CC BY
15
Kuldeep S. Chauhan
ESTIMATION AND TESTING PROCEDURES OF P(Y < X) FOR RT&A, No 3 (69)
THE INVERSE DISTRIBUTIONS Volume 17, September 2022
ESTIMATION AND TESTING PROCEDURES OF P(Y < X) FOR THE INVERSE DISTRIBUTIONS FAMILY UNDER TYPE-II CENSORING
Kuldeep Singh Chauhan •
Ram Lal Anand College, University of Delhi, New Delhi-110021 Kuldeepsinghchauhan.stat@rla.du.ac.in
Abstract
We recommended an inverse distributions family. The challenge of estimating R(t) and P in type-II censoring was measured to produce Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Maximum Likelihood Estimator (MLE). The estimators have been created for R(t) and P.Testing approaches for R(t) and P under type-II censoring have been constructed for hypotheses associated with various parametric functions. The author provides an alternate method for generating these estimators.A comparative assessment of two estimating techniques has been conducted. The simulation technique has been used to assess the performance of estimators.
Keywords: Inverse distributions family; testing procedur es; bootstrap sampling
1. Introduction
The reliability function describes the probability of a failure-free procedure until time t. The estimation of the stress-str ength [P(Y < X)] parameter , aimed at displaying system efficienc is one of the most important challenges in statistical inference, which can be applied to a wide variety of fields such as longe vity mechanical system dependability , statistics, and bio-statistics. In reliability, the P = P(Y < X) parameter , which define the lifetime for a specifi system, places the strength X against the stress Y. Several scholars have measur ed the problems of estimation of reliability functions under censoring.Lin et al. [13] illustrated the inverse gamma model's role in lifetime distribution.The inverse Weibull distribution produced a good fit discussed by Erto [11]. The inference reliability and P(Y < X) of a scaled Burr distribution were calculated by Surles and Padgett [16].Yadav et al. [18] estimated the P (Y< X) of the inverse Weibull distribution With a progressiv e type-II censoring technique.Chatur vedi and Kumari [7] conducted a reliable Bayesian study of the generalized inverted family of distributions. Enis and Geisser [10] acquire an estimate of the likelihood that Y< X. Weerahandi and Johnson [17] investigated testing reliability in P(Y < X) while X and Y are repeatedly distributed. Estimators of P(Y < X) in the gamma model are explor ed by Constatine et al. [9].A comparativ e study for Burr distribution is presented in R(t) and P estimated by Awad and Gharraf [1].Nigm and Amboeleneen [14] use progressive censoring to evaluate the parameters of the Inverse Weibull distribution. Chaudhar y and Chauhan [6] perfor med estimation and test appr oaches for P(Y < X) of the Weibull distribution with type-1 and type-II censoring.Chatur vedi and Kumari [3] developed estimate and analysis processes for the reliability of a broad range of distributions.
We consider a family of inverse distributions, which is reflecte in this paper .The UMVUES and MLES of R(t) and P are calculated using type-II censoring. A new approach for estimating the UMVUES and MLES of R(t) was invented, in which the expression of R(t) and P is not required. Initially, the estimators for R(t) are generated using this method.The R(t) derivative estimators are used to construct the p.d.f. at a certain point, and then determine P estimators.W e calculated P by considering instances in which X and Y are similar distributions but have dissimilar values. We
Kuldeep S. Chauhan
ESTIMATION AND TESTING PROCEDURES OF P(Y < X) FOR RT&A, No 3 (69)
THE INVERSE DISTRIBUTIONS Volume 17, September 2022
now have extended the findin to any distribution from the projected inverse distributions family where X and Y are members. The testing procedures are also being planned.A performance comparison performance of two estimating approaches was conducted. The simulation technique was used to examine the perfor mance of estimators.
