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54
Type 1 Topp-Leone q—Exponential Distribution and its
Applications
N icy Sebastian1, Jeena Joseph1 and Princy T2
•
department of Statistics, St Thomas College, Thrissur, Kerala, India- 680 001 nicycms@gmail.com, sony.jeena@gmail.com 2Department of Statistics, Cochin University of Science and Technology Cochin, Kerala, India 682 022 princyt.t@gmail.com
Abstract
The main purpose of this paper is to discuss a new lifetime distribution, called the type 1 Topp-Leone generated q-exponential distribution(Type 1 TLqE). Using the quantile approach various distributional properties, L—moments, order statistics, and reliability properties were established. We suggested a new reliability test plan, which is more advantageous and helps in making optimal decisions when the lifetimes follow this distribution. The new test plan is applied to illustrate its use in industrial contexts. Finally, we proved empirically the importance and the flexibility of the new model in model building by using a real data set.
Keywords: Type 1 Topp-Leone generated q-exponential distribution, Quantile density function, Quantile function, Hazar d quantile function, L—moments, Reliability Test Plan.
1. Introduction
There are many statistical distributions which plays an important role in modeling survival and life time data such as exponential, weibull, logistic etc. Almost all these distributions with unbounded support. But there are situations in real life, in which observations can take values only in a limited range such as percentages, proportions or fractions. Papke and Wooldridge [12] claims that in many economic settings, such as fraction of total weekly hours spent working, pension plan participation rates, industry market shares, fraction of land area allocated to agricultur e etc., the variable bounded between zero and one. Thus it is important to have models defined on the unit interval in order to have reasonable results.
A new distribution was introduced in 1955, called Topp Leone (TL) distribution, defined on finite support, proposed Topp and Leone [20] and used it as a model for failure data. A random variable X is distributed as the TL with parameter a denoted by x ~ TL(a), with a cumulativ e distribution function
Ftl(x) = xa (2 — x)a,0 < x < 1, a > 0. (1)
The corresponding probability function is
fTL (x) = 2axa—1 (1 — x)(2 — x)a—1. (2)
Topp Leone distribution provides closed forms of cumulativ e density function (cdf) and the probability density function (pdf) and describes empirical data with J-shaped histogram such as powered tool band failures, automatic calculating machine failure. The Topp Leone distribution
had been received little attention until Nadarajah and Kotz [10] discovered it. Further information and application of TL distribution can be obtained from Ghitany et al. [2], Kotz and Seier [7].
Lifetime data plays an important role in a wide range of applications such as medical, engineering an social sciences. When ther e is a need for mor e flexible distributions, almost all resear chers are about to use the new one with more generalization. An excellent review of Lee et al. [9] has provided through knowledge of several methods for generating families of continuous univariate distributions. They discussed some noticeable developments after 1980, are method of generating ske w distributions, beta generated method, method of adding parameters, transfor med-transfor mer method, and composite method. The beta generated (BG) family of distributions belongs to a parameter adding method (Lee et al. [9], Kumarasw amy [8]). In a similar manner the relation of a random variable X having the TLG distribution and a random variable T having TL distribution is X = G-1 (T), with T ~ TL(a). This relation demonstrates that the pdf of TL distribution, (2), is transfor med into a new pdf through the function G() The cdf of TL generated random variable X is defined as
r G(x)
Ftlg (x) = h(t)dt Jo
wher e h(t) is the pdf of TL variable and G(x) is the cdf of any arbitrar y random variable. Thus the cdf of TL generated random variable is
r G(x)
Ftlg(x) = 2a ta-1 (1 - t)(2 - t)a-1 dt = G(x)a(2 - G(x))a. (3)
0
By differentiating, we get the corresponding pdf,
fTLG(x) = 2ag(x)(1 - G(x))G(x)a-1 (2 - G(x))a-1, a > 0. (4)
In reliability analysis, a frequently used distribution is exponential distribution having the characterizing property of constant hazar d function. Due to this, exponential distribution is sometimes not suitable for analyzing data. This implies the need for more generalization. In such situations we use distribution called Topp-Leone Exponential distribution (TLE). TLE distribution comes as the combination of TL distribution and exponential distribution. Her e TL distribution is the generator and exponential is the parent distribution. Sangsanit and Bodhisuw an [16] presented the Topp-Leone generated exponential (TLE) distribution as an example of the ToppLeone generated distribution. A random variable X possessing TLE distribution having cdf and probability function defined respectiv ely as
Ftle(x) = (1 - exp(-Ax))a(2 - (1 - exp(-Ax)))a = (1 - exp(-2Ax))a
and
fTLE(x) = 2aAexp(-2Ax)(1 - exp(-2Ax))a-1,
wher e a is the shape parameter and A is the scale parameter .
Various entropy measur es have been developed by mathematicians and physicists to describe several phenomena, depending on the field and the context in which it is being used. Tsallis [19], introduced a generalization of the Boltzmann-Gibbs entropy. Tsallis statistics have found applications in many areas such as physics, chemistr y, biology, medicine, economics, geophysics,
etc. By maximizing Tsallis entropy, subject to certain constraints, leads to the Tsallis distribution,
1
also known as q-exponential distribution, which has the form f (x) = c[1 - (1 - q)x](1-q) wher e c is the normalizing constant. Various applications and generalizations of the q-exponential distribution are given in Picoli et al. [13]. In the limit q ^ 1, q-entropy converges to Boltzmann-Gibbs entropy.
