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統計學研究所

高 密 度 顯 著 性 檢 定 以 複 合 式 假 設 為 例

Highest Density Significance Test for Composite

Hypothesis

研 究 生:曾義家

指導教授:陳鄰安 教授

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高 密 度 顯 著 性 檢 定 以 複 合 式 假 設 為 例

Highest Density Significance Test for Composite

Hypothesis

研 究 生:曾義家 Student : Yi-Chia Tseng 指導教授:陳鄰安 Advisor : Dr. Lin-An Chen

國 立 交 通 大 學

統計學研究所

碩 士 論 文

A Thesis

Submitted to Institute of Statistics College of Science

Nation Chiao-Tung University in partial Fulfillment of the Requirements

for the degree of Master in

Statistics June 2006 Hsinchu, Taiwan

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高 密 度 顯 著 性 檢 定 以 複 合 式 假 設 為 例

學 生 : 曾 義 家 指 導 教 授 : 陳 鄰 安

國 立 交 通 大 學 統 計 學 研 究 所 碩 士 班

摘 要 延 伸 Chen(2005)的 想 法,我 們 提 出 針 對 複 合 式 虛 無 假 設 的 高 密 度 顯 著 性 檢 定。且 推 導 出 這 個 檢 定 的 存 在 性 以 及 它 的 最 佳 化 的 性 質,並 且 針 對 常 態 分 配 的 參 數,將 此 方 法 應 用 於 其 上。對 於 一 些 較 為 複 雜 的 高 密 度 顯 著 性 檢 定 問 題,我 們 利 用 近 似 性 的 高 密 度 顯 著 性 檢 定 加 以 解 決 。

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Highest Density Significance Test for Composite Hypothesis

student : Yi-Chia Tseng Advisors : Dr. Lin An Chen

Institute of Statistics National Chiao Tung University

ABSTRACT

Extending from the idea of Chen(2005), we proposed highest density significance (HDS) test for composite null hypothesis. Existence and optimality of this test are derived. Examples of HDS test for normal parameters are provided. For problems that HDS tests are complicated to derived, we propose the approximate HDS tests.

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誌 謝

時光飛似,轉眼間研究所求學階段即將結束,謝謝師長們的教導以及同學們 的陪伴,讓我經歷了兩年愉快且充實的碩士生涯。 首先,我要感謝我的指導老師 陳鄰安教授,謝謝老師在忙碌之餘還能撥冗 教導我,不厭其煩的指導我在研究時所遇到的問題,使我能如期完成論文。還要 感謝博士班 陳弘家學長,謝謝學長在我有困惑時為我解惑,使我釐清學長論文 中的觀念及問題。還要謝謝口試委員對我論文的指導與建議。當然還要感謝研究 室的同學給予我協助,以及閒暇之餘一起運動娛樂,感謝你們豐富了這兩年的生 活,留下美好的回憶。 最後,要感激父母及家人這麼多年來的栽培與支持,你們的關心與照顧,我 才能在毫無顧慮下順利完成學業,在此,將以本篇論文獻給曾經給我鼓勵協助的 家人、師長、朋友以及同學們,並致上我最誠摯的謝意。 義家 謹誌于 國立交通大學統計學研究所 中華民國九十五年六月

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Contents

中文提要 ……… i Abstract ……… ii 誌謝 ……… iii Content ……… 1 Chapter.1 Introduction……… 1

Chapter.2 Motivation for Highest Density Significance Test………… 3

Chapter.3 Existence and Optimality……… 6

Chapter.4 Testing Hypothesis for Normal Parameter……… 8

Chapter.5 Testing Hypothesis for Standard Deviation and for Both Mean and Standard Deviation……… 10

Chapter.6 Some Other Distributions……… 11

Chapter.7 Approximate Highest Density Significance Test……… 11

Appendix ……… 16

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Highest Density Signicance Test for Composite Hypothesis

Abstract

Extending from the idea of Chen (2005), we proposed highest density signi cance (HDS) test for composite null hypothesis. Existence and optimality of this test are derived. Examples of HDS test for normal parameters are provided. For problems that HDS tests are complicated to derive, we propose the approximate HDS tests.

Key words: Fisherian signi cance test Hypothesis testing signi cance test

1. Introduction

In the hypothesis testing, there are two important categories of hypothesis speci ca-tion, the sigini cance test and the Neyman-Pearson formulation. The Neyman-Pearson formulation considers a decision problem that we want to choose one from the null hy-pothesisH0 and an alternative hypothesisH1. On the other hand, the signi cance test

considers only one hypothesis, the null hypothesisH0. The signi cance test may occurs

that H0 is drawn from a scienti c guess and we are vague about the alternative, and

cannot easily parameterize them. Another case is that the model when H0 is true is

developed by a selection process on a subset and is to be checked with new data. Then the problem for signi cance test is more general than the Neyman-Pearson formulation in that when H0 is not true there are many possibilities for the true alternative.