2. Inverse Distributions Family
Suppose a random variable (r.v.)Y having pdf
f (y Y,p, i) = Y^ ^if ^ exp (~YG (y-1; i)) (1)
y > 0, Y > 0, p > 0
Where, G (y-1; i), is depend on i and a function of y. Furthermore, G (y-1; i) real-valued, rigorously reducing function of y with G(to; p) = to and G' (y-1; i) stances for the derivative of G (y-1; i) by y-1.Let p and i are known and y is unknown during this whole section. The (1) demonstrates that the inverse distributions family can be transfor med into the inverse distributions listed below as special cases:
1. If G(y; i) = yp, p > 0, p > 0, we obtained the inverse generalized gamma distribution.
2. If G(y; i) = y2, p = k + 1, (k = 0), we achieved the inverse Rayleigh distribution.
3. If G(y; i) = log 1 +
Vb ) , b > 0, V — 1, P > 1, we achie ved the inverse Burr distribUtion.
4. If G(y; i) = log (1 + l^) , b = 1, V > 1, p > 1,we obtained the inverse Lomax distribution.
5. If G(y; i) = log (y) and p = 1, we achieved the inverse Pareto distribution.
6. If G(y; i) = yr exp (fly), r > 0, a > 0, p = 1, we obtained the inverse modifie Weibull distribution.
2
7. If G(y; i) = iy + v2~, a = p = 1,we obtained the inverse linear exponential distribution.
8. If G(y; i) = log y, we achieved the inverse log-gamma distribution.
3. UMVUES of y and Reliability Functions
We investigate estimation with censored type-II data. Suppose Y(1) < Y(2) <................Y(n).
Assume n objects are subjected to a test, when the firs r observations are noted, the test is stopped. Supposing, 0 < r < n, be the lifespans of leading r values. Noticeably , (n - r) objects stay alive awaiting Y(r).
Lemma 1. Suppose Sr = ^r=1 G (y-)1, i) + (n - r)G ^y^Tj, ij. Then, for the inverse distributions family, Sr is complete and sufficien indicated as (1). Additionally , the pdf of Sr is
Yrp srp-1
k (sr; i) = r(rrp) exp(-Ysr) (2)
Proof. From (1), the joint pdf is
f * (% i = 1,2,... n; Y, p, i) = n! JIG' (y--)1; i) exJ -y ¿G (y--)1; i)\ (3)
i=1 ^ i=1 )
When we integrate y(r+1), V(r+2),........., V(n) throughout the region y(r) < y(r+1) < ... < y(n),
we get the likelihood as
r
h(i(iy(i =1,2.......r);7, v) =n(n -1).............(n - r +1)j n G' (y-)1, v) exp (-Ysr)
i=1
(4)
Sr is sufficien by Fisher-Neyman factorization theorem [15]. In (1), put A = G (y-1; v), the pdf is
dP-1
k(a; j, p, 9) = rp) exp (-ja); a > 0
Sr go by Johnson and Kotz [12] discovered the additiv e property of gamma distribution. Meanwhile, Sr is associated with the exponential distributions family, and it is still complete. ■
Theorem 1. The UMVUE of j-p is, for p G (-to, to),
j-p = { ripffesp, np + p > 0
0, other wise
Proof. From Lemma 1,
Es
Ynß f M
Y I *nß+p-
1 exP ("^) *r = { № } Y-p
T(np) Jo
and the theorem observes from Lehmann-S cheffe theorem [15]. Remark 1. We can write (1) as
■
f (y; Y, ß v)
Gß-1 (y-1; G' (y-1; M (-1)
y2 r(ß)
E ^G' (y-1; p) ■ Y+
i=0
