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41
Classical and Bayesian Estimation of Parameter of SSE(e)-distribution Under Type-II Censored Data
P. KuMAR1, D. KuMAR2, P. KuMAR3 & U. SlNGH4
•
department of Statistics, Faculty of Science and Technology, Mahatma Gandhi Kashi Vidyapith, Varanasi, India-221002 2,4Department of Statistics, Banaras Hindu University, Varanasi, India-221005 3Department of Statistics, Udai Pratap Autonomous College, Varanasi, India-221003 pawanchauhanstranger@gmail.com
Abstract
In this present piece of work, we have considered a lifetime distribution based on trigonometric function called SSE(e)-distribution and discuss its various properties which have not been added previously by host as well as any other authors. This distribution is useful and a good contribution in research under trigonometric function. We are deriving some more useful properties such as moments, conditional moments, mean deviation about mean, mean deviation about median, order statistics etc. Estimation of parameter has been done for both classical and Bayesian paradigms under Type-II censored sample. Simulation study has also been carried out to know the progress of the estimators in the sense of having smallest risk (over the sample space) at the long-run use.
Keywords: SSE(e)-distribution, Type-II censoring, Bayes estimator, MLE, Gauss-Laguerre method, risk function
1. Introduction
In statistical literature, there are several lifetime distributions available, for example exponential, gamma, Weibull, Lindley distribution etc. In past studies, calculations can only be handled when the expressions corresponding to various properties obtained in the nice closed form and when this was not achieved then rarely preferred. But in this modern era due to the advancement of computational facilities this problem have been resolved almost. Mostly, algebraic and exponential functions have been used to develop the new transformation and sometimes authors see gap in trigonometric, inverse and logarithmic type transformations. Keeping this in mind, the considered transformation is the good contribution in support of filling such gap. As we aware that the use of a single model is not found suitable in every aspect, therefore to adopt a suitable baseline model is also a quite tedious job. Study explores that exponential distribution is preferably used as a lifetime distribution but the extensive use of it is restrictive in the sense of its constant hazard rate. For simplicity and flexibility, we are also using here exponential distribution as a baseline distribution In these days, many authors are introducing transformation techniques to get a new lifetime distribution with the help of available baseline distributions some of which are popular as power transformation proposed by [6], sine square distribution by [1], [20] introduced quadratic rank transmutation map (QRTM), sinofarm distribution by [23], DUS transformation proposed by [10], minimum-guarantee distribution proposed by [11], CS transformation by [3], new Sine-G family based on [13] proposed by [16], new extension of Lindley distribution given by [17], PCM transformation by [12] and many more. In such continuation, [13] have proposed a
new transformation known as SS-transformation by using sine function which is given by
F(x) = sin (|G(x)) (1)
Where G(x) is the baseline's cumulative distribution function (cdf) and the accompanying probability density function (pdf) are
f (x) = |g(x) cos (|G(x)) (2)
They have utilized baseline distribution as exponential distribution and named as SS exponential (SSE (e))-distribution and having the following form of its pdf is
f (x) = ^2e x e~ex sin (|e-ex) ; (x,e) > 0 (3)
and its cdf in compact form is
F(x) = cos (f e-ex^ ; (x,e) > 0 (4)
The reported compact forms of reliability function and hazard rate function respectively are
R(x) = 1 - cos (2e-ex) (5)
and ( )
h(x) = ^e x e^£X cot (2e-ex) (6)
Figures 1, 2 and 3 presents the shape of pdf, cdf and hazard rate function of SSe (e)-distribution. And Figure 3, claims that the nature of hazard rate function of the SSe (e)-distribution is increasing which is different from baseline distribution.
The article is constructed as follows, introductory part have been shown in Section (1), statistical properties discussed in Section (2), estimation of parameter presented in Section (3), comparison of estimators in Section (4) and concluding remarks regarding the work quoted in Section (5).
0.0 0.5 1.0 1.5 2.0 2.5 3.0
X
Figure 1: Plots of pdf of SSe (e)-distribution for various choices of parameter e.
Figure 2: Plots of cdf of SSE(e)-distribution for various choices of parameter e.
— E =1.0 — E =2 0 -E -5.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 3: Plots of hazard rate function of SSe (e)-distribution for various choices of parameter e.