Over 200 years of development of signi cance test, it has been used in many branches of applied sicences. Some earliest use of signi cance test include that, for examples, Armitage (1983) claims to have found the germ of the idea in a medical discussion from 1662 and Arbuthnot (1710) observed that the male births exceeded female births in Lonton for each of the past 82 years that violates the assumption of equal chance of male birth. Some important signi cance tests latter developed include the Karl Pearson's (1900) chi-squares test and W. S. Gosset's (1908) student paper that pro-posed the rst solution to the problem of small-sample tests. Signi cance tests were given their modern justi cation and then popularized by Fisher that he derived most of the test statistics that we now use, in a series of papers and books during 1920s and 1930s. Traditionally the sigini cance test is to examine how to decide whether or not a given set of data is consistent withH0 it is conducted through several key steps:

TypesetbyA M S-T E X 1

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1. stimulating a suitable null hypothesis H0, 2. choosing a test statistic T to rank

possible experimental outcomes, 3. determining the p-value which is the probability

of the set of values of T at least as extreme as the value observed when H0 is true.

4. H0 being accepted (rejected) if the p-value is large (small) enough. This classical

way that the user have to select a test statistic, although often been recommended a sucient statistic, to formulate the test is generally called the Fisherian signi cance test since R. A. Fisher made the contribution on signi cance test the most.

Yes, over 200 years development, however, the concept and theory of Fisherian signi cance test are now scarcely introduced in modern texts of statistical inferences. There are some reasons reecting this fact. (a) The Fisherian signi cance tests are questioned about how to use test statistics and which test statistic to use under which circumstances. For one speci c hypothesis, there may have several test statistics for use such as the Pearson's (1900) chi-square test and the normal approximation method. With this diculty, people may be bothered for making decision when it is happened

that the null hypothesis H0 is rejected by using one test statistic inducing strong

evidence against H0 and accepted by using another test statistic inducing no real

evidence against H0. (b) On the other hand, there is also the problem of choosing

the one sided Fisherian signi cance test or the two sided Fisherian signi cance test. In practice, it often depends only on practicer's convenience. It is quite common that if the distribution of the test statistic is available and is symmetric then a two sided Fisherian signi cance test is often implemented and a one sided Fisherian signi cance test is implemented when it has an asymmetric distribution. However, if approximation for the distribution of the test statistic such as the normal aprroximation and the Pearsone's chi-square test has been used then they usually implement a two sided one.

For example, for testing H0 : p = p0 with X obeying a binomial distribution, the

Fisherian signi cance test may be conducted by the two sided Pearson's chi-square

and normal approximation tests and by choosing X as a test statistic for a one sided

test (see the latter one in Garthwaite et al. (2002)). For one hypothesis problem, a

two sided Fisherian signi cance test may have p-value approximated twice as it for

an one sided one. Without a fair justi cation in deciding a one sided or two sided

test, the conclusion based on p-value may be misleading. (c) There is absence of

a direct indication of any departure from H0. This happens mainly because that

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of the parameter assumed in null hypothesis. (d) There is lack of desired optimal property as a justi cation to support the calssically used Fisherian signi cance tests. In the late 1920s, E. S. Pearson, son of Karl Pearson, approached Jerzy Neyman with a question that bother him. If you test whether data t a particular probability distribution and the test statistic is not large enough to reject the distribution, how do you know that this is the best that you could done? How do you know that some other test statistics might not have rejected that probability distribution? The resulting collaboration (see this in a series of papers collected in Neyman and Pearson (1967)) between them produced the Neyman-Pearson formulation that requires tests selecting one between the null hypothesisH0and a well speci ed alternative hypothesis

H1. With Neyman-Pearson formulation, a mathematical theory of hypothesis testing

in which tests are derived as solutions of clearly stated optimum problems has been developed. The optimal property does made the Neyman-Pearson theory popular in statistical inferences.

In this paper, we consider the hypothesis H0 :  = 0. By letting X = x0 be the

observed sample, our approach of HDS test sets all x's satisfying L(x0) L(x00)

as set of extreme points for computingp-value. This setting of determining the extreme points not only gets rid o the diculty of deciding a one sided or two sided test but also automatically determine the test statistic. The defects (a) and (b) ocuured for the Fisherian signi cance test are then solved. Traditionally the chosen test statistic used in Fisherian signi cance test involves information only related to the parameter considered inH0 which often is either a location or scale parameter. But, the likelihood

based HDS test classi es a point x if it is an extreme by measuring its corresponding

likelihood using all information involving the size of its likelihood. This often result in a test statistic that involve information with amount bigger than that contained in the traditionally used test statistic. Therefore, we may expect that likelihood function for constructing a signi cance test de nitely will gain some more advantages. First, we will show that this new signi cance test has a desired property of optimality which then does provide a justi cation for the use of this new test. The defect (d) for Fisherian signi cance test is also overcomed. Second, often using extra information

often provides indication of departure from H0. Furthermore, examples of HDS test

for both continuous distribution and discrete distribution will be provided associated with discussion of comparison with the Fisherian signi cance tests.