An important characteristic of q-exponential distribution is that it has two parameters q and A providing more flexibility with regard to its decay, differently from exponential distribution. The q exponential distribution is defined by its cdf and pdf as,
(2-q)
F1qE (x) = 1 - [1 - (1 - q)Ax] 1-q . (5)
f1qE(x) = (2 - q)A[l - (1 - q)Ax] 1-q ,1 - A(1 - q)x > 0, A > 0, q < 2, q = 0 (6)
The parameter q is known as entropy index. As q ^ 1, the q-exponential distribution becomes exponential distribution. In that sense q-exponential distribution is a generalization of exponential distribution. The parameters q and A determine how quickly the pdf decays. In the reliability context, an important characteristic of the q-exponential distribution is its hazar d rate, which is not necessarily constant as in exponential distribution.
The rest of the paper is organized as follows. In section 2 we will discuss the Type 1 Topp-Leone q-Exponential Distribution. In section 3 consists of the quantile properties of Type 1 TLqE distribution. Section 4, we described a new reliability test plan for type 1 TLqE distribution and its applications are also discussed. In Section 5, we apply the Type 1 TLqE distribution to a real data sets to sho w that it can be used quite effectiv ely in analyzing lifetime data. Finally, concluding remarks and futur e work are addr essed in Section 6.
2. Type 1 Topp-Leone q-Exponential Distribution
In this section we discuss the type 1 Topp-Leone generated q- exponential (TLqE) distribution introduced by combining the TL distribution with q- exponential distribution, for more details see Sebastian et al. [17]. Substituting (5) and (6) in (3) and (4) respectiv ely we will get the distribution function and density function of TLqE distribution as follows:
F1TLqE(x) = {1 - [1 - (1 - q)Ax] 1-q'}a,x > 0, A,a > 0, q < 2, q = 0.
and
3-q n(2-q\
f 1 TLqE(x) = 2aA(2 - q)[1 - (1 - q)Ax]^ {1 - [1 - (1 - q)Ax]2(^)}a-1,
2( 2-i)
where 1 - [1 - (1 - q)Ax] (1-q) > 0, A, a > 0, q < 2.
Figure 1: Plots ofF(x) of TLqE distribution for a = 1.1, A = 0.1 (left) and for A = 0.3, q = 1.1 (right).
Nicy Sebastian, Jeena Joseph and Princy T RT&A, No 3 (69) Type 1 Topp-Leone q-Exponential Distribution_Volume I7, September 2022
Figure 2: Plots off(x) ofTLqE distribution for a = 1.1, A = 0.1 (left) and for A = 0.3, q = 1.1 (right).
In Figure 1 and Figure 2, we can see the plots of cdf and pdf of TLqE for different values of the shape parameters a and q. The sur viv al function, the probability density function and the Hazar d function are the three important functions that characterize the distribution of the survival times. Here
S(x) = 1 - {1 - [1 - (1 - q)Ax]2{ 2-q)}a,
and
3-q
2( ) ■]2( 1-q '
h(x) = 2aA(2-q)[1-(1-q)Ax] 1-q {1-[1-(1-q)Ax] v 1-q 7}a-1
( ) 2( ) 1-{1-[1-(1-q)Ax] (1-q '}a
respectiv ely are the survival and the hazar d function of TLqE distribution. Figure 3, gives the plots of h(x) of TLqE distribution for different values of the shape parameters a and q.
Figure 3: Plots ofh(x) ofTLqE distribution for a = 1.1, A = 0.1 (left) and for A = 0.3, q = 1.1 (right). As q ^ 1 then f1TLqE (x) goes to
f3TLqE(x) = 2aAe-2Ax (1 - e-2Ax)A, a > 0, x > 0, and the correspond cdf is
(7)
F3TLqE(x) = (1 - e-1XxY , x > 0, Я, a > 0.
3. Quantile properties of type 1 Topp-Leone q-Exponential Distribution
3.1. Distributional characteristics
In modelling and analysis of statistical data, probability distribution can be specified either in terms of distribution function or by the quantile function. Quantile functions have several
interesting properties that are not shared by distributions, which makes it more convenient for analysis. For example, the sum of two quantile functions is again a quantile function. For a nonnegativ e random variable X with distribution function F(x), the quantile function Q(u) is defined by (see Nair et al. [11])
Q(u) = F-1 (u) = inf {x : F(x) > u}, 0 < u < 1 (8)
For every -ro < x < ro and 0 < u < 1, we have
F(x) > u if and only if Q(u) < x.
Thus, if there exists an x such that F(x) = u, then F(Q(u)) = u and Q(u) is the smallest value of x satisfying F(x) = u. Further , if F(x) is continuous and strictly increasing, Q(u) is the unique value x such that F(x) = u, and so by solving the equation F(x) = u, we can find x in terms of u which is the quantile function of X.
