Question

6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n = 5 is drawn from the distribution of Y and the observations are 0.3, 0.4, 0.5, 0.5, 0.7. Compute the estimates for 0 based on the moment and maximum likelihood estimators in (c) and (d), respectively. Which estimate is not an acceptable estimate? Explain briefly. [Total: 20 marks]

Answer 1

Grimen, 114)= otherurise eshere ANSWER a. R.P. F. Since ifly) is the 0-1 dy Oy 8-1 dy =D> =-> Therfore, b: AN SW ER ELY) = -0- dy

ElY) = dy dly 042 %3D %3D O+2 Vari amce (Y) = E(Y?) - E(V} O+12 (0+1)2 (e+1) (1041)- (0+1)?(0+2) B(BrL) (0+1) (0+2) 10+1) (0+2)

AN SW ER e: Fisst order Sample momnt, m,' = 1 E(Y) Let By method estinalor, 1 moment => => o (1- mi) = mi estimate 0) ANSWER dist joint The v i=1, 2,., n likelihood function, f lyil0) = 6n The EW.

eraluate, To find an estimate we equation kkelihood → The 2s the/ masimumg dikolihord (antimate A0. =-> likelihood estimate the maumum AN SWER 04, 05, 0.s, 0.7 "The 0.3, Sample, (0.3+04-0 s+05+0.3) = 0.43 fowblen, By frevions 0.9230 larrro) 0.48 1-ml 1-0.48

Also, n 0.3 + In 0.4 + In O.5 1.2942(appron ) - 3. 8632 Since, O moment estimate d 0) not acceptable,

ANSWER a:

k=1

ANSWER b:

E(Y)=$\dpi{100} \theta$/($\dpi{100} \theta$+1)

Variance of Y=$\dpi{100} \theta$/(($\dpi{100} \theta$+1)²($\dpi{100} \theta$+2))

ANSWER c:

$\dpi{100} \theta$ bar= m₁'/(1-m₁')

ANSWER d:

$\dpi{100} \theta$cap= -n/$\dpi{100} \tiny \sum$ln y_i

ANSWER e:

$\dpi{100} \theta$bar=0.9230(approx.)

$\dpi{100} \theta$cap=1.2942(approx.)

$\dpi{100} \theta$bar is not an acceptable estimate as $\dpi{100} \theta$ >1.