The Uniformly Most Powerful (UMP) Test Essay

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1. Introduction and Background

The job of _nding uniformly most powerful ( UMP ) trial for proving the

simple hypothesisHydrogen0:_=_0 against reversible alternateHydrogen1:_I?=_0 has been

an interesting job in proving statistical hypotheses. It is good known that, for

distributions belong to the exponential household, such UMP trials do non be ( see

[ 8 ] ) . On the other manus, there has been some efforts to _nd UMP trials for other

distributions and/or for the job of proving nonreversible hypotheses. Historically,

surveies on UMP trials started in the 1940 & A ; apos ; s. Lehmann and Stein [ 7 ] presented

some most powerful trials for the parametric quantity of a normal distribution. Birnbaum

[ 3 ] and Pratt [ 13 ] offered the solutions for UMP trials for composite hypotheses

in the unvarying distributions with both terminal points of the support dependant on

the parametric quantity. Takeuchi [ 16 ] constructed UMP trial for the location parametric quantity of

an exponential distribution despite the being of a nuisance parametric quantity. Kabe

and Laurent [ 5 ] exhibited a UMP trial for a reversible hypothesis in the two-base hit

exponential distribution with a nuisance parametric quantity. Edelman [ 4 ] showed that the

usual reversible trial is really uniformly most powerful within the category of trials

with critical parts that are symmetric about the void hypothesis. Nomakuchi

[ 12 ] studied the UMP trials in the presence of a nuisance parametric quantity for uniform

distribution. A new method to find UMP trials in distinct sample infinites was

proposed by Scherb [ 15 ] . Wang et Al. [ 17 ] constructed a UMPU trial for compar-

ing the agencies of two negative binomial distributions. Migliorati [ 11 ] demonstrated

that a UMP trial exists for proving reversible hypothesis in the household with sup-

port dependant on the parametric quantity. Besides, McDermott and Wang [ 10 ] investigated

a building of uniformly more powerful trials for hypotheses about additive in-

equalities. Aaberge and Zhang [ 1 ] presented a category of exact UMP unbiased trials

for proving conditional symmetricalness against assorted alternate diagonals-parameter.

The UMP reversible trial in the household with right and left utmost points of the

support dependant on the parametric quantity is obtained by Sayyareh et Al. [ 14 ] .

But, every bit far as the writers know, there has been neither general consequence about

the being and preparation of the reversible generalised likeliness ratio ( GLR )

trial in the household with support dependant to parametric quantity, nor about the formal

relationship between such trials and the related UMP trials in this household.

In this paper, we obtain the GLR trials for proving hypothesisHydrogen0:_=_0 versus

reversible optionHydrogen1:_I?=_0, in the household of distributions with support de-

pendant on the parametric quantity. We besides demonstrated that UMP trials for proving the

above hypotheses are tantamount to the GLR trials. In add-on, we derive the gen-

eral signifier of UMP trials for provingHydrogen0:_1____2 versusHydrogen1:_ & A ; lt ; _1 or_ & A ; gt ; _2

in the households with the right or the left utmost point of the support dependant

on the parametric quantity.

This paper is organized as follows: In Section 2, we obtain the general signifier

of the GLR trial for proving simple hypothesis versus reversible option in the

household with the right or left utmost point of the support dependant on the pa-

rameter. In Section 3, we obtain the UMP trial for provingHydrogen0:_1____2 versus

Hydrogen1:_ & A ; lt ; _1 or_ & A ; gt ; _2 in the households with the right or the left utmost point of

the support dependant on the parametric quantity.

In the undermentioned, we will utilize the notationsTen( 1 ) for min (Ten1; : : : ; XN) , andTen(N)

for soap (Ten1; : : : ; XN) .

2. GLR reversible trial in the household with right ( left ) extreme

point of the support dependant on the parametric quantity

Let us get down this subdivision with a Lemma.

2.1. Lemma.I ) If

( 2.1 )degree Fahrenheit_(ten) =a(_)B(ten) ;degree Celsius & A ; lt ; x & A ; lt ; _ ; a(_)& A ; gt ;0;

is a uninterrupted map, soa(:)is a diminishing map.

two ) If

( 2.2 )g_(ten) =P(_)Q(ten) ;_ & A ; lt ; x & A ; lt ; vitamin D ; P(_)& A ; gt ;0;

is a uninterrupted map, soP(:)is an increasing map.