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2. Motivation for Highest Density Signicance Test

LetX1:::Xn be a random sample drawn from a distribution having a probability

density function (pdf)f(x) with parameter space . Consider the simple hypothesis

H0 :  =0 for some 0

2 . By letting vector X with X

0 = (X

1:::Xn) and sample

space , let's denote the join pdf ofX asL(x), also called it the likelihood function.

What is generally done in classical approach for signi cance test when X = x0 is

observed, particularly inuenced by R. A. Fisher and being called the Fisherian signi -cance test, is to determine the extreme set based on the distribution (or approximated distribution) of a test statistic. With a test statistic T = t(X), it then de ne the

p-value as

px0 =P0(T at least as extreme as the value observed): (2.1)

While this approach is applicable in certain practical problems, it is scarcely of suf- cient generality to warrant trying to nd necessary and sucient conditions for its applicability. There are several reasons for this point. First, as questioned from E. P. Perason that, for given one observation x0, we may reject H0 for having small p-value

that provides evidence against H0 with one test statistic but accept H0 for having

large p-value that provides no real evidence against H0 with another test statistic.

How can we decide to choose the test statistic? Second, the extreme set E in (2.1)

varies in choosing the one sided or two sided Fisherian signi cance test. However, in application, it often is decided based on practicer's convenience where we should know

that the two sided Fisherian signi cance test may have p-value as large as twice the

p-value of the one sided Fisherian signi cance test. This increases the diculty in

understanding the pvalue. Third, so far, there is no justi cation of desired optimal

property for this test-statistic based Fisherian signi cance test.

Suppose that we set out to order points in the sample space  according to the

amount of evidence they provide for H0 :  = 0. We should naturally order them

according to the value of the probability L(x0) any x with small L(x0) revealing

evidence against H0. Then, when X =x0 is observed and if we must choose subset of

possible observations which indicates thatH0 is true, then it seems sensible to put into

this subset thosex's for which the probabilityL(x0) is large - in other words to choose

a subset of the form fx : L(x

0)> L(x00)

g. On the other hand, the subset which

indicates thatH0is not true seems sensible to be of the form

fx:L(x

0) L(x00) g.

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With expectation that each extreme point has to be at least as extreme as the observed value x0, this leads the following de nition for de ning a new type of signi cance test.

Consider the null hypothesisH0 : =0. The HDS test proposed by Chen de nes

the p-value as phd = Z L(x0)L(x 0 0) L(x0)dx:

The method of highest density for signi cance test is obviously appealing for the follwoing facts:

(1) Fisherian signi cance test chooses a test statistic T =t(X) that gives an ordering of the sample points as evidence against H0: t(x1)> t(x2) means that x1 is stronger

than x2 as evidence againstH0. This evidence may varies in the chosen test statistic.

On the other hand, the ordering based on HDS test means thatx1 is stronger thanx2

as evidence against H0 when it occurs L(x10)< L(x20).

(2) The inequality L(x0) L(x00) automatically decide two desirabilities: the

test statistic involving in the test and setting if it is a one sided or two sided test. This solves the defects (a) and (b) occurred in Fisherian signi cance test.

(3) For this test, we set the non-extreme set including the points x's in sample space the highest part of the joint density function L(x0) such a set includes relatively

more probable points whenH0 is true. On the other hand, this extreme setEhd =

fx :

L(x0)< L(x00)

gof the HDS test is weirder than its corresponding non-extreme set

~

Ehd = fx:L(x 0)

L(x 00))

g in the sense that L(x

10)< L(x20) for x1

2 Ehd

and x2

2 E~hd. None of the traditional Fisherian signi cance tests has this appealing

in determining the extreme set for computing p-value of given observation x.

Denition 2.1.

Consider the null hypothesisH0 : 2

0. The HDS test de nes the

p-value as phd = sup2 0 Z L(x)L(x 0 ) L(x)dx: (2.2)

We classify the hypothesis testing problems whose level  MP, UMP, or UMPU

tests are proposed in the literature into the following three categories: (A)H0 : 0 versus H1 :  > 0 and H0 :



0 versus H1 : < 0

(B) H0 :  1 or 

 

2 versus H1 : 1 <  < 2, where both 1 and 2 are known

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(C) H0 : 1  2 versus H1 :  < 1 or  > 2, where both 1 and 2 are known

real-valued constants with 1 2

(1) From any family of distributions with the monotone likelihood ratio property,

there exists a level UMP test for any hypothesis testing problem belonging to

Cat-egory A. See, e.g., Lehmann (1986, Theorem 2 in Chapter 3).