From Chatur vedi and Tomar [5] (Lemma 1) and theorem 1, for integer -yalued p, the UMVUE of f (y; a, p, v) for stipulated point y
f(y;Y,P,V) = GP-1 (y-y2^(rl;v) C^(y-1;v)^
= GP-1 (y-1; v) G2 (y-1; V) iC (_-iGl (y-1;}) T(rp) s+p
y2 r(ß)
i=0
r(r ß + i)
Theorem 2. The UMVUE of f (y; a, p, v) for a stipulated point y
fii^ Y, ß v)
Gß-1 (yy-';y)Gt(y-1;y) y2 Sß B((r-1)ß,ß)
1
G(y-U;F)
r-1)ß-1
, G (y 1; ^ ) < Sr
Proof. Using remark 1 and theorem 1, we acquire the required solution
Theorem 3. The UMVUE of R(t) Where
1 fZ
(s, q) = mx jO1*"1 (1 - 'V"d'
The incomplete beta function
Proof. No w, let us suppose the expectation
f TO
/ fu(y; Y, p v)dy
■
S
The integration to Sr
TO f z*TO
t IJt f'(y;Y,p,i)d^k(sr;Y,p,i)dsr
TO
ESr {f (y; Y, p, i)} dy
fII (y; Y, p, i)dy
Rii (t)
■
Suppose two independent rv's X and Y follow the inverse distributions families f1 (x9, Y1, p1, i1) and f2 (y; Y2,p2, i2), sequentially ,
(l a \ YpGp-1 (x-1; i1) G' (x-1; j) ( ( -1 f1 ii(x;Yl,pl,i1H-- x2rp)--exp (-Y1 G(x 1;i1
f2II (y;Y2, p2, i2 )
x2 r (p1)
x > 0, Y1 > 0, p1 > 0 Yp2 Hp2-1 (y-1; i2 ) H' (y-1; i2 )
y2 r (p2 )
y > 0, Y2 > 0, p2 > 0
exp (-Y2H (y 1; i2))
Wher e p1, p2, i1 and i2 are well-known, however Y1 and Y2 are unknown. Suppose n objects arranged X and m objects arranged Y are subjected to a lifespan test, and that the expiry
quantities for X and Y are r and r', separately . As well as notation by S = ¿=1 G (x-1; and
T = ¿r= 1H (y-1; i2)
Theorem 4. The UMUVE of P is The summations range from 0 to (r - 1)p1 - 1 in case (r - 1)p1 is an integer .
Proof. From theor em 3
Gp1-1 (x-1;i1 )G' (y-1;i1)
f1 ii (x; Y1, p1, i1) H x1 tpp((r- 1)p1,p1) 0,
1
G(y 1;i1)
S
, G (x-1; iO < S other wise
(r-1)p-1
(5)
Hp2-1 (y-1;i2 )H' (y-1;i2 )
f2II (y; Y2, p2, i2) = ^ y2Tp2p((r'-1)p2,p2) 0,
1 - ,H (y-1; i2) < T
other wise
(r'-1)p2-1
(6)
The UMVUES of f1 (x; Y1, p1, i1) and f2 (y; y2, p2, i2) for specifie pints 'x' and 'y' separately similarly , from theorem 4, we get the UMVUE of P
/•TO z*TO
Pii = / f1i (x; Y1, p1, i1) f2i (y; Y2, p2, i2) dxdy
Jy=0 J x=y
Using (5) and (6) we get
Pii '
B((r-1)p1,p1 )B((r'-1)p2,p2 )Sp1 Tp2
rTO rTO J Gp-1 (x-1; i1) G' (x-1; i)
•'y=[H* (T)]-1 ./x=y 1 x2
G (x 1; i1) 1 S
(r-1p-1
Hp2-1 (y-1; i2 ) H' (y-1; i2 )
y2
1
H (y-1; i2)
T
(r'-1)p2-1
dxdy
■
Kuldeep S. Chauhan
ESTIMATION AND TESTING PROCEDURES OF P(Y < X) FOR RT&A., No 3 (69)
THE INVERSE DISTRIBUTIONS Volume 17, September 2022
Corollary 1. If v1 = v2 = v, and G (x-1; v) = H (x-1; v)
S\p2 £ (-1) ( {(n - 1)p1}- 1
Pii
B ((n - 1)p1,p1) B ((m - 1)p2,p2)\T) =0 (p1 + i)
l—-)— ( (m - 1)p2 - 1) (T)', ) if s < T j=0 (p1 + p2 + i + ])\ j )
T)p £ (-1) ( {(n - 1)p1}- 1
B ((n - 1)p1, p1) B((m - 1)p2, p2 H S) =0 (p1 + i)\ i
(■T;)lB (p1 + p2 + i, (m - 1 )p2), if S > T
The summatio n over i, from 0 to {(n - 1)*p1} - 1, if (n - 1)p1 is an integer and the summation over j, from 0 to (m - 1 ) p2 , if (m - 1 ) p2 is an integer .