Proof.I )

& A ; int ;_

degree Celsiussa(_)B(ten)dx= 1, implies that

& A ; int ;_

degree CelsiussB(ten)dx= 1=a(_) . Now we differentiate

from both sided of the 2nd relation to obtainB(_) =oˆˆˆa& A ; premier ;(_)=(a(_) ) 2:But B ( . )

is a nonnegative map, hence a ( . ) is a diminishing map.

two ) The cogent evidence can be done in a similar manner to the instance ( I ) . _

Now, we obtain the general signifiers of the GLR trials for proving hypothesisHydrogen0:

_=_0 versusHydrogen1:_I?=_0 in households with denseness ( 2.1 ) and ( 2.2 ) .

2.2. Theorem.Suppose thatTen= (Ten1; Ten2; : : : ; XN)is a random sample, with

observed valueten= (ten1; ten2; : : : ; xN), from a distribution with denseness map ( 2.1 ) .

3

Then a GLR trial of size_for provingHydrogen0:_=_0versusHydrogen1:_I?=_0is given by

_

1 (ten) =

8 & A ; lt ;

:

1ten(N)& A ; gt ; _0 orten(N)& A ; lt ; aoˆˆˆ1

(

a(_0 )

(

_+

(

a(_0 )

a(degree Celsiuss)

)N

)oˆˆˆ1

N

)

;

0 otherwise:

Proof.Let _ =degree Fahrenheit_:_ & A ; gt ; degree Celsiussg;_0 =degree Fahrenheit_:_=_0g

Liter(_;ten) = (a(_) )N

& A ; Pi ;N

I=1

B(tenI)I(degree Celsius ; _) (tenI) =

{

(a(_) )N& A ; Pi ;N

I=1B(tenI)ten(N)

__ ;

0ten(N)& A ; gt ; _ :

It is obvious that,MLE(_)_2_ =Ten(N) andMLE(_)_2_0 =_0;and

_(ten) =

sup_0Liter(_;ten)

sup_Liter(_;ten)

=

(a(_0 ) )N& A ; Pi ;N

I=1B(tenI)

(a(ten(N) ) )N

& A ; Pi ;N

I=1B(tenI)

=

{ (

a(_0 )

a(ten(N) )

)N

ten(N)

__0;

0ten(N)& A ; gt ; _0:

So, the GLR trial cullsHydrogen0, ifften(N)& A ; gt ; _0 or

(

a(_0 )=a(ten(N) )

)N

_K_whenten(N)

__0 ;

and ifften(N)& A ; gt ; _0 orten(N)& A ; lt ; k& A ; premier ;

_whenten(N)

__0, whereK& A ; premier ;

_is chosen so that

Phosphorus_0 (Ten(N)& A ; lt ; k& A ; premier ;

_) =_. Using Lemma 2.1 and distribution ofTen(N) , we have

_=

& A ; int ;K

& A ; premier ;

_

degree Celsiuss

niobium(T)

(a(_0 ) )N

(a(T) )Noˆˆˆ1dt ; k

& A ; premier ;

_=a

oˆˆˆ1

(

a(_0 )

(

_+

(a(_0 )

a(degree Celsiuss)

)N

)oˆˆˆ1

N

)

:

Therefore the GLR trial cullsHydrogen0 iffTen(N)& A ; lt ; aoˆˆˆ1

(

a(_0 )

(

_+

(

a(_0 )

a(degree Celsiuss)

)N)oˆˆˆ1

N

)

for

Ten(N)

__0 orTen(N)& A ; gt ; _0. _

2.3. Theorem.Suppose thatTen= (Ten1; Ten2; : : : ; XN)is a random sample, with

observed valueten= (ten1; ten2; : : : ; xN), from a distribution with denseness map ( 2.2 ) .