(2) From any one-parameter exponential family, there exists a levelUMP test for

any hypothesis testing problem belonging to Category B. See, e.g., Lehmann (1986,

Theorem 6 in Chapter 3).

(3) From any one-parameter exponential family, there exists a level  UMPU test

for any hypothesis testing problem belonging to CategoryC. See, e.g., Lehmann (1986,

Section 4.2).

Theorem 2.2.

Let X = (X1:::Xn) be a random sample from f(x) with an

observation X = x. Consider the hypothesis H0 :  = 

2 

0 and we assume that

the family of densities ff(x) :  2 g has a monotone likelihood in the statistic

T =t(X):

(a) If the monotone likelihood is nondecreasing in a functiont(x), then the test with

p-value

phd= sup2

0P(t(X) t(x 0))

is a HDS test.

(b) If the monotone likelihood is nonincreasing in t(x), then the test with p-value

phd= sup2

0P(t(X) t(x

0))

is a HDS test.

3. Existence and Optimality

Theorem 3.1.

LetX1:::Xnbe a random sample drawn from a distribution with pdf

f. Suppose that there exists a partition,A0A1:::Ak, of the sample space of random

variable X such that P0(X 2 A

0) = 0 and f is continuous on each Ai. Further,

suppose there exist functions f1:::fk, de ned on A1:::Ak, respectively, satisfying

(i) f(x) =fi(x) for x2Ai,

(ii) fi is monotone onAi,

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(iv)f;1

i has a continuous derivative on B, for eachi= 1:::k.

Then, HDS test exists.

Proof. Conditions (i)-(iv) are set for variable transformation of continuous random

variable which provides a continuous pdf of new variable f(Xi0) (see, for

exam-ple Cassella and Berger ( p53)). As the fact that L(X1:::Xn0) is a product of

f(Xi0)i = 1:::n, then P(L(X1:::Xn0) a) is a continuous and monotone

increasing function of a. Then, from the intermediate theorem,

P(L(X) L(x0)) (3.1)

exists for  2 

0. The result is followed from the fact that the set of values of (3.1)

with  2

0 is bounded which indicating the existence of in mum.

There are three remarks for this theorem:

(a) Most continuous type distributions appeared in literature ful ll the conditions

(i)-(iv) in the theorem and then the HDS tests exist for any level . Example that this

theorem does not hold includes the uniform distribution.

(b) (i)-(iv) provide only a sucien conditions and this set is de nitely not necessary

conditions and then the HDS tests for any level  exist in a wider famimly of

distri-butions.

(c) We can ignore the exceptional set A0 since P

0(X 2 A

0) = 0. It is a technical

device that is used to handle endpoints of intervals. Example for this case includes the double exponential distribution.

Most distributions of continuous type we have seen in literature are having pdf's of monotone cases or unimodels and they full ll the conditions (i)-(iv) in Theorem 3.1

such that the HDS tests exist where the unimodels have ranges of the form (0xmod)

where xmod is the mode of the distribution.

From Theorem 3.1, the level  HDS test has acceptance region f(x

1:::xn) : L(x1:::xn0)  ag where a satis es 1; = P 0(L(X 1:::Xn0)  a)). The

key of deriving the HDS test based on this theorem is that we need to know the dis-tribution of the random form of the likelihood function under the assumption that

H0 : =0 is true.

Theorem 3.2.

Consider the hypothesis H0 :  2 

0 where  may be a vector

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test with set of non-extreme points B(), then for any  2 0 such that Z L(x)L(x 0 ) L(x)dx= Z B() L(x)dx (3.2) then we have volume(fx:L(x)L(x 0) g) volume(B()): (3.3)

Proof. Suppose taht (3.2) holds. Deleting the subset common to fx : L(x) 

L(x0) gand B() yields Z fx:L(x)L(x 0 )g\B()c L(x)dx= Z B()\fx:L(x)L(x 0 )gc L(x)dx: Now, forxa 2fx:L(x)L(x 0) g\B()c andxb 2B()\fx:L(x)L(x 0) g, we have L(xa)> L(xb). Thus, volume(fx:L(x)L(x 0) and x 2B()cg) volume(fx:L(x)< L(x 0) and x 2B()g): (3.4)

So, adding the volume offx :L(x)L(x 0)

g\B() to both sides of (3.4), we have

the theorem.