2. Statistical Properties
In this section, we are discussing some statistical properties of SSE (e)-distribution which have not derived yet namely mean deviation about mean, mean deviation about median, order statistics etc. Firstly, we have discussed two lemma which are-
Statement (Lemma-1)
(e, r, Z )
xre Zx x sin ( ^e dx
¿0 (2k + 1)! ^ 2 )
n \2k+1
L((2fc + 1)e + Z )r+1\
Proof:
&(e, r, Z )
xre Zx sin C2e ex) dx = T
(-1)2k+1 ( n \2k+1
k=0
(2k +1)! \2J
xre-((2k+1)e+Z)xdx
£ (-1)2k+1 x (n\2k+1
k=0 (2k +1)! v2
r!
L((2k + 1)e + Z )r+1\
The r order moment about origin of SSE (e)-distribution have already obtained by [13]. Here, we obtain the same by using lemma 1 and is
71
E(Xr ) = - e x ft (e, r, e)
(7)
on putting r = 1,2,3,4 in (7), we get the first four raw moments of SSe (e)-distribution and are
71
71
E(X) = 2e x ft(e,1,e) ; E(X2) = 2e x ft(e,2,e) E(X3) = 2e x ft(e,3,e) ; E(X4) = 2e x ft(e,4,e) And, first four central moments are calculated by the following relations,
2
U2 =U2 -
=^3 - 3^2+ 2m
/3
=^4 — 4^3 + 6^2^12 — 3^14
On using above relations of central moments, we can obtain the measures of skewness and kurtosis, viz., ^i, 7i and j82, y2 respectively by the following expressions
ß1 = 4 71 = /1 F2
F3 V22
ß = ^ ß2 = -2
72 = ß2 - 3 = - 3 ^2
Statement (Lemma-2)
£2 (e, r, Z, t)
xre Zx x
Jt
TO r
71
sin ( 2 e
dx
(-1)
2k+1
£ 1=0 (2k + 1)! V2;
7T\
2k+1
r! x e-((2k+1)e+Z)t x (((2k + 1)e + Z)t) l !((2k + 1)e + Z )r+1
Proof:
£2 (e, r, Z, t)
■re-Z x
xe
71
sin I 2 e
dx = E ( ? \
k=o (2k + 1)! V 2)
2k+1
xj xre-((2k+1)e+Z)xdx
f (-1)2k+1 (n\2k+\
k=o¿0 (2k + 1)! ^
r! x e-((2k+1)e+Z)t x [((2k + 1)e + Z)t]
l! [(2k + 1)e + Z]
r+1
r
TO
o
o
x
TO
I
2.1. Conditional Moments
The conditional moment of rth order is represented by E(Xr \X > r) then by using lemma 2, we get
E(Xr\X > x) = n X [f^KeP}, (8)
2.2. Quantile Function
Using (4), we get the quantile function of order p (Q(p)) is
cos (2e-eQ(P)) = p e-eQ(p) = ^cos-1 (p)
Therefore,
Q(p) = -1 ln (n2cos-1(p))
(9)
2.3. Median
On putting p = 1 in equation (9) we will easily get the median of SSE (e)-distribution and if Md be the median of SSE(e)-distribution then the expression is
1 /2 M =-1 lnU
which is same expression as obtained by [13].
Table 1: Mean, median, variance, skewness and kurtosis of SSe (e)-distribution for different values of e.
(10)
e Mean Median Variance Skewness(7x) Kurtosis(72)
0.2 0.25814 2.02733 0.25695 4.13316 19.53054
0.7 0.25379 0.57924 0.16079 2.73800 6.85106
1 0.23603 0.40547 0.11153 2.53429 5.44002
2 0.17377 0.20273 0.04342 2.28171 3.84498
5 0.09108 0.08109 0.00919 2.12107 2.91886
10 0.05023 0.04055 0.00253 2.06265 2.60720
15 0.03446 0.02703 0.00118 2.04642 2.45738
Table 1 shows that the mean, median, variance, skewness (71) and kurtosis (y2) of the SSE(e)-distribution with pdf (3) for different choices of parameter e. The values of mean, median and variance of SSE (e)-distribution are decreases as values of parameter e increases this shows that mean, median and variance of the SSe (e)-distribution inversely related to parameter e. Since, 71 > 0 and y2 > 0 so the nature of SSE(e)-distribution has positively skewed and leptokurtic distribution for considered choices of parameter.