Proof. we get From Theorem 4 for S ^ T,
(-1)1 ( {n - 1}p1 - 1
Fu 1 B ((n - 1)p1,p1) B ((m - 1)p2,p2)J E (p1 + i)\ i
■JqTwp2-1 (1 - w)(m-1)p2-1 ^^ dw
and for S > T, from Theorem 2,
P =_1_¡T\p £ (-1)'
11 p ((n- l)p R, ) R ((m— l)fi„ R„ ) I S / C
p ((n - 1)p1, p1) p ((m - 1)p2, p2) v S) =0 (p1 + i) {n - 1}.p1 - ^ (^J JQ1 wp1 fp2fi-1 (1 - w)(m-1)p2-1 dw
and the second contention proved. ■
Remark 2. (i) UMVUES of R(t) and P are calculated independently using sampling pdf under type II censoring of UMVUES R(t) and P, as proved in theorems 3 and 4. As a result, we identifie two estimation concer ns that indicated inter dependence.
(ii) The UMVUES of P was achieved using type-II censoring, wher eas X and Y followed a similar distribution, possibly with dissimilar parameters or possibly with similar parameters, also while X and Y follo wed distinct distributions under all three conditions.
(iii) In theorem 4, if n ^ to then Var (j) ^ 0. We know that, f(y; j, p, v)2 R (t) and P are consistent estimators of f (y; j, p, v), R(t) and P, respectiv ely because these are continuous functions. So, j is a consistent estimator of Y.
4. MLES of j and Reliability Functions
Using the lemma 1
=(i)-" (7)
Theorem 5. The MLE for a specifie point y
~ , , (j)pGp-1 (y-1; v) G' (y-1; v) ( r,( 1 )) fu (y; j, p, v) = —-fTp) 6Xp [-jG {y ; v))
Proof. We can obtain from (7) and use the MLE's one-to-one property. ■
Kuldeep S. Chauhan
ESTIMATION AND TESTING PROCEDURES OF P(Y < X) FOR RT&A, No 3 (69)
THE INVERSE DISTRIBUTIONS Volume 17, September 2022
Theorem 6. R(t) is the MLE of R(t)
1 i TO
R(0 = /iG(f-1;j)(P), and Jy(p) = I xp-1 e-xdx
This is an incomplete gamma function.