4. Testing Hypothesis for Normal Parameters

Suppose that we are willing to accept as a fact that the outcome of the variable of

a random experiment has a normal distribution with mean  and variance 2. Now,

let X1:::Xn be a random sample drawn from normal distributionN(

2). We rst

consider the hypothesis about mean  with assuming that is known.

Case 1:

2 =

0 is known constant. Consider the null hypothesis H0 : 

2. The likelihood function is L(x) = (2 2 0) ;n=2e ; P ni=1 (xi;) 2 2 2 0 :

This is a monotone decreasing function inPn

i=1(xi ;)

2. With the fact thatL(X)

L(x0) if and only if Pn i=1(Xi ;) 2  Pn i=1(xi ;) 2 and Pn i=1(xi ;) 2 = Pn i=1(xi ;  x)2+n(x ;)

2, the p-value for H 0 is phd = supP( 2(n)  1 2 0 n X i=1 (xi;x) 2+n(x ;) 2]):

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This p-value varies in observation x0 = (x1:::xn) and assumption on . We list the

corresponding p-values associated with several hypotheses in Tables 1 and 2.

Table 1.

p-values for some hypotheses H0 about normal mean

Hypothesis p-value phd H0 : 0 8 < : P( 2(n)  P ni=1 (xi; x) 2 2 0 ) if x 0 P( 2(n)  P ni=1 (xi; 0 ) 2 2 0 ) if x > 0 H0 :  0 8 < : P( 2(n)  P ni=1 (xi; 0 ) 2 2 0 ) if x < 0 P( 2(n)  P ni=1 (xi; x) 2 2 0 ) if x  0 H0 :0  1 8 > > < > > : P( 2(n)  P ni=1 (xi; 0 ) 2 2 0 ) if x < 0 P( 2(n)  P ni=1 (xi; x) 2 2 0 if 0  1 P( 2(n)  P ni=1 (xi; 1 ) 2 2 0 ) if x > 1

Example 1.

In a semiconductor manufacturering process CVD metal thickness was measured on 30 wafers obtained over approximately 2 weeks. A data of this experiment

has been given in Montgomery, Runger and Hubele (2004) where = 1 is assumed to

be known and we have x= 15:99 and Pn

i=1(xi ;x)

2 = 29:107. We have the following

tests:

(a) If we test H0 : 15  17, thep-value is

phd = sup(1319)P( 2(30)  Pn i=1(xi ;x) 2 2 ) = 0:5119:

(b) If we test H0 : 17  19, the p-value is

phd= sup(1719)P( 2(30)  Pn i=1(xi ;x) 2 2 + 30( x;17) 2 2 ) = 0:00099:

(c) If we test H0 : 13  15, thep-value is

phd= sup(1315)P( 2(30)  Pn i=1(xi ;x) 2 2 + 30( x;15) 2 2 ) = 0:00139:

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Hypothesis p-value phd H0 : 2( 01) ( 23) P( 2(n)  P ni=1 (xi; x) 2 2 0 + n( x;0) 2 2 0 ) if x 0 P( 2(n)  P ni=1 (xi; x) 2 2 0 ) if 0 x  1 or 2 x  3 P( 2(n)  P ni=1 (xi; x) 2 2 0 + n( x; 1 ) 2 2 0 ) if 1 x  2 with jx; 1 j jx; 2 j P( 2(n)  P ni=1 (xi; x) 2 2 0 + n( x; 2 ) 2 2 0 ) if 1 x  2 with jx; 1 j>jx; 2 j P( 2(n)  P ni=1 (xi; x) 2 2 0 + n( x; 3 ) 2 2 0 ) if x 3

Table 3.

p-values for some hypotheses H0 about normal mean

Hypothesis p-value phd H0 : 0 or   1 P( 2(n)  P ni=1 (xi; x) 2 2 0 + n( x; 0 ) 2 2 0 ) if x 0 P( 2(n)  P ni=1 (xi; x) 2 2 0 + n( x;0) 2 2 0 ) if 0 x  1 with jx; 0 j jx; 1 j P( 2(n)  P ni=1 (xi; x) 2 2 0 + n( x; 1 ) 2 2 0 ) if 0 x  1 with jx; 0 jjx; 1 j P( 2(n)  P ni=1 (xi; x) 2 2 0 + n( x; 1 ) 2 2 0 ) if x  1

5. Testing Hypothesis For Standard Deviation and For Both Mean and

Standard Deviation

Case 2:

=0 is known.