By using inversion method, we can generate a randon variate from TLqE distribution. We have alr eady seen that the relationship betw een a random variable X, having TLqE distribution, and a random variable T, having the TL distribution, is
X = G-1 (t)
1-q
= 1 - (1 - t) 2-q
(1 - q)A (9)
where G-1 (■) is related to inversion of the q-exponential cdf. The quantile function of the TL distribution is _
t = 1 1 - u a, (10)
wher e u is picked from the uniform distribution over (0,1). Then the quantile function of type 1 TLqE distribution is obtained by using equation (9) and (10),
—X ( ^ )
A(2-q)
1 - V 1 - u
la
Q(u)= (1 - q)A-' q < 2. (11)
The quantile-based measur es of the distributional characteristics like location, dispersion, skewness, and kurtosis are useful for estimating parameters of the model by matching population characteristics with corresponding sample characteristics. We can obtain the median as Median = Q(2). Dispersion is measur ed by the interquartile range, IQR = Q(3) - Q( 1). Skewness is
measur ed by Galton's coefficient, S = Q(3)+IQQ(j|) 2M. Moors proposed a measur e of kurtosis as,
T = Q( 7 )-Q( 8 )+Q( 3 )-Q( 1) Q
1 = IQR .
If f (x) is the probability function of X, then f (Q(u)) is called the density quantile function. The derivative of Q(u),
q(u) = Q' (u),
is known as the quantile density function of X. If F(x) is right continuous and strictly increasing, we have
F(Q(u)) = u (12)
so that F(x) = u implies x = Q(u). When f(x) is the probability density function (PDF) of X; we have from (12)
q(u)f (Q(u)) = 1 (13)
Nicy Sebastian, Jeena Joseph and Princy T RT&A, No 3 (69) Type 1 Topp-Leone q-Exponential Distribution_Volume 17, September 2022
Quantile function has several properties that are not shared by distribution function. See Nair et
al. [11] for details. Now the qunatile density of type 1 TLqE distribution is obtained as
1 q-3
"(u) = 2«Ä(2-"j-L'- u0 ^ (14)
For the proposed family of distribution, the density function f (x) can be written in terms of the distribution function as
f (x) = 2a\(2 - q)-U--. (15)
'1 - (F(x)) a)2(2-q)
For all values of the parameters, the density is strictly decr easing in x and it tends to zer o as x ^ to.
3.2. L—moments
The L-moments are often found to be more desirable than the conventional moments in describing the characteristics of the distributions as well as for infer ence. A unified theor y and a systematic study on L— moments have been presented by Hosking [3]. The rth L— moment is given by
f1 r-1
E (-1y-1-klr - Л (r - 1 + k\.,k
k=0
k
ukQ(u)du.
(16)
Theorem 1. For the type 1 TLqE distribution, the rth L- moment can be obtained by using the follo wing recurr ence relation,
r— 1
Lr = E (-1)
k=0
r-1-k r - 1 k
r - 1 + k\ a
k JW-V)
B(1, ka) - B(21-q.Î + 1, ka)
(17)
So we can evaluate the L—coefficient of variation (т2), analogous to the coefficient of variation based on ordinary moments is given by, т2 = j^. Similarly the L—coefficient of skewness, (т3) and kurtosis, (т4) of type 1 Topp-Leone generated q-expo nential quantile function respectiv ely can be obtained as т3 = and т4 =
3.3. Order statistics of type 1 Topp-Leone q-Exponential Distribution
If Xr :n is the rth order statistic in a random sample of size n, then the density function of Xr:n can be written as
fr(x) = B(r, n — r + 1) f (x)F(x)r—1 (1 — F(x))n—r (18)
From Eq.(15), we have
fr (x)
2aЯ(2 - q) F(x)r-1 (1 - F(x))n
. q-3
1 - (F(x))^ ^
B(r, n - r + 1)
(19)
Hence,
Цг.п = E(XrM ) = J xfr (x)dx
2aЯ(2 - q) B(r, n - r + 1)
xF(x)r-a (1 - F(x))n-r dx.
1 - (F(x))
q-3
2(2-q)
In quantile terms, we have
E(Xr:n )
2^A(2 - q) r1 Qu ur-a (1 - u)
lo (i - (u) a)
B(r, n - r + 1)
q-3 2(2-7
du.
For the type 1 TLqE distribution, the first-or der statistic X1:n has the quantile function
Qi(u) = Q(i - (1 - u)n)
A(1 - q)
1 - (1 - (1 - u)n
, (1-q) 11 2(2-q)
and the nth order statistic Xn:n has the quantile function
Qn (u) = Q(un)
A(1 - q)
1 - (1 - una )
(1-q) 2(2-q)
(21)
(22)
(23)
3.4. Hazar d quantile function
One of the basic concepts emplo yed for modeling and analysis of lifetime data is the hazar d rate. In a quantile setup, Nair et al. [11] defined the hazard quantile function, which is equivalent to the hazar d rate. The hazar d quantile function H(u) is defined as
H(u) = h(Q(u))) = (1 - u)-1 fQ(u) = [(1 - u)q(u)]-1. (24)
Thus H(u) can be interpreted as the conditional probability of failur e of a unit in the next small interval of time given the survival of the unit until 100(1 - u)% point of the distribution. Note that H(u) uniquely deter mines the distribution using the identity ,
"u dp
Q("> = Jo (1 - p)H(p).
