This document provides an overview of the monotone likelihood ratio property for families of probability mass functions or probability density functions. It defines the MLR property and provides examples of families that satisfy it, including the normal, Bernoulli, geometric, and exponential distributions. It also discusses how the MLR property can be used to derive uniformly most powerful tests for one-sided hypotheses. The document outlines applications of MLR related to hypothesis testing, uniformly most powerful tests, and invariance. It compares the monotone likelihood ratio test to the maximum likelihood ratio test. References are provided at the end.

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⊆ 𝑅 .We say that 𝑓(𝑥, 𝜃) has a monotone likelihood ratio (MLR)
statistic T(x) .if 𝜃1 < 𝜃2
Whenever 𝑓(𝑥, 𝜃1) and 𝑓(𝑥, 𝜃2) are distinct. The ratio
𝑓(𝑥,𝜃1)
𝑓(𝑥,𝜃2)
is a non
decreasing function of 𝑇(𝑥) for the set of values X for which at least one of
𝒇(𝒙, 𝜽 𝟏) and 𝒇(𝒙, 𝜽 𝟏) is 𝜽 > 𝟎

4. In the present module we define the monotone likelihood ratio (MLR) property for a
family of pmf or pdf denoted by 𝑓 𝑥, 𝜃 : 𝜃 ∈ Θ Θ ⊂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 𝑓 𝑥, 𝜃 : 𝜃 ∈ Θ Θ ⊂R is said to have MLR property in a real
valued statistic T(x) if, for any 𝜃1 < 𝜃2 ∈ Θ .
the following are satisfied.
(i) 𝑓(𝑥, 𝜃1)≠ 𝑓(𝑥, 𝜃2)
[Distribution are distinct corresponding to distinct parameter points]
(ii) The ratio R 𝒙 =
𝒇(𝒙,𝜽 𝟐)
𝒇(𝒙,𝜽 𝟏)
is non-decreasing in T(x) on the set
𝑥: max(𝑓 𝑥, 𝜃2 , 𝑓(𝑥, 𝜃2) .
Note If 𝑓 𝑥, 𝜃1 = 0 and 𝑓 𝑥, 𝜃2 > 0 , R(x) = 0.
𝑓 𝑥, 𝜃1 = 0 and 𝑓 𝑥, 𝜃2 > 0 , 𝑅(𝑥) = ∞.
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
𝑓 𝑥, 𝜃 : 𝜃 ∈ Θ Θ ⊂R : One parameter Exponential family. Then we can express
f(𝑥, 𝜃) in the form,
𝑓(𝑥, 𝜃) = 𝑢(𝜃)exp(𝑞(𝜃)𝑇(𝑥))𝑣(𝑥)
such that 𝑢(𝜃) and 𝑞(𝜃) depends only 𝑜𝑛 𝜃, 𝑣(𝑥) is independent of 𝜃 𝑎𝑛𝑑 𝑇(𝑥) depends
only on x. We set 𝑇(𝑥) such that 𝑄(𝜃) is a strictly increasing function of 𝜃.
Then we have for 𝜃1 > 𝜃2,
𝒇(𝒙,𝜽 𝟐)
𝒇(𝒙,𝜽 𝟏)
=
𝑢(𝜃2)
𝑢(𝜃2)
exp 𝑄 𝜃2 − 𝑄 𝜃2 𝑇 𝑥 ,
increasing 𝑖𝑛 𝑇(𝑥) because 𝑄(𝜃) is a strictly increasing function of θ .
Hence, {𝑓(𝑥, 𝜃), 𝜃 ∈ Θ} has MLR in T(x)
Note If (𝑥1 , 𝑥2 ………….. 𝑥 𝑛) is a random sample of size n from the population with
p.m.f or p.d.f. 𝑓(𝑥, 𝜃) then 𝑓(𝑥, 𝜃) has MLR in 𝑖=1
𝑛
𝑇(𝑥𝑖) .

7. Let (𝑥1 , 𝑥2 ………….. 𝑥 𝑛) , be a random sample from 𝑁(𝜃, 1), population.
Therefore,
𝑓 𝑥, 𝜃 = (2𝜋)
−𝑛
2 𝑒(−
1
2 𝑖=1
𝑛
(𝑥𝑖−𝜃)2)
=𝑒{
−𝑛𝜃2
2
}
𝑒{𝜃 𝑖=1
𝑛
𝑥 𝑖}(2𝜋)
−𝑛
2 𝑒(−
1
2 𝑖=1
𝑛
(𝑥𝑖−𝜃)2)
= 𝑢(𝜃)exp(𝑞(𝜃)𝑇(𝑥))𝑣(𝑥)
where 𝑢(𝜃) = 𝑒{
−𝑛𝜃2
2
}
, 𝑄 𝜃 = 𝜃, T(x) = 𝑖=1
𝑛
𝑥𝑖 and
𝑣(𝑥) = (2𝜋)
−𝑛
2 𝑒(−
1
2 𝑖=1
𝑛
(𝑥𝑖−𝜃)2)
𝑓(𝑥, 𝜃) has MLR in T(x) = 𝑖=1
𝑛
𝑥𝑖
Normal Distribution
To continued……..