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Hypothesis p-value phd H0 : 2 2 0 P( 2(n)  P ni=1 (xi;0) 2 2 0 ) H0 : 2 0 2 2 1 P( 2(n)  P ni=1 (xi; 0 ) 2 2 1 )

Table 5.

p-values for some hypotheses H0 about normal mean and variance

Hypothesis p-value phd H0 : 0 2 0 2 2 1 P( 2(n)  P ni=1 (xi;x ) 2 2 1 ) if x < 0 P( 2(n)  P ni=1 (xi;0) 2 2 1 ) if x 0 H0 :0  1 2 0 2 2 1 P( 2(n)  P ni=1 (xi;0) 2 2 1 ) if x < 0 P( 2(n)  P ni=1 (xi; x) 2 2 1 ) if 0 x  1 P( 2(n)  P ni=1 (xi; 1 ) 2 2 1 ) if x > 1

6. Some Other Distributions

Let's consider the testing hypothesis about the parameter in the following exponetial distribution

f(x) =e;xx >0:

Table 5.

p-values for some hypotheses H0 about exponential parameter

Hypothesis p-value phd H0 : 0 P(Gamma(n1)  n x 0) H0 : 2( 01) ( 23) P(Gamma(n1)  n x 3)

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In statistical inferences for some distributions, approximate techniques are often desirable. Sometimes there are due to that direct evaluation of exact statistical in-ference is overwhemingly dicult and sometimes the approximations are cheaper and

quiker. For example, consider that we have a random sample X1:::Xn drawn from

the Gamma distribution with pdf

f(x ) = 1;() x;1e

;xx >0

where  > 0 are parameters. We further assume that  is known constant and we

want to test the composite null hypothesis H0 :

2 . With this case the p-value

for the HDS test is

phd = sup2P f(ni =1Xi) ;1e ; P ni=1Xi  (ni =1xi) ;1e ; P ni=1xi  g: (6.1)

The p-value of (6.1) may be derived only if we have an explicit distribution of the

statistic ni=1Xi)

;1e ;

P

ni=1Xi

 under the distribution P. Howewver, this is

compli-cated to derive it. Then, an approximation technique for this topic of highest density signi cance tests is needed.

Thep-value of (2.2) for a highest density signi cance test may be reformulated as

phd= sup2 0P fni =1f(Xi) ni=1f(xi) g = sup2 0P f n X i=1 lnf(Xi) Xn i=1 lnf(xi)g: (6.2)

The asymptotic theory may be applied on the statistic Pn

i=1lnf(Xi) of (6.2) when

 is true for this test due the fact thatlnf(Xi)i= 1:::nare independent and

iden-tically distributed with further assumptions that its meanE ln f(X)] and variance

V ar ln f(X)] exist. With the central limit theorem, the p-value of an approximate

highest density signi cance test is

phd:app= sup2 0#( n;1 Pn i=1ln f(xi) ;E ln f(X)] p n;1V ar  ln f(X)] ) (6.3)

where # is the distribution function of the standard normal distribution.

It is interesting to see if the approximate highest density signi cance test is appro-priate to use when the exact one is not available. We consider the approappro-priateness

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based on the eciencies of the approximation technique. Let's consider a simulation when the underlying distribution is normal as an example. Suppose that now we have

a random sample X1:::Xn from normal distribution N(

2) and we consider

hy-pothesis H0 : ( ) 2

0. To formulate the test in (6.3), the logarithm of normal pdf

is ln f(x ) = ; 1 2ln(2 2) ; (x;) 2 2 2

which implies that E ln f(X)] =;

1 2ln(2 2) ; 1 2 and V ar ln f(X)] = 1 2. With

some arrangements, we have that the approximatep-value under the normal

distribu-tion is pN( 2 ) hd:app = sup()2 0#((2n) ;1=2 n X i=1 1; (xi;) 2 2 ]): (6.4)

For studying the eciencies of the approximatep-value of formula (6.4), we further

assume that is known to be value 1. Then the approximate p-value is reduced to

pN( 2 ) hd:app = sup2mu#((2n) ;1=2 n X i=1 1; (xi;) 2 2 0 ]): (6.5)

We generate random sample of size n from normal distribution N(1) and, from

this sample, we compute p-values from Table 1 and (6.5). This simulation is done

with replication 100000 and we compute the average p-values of the exact one and

approximate one. The following table display these results.