Then a GLR trial of size_for the simple hypothesisHydrogen0:_=_0versus reversible

optionHydrogen1:_I?=_0is given by

_

2 (ten) =

8 & A ; lt ;

:

1ten( 1 )& A ; lt ; _0 orten( 1 )& A ; gt ; Poˆˆˆ1

(

P(_0 )

(

_+

(

P(_0 )

P(vitamin D)

)N

)oˆˆˆ1

N

)

;

0 otherwise:

Proof.The cogent evidence is similar to the cogent evidence of Theorem 2.2. _

2.4. Remark.Birkes [ 2 ] shown that GLR trial for a nonreversible hypotheses in a

one-parameter exponential household of distribution coincide with the UMP trial in

the same job. He besides shown that under certain conditions on the likeliness

map, GLR trial for a reversible hypothesis in a one-parameter exponential

household is normally coincide with the UMPU trial. But as the writers know, there

has been no plant on the relationship between UMP trials and GLR trials for

reversible hypotheses.

2.5. Remark.The GLR trials obtained above for simple hypothesis against two-

sided hypothesis in the household with right ( left ) utmost point of the support de-

pendant on the parametric quantity are coincided with the UMP trials in the same job

( see Theorems 4.3 and 4.4 in [ 14 ] ) .

4

3. UMP trial for proving composite hypotheses

In this subdivision, we will get UMP trials for proving hypothesisHydrogen0:_1____2

versusHydrogen1:_ & A ; lt ; _1 or_ & A ; gt ; _2 in the household with support dependant on the

parametric quantity. Note that, such trials don & amp ; apos ; t be for the exponential household ( see [ 8 ] ) ,

but it is shown that there exist a degree_UMPU trial for such hypotheses for

one-parameter exponential household ( see [ 8 ] ) .

3.1. Theorem.Suppose thatTen= (Ten1; Ten2; : : : ; XN)is a random sample, with

observed valueten= (ten1; ten2; : : : ; xN), from a distribution with denseness map ( 2.1 ) .

Then a UMP trial of size_for a hypothesisHydrogen0:_1____2versusHydrogen1:_ & A ; lt ;

_1 or_ & A ; gt ; _2is given by

( 3.1 )I•

_

(ten) =

{

1

0

ten(N)& A ; gt ; _2 orten(N)

_K ;

otherwise;

whereKis chosen so that_=Tocopherol_1 [I•_(Ten) ] =Tocopherol_2 [I•_(Ten) ].

Proof.First we divideHydrogen1 intoHydrogen& A ; premier ;

1:_ & A ; lt ; _1 andHydrogen& A ; prime ; & A ; premier ;

1:_ & A ; gt ; _2. Then we _nd a Umpire

trial forHydrogen& A ; premier ;

1 andHydrogen& A ; prime ; & A ; premier ;

1. Suppose that_02[_1; _2 ] .

1. To _nd a UMP trial for

( 3.2 )

{

Hydrogen0:_=_0;

Hydrogen& A ; premier ;

1:_ & A ; lt ; _1& A ; lt ; _0;

_rst see the job of proving the undermentioned hypotheses

( 3.3 )

{

Hydrogen0:_=_0;

Hydrogen& A ; premier ;

11:_=_& A ; premier ;

1 (_& A ; premier ;

1& A ; lt ; _1__0 ):

Harmonizing to Neyman-Pearson Lemma, each trial conforming with

( 3.4 )I•1 (ten) =

8 & A ; lt ;

:

1R1 (ten)& A ; gt ; k1;

1R1 (ten) =K1;

0R1 (ten)& A ; lt ; k1;

is a MP trial of size_for proving hypotheses ( 3.3 ) on the status that ( see [ 6 ] )

( 3.5 )_=Tocopherol_0 [I•1 (Ten) ] =Phosphorus_0 (R1 (Ten)& A ; gt ; k1 ) + 1Phosphorus_0 (R1 (Ten) =K1 );

where

R1 (ten) =

degree Fahrenheit_& A ; premier ;

1

(ten)

degree Fahrenheit_0 (ten)

=

{ (

a(_

& A ; premier ;

1 )

a(_0 )

)N

ten(N)

__& A ; premier ;

1;

0ten(N)& A ; gt ; _& A ; premier ;

1:

So,I•1 consists of

I•1 (ten) =

{

1

0

ten(N)

__& A ; premier ;

1;

ten(N)& A ; gt ; _& A ; premier ;

1;

is a MP trial of size_, for the same hypotheses, on the status that_=

1Phosphorus_1 (Ten(N)