2.4. Mean deviation about mean and median
The mean deviation (MD) about mean is another measure of dispersion and is defined as,
$i(x)
\x - f (x)dx
where ^ is the mean of SSE(e)-distribution, then
r i1 r œ
fa (x)= (p — x)f (x)dx + (x - y)f (x)dx
JO Ju
fa (x) = 2p x F(u) — 2p + 2
u
x f( x) dx
0
where notations have their usual meanings then by using lemma 2, we get
fM n
xf (x)dx = — e x £2(e, 1,e, p) Jp 2
Therefore,
< (x) = 2p x F(p) - 2p + ne x £2(e, 1,e, p) (11)
In the similar way, MD about median is
(• to f-M [" (• to
<2(x) = \x - M\f (x)dx = (M - x)f (x)dx + (x - M)f (x)dx Jo Jo \_Jm
!• TO
= -p + 2 xf (x)dx
M
Now, by lemma 2, we have
f to n
xf (x)dx = — e x £2(e,1,e,M) M2
Finally,
(x) = -U + ne x £2 (e, 1, e, M) (12)
2.5. Order Statistics
Let us take random sample of size n from the SSE(e)-distribution say, {X!,X2,...,Xn} and associated order statistics is X(X) < X(2) < ... < X(r), then pdf of rth order statistics is
n' 1
fr(x) = {r - 1)n(n - r)' X Fr-1(x) x f (x) x [1 - F(x)]n-r
=* fr(x) = (r - 4 - r)! X nL(-1)l (n -r)Fr+i+1 (x) X f (x) (13) Now, using (3) and (4) in 13), we have
fr W = (r - m„ - r)! X f D"1)'' (n-r) X sin (ne--) [cos (fe-»)]'+' (14)
and corresponding cdf of rth order statistics is
Fr(x) = t (x) x [1 - F(x)]n-i = EE^) C - 0 (-1)^(x) (15)
Using equation (4) in (15), we obtain the expression of cdf of rth order statistic of SSE(e)-distribution as follows
Fr (x) = EE (n )(n -i)(-1)' {cos ( 2 e-")} + (16)
3. Estimation of Parameter
In this section, we have discussed the estimation of parameter e of SSE (e)-distribution for Type-II censored data under both Classical and Bayesian paradigms. It is observed that, it is not possible to obtain the failure times of all the test units placed on a life testing experiment because of the associated costs such as cost of per unit is high or limitations on experimental time etc. Therefore, such situations are handled by removal of test units before the actual failure occurs
and are termed as censoring scheme. Since, the removal of these units can be done in various possible ways, these are further known as various type of censoring scheme. The two widely used censoring schemes are Type-I and Type-II censoring schemes. Here, we consider Type-II censoring only. Let x^), x(2),..., x(r) be r-ordered Type-II right censored random observations obtained from n units placed on a life testing experiment where each unit has its lifetime and follows SSE(e)-distribution having pdf (3) with largest (n — r) lifetimes have been censored, then the likelihood function is given by [4] is
n' ■ / \ n
LC(e|X) = T—-y nf (x(i);e) (l — F(x(r);e)J ( )' i=1
(17)
Several authors have been done their work in this direction, [18] have been discussed Bayesian estimation of parameter under Type-II censored data, [9] worked on classical and Bayesian estimation of reliability estimation of Maxwell distribution under Type-II censored data, [22] have discussed the Bayesian estimation of exponentiated gamma parameter and reliability function under Type-II censored data for asymmetric loss function, [7] presents the statistical evidences of Type-II censored data, [19] have been derived the Bayesian estimation techniques of system reliability for Weibull distribution under Type-II censored data, [8] discussed the comparison between same Bayesian estimation methods for the parameter of exponential distribution based on Type-II censored data. [5] have been discussed the estimation procedure for new lifetime models under classical and Bayesian set-up in the presence of Type-II censored sample. [2] studied the various properties of Pareto distribution using Type-II hybrid censored sample data.