The hazar d quantile functions of type 1 TLqE distribution is
H(u) = (1 - u)
1
2aA(2 - q)
u a
1 — u a
q-3
-1
(25)
(26)
with H(0) = ro and H(1) = 0. Plots of hazar d quantile function for different values of parameters are given in figur e 4.
nr
1
1
Figure 4: Plots of hazard quantile function
No. Parameter region Shape of hazar d quantile function
1 0 < a < 1 and q < 2 Decr easing hazar d rate (DHR)
2 a > 0 and q < 2 Upside-do wn Bathtub
3 a = 1, q < 2 and A = 0 Constant
4 a = 1 and q < 2 DHR
Table 1: Behavior of the hazard quantile function for different regions of parameter space.
3.5. Mean residual quantile function
Another concept used in reliability is that of residual life Xt = (X — t|X > t) with survival function
Ft(x) = F(t + x)/ F(t), x > 0,0 < t < T. The mean residual life function is then
/TO
F(x)dx.
Accor dingly, the mean residual quantile function is defined by Nair et al. [11] as
M(u) = mQ(u) = (l — u)—1 jl (Q(t) — Q(u))dt (27)
Ju
which is the average remaining life beyond the 100(1 — u)% point of the distribution. For the type 1 TLqE distribution, M(u) has the form
M(u)
) (+ «) + (' - u'")»
A(1 — q)
where Bu (a, b) = f^ xa—1 (1 — x)b—1 dx is the incomplete beta function.
3.6. Reversed hazar d quantile function
The reversed hazard quantile function is (Nair et al. [11]) defined by
(28)
A(u) = (29)
uq(u)
and it deter mines the distribution through the formula
{u 1
Q(u)=J0 pmdv. (30)
For type 1 TLqE distribution,
1 q—3
A(u) = q(u) = 20!^ u *——u ^ . (31)
4. Reliability Test Plan
Acceptance sampling plan is an inspection procedur e used to determine whether to accept or reject a specific quantity of material. (See Kantam et al. [6], Rao et al. [15], Jose and Joseph [4], Joseph and Jose [5] etc.) If it is applied to a series of lots, it prescribes a procedur e that will give a specified pr obability of accepting lots of giv en quality .
In statistical quality contr ol, acceptance sampling plan is concer ned with the inspection of a sample of products taken from a lot and the decision whether to accept or reject the lot based on the quality of the product. Here we discuss the reliability test, with its operating characteristic function plan for accepting or rejecting a lot wher e the lifetime of the product follows type 1 Topp - Leone q— exponential distribution. In a life testing experiment, the procedur e is to terminate the test by a pr edeter mined time 't' and note the number of failur es. If the number of failur es at the end of time 't' does not exceed a given number 'c', called acceptance number then we accept the lot with a given probability of at least 'p'. But if the number of failures exceeds 'c' before
1
Nicy Sebastian, Jeena Joseph and Princy T RT&A, No 3 (69) Type 1 Topp-Leone q—Exponential Distribution_Volume I7, September 2022
time 't', we reject the lot. For such truncated life test and the associated decision rule, we are interested to obtain the smallest sample size to make at a decision. Even though a large number of distributions belonging to Topp-Leone generated family have been developed with wide range of applications, none of these have been applied in acceptance sampling to develop reliability test plans.This motivated the present study.
Assume that the lifetime of a product T follows the type 1 Topp-Leone q- exponential distribution with cumulativ e distribution function (cdf)
t 2(2-)
F(t) = {1 - [1 - (1 - q) -P 1-qj }a, t > 0, A, a > 0, q < 2. (32)
Let A0 be the required minimum average life time and the shape parameters a and q are known. Then
FTLqE (t; a, q, A) < GjiqE(t; a, q, A0) ^ A > A0. (33)
A sampling plan is specified by the number of units n on test, the acceptance number c, the maximum test duration t and the minimum average lifetime represented by A0.
The probability of accepting a bad lot (consumer 's risk) should not exceed the value 1 - p*, wher e p* is a lower bound for the probability that a lot of true value A below A0 is rejected by the sampling plan. For fixed p* the sampling plan is characterized by (n, c, t/ A0). Binomial distribution can be used to find the acceptance probability for sufficiently large lots. The aim is to deter mine the smallest positiv e integer n for giv en values of c and t/ A0 such that
n
L(Po) = £(nVi(1 - Po)n-i - 1 - P* (34)
i=o V'/
wher e p0 = FiLqE (t; a, q, A0) given by (32) which indicates failur e probability before time 't' which depends only on the ratio t/ A0. The operating characteristic function L(p) is the acceptance probability of the lot as a function of the failure probability p(A) = FiLqE (t; a, q, A)
The average life time of the product is increasing with A and the failur e probability p(A) decr eases implying that the operating characteristic function is increasing in A. The minimum values of n satisfying (34) are obtained for a = 2, q = 1.1 and p*=0.75, 0.95, 0.99 and t/ A0 = 0.248, 0.361, 0.482, 0.602, 0.903, 1.204, 1.505 and 1.806. The results are displayed in Table 2.
If p0 = FiLqE (t; a, q, A0) is small and n is very large, the binomial probability may be approximated by Poisson probability with parameter 0 = np0 so that (34) becomes
c 0'
L1 (p0) = E 0!e-0 < 1 - p* (35)
i=0
The minimum values of n satisfying (35) are obtained for the same combination of values of a, q, p* and t/ A0 and are displa yed in Table 3.