Proof. Using MLE's invariance property and theorem 5
/TO
fll(y; Y, p, i)dy
= ^ LTO Gp-1 (y-1;£ G'(y-1;- (-^ (y-1;„)) dy
1 i' to
xp-1e-xdx
r(fi) ./ Si; G(t-l;^)
Corollary 2. When fi = 1,
R(t) = exp (-¿G (i-1; «))
Theorem 7. P is the MLE of P
P = (-72 )fe r Y(v)
P r(fii )r(fi2) Jy=0
/7l((y-(1;":) )) e-zzfi-idz z-(7g(*-T) ;«4))
Hfi2-1 (y-1; «2) H' (y-1; «2) ( ( 1 )) , --exp (--72H(y-1; «2)) dy
Proof. Using the MLE's one-to-one condition and theorem 5,
= (71 )fi1 (-72)fe [y(m) rX(n) i Gp-1 (x-1; «1 ) G' (x-1; «1 ) 1 r (fi ) r (fi2 ) Jy=0 Jx=y \ x2 f
exp (-71G (x-1;«1 )){ H2-1 (y-1;«y22)H'(y-1;«2^ exp (--72H (y-1;«2)) dxdy = (-71 )fi1 (72)fi2 fy'r Hp2-1 (y-1; «2) H' (y-1; «2) exp ( 7 H (y-1. « ))
= r (fi1 ) r (fi2 ) X=0-^-exp l"-72 Hly ; «2JJ
f71 G(y;1;H1 ) ) e-z f^dzU /z=-È1 ^X-1;«^ VW -T1 J
■
■
Remark 3. (i) UMVUES are acceptable for the MLES under remarks 2.
(ii) There is no need to use reliability function expressions to obtain UMVUES and MLES.
5. H ypotheses Testing
Putting the hypothesis to the test H0 : y = Yo versus H1 : y = Y0, from eq.(1), The likelihood function for Y
L(y/ y) = n ■ (n - 1).---(n - r - 1) ■ Yrp n{ G(y-1;i G(y-1; ^ I exp (-YSr) (8)
Kuldeep S. Chauhan
ESTIMATION AND TESTING PROCEDURES OF P(Y < X) FOR RT&A, No 3 (69)
THE INVERSE DISTRIBUTIONS Volume 17, September 2022
For H0
Sup L(y/ y) = n ■ (n - 1)----.(n - r - 1) ■ Y0p 0=1 { Gp-1 (y-1i)G(y-1;i^ exp (-Y0Sr)
©0 = {Y : Y = Y0}
SupL(Y/ y) = n.(n - 1).---.(n - r - 1). (Sr)r nr=1 { Gp-1 (y-1;i)G'(y-1;i^ exp (-r)
© = {y : Y = Y0} Likelihood ratio
*(x) = {ip^ } = (^) exp (-(r + Y0Sr)) (9)
And if p = 1 ( )
*(*)=(Y0^) exp (-(r + Y0Sr))
We note from (10) we get 2y0Sr ~ x2r, the rejection region is given by
{0 < Sr < m0} U {m0 < Sr < to} Wher e m0 and m' obtained from
(10)
P
2 p p 2 X2r < 2Y0 m0 or 2y0m0 < X2r
a
Thus ( ) ( )
m0 = 2YpX2r (1 - f) and m0 = 2YpxL (
For H0 : y < Y0 against H1 : y > Y0, It follows from (8) that, for Y1 < Y2
k (y(D, y(2),...............y(r); Y2, V) (Y2 )r ( ( ) S ) (11)
-7-) = — exp (- (Y2 - Y2) Sr) (11)
k(y(1), y(2),...............y(r); Yl, ij VY1/
It follows that from (11) that ^y(1),y(2),................y(r);Y2,i) has a maximum likelihood
ratio in Sr. Thus, the UMPCR for analysis H0 : y < Y0 against H1 : y > Y0 is given by
* (y(1),y(2),.................y(r))
Wher e m' obtained from
1, s, < m'0 0, otherrise
P
Ther efor e
2p X2r < 2Y0 m0
m0 = 2YpxL (1 - f)
Now for H0 : P = P0 against H1 : P = P0 under type II censoring. Then H0 is equivalent to Y1 = mYo, under H0
m (r + r') (r + r')
71 = Je , T 72 =
mS + T 2 mS + T
For m the likelihood of y1 and y2 for, x(i); i = 1, 2,........r and y^; j = 1, 2,........r is given by
Then
L (y^ Y1/ x(i), j = mYr Yi exp (- (Y1 s - Y2 t))
( ) mm' exp (- (r + r')
Sup L (Y1, Y1/ X(i), yj = (mS + T)r+r'
Sup L ( Y1, Y1/ x(i), j = m exp Sr^rr + r')
a
Then likelihood ratio
A (?i> Yi/y_U)) = m
1 + mS
r+r
The F-statistic and using the statistic that
S ri Yi
T r' Yi'(2r" m2 and m2 obtained from the given condition
( S S ]
;F(2r,2r)(.) < m2 and t > ™2j
P
r' mS r' mS
< F2r,2r' U rT > F2r',2 r'
rT
m2
r „ ( a \ , ,
— F(2r,2r) y1 - 2 1 and m
r m
2 = rmF(2r,2r')
6. Result
We can see in remarks 2(iii) wher e Y,f (x; Y, ft, p), R(t) and P are consistent estimators.
