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Two-sided tests
Now consider testing H0 : θ = θ0 vs. H1 : θ ̸= θ0. It can be shown
(proof omitted) that if the family has monotone likelihood ratio
then the power functions of the UMP 1-sided tests are strictly
monotone.
Each of the 1-sided UMP test works well for their stated 1-sided
H1. However, for the 2-sided H1, there exists θ such that the tests
are biased in the sense that
Eθ [δ0(X)] < Eθ0 [δ0(X)]. 1
Simple vs. composite: UMPU tests
An unbiased test has power function no small than α for all θ
under H1.
For 1-parameter exponential families with PDF
fθ(x) = h(x)eθT(x)−A(θ), a UMP-Unbiased (UMPU) test exists.
Theorem: For a 1-parameter exponential family with sufficient
statistic T(X), a UMPU test at level α for testing H0 : θ = θ0 vs.
H1 : θ ̸= θ0 exists and is given by:
δ(X) =

1, T(X) > c2 or T(X) < c1
γi, T(X) = ci, i = 1, 2
0, c1 < T(X) < c2
where ci,γi, i = 1, 2, are chosen such that Eθ0 [δ(X)] = α and
Eθ0 [T(X)δ(X)] = Eθ0 [T(X)]Eθ0 [δ(X)] = αEθ0 [T(X)].
2
Simple vs. composite: UMPU tests
Example: Suppose we can observe a sample X ∼ exp(θ), with
PDF fθ(x) = 1θ e
−x/θ, x > 0. Determine the UMPU test for testing
H0 : θ = 1 vs. H1 : θ ̸= 1.
3
Simple vs. composite: UMPU tests
Example: Suppose X ∼ exp(θ), fθ(x) = 1θ e−x/θ, x > 0. Determine
the UMPU test for testing H0 : θ = 1 vs. H1 : θ ̸= 1.
4
Simple vs. Composite
We have mainly presented methods in this case applicable to
1-parameter exponential families.
The general properties of MLE also lead to generally
applicable testing methods.
Suppose we have X = (X1, . . . ,Xn) iid with common fθ(·) for
the parametric family {fθ(·) : θ ∈ Θ}, Θ ∈ R.
Recall the most power test for H0 : θ = θ0 vs. H1 : θ = θ1 is
given by the Neyman-Pearson likelihood ratio (NPLR) test
statistic
∏ni=1 fθ1(Xi)
∏ni=1 fθ0(Xi)
,
which requires knowing both θ0 and θ1.
If we are testing H0 : θ = θ0 vs. H1 : θ ∈ Θ\{θ0}, we could
try to first “estimate” a θ1 value and plug it into the NPLR
statistic. 5
Simple vs. Composite: GLRT
Definition: The Generalised likelihood ratio test (GLRT) for
testing H0 : θ = θ0 vs. H1 : θ ∈ Θ\{θ0} uses the statistic:
∏ni=1 fθˆ(Xi)
∏ni=1 fθ0(Xi)
,
where θˆ = argmaxθ∈Θ∏
n
i=1 fθ(Xi).
6
Simple vs. Composite: GLRT
The limiting distribution of the log-GLRT statistic under H0 is 12χ
2
1
(under some regularity conditions).
Sketch proof:
7
Simple vs. Composite: GLRT
(Proof continued)
8
Simple vs. Composite: GLRT
(Proof continued)
9
Composite vs. Composite
Suppose X ∼ fθ(x) for a 1-parameter family {fθ(·) : θ ∈ Θ},
Θ ∈ R. We are testing H0 : θ ∈ Θ0 vs. H1 : θ ∈ Θ\Θ0, Θ0 ⊆ Θ.
For certain composite H0, optimal tests exist.
Proposition: If the family has monotone likelihood ratio in a
statistic T(X), then the UMP test of H0 : θ ≤ θ0 vs. H1 : θ > θ0 is
of the same form as for H0 : θ = θ0 vs. H1 : θ > θ0:
δ(X) =

1, T(X) > c
γ, T(X) = c
0, T(X) < c,
where c,γ are chosen such that Eθ0 [δ(X)] = α.
10
Composite vs. Composite
Proposition: For a 1-parameter exponential family:
fθ(x) = h(x) exp(w(θ)T(x)−A(θ))
where w(θ) is strictly increasing in θ. We are testing
H0 : θ ≤ θ1 or θ ≥ θ2 (θ1 < θ2) vs. H1 : θ1 < θ < θ2. The UMPU
test exists and is of the form:
δ(X) =

1, c1 < T(X) < c2
γi, T(X) = γi, i = 1, 2
0, T(X) < c1 or T(X) > c2,
where ci,γi, i = 1, 2, are selected such that
Eθ1 [δ(X)] = Eθ2 [δ(X)] = α. (Such tests are of interest when trying
to show a new drug is “effectively equivalent” to some standard.) 11
Multivariate case
Suppose we have a family of distributions indexed by more than 1
parameter. The GLRT provides a useful general method of testing
H0 : θ ∈ Θ0 vs. H1 : θ ∈ Θ\Θ0. Compute the test statistic:
log
(
L(θˆ;X)
L(θˆ0;X)
)
where
θˆ = argmax
θ∈Θ
L(θ;X)
is the “unrestricted” MLE, and
θˆ0 = argmax
θ∈Θ0
L(θ;X).
12
Multivariate case
Many commonly used statistical tests are equivalent to the GLRT.
Example: 1-way ANOVA F-test. Consider Xij ∼ N(µi, σ2), for
i = 1, . . . , g being indexes for groups and j = 1, . . . ,ni, and all Xij
are independent. The total sample size is denoted by N = ∑gi=1 ni.
Consider testing H0 : µ1 = µ2 = · · · = µg vs. H1 : µi’s are not all
equal.
13
Multivariate case
(Example continued)
14
Multivariate case
(Example continued)
15
Multivariate case
(Example continued)
16
Multivariate case
Example: one-sided t-test. Let X1, . . . ,Xn be iid N(µ, σ2), and
Θ = {(µ, σ2) : µ ≥ 0, σ2 > 0}. Consider testing H0 : µ = 0 vs.

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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