CHAPTER 14 Hypothesis testing and confidence regions The current framework of hypothesis testing is largely due to the work of Neyman and Pearson in the late 1920s, early 30s, complementing Fisher’s work on estimation As in estimation, we begin by postulating a statistical model but instead of seeking an estimator of 8@in @ we consider the question whether @€@, R (see Fig 11.4) In order to illustrate the concepts introduced so far let us consider the following example Let X be the random variable representing the marks achieved by students in an econometric theory paper and let the statistical model be: ()i O=< (ii) lớn f(X; = A= Tam ep 1/X -0\? —~|—— ( : ) | 0e© = =[0, 100]; : X=(X¡.X; ,X/,, n=40 is a random sample from ƒ(x; Ø) The hypothesis to be tested is against Hy: 0=60 (i.e X ~N(60,64)), H,: 0460 (ie X~N(u, 64), #60), Common ©, ={60} ©, =[0, 100] — {60} sense suggests that if some ‘good’ estimator of 0, say X,,= (1/n) $7 X;, for the sample realisation x takes a value ‘around’ 60 then we will be inclined to accept H, Let us formalise this argument: The acceptance region takes the form 60-—e 1.96; 0= 56) =0.8849, Pr(|t(X)| > 1.96; @= 58) = 0.3520, Pr(|r(X)| > 1.96; 0=60)=0.05, Pr(|t(X)| > 1.96; 6= 62) =0.3520, Pr(|t(X)| > 1.96; = 64) = 0.8849, Pr(|t(X)| > 1.96; 0=66)=0.9973 As we can see, the power of the test increases as we go further away from 0=60 (Hạ) and the power at 6=60 equals the probability of type I error This prompts us to define the power function as follows: Definition A0)=Pr(xeEC,), G€O is called the power function of the test defined by the rejection region C, Definition œz=maXạ,e, A(8) is defined to be the size (or the significance level) of the test In the case where Hy is simple, say 6= 6, then « = A(6,) These definitions enable us to define a criterion for ‘a best’ test of a given size « to be the one (if it exists) whose power function A(@), 8€@, is maximum at every Definition A test of Hạ: c©g against H,: €O, as defined by some rejection region C, is said to be uniformly most powerful (UM P) test of size x if (i) max A(8) =a; (ii) 2(0)>Z*\0)_ GEO, for all 0e©;; where Z2*() ¡is the power function oƒ any other test 0ƒ size a As we saw above, in order to be able to determine the power function we need to know the distribution of the test statistic t(X) (in terms of which C, is defined) under H, (i.e when Hg is false) The concept of a UMP test provides us with the criterion needed to choose between tests for the same H o Let us consider the question of optimality for the size 0.05 test derived 292 Hypothesis testing and confidence regions f (z) Cy ={x:1(X) = 1.645} (14.16) 1.645 f (z) C‡ * ={x+(X) 60 but A* (6) c,) = Pr( c‡ TH: where =1—, (14.39) under H) (14.40) _ £0) =O) 9, 1) In this case we can control f if we can increase the sample size since — B =Pr(t,(X) -u —Hm/2 AX) = > (X;— Ho)? y (x,-X,)? i=] ¡=1 At first sight it might seem an impossible task to determine the distribution of A(x) Note, however, that H t > (x; —Ho)? = ¥ i=t ¡=1 (x¡— X„)*+n(X„— , which implies that Vv Ax)= 14+ An Hol > (x¡— —n/2 X„? 2N -(1455) -H/2 " where W= /n[(Ý„— mạ)/s]~ tín— 1) under Ho, W~t(n—1;8) under H,, ja MH) uu, €O) Since A(x) isa monotone decreasing function of W the rejection region takes the form C,{x:|W|>c,}, 302 Hypothesis testing and confidence regions and z, c, and Z0) can be derived from the distribution of W Example In the context of the statistical model of example against and consider Hy: 07 =08 H,:0°?403, =í§ =‡¡0 5l O=RxR, =(, 05), HER} (aX nj n exp n > (x, Ơ,)? ~y7(n1) Me Sh] = The inequality A(x)