8. continued……..
Let (𝑥1 , 𝑥2 ………….. 𝑥 𝑛) , be a random sample from 𝑁(0, 𝜃2
), population. Therefore,
𝑓 𝑥, 𝜃 = (2𝜋)
−𝑛
2 𝜃−𝑛
𝑒(−
1
2 𝑖=1
𝑛
(𝑥𝑖)2)
= 𝑢(𝜃)exp(𝑞(𝜃)𝑇(𝑥))𝑣(𝑥)
where 𝑢(𝜃) = 𝜃−𝑛
, 𝑄 𝜃 =
−1
2𝜃2 , T(x) = 𝑖=1
𝑛
𝑥𝑖 and
𝑣(𝑥) = (2𝜋)
−𝑛
2
𝑓(𝑥, 𝜃) has MLR in T(x) = 𝑖=1
𝑛
𝑥𝑖2

9. Let (𝑥1 , 𝑥2 ………….. 𝑥 𝑛) , be a random sample of size n from
Bernoulli(𝜃), population.
𝑓(𝑥, 𝜃) = θ
𝑖=1
𝑛
𝑥𝑖 (1 − 𝜃) 𝑛− 𝑖=𝑖−1
𝑛
𝑥 𝑖
= (1 − 𝜃) 𝑛[ln(
𝜃
1−𝜃
) 𝑖=1
𝑛
𝑥𝑖]
= 𝑢(𝜃)exp(𝑞(𝜃)𝑇(𝑥)𝑣(𝑥)
Bernoulli Distribution

10. Let (𝑥1 , 𝑥2 ………….. 𝑥 𝑛) , be a random sample of size n from geometric distribution with p.m.f.
𝑓(𝑥, 𝜃) = θ (1 − 𝜃) 𝑥,𝑥 = 0,1,2,3, … … . 0 < 𝜃 < 1
Then
𝑓(𝑥, 𝜃) = θ
𝑖=1
𝑛
𝑥𝑖 (1 − 𝜃) 𝑛− 𝑖=𝑖−1
𝑛
𝑥 𝑖
= 𝜃 𝑛
[ln(1 − 𝜃)
𝑖=1
𝑛
𝑥𝑖]
= 𝑢(𝜃)exp(𝑞(𝜃)𝑇(𝑥)𝑣(𝑥)
Where c(𝜃)= 𝜃 𝑛,q(𝜃)=− ln(1 − 𝜃), T(x) = − 𝑖=1
𝑛
𝑥𝑖
And v(x)=1
𝑓(𝑥, 𝜃) has MLR in T(x) = − 𝑖=1
𝑛
𝑥𝑖
Geometric Distribution

11. Let (𝑥1 , 𝑥2 ………….. 𝑥 𝑛) , be a random sample of size n from the
exponential distribution with p.d.f.
𝑓(𝑥, 𝜃) = θ𝑒[−𝜃𝑥],𝑥 > 0, 𝜃 > 0
Now
𝑓(𝑥, 𝜃) = θ 𝑛
𝑒[−𝜃𝑥]
𝑓(𝑥, 𝜃) = θ 𝑛 𝑒[−𝜃 𝑖=1
𝑛
𝑥𝑖]
= 𝑢(𝜃)exp(𝑞(𝜃)𝑇(𝑥)𝑣(𝑥)
Where 𝑢(𝜃)= 𝜃 𝑛,q(𝜃)=𝜃, T(x) = − 𝑖=1
𝑛
𝑥𝑖
And v(x)=1
𝑓(𝑥, 𝜃) has MLR in T(x) = − 𝑖=1
𝑛
𝑥𝑖
exponential distribution
exponential distribution continued……..

12. Let (𝑥1 , 𝑥2 ………….. 𝑥 𝑛) , be a random sample of size n from the exponential distribution with p.d.f.
𝑓(𝑥, 𝜃) = 1/θ𝑒[−𝜃/𝑥]
,𝑥 > 0, 𝜃 > 0
Now
𝑓(𝑥, 𝜃) =
1
θ
𝑛
𝑒[−𝜃/𝑥]
𝑓(𝑥, 𝜃) = (1/θ) 𝑛 𝑒[−𝜃/ 𝑖=1
𝑛
𝑥𝑖]
= 𝑢(𝜃)exp(𝑞(𝜃)𝑇(𝑥)𝑣(𝑥)
Where 𝑢(𝜃)= (1/𝜃) 𝑛
,q(𝜃)=(−
1
𝜃
), T(x) = − 𝑖=1
𝑛
𝑥𝑖
And v(x)=1
𝑓(𝑥, 𝜃) has MLR in T(x) = − 𝑖=1
𝑛
𝑥𝑖

13. X ∼ Cauchy(θ,1)
𝑓(𝑥, θ)=
1
𝜋
1
1+(𝑥−θ)2
For any θ2 > θ1
𝑓(𝑥,θ2)
𝑓(𝑥,θ1)
=
1+(𝑥−θ1)2
1+(𝑥−θ2)2
Thus Cauchy(θ,1) is not a member of MLR family
Non-exponential family
Non exponential distribution
continued……..

14. X ∼ Cauchy(θ,1)
𝑓(𝑥, θ)=
1
𝜋
θ
ߠ2+𝑥2
For any 𝜃1 < 𝜃2
𝑓(𝑥,θ2)
𝑓(𝑥,θ1)
= (
𝜃2
2
𝜃1
2)
𝜃1
2
+𝑥2
𝜃2
2
+𝑥2
increasing in 𝑥2 or in |x|, Thus Cauchy(0,θ) is a member of MLR family in |𝑥 |

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 μ1, but only on the
fact that  0 >  1. So this is a UMPT of H0:  = 0 vs H1:  <  0.


Z
n
X 0 


16. If L is a decreasing function of y for every given 0>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(0)/L(1) depends on (x1,x2,…,xn) only through the statistic y=u(x1,x2,…,xn)
and L is an increasing function of y for every given 0>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
1. 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 MTP2 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