3.1. Classical Estimation
Using (17), the likelihood function of the SSe(e)-distribution under Type-II censoring scheme is
Lc (e|X)
n'
(n - r)
-ere-e Er=1 x(i) n sin(^2e-ex('')
i=1
1 - cod \e-ex(r)
(18)
and taking logarithm on both sides of (18), we get
ln LC = ln
n'
(n - r)
+ r ln e + e£ x(i) + £)ln
(i)
i=1 i=1
sin ( 2 e-ex('')
+ (n - r) ln
On differentiating (19) w.r.to e and equate the resultant to zero, we get d ln LC r r
n
1 - cos —e-ex(r)
(19)
de e
i=1
e + Exw- 7 Ex(i)e-ex(i)
i=1
cot ( 2e-x(")
+ (n - r)
n x(r)e-ex(r) sini fe-ex(r)
2 X
1 - cos (f e-ex(r))
(20)
The above equation cannot be solved analytically. So, we use numerical approximation technique through R software to solve them numerically in terms of e i.e. £mc which maximizes the equation (18).
n—r
0
3.2. Bayesian Estimation
In Bayesian paradigm, posterior probability is an effect of two components prior probability and likelihood function, and calculated from the statistical model for the observed data. The prior distribution of the parameters is assumed before the data observed. There are different kinds of prior distribution of parameters defined as proper and improper priors. Another way to define the priors based on available advanced information is known as informative and non-informative priors.
Here, we use informative prior as a Gamma(a, b) prior for e of SSE (e)-distribution and having the following form
ba _ ,
(21)
ba
n(a, b) = b-ea-1e-be ; e > 0, a, b > 0 ra
where, hyper-parameters are a and b. If two information's which are independent in nature on e (say prior mean and prior variance are known) are provided, they can be obtained, for more details see [21], [14], [15]. The mean and variance of the prior distribution (21) are | and |2
respectively. Thus, we take M = f and V = |2 giving b = M and a = M. The informative gamma prior behaves like non-informative prior if hyper-parameters changes i.e. if we fixed prior mean and taking large prior variance.
The posterior density of e given the sample observations X is given below
Vc (e|X)
Lc (e|X) x n(a, b) J0TO LC(e|X) x n(a, b)de
(22)
By using equation (21) and (18) in (22), posterior density of e given X under Type-II censoring is
Vc (e|X)
er+a-1e-e(Tr=1 x(i)+b) jnr=1sin (f e-ex(')\
1 - cod f e-ex(r)
/0TO er+a-1e-e(Tr=1 x«+b) jnr=1sin (f e-ex(') \
1 - cod 2e-ex(r)
(23)
de
The expressions for considered loss functions namely squared error loss function (SELF) and general entropy loss function (GELF) having the following forms
Ls (eSC, e) = (^sc - e)2
Lg(eGC,e) = (^)c - cln (^) - 1
If eGc is a Bayes estimator of e for Type-II censoring under GELF then, we get
(24)
(25)
eGC
j0TO er+a-c-1 e-e(Tr=1 x(i)+b) x nr=1sin (2e-ex(') \ [1 - cos (f e-ex(r) \
de
/0TO er+a-1 e-e(Tr=1 x(i)+b) x nr=1 sin (2e-ex(i)) [1 - co^f e-ex«)
de
(26)
Putting c = -1 in equation (26), we get the Bayes estimator £sc of e for Type-II censoring, we get
eSC
J0TO er+ae-e(Tr=1 x(i)+b) x nr=1si^2e-ex(i)
1 - cos ( 2e-ex(r)
de
/0TO er+a-1e-e(Tr=1 x(0+b) x HUsin (2e-ex(')
1 - cos ( 2 e-ex(r)
de
(27)
n—r
n—r
The above equations (26) and (27) are not solvable analytically. Therefore, we propose some numerical approximation technique to get the solution. Basically, we have used here Gauss-Laguerre quadrature formula to obtain the solution.
