Question

3. The water quality of a river was investigated by taking 15 water samples and recording the counts of a particular bacteria for each sample. The data can be represented as {21,..., X15}, where z; is the number of bacteria in sample i, and the xi's are assumed to be independent and follow a Poi(21) distribution. (a) Write down the log-likelihood function for 01. Use this to find the maximum likeli- hood estimate of 0, in terms of the observed values 21,...,015, making sure to check that you have found a maximum likelihood estimate. (5 Marks] (b) Show that the maximum likelihood estimator of 01, 1 = ī, is a consistent estimator of 01, assuming E(1) = 01, as n +0. [3 Marks] (c) Using the information that iDi = 362 and k(x) = 1.61 find an approximate 95% confidence interval for 0, by the Wald method. Comment on what this confidence interval tells you. [4 Marks)

Answer 1

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Since Xia PCON - few e904 0,20 8,X=0,152. x = 0, 1, 2... then the likelihood function B, ((21.8 . Ris) be the sample7 L(x, 0) = 9:00 - +215 e 21+22 - B * I @ log Likelihood Tim., logwa log c = 150, + h - 4 xi logo, = log c - 150,+ logo, .. logor = x logo (loge=1 to find Mince of a differentiate loge with respect to o, and gloge zo o n a o-15*14 &x; # 1 = 0 equating it to Zerd 15 Wu - =15 So Sample mean is the MLE of o = 15%* - sxi= Exi=csă -1 22 lie fle =0 <0 e "com) 2010 01=ã, → Lokeb hood fumetton is ma eimized ab o x = iş logL 1B maximized at ►

MLE of a on = 7 in Oades to show that it is consistent, sufficiency conditions, E[īJEEEH EXT] (4) =+*+5 01 = 0, ECPD)= 0, Xiup con VCXL)- , by the moments air of Poisson o, as no mi) v CO )= V(4 [vcax) = a²v(x) = 19** 150 Xin PCO) Elin P(450) vleri)150, o here. n=15 as n 00 VC-500 from © ¢ ® ie ă Po, An I is I estimentos To of parameter o said to be consistent iff To bo ie, ł is consistent for o, o giver i = 362 X = 1 Exi = 362 = 24.133 then wald confidence intesval Jos mean is, ē u ham = 362 ł 2 Zala koa) 196 Itt 362 LS R(x) = 1.608 - (362, 196*1.61, 362 +1.96*1.6) = 1.61 (givens = (20.977, 27. 2889) - (20.98 , 27.29) il, we are about 95% Cestain that unknown parameter o, should some where In the interval (20.98, 21.29) Il