The operating characteristic function of the sampling plan (n,c,t/ A0) gives the probability L(p) of accepting the lot with
L(p) = E(nV0!(1 - p0)n-i (36)
i=0 V
wher e p = F(t, A) is consider ed as a function of A.
Table 2: Minimum sample size using binomial approximation
p* c t/ Ac
0.248 0.361 0.482 0.602 0.903 1.204 1.505 1.806
0 11 6 4 3 2 1 1 1
1 22 12 8 6 4 3 3 2
2 32 17 11 9 6 4 4 4
3 41 22 15 11 7 6 5 5
4 51 27 18 14 9 7 6 6
0.75 5 60 33 22 17 11 9 8 7
6 70 38 25 19 13 10 9 8
7 79 43 29 22 14 12 10 9
8 88 48 32 24 16 13 11 11
9 97 52 35 27 18 14 13 12
10 106 57 39 30 20 16 14 13
0 24 12 8 6 4 3 2 2
1 38 20 13 10 6 4 4 3
2 50 27 17 13 8 6 5 5
3 62 33 22 16 10 8 7 6
4 73 39 26 19 12 9 8 7
0.95 5 84 45 30 22 14 11 9 8
6 95 51 34 25 16 12 11 10
7 106 56 37 28 18 14 12 11
8 116 62 41 31 20 15 13 12
9 126 68 45 34 22 17 15 13
10 136 73 49 37 24 18 16 14
0 36 19 12 9 5 4 3 2
1 52 27 18 13 13 6 5 4
2 66 35 23 17 17 8 6 5
3 80 42 27 20 20 9 8 7
4 92 49 32 24 24 11 9 8
0.99 5 104 55 36 27 27 13 11 9
6 116 61 41 30 30 14 12 11
7 128 68 45 33 33 16 13 12
8 139 74 49 36 36 17 15 13
9 150 80 53 39 39 19 16 14
10 162 86 57 42 42 21 17 16
Table 3: Minimum sample size using poisson approximation
p* c t/ A0
0.248 0.361 0.482 0.602 0.903 1.204 1.505 1.806
0 12 7 5 4 3 2 2 2
1 23 13 9 7 5 4 4 3
2 33 18 13 10 7 6 5 5
3 43 23 16 13 9 7 7 6
4 52 29 20 15 11 9 8 7
0.75 5 62 34 23 18 12 10 9 9
6 71 39 27 21 14 12 11 10
7 80 44 30 23 16 13 12 11
8 89 49 34 26 18 15 13 12
9 99 54 37 29 20 16 15 14
10 108 59 40 31 21 18 16 15
0 25 14 10 8 5 4 4 4
1 40 22 15 12 8 7 6 6
2 52 29 20 15 11 9 8 7
3 64 36 24 19 13 11 10 9
4 76 42 29 22 15 13 11 11
0.95 5 87 48 33 25 17 14 13 12
6 98 54 37 28 20 16 14 14
7 109 60 41 32 22 18 16 15
8 119 65 45 35 24 20 18 17
9 130 71 49 38 26 21 19 18
10 140 77 52 41 28 23 21 19
0 38 21 15 11 8 7 9 6
1 55 30 21 16 11 9 12 8
2 70 38 26 20 14 12 15 10
3 83 46 31 24 17 14 18 12
4 96 53 36 28 19 16 21 13
0.99 5 109 59 41 31 22 18 24 15
6 120 66 45 35 24 20 26 17
7 132 72 49 38 26 22 29 18
8 144 79 54 42 28 23 31 20
9 156 85 58 45 31 25 34 21
10 167 91 62 48 33 27 36 23
Nicy Sebastian, Jeena Joseph and Princy T RT&A, No 3 (69) Type 1 Topp-Leone q-Exponential Distribution_Volume I7, September 2022
Table 4: Values of the Operating Characteristic function for the sampling plan (n,c,t/A0)
A/ A0
p* n c t/ A0 2 2.5 3 3.5 4 4.5 5
32 2 0.241 0.8821 0.9541 0.9804 0.9909 0.9954 0.9975 0.9986
17 2 0.361 0.8665 0.9453 0.9758 0.9884 0.9940 0.9967 0.9981
11 2 0.482 0.8594 0.9403 0.9728 0.9866 0.9930 0.9961 0.9977
0.75 9 2 0.602 0.8113 0.9144 0.9591 0.9792 0.9888 0.9937 0.9962
6 2 0.903 0.7429 0.8711 0.9334 0.9640 0.9797 0.9880 0.9927
4 2 1.204 0.7926 0.8952 0.9449 0.9696 0.9825 0.9895 0.9935
4 2 1.505 0.6387 0.7926 0.8801 0.9291 0.9568 0.9729 0.9825
4 2 1.806 0.4854 0.6703 0.7926 0.8688 0.9158 0.9449 0.9632
50 2 0.241 0.7096 0.8689 0.9389 0.9699 0.9843 0.9913 0.9949
27 2 0.361 0.6638 0.8383 0.9208 0.9595 0.9782 0.9877 0.9928
17 2 0.482 0.6579 0.8306 0.9149 0.9555 0.9756 0.9861 0.9917
0.95 13 2 0.602 0.6111 0.7968 0.8936 0.9425 0.9677 0.9812 0.9886
8 2 0.903 0.5499 0.7448 0.8567 0.9181 0.9518 0.9708 0.9817
6 2 1.204 0.4955 0.6944 0.8184 0.8911 0.9334 0.9582 0.9731
5 2 1.505 0.4409 0.6410 0.7756 0.8595 0.9109 0.9423 0.9619
5 2 1.806 0.2792 0.4788 0.6410 0.7572 0.8359 0.8883 0.9231
66 2 0.241 0.5459 0.7689 0.8838 0.9398 0.9675 0.9817 0.9892
35 2 0.361 0.4989 0.7292 0.8572 0.9232 0.9573 0.9753 0.9852
23 2 0.482 0.4582 0.6921 0.8307 0.9059 0.9462 0.9683 0.9806
0.99 17 2 0.602 0.4242 0.6588 0.8055 0.8887 0.9349 0.9608 0.9757
17 2 0.903 0.3109 0.5421 0.7126 0.8221 0.8892 0.9298 0.9546
8 2 1.204 0.2681 0.4858 0.6596 0.7792 0.8567 0.9061 0.9374
6 2 1.505 0.2866 0.4955 0.6612 0.7764 0.8523 0.9015 0.9334
5 2 1.806 0.2792 0.4788 0.6410 0.7572 0.8359 0.8883 0.9231
The values of n and c are determined by means of operating characteristics (OC) function for given value of p* and t/ A0 are displa yed in Table 4 by considering the fact that p = F(/ A^).