4. Comparison of Estimators
In this section, we compare the performance of the considered estimators (£mc, £sc, £gc ) of parameter e of SSE(e)-distribution in the presence of Type-II censoring scheme in terms of lowest risks (expected loss over Q) under GELF. It is clear that, the expressions of risk function are not obtained in implicit form. So, we use Gauss-Laguerre quadrature formula to obtain the estimators (£sc and £gc) of parameter e for computing the risks under Type-II censored data. To know the performance of estimator in long run use, we simulate 20,000 samples for different sample
size n and different effective sample size r (for Type-II censoring) from SSe(e)-distribution with different choices of values of parameter (e = 1.5, 2.0, 3.0) and loss parameter c = ±2.
Tables 2, 3 and 4 represents the risks for the variations in hyper-parameters (variation in prior variance (V = 0.5, 1.0, 2.0, 5.0, 80) for fixed prior mean (M = 1.0, 2.0, 3.0)) when true value of the parameter e = 2 for sample size n = 30 with different censoring schemes r = 12, 18, 24 and 30.
Tables 5 and 6 shows the variation in n and r with minimum prior variance (high confidence level V = 0.5) and prior mean (M = 2.0) for the true value of e = 1.5 and 3, respectively.
Tables 2, 3 and 4 presents the simulated risks under GELF for variation in prior variance (high to low confidence level) with fixed prior mean (M). We see that the risks under GELF for the Bayes estimators of e under SELF and GELF are increases as values of prior variance increases (high to low confidence level) and if prior mean increases then the risks under GELF decreases for the Bayes estimators under SELF and GELF for Type-II censored samples. Bayes estimator under GELF eGC outperforms MLEs (eMC) and SELF (eSC) under Type-II censored sample when under estimation is more serious as compared to over estimation (c = -2) and when over estimation is more serious as compared to under estimation (c = +2), then Bayes estimator under SELF (esc) outperforms MLEs (eMC) and GELF (eGC) under Type-II censored sample. It is also noted that when prior variance is large (low confidence level i.e. very weak information about the parameter e) then classical estimator MLEs (eMC) performs better than the Bayes estimators under SELF (esc) and GELF (eGc).
Tables 5 and 6 shows that the variation in sample size n and corresponding different Type-II censoring schemes r for the true values of parameter e = 1.5 and 3. Table 5 provides simulated risks of the Bayes estimators (eSC) of e under SELF outperforms MLEs (eMC) and Bayes estimators of e under GELF (eGc) in both cases under estimation is more serious than over estimation and vice-versa. While Table 6 provides Bayes estimator under GELF (eGC) outperforms the Bayes estimator under SELF (e$c) and MLE (emc) for the situation when under estimation is more serious than over estimation but in reverse case, Bayes estimator under SELF (eSC) outperforms MLEs (eMC) and Bayes estimator under GELF (eGC) for the true value of parameter e = 3. It is also observed that the risks of all estimators of e for Type-II censored sample decreases with increase in the value of n and r for all considered values of the parameter e.
Table 2: Risks of the estimators of e under GELF when prior variance varies for fixed n = 30, r(r = 12,18,24), e = 2.0, M = 1 andc = ±2.
V scheme r c =-2 c=2
MLE SELF GELF MLE SELF GELF
12 0.15180 0.13386 0.11519 0.20147 0.09545 0.13667
0.5 18 0.09915 0.09065 0.08189 0.12088 0.07236 0.07057
24 0.07407 0.06885 0.06373 0.08397 0.05817 0.05724
30 0.05938 0.05558 0.05214 0.06572 0.04751 0.04789
12 0.15180 0.13616 0.11989 0.20147 0.12083 0.13550
1 18 0.09915 0.08988 0.08482 0.12088 0.08641 0.08943
24 0.07407 0.06883 0.06587 0.08397 0.06675 0.06850
30 0.05938 0.05605 0.05410 0.06572 0.05494 0.05612
12 0.15180 0.14083 0.13321 0.20147 0.15280 0.16792
2 18 0.09915 0.09324 0.08994 0.12088 0.09933 0.10516
24 0.07407 0.06992 0.06806 0.08397 0.07367 0.07693
30 0.05938 0.05718 0.05594 0.06572 0.05952 0.06165
12 0.15180 0.14373 0.13906 0.20147 0.17322 0.17442
10 18 0.09915 0.09650 0.09440 0.12088 0.10864 0.11646
24 0.07407 0.07278 0.07155 0.08397 0.07953 0.08381
30 0.05938 0.05919 0.05836 0.06572 0.06330 0.06606
12 0.15180 0.15085 0.14829 0.20147 0.18860 0.19630
80 18 0.09915 0.09847 0.09735 0.12088 0.11827 0.12777
24 0.07407 0.07445 0.07382 0.08397 0.08530 0.09051
30 0.05938 0.06022 0.05979 0.06572 0.06677 0.07010
Table 3: Risks of the estimators of e under GELF when prior variance varies for fixed n = 30, r(r = 12,18,24), e = 2.0, M = 2 andc = ±2.
