Question

Suppose n independent, identically distributed observations are drawn from an exponential ($\lambda$) distribution, with pdf given by f(x,$\lambda$)=$\lambda$​​​​​​​$e^{-\lambda x}$, 0 < x < $\infty$ .

The data are x1, x2, .. , xn

Construct a likelihood ratio hypothesis test of Ho : $\lambda =\lambda 0$ vs H1: $\lambda =\lambda 1$ (where $\lambda 0$ and $\lambda 1$ are known constants, with $\lambda 0 < \lambda 1$ ), where the critical value is taken to be a constant c

Answer 1

sim: To construct o likelihood batió test. 1 1 0 Giveo 112.1 = he-12 0<xco = 0 ; otherwisé...!! ! Hoi aslo vs H1: 1 = 11! Where 10 <N. critical value ? ce constant.

esolution : Here we find the most poioer fw test I the potef_exponential distribution is given by - Now, the likelihood funétion can be given asa! ILX, N = "7 fcxis. = F xe-xa 11 1x, 1) =(ne-a zai: A T Now, let the likelihood under Ho : A=%o be." HIX, 10) = Chene -to Exi. If the likelihood under He: 1 = go be !! 16(X, 12) = () ne-li zai. Now, by using Neyman-pearson lemma, most powerful test is L1(x) z kol:(.50474 101x) caune -1 EXi k a sih "Cone -10 Exim dish wit to it by heinen D-Xi (11-ro 10 > K. it will all Γλι η I no log I MI-EXilm-o) , logkut-taking loa Lon both sides. bee >10. 1lo Isoils. I 1. no log ( A segre) log / 11 so "отло / -

- Ex(1 -10) > 19k – nlog At) -Exi( 11-10) ? $ ; ' = loge - nlog tly EXi (11-no) <K'. Zxi ki (11-10) Idi <" IX i <C! where cl = k) =_ke! (11-10) ; lell = kl. ai-10) The most powerful critical region or best critical region - Expcc' or Xce where cach where cip the critical value.