The producer's risk is the probability of rejecting a lot when A > Ao.We can compute the producer's risk by first finding p = F(t; A) and then using the binomial distributi on function. For the given value of producer's risk say 0.05 we obtain p from the sampling plan given in Table 1 subject to the condition that
E(T)po!(1 - po)n-i ^ 0.95 (37)
i=o Vi '
The minimum value of A/ A0 satisfying (37) for the sampling plan (n,c,t/ A0) and for the given p* are listed in Table 5.
4.1. Explanation of the tables
Assume that the lifetime follows type 1 TLqE distribution with a=2 and q=1.1. Suppose that the experimenter is interested in establishing that the true unknown average life is at least 1000 hours with confidence p* = 0.75. It is desir ed to stop the experiment at t = 602 hours. Then, for an acceptance number c = 2, the required n is 9 (Table 2) . If during 602 hours, no more than 2 failures out of 9 are observed, then the experimenter can assert that the average life is at least 1000 hours with a confidence level of 0.75. If the Poisson appr oximation to binomial probability is used, the value of n is 10 ( Table 3) . For this sampling plan (n = 9, c = 2, t/ A0=0.602) under the type 1 TLqE distribution, the operating characteristic values from Table 3 are given below. Comparing with Reliability Test Plans for Marshall- Olkin Extended Exponential distribution (see Rao et al. [15]), for a=2, acceptance number c=9, for the specified ratio t/ A0=0.482 and confidence level p* =0.75, the minimum sample size is 49 using binomial approximation, whereas for type 1 TLqE distribution it is 35. Similarly , if we are considering each value of c and each vale of t/ A0, the scaled termination time is unifor mly smaller than those for the present reliability test plans.This improvement makes the new test plan more advantageous and helps in making optimal decisions.
Table 5: Minimum ratio of true A and required A0 for the acceptability of a lot with producer's risk of 0.05 for a = 2, q = 1.1
p* c t/ Ac
0.241 0.361 0.482 0.602 0.903 1.204 1.505 1.806
0 6.6277 6.9167 7.4538 8.5749 10.0789 9.02959 10.8794 12.9175
1 3.3083 3.3984 3.6148 3.8266 4.5121 4.9581 6.1036 5.0452
2 2.5084 2.6096 2.6871 2.9107 3.3462 3.1597 3.8907 4.6549
0.75 3 2.1518 2.7074 2.3305 2.3933 2.5223 2.9791 3.0960 3.6261
4 1.9886 2.0125 2.0707 2.2308 2.3344 2.4468 2.5469 3.0898
5 1.8547 1.9365 1.9647 2.0913 2.2343 2.4174 2.6911 2.7487
6 1.7632 1.8542 1.8795 1.9327 2.0847 2.1529 2.3735 2.4319
7 1.7019 1.7454 1.8105 1.8628 1.9042 2.1529 2.1756 2.2671
8 1.6283 1.6914 1.7471 1.7379 1.8576 2.0156 2.0471 2.4124
9 1.5945 1.6413 1.6744 1.7379 1.8576 1.8908 2.1317 2.3129
10 1.5624 1.5946 1.6406 1.7093 1.8130 1.8908 2.0063 2.1783
0 9.2357 10.6374 11.5682 12.5609 13.9643 16.5278 16.0189 19.2227
1 4.3401 4.7867 4.9101 5.1202 5.8541 6.0161 7.2767 7.3615
2 3.1957 3.3984 3.5699 3.7542 4.1055 4.3640 4.8318 5.6912
0.95 3 2.6899 2.8940 3.0059 3.1139 3.2874 3.8654 4.2043 4.4687
4 2.4550 2.5494 2.6871 2.7369 2.9548 3.0668 3.5219 3.7152
5 2.2687 2.3403 2.4084 2.5180 2.6971 2.8961 3.0217 3.2294
6 2.1163 2.1899 2.2584 2.3362 2.4825 2.5905 3.0217 3.2294
7 2.0186 2.0965 2.1290 2.1820 2.3698 2.5389 2.7524 2.9591
8 1.9320 2.0125 2.0707 2.0913 2.3001 2.3603 2.5195 2.7204
9 1.8547 1.9365 1.9647 2.0490 2.1721 2.3055 2.5195 2.5581
10 1.8074 1.8806 1.9163 2.0086 2.1721 2.2019 2.3635 2.4565
0 12.2925 13.5082 13.2554 14.7459 17.3020 20.5092 20.6597 21.2298
1 5.3685 5.5826 5.9984 6.0065 9.0097 7.6532 8.7403 9.0242