Table 6.

p-values for exact and approximate highest density signi cance tests when

H0 is not true

Sample size Exact test Appro. test Exact test Appro. test

H0 : 2(;2;1) H 0 : 2(;11) n= 10 2:47E;09 1:35E;17 0:1216 0:1243 n= 20 2:62E;18 7:39E;50 0:0492 0:0467 n= 30 6:60E;28 3:48E;92 0:0204 0:0179 n= 50 1:44E;42 1:70E;149 0:0040 0:0031 n= 100 3:29E;112 0:0000 0:0000 0:0000 H0 : 2(34) H 0 : 2(56) n= 10 0:1202 0:1227 4:76E;09 2:52E;15 n= 20 0:0479 0:0453 1:29E;18 6:60E;53 n= 30 0:0208 0:0182 5:03E;30 4:33E;105 n= 50 0:0044 0:0034 3:23E;51 2:29E;206 n= 100 0:0001 0:0000 2:71E;110 0:0000

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Table 7.

p-values for exact and approximate highest density signi cance tests when

H0 is true

Sample size Exact test Appro. test Exact test Appro. test

H0 : 2(13) H 0 : 2(04) n= 10 0:5688 0:5864 0:5666 0:5833 n= 20 0:5473 0:5606 0:5477 0:5609 n= 30 0:5347 0:5460 0:5381 0:5492 n= 50 0:5305 0:5398 0:5246 0:5337 n= 100 0:5157 0:5224 0:5214 0:5280

In the application of approximate highest density signi cance test, we rst consider

the approximatep-value of (6.1) with random sample from Gamma distribution.

Theorem 6.1.

Thep-value of an approximate highest density signi cance test for the Gamma distribution is

pGammahdapp = sup2#((

;1) n ;1 Pn i=1lnxi ;(ln( ) +PG 0])];(x= ;) p n;1V ar ln f(X )] ) where PGnz] is the nth derivative of the digamma function (z) = ;

0 (z) ;(z) and V arln f(X )] = 3(ln )2+ 6ln PG 0] + 3(PG 0])2+PG 1]] ;2(PG 0]) 2 ;2(ln ) 2 ;PG 1];2(;1)PG 0+ 1] ;2(;1)(ln ;1)PG 0] +(+ 1)  ;1 ;(ln ) 2+ 22PG 0]:

Let's consider a case for simulation. Suppose that we have known  = 2. In this

situation, we see thatPG 02] = 1;PG 12] = 2 6 ;1PG 03] = 3 2 ; indicating that E ln f(X )] =;1; ;ln and V arln f(X )] = 2 6

;1. Then, thep-value

of the approximate highest density signi cance test is

pGammahdapp = sup

2#( n;1 Pn i=1lnxi ;x=  ;ln + 1 + p (2=6 ;1)=n ): (6.6)

For comparison, we also consider the following two Fisherian signi cance tests with

p-values pclaA= sup2P fGamma(2n1) Pn i=1xi g

andclaB = sup2P

fGamma(2n1)

Pn

i=1xi

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The following table displays the simulation results.

Table 7.

p-values for exact and approximate highest density signi cance tests for

H0 : 1:5 2:5

Sample size pGammahdapp pclaA pclaB

True = 2 n= 10 0:7255 0:7547 0:8297 n= 20 0:7912 0:8362 0:9065 n= 30 0:8372 0:8870 0:9406 n= 50 0:8973 0:9451 0:9784 n= 100 0:9611 0:9873 0:9976 True = 3 n= 10 0:3218 0:2906 0:9833 n= 20 0:2338 0:2080 0:9989 n= 30 0:1846 0:1573 0:9999 n= 50 0:1188 0:0989 1 n= 100 0:0472 0:0335 1 True = 5 n= 10 0:0160 0:0151 0:9999 n= 20 0:0017 0:0012 1 n= 30 8:260e;05 4:657e;05 1 n= 50 4:231e;08 4:199e;08 1 n= 100 3:248e;24 2:201e;14 1

In the next, we consider the Weibull distribution which has been very useful in

monitoring the lifetime data. Consider that we have a random sample X1:::Xnfrom

the one parameter Weibull distribution with pdf of the form

f(x ) = x;1e;xx >0: (6.4)

Theorem 6.2.

Thep-value of an approximate highest density signi cance test for the Weibull distribution of (6.4) is

pWeibullhd:app = sup2#( p nf( ;1) n ;1 Pn i=1ln xi+ (= )] ; n ;1 Pn i=1xi ;1]g (( ;1)) 2=(6 2) ;2( ;1)= + 1 ): where  =R 1 0 ln te

;tdt is the Euler's constant approximately equaled 0:57722.

Table 8.

p-values for approximate highest density signi cance tests for Weibull dis-tribution when true is 2

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Sample size H0 : 2(2:53) H 0 : 2(1:52:5) n= 10 0:7180 0:9673 n= 20 0:8156 0:4696 n= 30 1:4339E;07 0:4596 n= 50 0:0001 0:3519 n= 100 8:1764E;07 0:7988

In the next example we consider the extreme value distribution with pdf

f(x ) = 1 ex;

 e;e

x;

 x

2R (6.5)

where parameters  2 R and > 0. We assume that we have a random sample

X1:::Xn drawn from this distribution. We have p-value for an approximate highest

density signi cance test.

