Monotone likelihood ratio test

The monotone likelihood ratio test is the important part of statistics.
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Md.Sohel Rana
Jahangirnagar University

1. Monotone likelihood ratio Presenting By Md. Sohel Rana Class ID-277 Department of statistics Jahangirnagar University Savar , Dhaka

2. Outline of the presentation Definition Monotone Likelihood Ration (MLR) family of distribution Some examples on MLR families Reference Monotone Likelihood ratio and Maximum Likelihood ratio

3. Definition Let {f(x, θ) : θ ∈ Θ } be a family of PDF (PMF’s) Θ .We say that has a monotone likelihood ratio (MLR) statistic T(x) .if   Whenever and are distinct. The ratio is a non decreasing function of for the set of values X for which at least one of and is  

4. In the present module we define the monotone likelihood ratio (MLR) property for a family of pmf or pdf denoted by ⊂R. we exploit this property to derive the UMP level α tests for one-sided null against one-sided alternative hypotheses in some situations.   A real parametric family ⊂R is said to have MLR property in a real valued statistic T(x) if, for any . the following are satisfied. (i) ) [ Distribution are distinct corresponding to distinct parameter points] The ratio R is non-decreasing in T(x) on the set . Note If and , R(x) = 0. and , = .   Monotone Likelihood Ration (MLR) family of distribution

5. Some examples on MLR families One parameter and n parameter Exponential family Normal Distribution Bernoulli Distribution Geometric Distribution

6. One parameter Exponential family ⊂R : One parameter Exponential family. Then we can express in the form, such that and depends only is independent of depends only on x. We set such that is a strictly increasing function of . Then we have for , = increasing because is a strictly increasing function of θ . Hence, has MLR in T(x)   Note If ( , ………….. ) is a random sample of size n from the population with p.m.f or p.d.f . then has MLR in  

7. Let ( , ………….. ) , be a random sample from population. Therefore, = where , and has MLR in   Normal Distribution To continued……..

8. continued……. . Let ( , ………….. ) , be a random sample from population. Therefore, where , and has MLR in  

9. Let ( , ………….. ) , be a random sample of size n from Bernoulli population. =   Bernoulli Distribution

10. Let ( , ………….. ) , be a random sample of size n from geometric distribution with p.m.f . , Then   Where c( )= ,q( )= , And v(x)=1 has MLR in   Geometric Distribution

11. Let ( , ………….. ) , be a random sample of size n from the exponential distribution with p.d.f. , Now Where ( )= ,q( )= , And v(x)=1 has MLR in   exponential distribution exponential distribution continued……..

12. Let ( , ………….. ) , be a random sample of size n from the exponential distribution with p.d.f. , Now (1/ Where ( )= ,q( )= , And v(x)=1 has MLR in  

13. X ∼ Cauchy ( θ, 1) )= For any θ 2 > θ 1 Thus Cauchy(θ,1) is not a member of MLR family   Non-exponential family Non exponential distribution continued……..

14. X ∼ Cauchy ( θ, 1) )= For any increasing in or in |x|, Thus Cauchy(0,θ) is a member of MLR family in  

15. UNIFORMLY MOST POWERFUL (UMP) TEST If a test is most powerful against every possible value in a composite alternative, then it will be a UMP test. One way of finding UMPT is to find MPT by Neyman -Pearson Lemma for a particular alternative value, and then show that test does not depend on the specific alternative value. Example: X~N( ,  2 ) , we reject Ho if Note that this does not depend on particular value of μ 1, but only on the fact that  >  1. So this is a UMPT of H :  =  vs H 1 :  <  0.

16. If L is a decreasing function of y for every given  > 1 , then we have a monotone likelihood ratio (MLR) in statistic − y . To find UMPT, we can also use Monotone Likelihood Ratio (MLR). UNIFORMLY MOST POWERFUL (UMP) TEST If L=L(  )/L( 1 ) depends on (x 1 ,x 2 ,…, x n ) only through the statistic y=u(x 1 ,x 2 ,…, x n ) and L is an increasing function of y for every given  > 1 , then we have a monotone likelihood ratio (MLR) in statistic y .

17. Monotone likelihood ratio with hypothesis test ,UMP And Others Presenting: Md.Sohel Rana

18. Agenda Basic concepts Neyman -Pearson lemma UMP Invariance CFAR 18

19. Monotone Likelihood Vs Maximum Likelihood Ratio Test

20. Reference Anderson, G. 1996. “Nonparametric Tests of Stochastic Dominance in Income Distributions.” Econometrica 64(September): 1183–93 2.https://en.wikipedia.org/wiki/Monotone_likelihood_ratio(10.30 pm,25/06/2018 ) 3.Athey, S. 2002. “Monotone Comparative Statics Under Uncertainty.” Quarterly Journal of Economics 117(February): 187–223 4. Athey , S. 2002. “Monotone Comparative Statics Under Uncertainty.” Quarterly Journal of Economics 117(February): 187–223 5. Bartolucci , F., and A. Forcina . 2000. “A Likelihood Ratio Test for MTP 2 within Binary Variables.” Annals of Statistics 28(4): 1206–18. 6. Beach, C. M., and R. Davidson. 1983. “Distribution-Free Statistical Inference with Lorenz Curves and Income Shares.” Review of Economic Studies 50(October): 723–35.

21. Reference 6.Beach, C. M., and J. Richmond. 1985. “Joint Confidence Intervals for Income Shares and Lorenz Curves.” International Economic Review 26(June): 439–50. 7.Chambers, R. G. 1989. “Insurability and Moral Hazard in Agricultural Insurance Markets.” American Journal of Agricultural Economics 71(August): 604–16. 8.Chow, K. V. 1989. “Statistical Inference for Stochastic Dominance: A Distribution Free Approach . ” PhD thesis, Department of Finance, University of Alabama