Figure 1: Uniformly Minimum Variance Unbiased Estimator
r
a
r
Kuldeep S. Chauhan
ESTIMATION AND TESTING PROCEDURES OF P(Y < X) FOR RT&A, No 3 (69)
THE INVERSE DISTRIBUTIONS Volume I7, September 2022
Table 1: Estimate o/Rft) Us/ng S/mw/at/on Approach
10
15
50
t R(t) R (t) R (t) R (t) R (t) R (t) R (t)
15 0.988256 0.982199 0.987858 0.985162 0.989151 0.987757 0.988996
- 0.006058 - 0.000399 - 0.003095 0.000894 - 0.000500 0.000740
0.000342 0.000271 0.000167 0.000134 0.00004 0.000038
0.050549 0.042173 0.039335 0.034016 0.020675 0.019854
79.2216 73.58140 83.6338 80.0726 87.3715 86.9576
r
20 0.917915 0.910388 0.914626 0.915237 0.918637 0.918618 0.919682
- 0.007528 - 0.003289 - 0.002678 0.000722 0.000703 0.001767
0.003464 0.003984 0.002017 0.002234 0.000565 0.000585
0.178809 0.188914 0.145604 0.152525 0.079166 0.080517
86.77420 85.8389 89.1520 88.8500 89.8782 89.8563
25 0.798104 0.79897 0.793371 0.801307 0.797564 0.80119 0.799987
0.000866 - 0.004733 0.003204 - 0.000540 0.003086 0.001883
0.008834 0.010559 0.005178 0.005855 0.001404 0.001456
0.295848 0.323471 0.237045 0.252153 0.125086 0.127418
88.3448 88.3294 90.0189 90.0201 90.3765 90.3809
30 0.670807 0.680869 0.666155 0.679482 0.669284 0.675551 0.672371
0.010061 - 0.004652 0.008675 - 0.001523 0.004743 0.001564
0.012591 0.014486 0.007144 0.007823 0.001827 0.001874
0.356619 0.38318 0.278715 0.291637 0.14255 0.144326
88.0549 87.9343 89.7265 89.6547 90.4366 90.4364
45 0.389714 0.409555 0.387121 0.402778 0.387598 0.394959 0.390400
0.019841 - 0.002593 0.013064 - 0.002116 0.005245 0.000686
0.011188 0.011276 0.005701 0.005672 0.001285 0.001279
0.331236 0.331191 0.245934 0.244903 0.118951 0.118657
85.906 85.5138 88.3818 88.2268 90.2805 90.2736
50 0.32968 0.349176 0.32759 0.342179 0.327687 0.334531 0.330208
0.019486 - 0.002090 0.012499 - 0.001993 0.004851 0.000528
0.009403 0.009186 0.004661 0.004541 0.001024 0.001013
0.301375 0.296388 0.221558 0.218317 0.106079 0.105504
85.3279 84.9272 88.0684 87.9203 90.2329 90.2261
55 0.281492 0.300028 0.279791 0.293153 0.279655 0.285911 0.281906
0.018536 - 0.001701 0.011661 - 0.001837 0.004419 0.000414
0.007747 0.007379 0.003752 0.003595 0.000808 0.000795
0.27164 0.263625 0.198166 0.193649 0.094143 0.093416
84.8402 84.4478 87.8163 87.6789 90.1936 90.1871
60 0.242535 0.25984 0.241133 0.253268 0.24086 0.246532 0.242865
0.017305 - 0.001402 0.010733 - 0.001675 0.003997 0.00033
0.006327 0.005901 0.003004 0.00284 0.000636 0.000625
0.243921 0.23421 0.176869 0.171694 0.083515 0.082718
84.4303 84.0555 87.6127 87.4872 90.1612 90.1552
70 0.184604 0.199307 0.183617 0.193547 0.183228 0.187856 0.184825
0.014703 - 0.000987 0.008943 - 0.001377 0.003252 0.000221