Theorem 6.3.

The p-value for an approximate highest density signi cance test for the extreme value distribution of (6.5) is

pExtremehd:app = sup()2#((

x;)= ;n ;1 Pn i=1e (xi;)= + 1:57722 p n;1(2=6 ;1) )

7. Appendix

Proof of Theorem 6.1. With density f(x ), we have

ln f(x ) = (;1)ln x; x ;ln ;ln ;() (ln f(x ))2 = ( ;1) ln x] 2 ;2(;1) xln x ;2(;1)(ln +ln;())ln x + x2 + 2(ln + ln ;())x+ (ln )2+ 2ln;()ln +ln;2():

It is easy to check that E X] =  and E ln X] = ln +PG 0] which indicates

that

E ln f(X )] = (;1)(ln +PG 0]);(1 +ln );ln;(): (6.6)

Furthermore, we also have

E X2] = ;(+ 2)

;() 

E (ln X)2] = (ln +PG 0])2+PG 1]

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With the above results, we have E (ln f(X ))2] = ( ;1)f(ln +PG 0]) 2+PG 1] g (6.7) ;2(;1);( + 1) ;() (ln+ +PG 0+ 1]);2(;1)(ln +ln ;()) (ln +PG 0]) + ;(;(+ 2)) ;1 + 2(ln +ln ;()) + (ln )2+ln ;()2:

Then V arln f(X )] is induced from (6.6) and (6.7) and then the theorem is

fol-lowed.

Proof of Theorem 6.2. With pdf f(x ) of (6.4), its logarithm is ln f(x ) = ln + ( ;1)ln x;x. For obtaining the mean and variance of ln f(X ), we may derive

the followings E ln(X)] =;   E ln(X)X] = 1;  E ln(X)2] = 6 2+2 6 2  E X ] = 1 E X2] = 2:

From the above results, we then have

E ln f(X )] =ln ; ( ;1) ;1 V arln f(X )] =;2 ;1 + ( ;1 )2 2 6 + 1: Imposing these results in (6.1), we then have the theorem.

Proof of Theorem 6.3. The logarithm of the pdf isln f(x ) = x;

 ;ln ;e x;  . we also have, E ln f(X )] =E X; ];ln ;E e X;  ] E (ln f(X ))2] =E  X;  2 ];2ln E X; ];E 2( X; )eX ;  ] + 2ln EeX;  ] +E e2(X ;  )] + (ln ) 2 whereE X;  ] =;0:57722E  X;  2 ] = 2 6 +(0:57722) 2E eX ;  ] = 1E e2(X ;  )] = 2E 2(X;  )eX;  ] = 0:84557:Henceful, we have E ln f(X )] = ;0:57722;ln V arlnf(X )] = 2 6 ;1:

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The theorem is followed with implementing the above results in (6.3).

References

Arbuthnott, J. (1710). An argument for Divine Providence, taken from the

con-stant regularity observ'd in the birth of both sexes. Philos. Trans. 27, 186-190.

Reprinted in Kendall, M. G. and Plackett, R. L.,Studies in the History of Statis-tics and Probability, Vol II. London: Charles Grin, 1977, 30-34.

Armitage, P. (1983). Trials and errors: the emergence of clinical statistics. Journal of the Royal Statistical Society A. 146, 321-334.

Christensen, R. (2005). Testing Fisher, Neyman, Pearson, and Bayes. The American

Statistician. 59, 121-126.

Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. London A222, 309-368.

Fisher, R. A. (1925). Statistical Methods for Research Workers. Edinburg: Oliver and

Boyd.

Garthwaite, P. H., Jollie, I. T. and Jones, B. (2002). Statistical Inference. Oxford

University Press: Oxford.

Gossett, W. (1908). The probable error of the mean. Biometrika 6, 1-25.

Lehmann, E. L. (1986). Testing Statistical Hypotheses, 2nd ed. John Wiley and Sons:

New York.

Mood, A. M., Graybill, F. A. and Boes, D. C. (1974). Introduction to the Theory of

Statistics. McGraw-Hill, Inc.

Neyman, J. and Pearson, E. S. (1967). Joint Statistical Papers. Cambridge University

Press.

Pearson, K. (1900). On the criterion that a given system of deviations from the proba-ble in the case of a correlated system of variaproba-bles is such that it can be reasonably

be supposed to have arisen from random sampling. Philosophical Magazine 5,

157-175.

數據

Table 1. p -values for some hypotheses H 0 about normal mean
Table 3. p -values for some hypotheses H 0 about normal mean
Table 5. p -values for some hypotheses H 0 about normal mean and variance
Table 6. p -values for exact and approximate highest density signi	cance tests when H 0 is not true
+3

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