__& A ; premier ;

1 ) . But,I•1 depends on_& A ; premier ;

1 merely by_0__1& A ; gt ; _& A ; premier ;

1, thereforeI•1 is a

UMP trial for proving hypotheses ( 3.2 ) . We can de_ne 0& A ; lt ;1 (ten)& A ; lt ;1 at any signifier

of desirable on the status of equation ( 3.5 ) . We take

1 (ten) =

{

1ten(N)

_degree Celsiuss1;

0ten(N)& A ; gt ; degree Celsiuss1 (degree Celsiuss1& A ; lt ; _& A ; premier ;

1& A ; lt ; _1 ):

5

By replacing 1 (ten) inI•1, we have the undermentioned UMP trial for hypotheses ( 3.2 )

I•

_

1 (ten) =

{

1

0

ten(N)& A ; gt ; _2 orten(N)

_degree Celsiuss1;

otherwise;

which is a UMP trial of size_on the status that

_=Phosphorus_0 (Ten(N)& A ; gt ; _2 ) +Phosphorus_0 (Ten(N)

_degree Celsiuss1 ) =Phosphorus_0 (Ten(N)

_degree Celsiuss1 ):

Note thatI•_

1 depends on_& A ; premier ;

1 merely by_0& A ; gt ; _1& A ; gt ; _& A ; premier ;

1, thereforeI•_

1 is the UMP trial for

proving hypotheses ( 3.2 ) .

2. Now, we show that trial ( 3.1 ) is besides the UMP trial for proving hypotheses

( 3.6 )

{

Hydrogen0:_=_0;

Hydrogen& A ; prime ; & A ; premier ;

1:_ & A ; gt ; _2& A ; gt ; _0;

For this intent, _rst consider the job of proving the simple hypotheses

( 3.7 )

{

Hydrogen0:_=_0;

Hydrogen& A ; prime ; & A ; premier ;

12:_=_& A ; premier ;

2 (_0__2& A ; lt ; _& A ; premier ;

2 ):

Harmonizing to Neyman-Pearson Lemma, each trial conforming with ( 3.4 ) is a Military policeman

trial of size_for hypotheses ( 3.7 ) on the status that ( see [ 6 ] )

_=Tocopherol_0 [I•2 (Ten) ] =Phosphorus_0 (R2 (Ten)& A ; gt ; k2 ) + 2Phosphorus_0 (R2 (Ten) =K2 );

where

R2 (ten) =

degree Fahrenheit_& A ; premier ;

2

(ten)

degree Fahrenheit_0 (ten)

=

(

a(_& A ; premier ;

2 )

a(_0 )

)NI(degree Celsius ; _& A ; premier ;

2 ) (ten(N) )

I(degree Celsius ; _0 ) (ten(N) )

:

On the other manus, we know that_02[_1; _2 ] , so we can take each value in this

interval for_0. Now, allow_0 =_2, therefore

R2 (ten) =

{ (

a(_

& A ; premier ;

2 )

a(_0 )

)N

ten(N)

__0;

1ten(N)& A ; gt ; _0:

So,I•2 consists of

I•2 (ten) =

{

1

2

ten(N)& A ; gt ; _0;

ten(N)

__0;

is a MP trial of size_for the related hypotheses. But,I•2 depends on_& A ; premier ;

2 merely by

_0__2& A ; lt ; _& A ; premier ;

2, thereforeI•2 is a UMP trial for proving hypotheses ( 3.6 ) . Now we de_ne

2 (ten) =

{

1ten(N)

_degree Celsiuss2;

0ten(N)& A ; gt ; degree Celsiuss2 (degree Celsiuss2& A ; lt ; _1& A ; lt ; _2 ):

By replacing 2 (ten) inI•2, we have the undermentioned UMP trial for hypotheses ( 3.6 )

I•

_

2 (ten) =

{

1

0

ten(N)& A ; gt ; _2 orten(N)

_degree Celsiuss2;

otherwise:

This trial depends on_& A ; premier ;

2 merely by_0__2& A ; lt ; _& A ; premier ;

2, therefore it is UMP trial for proving

hypotheses ( 3.6 ) .