V scheme r c =-2 c=2
MLE SELF GELF MLE SELF GELF
12 0.15180 0.05785 0.05776 0.20147 0.06075 0.06133
0.5 18 0.09915 0.05112 0.05095 0.12088 0.05416 0.05772
24 0.07407 0.04445 0.04421 0.08397 0.04712 0.04948
30 0.05938 0.03881 0.03858 0.06572 0.04070 0.04237
12 0.15180 0.08699 0.08659 0.20147 0.09806 0.09889
1 18 0.09915 0.06802 0.06764 0.12088 0.07527 0.08079
24 0.07407 0.05566 0.05534 0.08397 0.06073 0.06412
30 0.05938 0.04725 0.04700 0.06572 0.05079 0.05314
12 0.15180 0.11490 0.11311 0.20147 0.13630 0.14083
2 18 0.09915 0.08291 0.08192 0.12088 0.09455 0.10148
24 0.07407 0.06399 0.06330 0.08397 0.07033 0.07421
30 0.05938 0.05327 0.05274 0.06572 0.05719 0.05974
12 0.15180 0.13415 0.13238 0.20147 0.17091 0.17531
5 18 0.09915 0.09104 0.09008 0.12088 0.10704 0.11543
24 0.07407 0.06982 0.06916 0.08397 0.07840 0.08300
30 0.05938 0.05707 0.05665 0.06572 0.06254 0.06558
12 0.15180 0.15018 0.14771 0.20147 0.19933 0.19672
80 18 0.09915 0.09963 0.09857 0.12088 0.11883 0.12845
24 0.07407 0.07551 0.07482 0.08397 0.08598 0.09119
30 0.05938 0.06064 0.06018 0.06572 0.06703 0.07033
Table 4: Risks of the estimators of e under GELF when prior variance varies for fixed n = 30 and r(r = 12, 18, 24), e = 2.0, M = 3 andc = ±2.
V schemes r c =-2 c=2
MLE SELF GELF MLE SELF GELF
12 0.15180 0.11285 0.12353 0.20147 0.18003 0.18618
0.5 18 0.09915 0.08463 0.09185 0.12088 0.12879 0.14174
24 0.07407 0.06730 0.07252 0.08397 0.09833 0.10731
30 0.05938 0.05613 0.06017 0.06572 0.07954 0.08627
12 0.15180 0.10204 0.11257 0.20147 0.16348 0.16861
1 18 0.09915 0.07569 0.08181 0.12088 0.11215 0.12486
24 0.07407 0.06083 0.06489 0.08397 0.08519 0.09338
30 0.05938 0.05096 0.05389 0.06572 0.06833 0.07411
12 0.15180 0.10845 0.11640 0.20147 0.16802 0.16768
2 18 0.09915 0.07780 0.08184 0.12088 0.10803 0.11971
24 0.07407 0.06115 0.06362 0.08397 0.07953 0.08651
30 0.05938 0.05096 0.05263 0.06572 0.06339 0.06810
12 0.15180 0.12592 0.12934 0.20147 0.17881 0.17903
5 18 0.09915 0.08871 0.09023 0.12088 0.11345 0.12417
24 0.07407 0.06864 0.06947 0.08397 0.08334 0.08942
30 0.05938 0.05580 0.05640 0.06572 0.06568 0.06971
12 0.15180 0.14540 0.14786 0.20147 0.18968 0.19429
80 18 0.09915 0.09905 0.09951 0.12088 0.12069 0.13089
24 0.07407 0.07306 0.07313 0.08397 0.08471 0.09015
30 0.05938 0.05921 0.05997 0.06572 0.06597 0.06942
Table 5: Risks of the estimators of e under GELF when V = 0.5, M = 2 and c = ±2 for true value of e = 1.5.