0.004203 0.003797 0.001936 0.001794 0.000401 0.000391
0.196726 0.185941 0.141401 0.135939 0.066187 0.065382
83.7942 83.4651 87.3111 87.2079 90.1126 90.1076
Table 2: Estimation ofP Using Simulation Approach
r, r' (10, 10) (10, 15) (15,15) (25, 25)
(m, n) P P P P P P P P
(5, 5) 0.66625 0.55623 0.79245 0.4739 0.85131 0.38238 0.88938 0.30929
-0.00042 -0.11044 -0.00755 -0.32610 -0.00583 -0.47476 0.0005 -0.57960
0.0131 0.00542 0.00825 0.0149 0.00521 0.01713 0.00226 0.01262
0.37465 0.22246 0.29287 0.37771 0.21917 0.42422 0.15231 0.36142
89.4513 80.3704 88.3766 85.7414 85.8522 89.4548 87.8912 88.9059
(5, 10) 0.66939 0.536 0.80224 0.46939 0.853 0.3872 0.88923 0.3223
0.00272 -0.13067 0.00224 -0.33061 -0.00415 -0.46995 0.00034 -0.56659
0.01343 0.00452 0.00617 0.01127 0.00525 0.01481 0.00244 0.0124
0.38062 0.1954 0.25112 0.31962 0.23214 0.40524 0.15582 0.35937
89.8872 78.3713 88.461 84.4431 87.5116 90.3086 87.722 89.3048
(10, 10) 0.66096 0.65675 0.79749 0.70795 0.84474 0.65375 0.88476 0.5824
-0.00591 -0.00991 -0.00251 -0.09205 -0.01241 -0.20340 -0.00413 -0.30649
0.00709 0.0055 0.0032 0.00192 0.00329 0.00603 0.00155 0.00975
0.26423 0.21508 0.17838 0.119 0.18275 0.24016 0.12176 0.32224
88.238 82.9781 87.8595 74.2657 86.8354 84.0915 87.1155 89.1193
(15, 15) 0.66441 0.66743 0.80245 0.77868 0.85492 0.76714 0.88795 0.71988
-0.00226 0.00076 0.00245 -0.02132 -0.00223 -0.09000 -0.00094 -0.16901
0.00604 0.00588 0.00166 0.0005 0.00127 0.00126 0.00092 0.00398
0.26138 0.25845 0.13777 0.06267 0.11497 0.10138 0.09689 0.19464
89.9864 89.6238 90.897 75.9468 88.7437 77.2489 88.1403 85.3064
(15, 25) 0.66978 0.66943 0.79561 0.76784 0.85674 0.76313 0.88892 0.72191
0.00311 0.00277 -0.00439 -0.03216 -0.00040 -0.09401 0.00003 -0.16698
0.00324 0.00312 0.00234 0.00088 0.00111 0.00131 0.00065 0.00287
0.18845 0.18624 0.15194 0.08226 0.10749 0.10513 0.08171 0.16528
90.1082 90.2303 87.77 75.2671 89.2306 75.7539 88.6398 85.857
(25, 25) 0.66432 0.66728 0.79942 0.79972 0.85627 0.84052 0.88749 0.84112
-0.00234 0.00061 -0.00058 -0.00028 -0.00088 -0.01662 -0.00140 -0.04777
0.00296 0.00304 0.00182 0.00149 0.00084 0.00032 0.00054 0.00027
0.17797 0.18026 0.14352 0.12584 0.09115 0.04498 0.0799 0.05119
89.8014 89.7715 90.4715 88.2366 88.2113 73.121 90.712 76.8041
(25, 30) 0.66545 0.66744 0.79985 0.7996 0.85393 0.83576 0.88492 0.84001
-0.00122 0.00077 -0.00015 -0.00040 -0.00322 -0.02138 -0.00397 -0.04888
0.00278 0.00285 0.00145 0.0012 0.0013 0.00054 0.00062 0.00031
0.17136 0.1736 0.1295 0.11518 0.11215 0.05916 0.07973 0.04987
89.5562 89.6143 90.8001 89.2747 87.3483 72.9983 88.8342 74.4235
(30, 30) 0.66466 0.67086 0.79943 0.80168 0.85183 0.84563 0.88852 0.86285