2 3.7269 3.9328 4.1647 4.3518 6.5277 5.3182 5.4550 5.7982
0.99 3 3.1957 3.2789 3.4039 3.4961 5.2442 4.1826 4.8318 5.0452
4 2.7589 2.8940 3.0059 3.1819 4.6708 3.7256 3.8335 4.2262
5 2.5084 2.6096 2.7086 2.8206 4.2051 3.3635 3.7239 3.6261
6 2.3569 2.4134 2.5856 2.6591 3.9886 2.9791 3.2546 3.6261
7 2.2279 2.3172 2.4084 2.4538 3.6622 2.8961 2.9504 3.3029
8 2.1163 2.2099 2.2865 2.3251 3.4877 2.6037 2.9504 3.0234
9 2.0498 2.1326 2.1914 2.2308 3.3462 2.5389 2.7524 2.8362
10 1.9886 2.0619 2.1290 2.1820 3.2308 2.5389 2.5748 2.8362
Table 6: Values of the operating characteristic function L(p) for values of A/ A0 .
A/ A0 2 2.5 3 3.5 4 4.5 5
L(p) 0.8113 0.9144 0.9591 0.9792 0.9888 0.9937 0.9962
4.2. Application
Consider the following ordered failur e times of the release of a softw are given in terms of hours from starting of the execution of the software up to the time at which a failur e of the software is occurred (see Wood [21]). This data can be regarded as an ordered sample of size n = 9 consisting of the observations {254, 788, 1054, 1393, 2216, 2880, 3593, 4281, 5180}. Let the required average lifetime be 1000 hours and the testing time be t = 602 hours, which leads to ratio of t/ A0 = 0.602 with a corresponding sample size n = 9 and an acceptance number c = 2, which are obtained from Table 1 for p* = 0.75. Therefore, the sampling plan for the above sample data is (n =9, c = 2, t/ A0 = 0.602). Based on the observations, we have to decide whether to accept the product or reject it. We accept the product only if the number of failures before 602 hours is less than or equal to 2. However, the confidence level is assur ed by the sampling plan only if the given life times follow type 1 TLqE distribution. In order to confirm that the given sample is generated by lifetimes following the type 1 TLqE distribution, we have compared the sample quantiles and the corresponding population quantiles and found a satisfactor y agreement. Thus, the adoptio n of the decision rule of the sampling plan seems to be justified. In the sample of 9 units, there is only one failur e at 254 hours before t = 602 hours. Therefore we accept the product.
5. N umerical Illustration
The data consists of the number of successiv e failur e for the air conditioning system reported of each member in a fleet of thirteen Boeing 720 jet air planes. The pooled data with 214 observations was consider ed by Proschan [14]. 50, 130, 487, 57, 102, 15, 14, 10, 57, 320, 261, 51, 44, 9, 254, 493, 33, 18, 209, 41, 58, 60, 48, 56, 87, 11, 102, 12, 5, 14, 14, 29, 37, 186, 29, 104, 7, 4, 72, 270, 283, 7, 61, 100, 61, 502, 220, 120, 141, 22, 603, 35, 98, 54, 100, 11, 181, 65, 49, 12, 239, 14, 18, 39, 3, 12, 5, 32, 9, 438, 43, 134, 184, 20, 386, 182, 71, 80, 188, 230, 152, 5, 36, 79, 59, 33, 246, 1, 79, 3, 27, 201, 84, 27, 156, 21, 16, 88, 130, 14, 118, 44, 15, 42, 106, 46, 230, 26, 59, 153, 104, 20, 206, 5, 66, 34, 29, 26, 35, 5, 82, 31, 118, 326, 12, 54, 36, 34, 18, 25, 120, 31, 22, 18, 216, 139, 67, 310, 3, 46, 210, 57, 76, 14, 111, 97, 62, 39, 30, 7, 44, 11, 63, 23, 22, 23, 14, 18, 13, 34, 16, 18, 130, 90, 163, 208, 1, 24, 70, 16, 101, 52, 208, 95, 62, 11, 191, 14, 71.
Table 7: The values of estimated parameters ofDataset 1
The model Estimate and Standar d Error (in paranthesis) of Dataset 1
Type 1 TLqE TLE E a=1.2150 (0.2422),A=0.0149 (0.0079), q=1.3883(0.1279) a=0.9036 (0.0884), A=0.0052 (0.0005) A =0.0112 (0.0008)
Table 8: Goodness of fit of collection of different distributions for the data set.