V scheme r c =-2 c=2
MLE SELF GELF MLE SELF GELF
6 0.30733 0.11180 0.12843 0.58496 0.17749 0.19775
15 9 0.20172 0.09625 0.10710 0.30346 0.14726 0.17215
12 0.14916 0.08295 0.09059 0.20110 0.12254 0.14004
15 0.12036 0.07335 0.07906 0.15104 0.10453 0.11761
8 0.22597 0.09958 0.11208 0.35915 0.15420 0.15351
20 12 0.14976 0.08279 0.09040 0.20231 0.12191 0.13939
16 0.11135 0.06989 0.07502 0.13774 0.09803 0.10978
20 0.09013 0.06117 0.06495 0.10558 0.08244 0.09104
12 0.15112 0.08386 0.09189 0.20662 0.12435 0.12456
30 18 0.09839 0.06540 0.06998 0.11985 0.09049 0.10073
24 0.07255 0.05290 0.05584 0.08325 0.06937 0.07586
30 0.05884 0.04529 0.04738 0.06545 0.05727 0.06186
24 0.07548 0.05425 0.05723 0.08712 0.07144 0.07669
60 36 0.04909 0.03924 0.04075 0.05379 0.04825 0.05158
48 0.03676 0.03096 0.03186 0.03932 0.03655 0.03855
60 0.02966 0.02581 0.02643 0.03120 0.02961 0.03098
Table 6: Risks of the estimators of e under GELF when V = 0.5, M = 2 and c = ±2 for true value of e = 3.0.
V scheme r c =-2 c=2
MLE SELF GELF MLE SELF GELF
6 0.30882 0.18838 0.14822 0.57630 0.11627 0.20923
15 9 0.20002 0.14235 0.11626 0.29734 0.09092 0.07627
12 0.14878 0.11418 0.09590 0.19835 0.07558 0.06526
15 0.12139 0.09769 0.08385 0.14948 0.06649 0.05869
8 0.22674 0.15471 0.12462 0.35157 0.09761 0.16869
20 12 0.14766 0.11417 0.09571 0.19338 0.07540 0.06492
16 0.11053 0.09059 0.07813 0.13388 0.06245 0.05537
20 0.08972 0.07659 0.06737 0.10351 0.05458 0.04936
12 0.15287 0.11776 0.09863 0.20241 0.07752 0.12301
30 18 0.09955 0.08390 0.07307 0.11933 0.05891 0.05277
24 0.07382 0.06505 0.05808 0.08374 0.04801 0.04407
30 0.05927 0.05340 0.04848 0.06563 0.04099 0.03822
24 0.07358 0.06458 0.05742 0.08556 0.04785 0.06514
60 36 0.04797 0.04366 0.04011 0.05294 0.03478 0.03278
48 0.03608 0.03355 0.03141 0.03875 0.02795 0.02676
60 0.02952 0.02785 0.02637 0.03116 0.02385 0.02303
5. Conclusion
In this paper, we have been consider a lifetime distribution by using sine function which has proposed by [13]. We have also discussed some statistical properties of the considered distribution such as conditional moments, mean deviation about mean, mean deviation about median and derived expressions of the pdf and cdf of rth order statistics. Mean, median and variance are inversely related to the parameter e of SSE(e)-distribution and the distribution has positively skewed and leptokurtic nature. We have developed classical and Bayesian estimation procedure for estimation of parameter e under Type-II censored data. And also check the workout of the estimators at the long-run by performing simulation study. The Bayes estimator under SELF (£sc ) outperforms MLE (eMC) and Bayes estimator under GELF (£gc) for the true value of parameter e = 1.5 whatever the seriousness i.e. over estimation is more serious than under estimation and reversely. In all other considered cases, Bayes estimator under GELF (£gc) outperforms MLE (emc) and Bayes estimator under SELF (£sc) when under estimation is more serious than over estimation but in reverse case Bayes estimator under SELF (£sc ) outperform MLE (eMC) and Bayes estimator under GELF (£gc). Finally, we see that risks under GELF decreases as sample informations (n & r) increases.
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