-0.00201 0.00419 -0.00057 0.00168 -0.00531 -0.01151 -0.00037 -0.02604
0.00223 0.0019 0.00127 0.00117 0.00115 0.00067 0.00048 0.00014
0.16182 0.14227 0.11788 0.11381 0.10656 0.07456 0.0739 0.02872
91.3111 89.3985 89.6646 89.5152 87.8413 80.5097 90.0725 71.648
Figure 1 is plotted /(y, y, p, i) under type II censoring for various values of r = 5(5), 10, 15,20, 30 and 50 and concludes that the curves of /(y, y, p, i) getting closer to the curve of f (y; y, p, i) as r increases.For r = 30, validates the consistency property of the estimators, because the curves overlap.W e have presented a simulation study when y is unkno wn with the bootstrap re-sampling procedur e for r = 10(5)15 and 50 while other parameters are known. If G (y-1; i) = y2, p = 1, and y = 1. Table 1 shows computation using 500 bootstrap replications with a 95% confidenc coefficien to obtain the estimated value of UMVUES and MLES for R(t),bias, variance, and MSES, for different values of t.Also displa y simulation trials using the bootstrap re-sampling procedure for (n, m) = (5,10),(10, 10), (15,15), (15,25), (25,25), (25,30), (30,30)) across different (r', r') = (10, 10), (10, 15), (15, 15) and (25,25), while Y1 and y2 are unidentifie but the other parameters are identifie to estimate P. The free sample is produced as of (1), if G (x-1; i) = log (x), p1 = p2 = 1, G (y-1, i) = log (y), Y1 = 1 and y2 = 5,5,5 and 8. Table 2 shows computations using 500 bootstrap replications with a 95% confidenc coefficien to obtain the estimated value of UMVUES and MLES P, bias, variance, and mean sum of squar es (MSES).
7. Discussion
We established estimation algorithms for the inverse distributions family based on type-II censoring in this paper .The point estimates are taken into consideration. Hypotheses were generated for many parametric functions, and UMPCR was achieved. Simulation techniques are used to study the efficiency of the UMVUES and MLES of reliability functions, as well as other parameters. For type-II censorship, the UMVUE of R(t) is superior to the MLE of R(t) for different t. Further more, for all values of (r, r'), the MLE of P outperfor ms the UMVUE of P. On the other hand for large t, UMVUE comes to be more effective than MLE of R(t). From the study of P it has been deter mined that for m < n, UMVUE is superior to MLE for P. Alter nativ ely, for n < m, it is concluded that MLE is superior to UMVUE for P. As n and m rise both estimators yield equally effectiv e. Using Figur e 1, we validated the consistency property of the estimators under censoring appr oaches.
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