Question

Let X and Y have similar truncated conditional exponential distributions with densities: f(2Y = y) < ye for 0<x<a, and f(y|X = ) x zety for 0 <y<b, where a, b are constants. Write a function Gibbs (a,b) which takes (a, b) as input to generate n = 1000 samples of (X,Y).

Answer 1

Solution : - Let X and X have similax Huncated conditional exponential distributions with densities; le (24.0=X) = =Y - Exy (alrey) x y ex olaca and : DEPO-X,!=819Carina -frix (4) X=2) a 0 x45b -_Where 9b are constants. We will use Gibbs sampling to generate a sample tom (X,Y). To use Gibbs_sampling method - he need conditional distribution of ºx|=4 and Ylx=x. The Gibbs Sampling procedure is as follows; - Fitst initialise the sample as 10,40) - Then we generate next Sample point (44, 42) as follows: Generate a fandom sample of_size_1 from pdf fxit Cal Y=40) say ng > Generate a Zandom sample of size 1 from pdf fyly (41 X=29)

and hence we get (19,47 - We generate a sandom sample point (12392) asa > generate de from polf' fxt (QUYE44) md y from pdf frix (41 X= x2) 7 we get (2272) - Continue the procedure of generating di from fxx (xl Ya Viet) and" generating Yi from polf tylx (Y LX=X; antil we get the !, sample of desized size 'y? + Now we obtain the conditional pdfs of x ly=y and Y1X=2 as follows: We know that Exiy exlYz4) dx = ta GyI 8471 9 z 1 protein 4 Jo 6 6 -2 ) 22 txt (alY24) Oluca ay

Similarly we get bob svih vas mot doimo s & Glebe za it. mod zz. 1 ne priroda to viv? trix. cY1x=x) = 1 982, Oly<b a n-obe Now to generate a Zandom sample from conditional distaibution. xlszy we use_inverse transformation method as folbusz 1.) We know that, for a continuous distributimi _it's cof_Ex (2) follows standard uniform distribution V160,1). " The That is ExGN V(0,13 -

Similarly we get bob svih vas mot doimo s & Glebe za it. mod zz. 1 ne priroda to viv? trix. cY1x=x) = 1 982, Oly<b a n-obe Now to generate a Zandom sample from conditional distaibution. xlszy we use_inverse transformation method as folbusz 1.) We know that, for a continuous distributimi _it's cof_Ex (2) follows standard uniform distribution V160,1). " The That is ExGN V(0,13 -

Važiable _Exiy ()= cdf of conditional tandem XX - Ext (u) dy berhand annat ful domy e dy , en und 1-ě a yo t nili rinting HI . * Frer (9) 1-2% 1- 79%. 064<a Ev Toit for ut (0,1) Since Exly (2) À V (61) let us assume, Loir Exox (2) - U 1-e olyo - 4 z 1-8 16 = u [ 3-946] 1--4. [1-204] = % 7 dn (1-4 [1- 84 ) = -ty

2: 51 * Jn (14 4[-edyo)) -- ⓇO To draw the fandom Semple_from_xlyzYo te draw first Land.com Sample lu” from standard _unitorm. distzibution . Then put this value u in above equation ④ to get uz". Now since Y | X=80 has similar distribution as Xlr=% in some way we will get the following equation (10) ya 13 * Jo (9=u [1–pbloy) - 0B No Using this equation, - We first draw the fandom sample from U1(0,1) sau "U Then put this value u in the equation ) to get_sample from Y | X=X0 * To get cand am Sample tcom joint distzibution of ext) we use Gibbs sampling stated above.

R-Script Code for Gibss Sampling :

Trextester Language: R #R version 3.3.2 Gibbs=function(a,b){ #The function that returns the X value of conditional pdf of X given Y. XgivenY=function(y) { u=runif(1) K=1-exp(-a*y) L=(-1/y)*(log(1-u*K)) return (L) #The function that returns the Y value of conditional pdf of Y given X YgivenX=function() { u=runif(1) K=1-exp(-b*x) L=(-1/x)*(log(1-u*K)) return (L) n=1000 X=CO) Y=c0) X[1]=runif(1,0,a) Y[1]=runif(1,0,1) #initial value of X #initial value of Y for(i in 2:n){ X[1] =Xgiven Y(Y[i-1]) Y[i] = Ygivenx(x[i]) Data=data.frame(X,Y) return(Data) D=Gibbs(2,3) head(D) dim(D) Run it (Shift+Enter) Save it Put on a wall [ + ] Show

Output :

Y 1 1.9028030 0.03169758 2 1.0320945 0.18805332 3 1.7110112 0.51020021 4 1.1628591 0.25437465 5 0.3996714 0.10710690 6 0.6911116 0.01476928 [1] 1000 2