AIC CAIC BIC HQIC A* w* K-S p value -log L
Type 1 TLqE 1964.01 1964.18 1973.60 1967.92 0.46 0.06 0.04 0.78 979.02
TLE 1968.37 1968.44 1974.74 1970.95 1.37 0.25 0.07 0.28 982.18
E 1967.47 1967.49 1970.65 1968.76 2.08 0.40 0.08 0.13 982.73
Figure 5: The P - P plot and estimated pdf of data set.
According to the Table 8 the type 1 TLqE model is more appropriate as compared to the TLE and exponential distribution.
6. Conclusion and future work
Introduced a quantile function associated with the type 1 Topp-Leone q-exponential distribution. The estimation of parameters of the model using L—moments is studied. Also, a reliability test plan was derived on the basis that the lifetime distribution of the test item follows the type 1 TLqE distribution. Besides, we find the minimum sample size needed for the acceptance or rejection of a lot based on percentiles. Some useful tables were provided and applied to establish the test plan. The new test plan is applied to illustrate its use in industrial contexts. We proved empirically the importance and flexibility of the new model in the model building by using a real data set. One can develop a parallel theory for type 2 TLqE distribution using the type 2 beta generated form given in Sebastian et al. [18].
Acknowledgement
The authors have no conflict of interest to declare. The first author would like to thank the Resear ch Council, St Thomas College, Thrissur for the financial assistance for this work under project number F.No.STC/SANTHOME/SEEDMONEY/2020-21/09.
References
[1] Gell-Mann, M. and Tsallis, C. (Eds.) Nonextensiv e Entropy: Inter disciplinar y Applications, Oxford University Press, New York, 2004.
[2] Ghitany M. E. , Kotz, S. And Xie, M. (2005). On some reliability measur es and their stochastic orderings for the Topp-Leone distribution, Journal of Applied Statistics, 32(7): 715-722.
[3] Hosking, J. R. (1990). L—Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics. Journal of the Royal Statistical Society: Series B (Methodological), 52: 105-124.
[4] Jose, K. K., Joseph, J. (2018). Reliability test plan for the gumbel-unifor m distribution. Stochastics and Quality Control, 33(1): 71-81.
[5] Joseph, J., Jose, K. K. (2021). Reliability Test Plan for Gumbel-Par eto Life Time Model. International Journal of Statistics and Reliability Engineering, 8(1): 121-131.
[6] Kantam, R. R. L., Rosaiah, K. and Rao, G. S. (2001). Acceptance sampling based on life tests: log-logistic model, Journal of Applied Statistics, 28 (1): 121-128.
[7] Kotz, S. and Seier, E. (2007). Kurtosis of the Topp-Leone distributions. Interstat, 1: 1-15.
[8] Kumarasw amy, P. (1980). Generalized probability density-function for double-bounded random processes. Journal of Hydrology, 46: 79-88.
[9] Lee, C., Famoye, F. and Alzaatr eh, A. Y. (2013). Methods of generating families of univariate continuous distributions in the recent decades. WIREs Comput Stat, 5: 219-238.
[10] Nadarajah, S. and Kotz, S. (2003). Moments of some J-shaped distributions. Journal of Applied Statistics, 30: 311-317.
[11] Nair, N. Unnikrishnan, Sankaran, P. G. and Balakrishnan, N. Quantile-Based Reliability Analysis: Statistics for Industr y and Technology . Springer , New York, 2013.
[12] Papke, L. and Wooldridge, J. (1996). Econometric Methods for Fractional Response Variables with an Application to 401(K) Plan Participation Rates, Journal of Applied Econometrics, 11( 6): 619-632.
[13] Picoli, S., Mendes, R. S. and Malacarne, L. C. (2003). q-exponential, Weibull and q- Weibull distributions: an empirical analysis, Physica A, 324(3): 678-688.
[14] Proschan, F. (1963). Theoretical explanation of observed decreasing failur e rate. Technometrics, 5(3): 375-383.
[15] Rao, S. G., Ghitany, M. E., Kantam, R. R. L. (2009). Reliability Test Plans for Marshall-Olkin Extended Exponential Distribution, Applied Mathematical Sciences, 3 (55): 2745-2755.
[16] Sangsanit, Y. and Bodhisuw an, W. (2016). The topp-leone generator of distributions: properties and inferences. Songklanakarin Journal of Science and Technology, 38(5): 537-548.
[17] Sebastian, N. Rasin R. S. and Silviy a P. O. (2019). Topp-Leone Generator Distributions and its Applications, Proceedings of National Conference on Advances in Statistical Methods, 127-139.
[18] Sebastian, N., Nair, S. S. and Joseph, D. P. (2015). An Overview of the Pathw ay Idea and Its Applications in Statistical and Physical Sciences, Axioms, 4(1): 530-553.
[19] Tsallis, C. (1988). Possible generalizations of Boltzmann-Gibbs statistics, Journal of Statistical Physics, 52: 479-487.
[20] Topp, C. W. and Leone, F. C. (1955). A family of J-shaped frequency function. Journal of the American Statistical Association, 50(269): 209-219.
[21] Wood, A. (1996). Predicting softw are reliability , IEEE Transactions on Software Engineering, 22: 69-77.
