints and, where the constraint is an empirical version of E(X) = θ 0.

14. 3. Research Methodology  Empirical Likelihood Method (cont.) The empirical log-likelihood ratio statistic to test θ = θ 0 is given by where R(θ) is the empirical log-likelihood ratio function defined through the definition of the empirical likelihood ratio function by Owen (1988).

15. 3. Research Methodology  Goodness-of-Fit Test The goodness-of-fit test is a statistical test to determine whether the observations are consistent with the particular statistical model. It describes how well the particular model fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under a statistical model.

16. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio The hypothesis to be tested is where f H 0 and f H 1 are both unknown.

17. 3. Research Methodology  Goodness-of-Fit Test When density functions f H 0 and f H 1 are completely known, the most powerful test statistics is the likelihood ratio where under the null hypothesis X 1, X 2, …, X n follows a Weibull distribution with parameters β and .

18. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio In this study, forms of f H 0 and f H 1 are both unknown, but are estimable. We follow the similar idea by Vexler and Gurevich (2010) and Ning and Ngunkeng (2013) to construct a test statistic in forms of estimated likelihood ratios based goodness-of-fit test for the two-parameter Weibull distribution. Apply the maximum empirical likelihood method to estimate of the numerator of the ratio (3). Rewrite the likelihood function in the form of where X (1) ≤ X (2) ≤ ⋯ ≤ X (n) are the order statistics based on the observations X 1, X 2, …, X n.

19. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio Following the maximum empirical likelihood method, we can derive values of f i that maximize L f and satisfy the empirical constraints under the alternative hypothesis H 1. Obviously, values of f i should be restricted by the equation ∫ f(s)ds = 1. Thus, we need an empirical form of the constraint ∫ f(s)ds = 1. We first give the following lemma by Vexler and Gurevich (2010) to obtain this empirical constraint. Lemma 1 Let f(x) be a density function. Then where X (j-m) = X (1) if j-m ≤ 1 and X (j+m) = X (n), if j+m ≥ n.

20. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio It is obvious that since and we denote, using the empirical approximation to the remainder term in Lemma 1, we have From Lemma 1,we can empirically estimate δ m via Notice that δ m → 1 when m ⁄ n → 0 as m, n → ∞.

21. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio By applying the mean value theorem to the term of, we have Thus, the empirical constraint under the alternative hypothesis H 1 is given by

22. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio Apply the Lagrange multiplier method to maximize that subject to the constraint. The Lagrange function defined by where λ is a lagrange multiplier. By taking the derivative of the above equation with respect to each f j, j = 1, 2, …, n, and λ, we obtain

23. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio and respectively. From the equation (5), we have Then multiply equation (4) by f j and taking summation, we have

24. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio Since, we have λ = -n. Finally, we will obtain the estimate value of f j to maximize, which also maximizes as where X (j-m) = X (1) if j-m ≤ 1 and X (j+m) = X (n), if j+m ≥ n. Thus, using the maximum empirical likelihood method, the empirical likelihood ration based goodness-of-fit test for the two- parameter Weibull distribution can be constructed as

25. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio where θ = (β, α) ' is the parameter vector of a two-parameter Weibull distribution. To maximize the denominator, since the parameters β and α are unknown, the maximum likelihood estimate of α based on the observations can be applied. The maximum likelihood estimators and of β and α, respectively, are solutions of the equations: and

26. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio We notice that the distribution of the test statistic WB nm strongly depends on the integer m. Thus, the optimal values of m should be evaluated to make the test more efficient. We follow the same argument by Vexler and Gurevich (2010) to reconstruct the test statistic according to the properties of the empirical likelihood method. We adopt their idea here to reconstruct the test statistic in (7) as where δ ∈ (0, 1).

27. 3. Research Methodology  Goodness-of-Fit Test Based on Empirical Likelihood Ratio Similar to the argument of Vexler et al. (2011) and Ning and Ngunkeng (2013), we take δ =0.5 in the equation (8). Thus, the final form of the test statistic is

28. 3. Research Methodology  Asymptotic Properties of the Proposed Test Statistic Denote and We assume the following conditions hold: (C1) (C2) Under the null hypothesis, in probability. (C3) Under alternative hypothesis, in probability where θ 0 is a constant vector with finite components. (C4) There are open intervals and containing θ and θ 0 respectively. There also exists a function s(x) such that for all x ∈ R and.

29. 3. Research Methodology  Asymptotic Properties of the Proposed Test Statistic (cont.) Proposition 1 Assume that the condition (C1)–(C4) hold. Then, under H 0, in probability as → ∞, while, under H 1, in probability as → ∞. Given condition (C1)–(C4), Proposition 1 shows that the power of the test goes to 1 as → ∞ under the alternative hypothesis. Thus, the proposed test is consistent.

30. 3. Research Methodology  Calculation of Critical Values and Evaluation of Type I Error Control To calculate the critical values for fixed sample sizes n = 10, 20, 30, 40, 50, 100, 200, 500, we simulate 5,000 samples from WB(β,  ) with different values of (β,  ) = (1, 0.5), (1, 2), (1, 4), (1, 8). For each simulated sample, we use R package MASS to estimate parameters β and . Then we can calculate a statistic for each sample based on equation (9). After we obtain all 5,000 test statistics, we order them and choose 90 th, 95 th and 99 th percentiles to be the critical values corresponding to the significance level  = 0.1, 0.05 and 0.01, respectively.

31. 3. Research Methodology  Calculation of Critical Values and Evaluation of Type I Error Control (cont.) Consequently, to investigate the performance of the proposed test in controlling the Type I error with the significance level  = 0.1, 0.05 and 0.01, we conduct simulations 5,000 times under WB(β,  ) with different values of (β,  ) = (1, 0.5), (1, 2), (1, 4), (1, 8) and sample sizes n = 20, 50, 100, 200, 500, 1000. For each sample, we calculate a sample statistic based on equation (9) and compares to the critical value. The percentage of rejecting the null hypothesis will be the size of the proposed test.

32. 3. Research Methodology  Evaluation of the Power of the Proposed Test In order to study the power of the proposed test, we simulate 10,000 samples with sample size sizes n = 20, 50, 100, 200, 500, 1000 from Beta(0.25, 0.25), Beta(2, 2), N(0, 1) TruncN(-1,1). Then we compute the powers of Kolmogorov- Smirnov test, Cramér-von Mises test, Anderson-Darling test and the proposed test WB n at the nominal level 0.05.

34. Probability density function

35. Cumulative distribution function