As it can see from above treatments, the UMP trials for proving ( 3.2 ) and ( 3.6 )

are tantamount and both are given by ( 3.1 ) . So,I•_(ten) is the UMP trial for the

hypothesisHydrogen0:_1____2 versusHydrogen1:_ & A ; lt ; _1 or_ & A ; gt ; _2. _

6

3.2. Theorem.Suppose thatTen= (Ten1; Ten2; : : : ; XN)is a random sample, with

observed valueten= (ten1; ten2; : : : ; xN), from a distribution with denseness map ( 2.2 ) .

Then a UMP trial of size_for a hypothesisHydrogen0:_1____2versusHydrogen1:_ & A ; lt ;

_1 or_ & A ; gt ; _2is given by

& A ; phi ;

_

(ten) =

{

1

0

ten( 1 )& A ; lt ; _1 orten( 1 )

_m ;

otherwise;

wheremis chosen so that_=Tocopherol_1 [& A ; phi ;_(Ten) ] =Tocopherol_2 [& A ; phi ;_(Ten) ].

Proof.The cogent evidence is similar to the cogent evidence of Theorem 3.1. _

3.3. Remark.Note that, if in Theorems 3.1 and 3.2,_1 =_2, so, as an exceptional

instance, the rejection part of these trials would be tantamount to the rejection part

of Theorems 4.3 and 4.4 in [ 14 ] .

Mentions

[ 1 ] Aaberge, R. and Zhang, L.C.A category of exact UMP indifferent trials for conditional symmetricalness

in small-sample square eventuality tabular arraies, J. Appl. Statist.32, 333-339, 2005.

[ 2 ] Birkes, D.Generalized likeliness ratio trials and uniformly most powerful trials, Amer.

Statist.44( 2 ) , 163-166, 1990.

[ 3 ] Birnbaum, A.Admissible trial for the mean of a rectangular distribution, Ann. Math. Statist.

25, 157-161, 1954.

[ 4 ] Edelman, D.A note on uniformly most powerful reversible trials, Amer. Statist.44( 3 ) , 219-

220, 1990.

[ 5 ] Kabe, D.G. and Laurent, A.G.On some nuisance parametric quantity free uniformly most powerful

trials, Biom. J.23, 245-250, 1981.

[ 6 ] Kale, B.K.A First Course on Parametric Inference, ( Narosa Publishing House, New Delhi,

1999 ) .

[ 7 ] Lehmann, E.L. , Stein, C.Most powerful trials of composite hypotheses I. normal distribution,

Ann. Math. Statist.19( 4 ) , 495-516, 1948.

[ 8 ] Lehmann, E.L. and Romano, J.P.Testing Statistical Hypothesiss, ( Third Ed. , Springer, New

York, 2005 ) .

[ 9 ] Majerski, P. and Szkutnik, Z.Estimates to most powerful invariant trials for multi-

normalcy against some irregular options, Trial19, 113-130, 2010.

[ 10 ] McDermott, M.P. , Wang, Y.Construction of uniformly more powerful trials for hypotheses

about additive inequalities, J. Statist. Plann. Inference107, 207-217, 2002.

[ 11 ] Migliorati, S.Uniformly most powerful trials for reversible hypotheses, Metron,LX-N, 107-

114, 2002.

[ 12 ] Nomakuchi, K.A note on the uniformly most powerful trials in the presence of nuisance

parametric quantities, Ann. Inst. Statist. Math.44, 141-145, 1992.

[ 13 ] Pratt, J.W.Admissible nonreversible trials for the mean of a rectangular distribution, Ann.

Math. Statist.29, 1268-1271, 1958.

[ 14 ] Sayyareh, A. , Barmalzan, G. , and Haidari, A.Two sided uniformly most powerful trial for

Pitman household, Appl. Math. Sci.74( 5 ) , 3649-3660, 2011.

[ 15 ] Scherb, H.Determination of uniformly most powerful trials in distinct sample infinites,

Metrika53, 71-84, 2001.

[ 16 ] Takeuchi, K.A note on the trial for the location parametric quantity of an exponential distribution,

Ann. Math. Statist.40, 1838-1839, 1969.

[ 17 ] Wang, Y. , Young, L.J. and Johnson, D.E.A UMPU trial for comparing agencies of two negative

binomial distributions, Commun. Stat. Simulat.30, 1053-1075, 2001.

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