STATISTICAL ANALYSIS TECHNIQUE OF TWO-PARAMETER GENERALIZED EXPONENTIAL SUM DISTRIBUTION∗

A new life distribution is proposed, known as “two-parameter generalized exponential sum distribution”. We study the density function and failure rate function, the average failure rate function, the image features and the numerical characteristics of the mean residual life of the distribution. Several methods of calculating point estimation of parameters are discussed. Through the Monte-Carlo simulation, we compare the precision of the point estimations. In our opinion, the best linear unbiased estimation is the most optimal solution of these methods. At the same time, several methods of calculating parameters of interval estimations are given. We also discuss the precision of interval estimations by Monte-Carlo simulation and use the best linear unbiased estimation and the best linear invariant estimation to construct interval estimations which are better than other estimation method. Finally, several simulation examples and a case of maintaining tanks is used to illustrate the application of the methods presented in this paper.


Introduction
A function has the following form: where β > 0 is the scale parameter and m > 0 is the shape parameter.
Conspicuously, if X ∼ GES(β; m), the density function f (x), the reliability functionF (x), the failure rate function λ(x), the average failure rate function G(x) and the mean residual life function m(x) of X are as follows, respectively.
Obviously, if X ∼ GES(β; m), let Y = (X/β) m , then Y ∼ GES(1; 1). This paper proposes a new life distribution, known as the "two-parameter generalized exponential sum distribution", and its typical characteristics is that the failure rate function takes the shape of "inverted bathtub" for 1/2 < m < 1. We study the density function and failure rate function, the average failure rate function, the image features and the numerical characteristics of the mean residual life of the distribution. Through the Monte-Carlo simulation, we compare the precision of the point estimations. In our opinion, the best linear unbiased estimation is the most optimal solution of these methods. At the same time, several methods of calculating parameters of interval estimations are given. We also discuss the precision of interval estimations by Monte-Carlo simulation and use the best linear unbiased estimation and the best linear invariant estimation to construct interval estimations which are better than other methods. Finally, several simulation examples and a case of maintaining tanks are used to illustrate the application of the method in this paper.
2. "Two-parameter generalized exponential sum distribution", "Three-parameter generalized exponential sum distribution" and their application background First of all, we explain the origin of "two-parameter generalized exponential sum distribution" and "three-parameter generalized exponential sum distribution".
Definition 2.1. Suppose random variables X, Y are independent of each other, and if X ∼ Exp(λ 1 ), Y ∼ Exp(λ 2 ), then the distribution of random variable Z = X + Y is an "exponential sum distribution".
Obviously, the distribution function F Z (z) and the density function f Z (z) are Particularly, when λ 1 = λ 2 = λ, F Z (z) = 1 − e −λz − λze −λz , f Z (z) = λ 2 ze −λz . It is noteworthy that "exponential sum distribution" actually corresponds to the life distribution of cold-standby system which is made up of two independent exponential distribution units. But Ding Yong in the literature [2] names the distribution of the above referred to as "exponential difference distribution", the main reason is that its density function contains the difference between two exponential functions. The literature [7] calls it "sub-exponential distribution" and points out that the computer system's I/O service time of the operation usually follows the "sub-exponential distribution". In this paper we think that the definition: exponential sum distribution is more appropriate. The literature [2] also discusses the "exponential sum distribution" of extremum, inflection point, mathematical expectation and variance, etc. The moment estimation of parameters is given, as well as the relationship between the "exponential sum distribution" and exponential distribution.
From the above, we introduce a shape parameter to the "exponential sum distribution" to get another distribution and name it as "three-parameter generalized exponential sum distribution". Definition 2.2. If the distribution function of a non-negative continuous random variable X obeys "three-parameter generalized exponential sum distribution", denoted as X ∼ GES(β 1 , β 2 ; m), its distribution function F (x) and density function f (x) have the following forms: where m > 0 is the shape parameter and β 1 > 0, β 2 > 0 are the scale parameters.
This distribution is called "three-parameter generalized exponential sum distribution", if λ 1 = 1/β m 1 , λ 2 = 1/β m 2 , the forms of distribution function F (x) and density function f (x) will become: If m = 1, then the distribution is "exponential sum distribution". Furthermore, it is worth noting that F (x) = 1 − e −(x/β2) m for β 1 → 0, which is two-parameter Weibull distribution with shape-scale parameters. That is to say, "three-parameter generalized exponential sum distribution" can also be regarded as the generalization of two-parameter Weibull distribution.
If β 1 = β 2 = β, X obeys "two-parameter generalized exponential sum distribution", X ∼ GES(β; m). In fact, if λ 2 = λ 1 = λ = 1/β m , then We can see from the above, if the shape parameter is m = 1, the "threeparameter generalized exponential sum distribution" is the "exponential sum distribution" at the moment. Next, we will illustrate the application of "three-parameter generalized exponential sum distribution" in pharmacokinetics by using this special case of "exponential sum distribution".
The classic pharmacokinetic is based on the compartment model. According to the literature [5] and [13], the linear orally or intramuscular injection -the compartment model, the relationship between drug concentration and time in the body can be modeled by: where D stands for dosage of drugs, F is the fraction that is absorbed by the body, V is the apparent volume of distribution of the body, λ a stands for constant rate of drug absorption, λ is the constant rate of elimination of drugs. Under normal circumstances, drug absorption is faster than elimination, hence λ a > λ.
To avoid the influence due to selections of compartment numbers on the model, the statistical moment theory is applied to the pharmacokinetic study. The theory tells us that when drugs enter the organisms, the individual differences, biochemical drugs, pharmacology and other random factors affect the length of residence time of each drug molecule in vivo, which can be treated as a random variable and it reflects the drug in the body of the absorption, distribution and elimination of the corresponding overall effect. To convert the medicine-time curve to the probability density curve, f (t) = c(t) AU C is defined as the probability density of the retention time of drug in the body, where AU C = +∞ 0 c(t)dt is the area under the medicine-time curve. Then, which is exactly the "exponential sum distribution". The average retention time of drug in the body is its mathematical expectation +∞ 0 tf (t)dt = 1/λ a +1/λ, which is the sum of the average absorption time 1/λ a and the mean time to eliminate drugs 1/λ. The variance of residence time is 1/λ 2 a + 1/λ 2 , and the drug concentration reaches their peak (maximum point) is (ln λ a − ln λ)/(λ a − λ), which is exactly the reciprocal of logarithmic average of two rate constants λ a and λ.
Next, we illustrate its application in the maintenance theory by a particular case of the "two-parameter generalized exponential sum distribution".
For the "two-parameter generalized exponential sum distribution", let m = 1, β = β 0 /2, that is the Zrlanga distribution, and the distribution function and density function are: Russia introduced the Zrlanga distribution when they studied the time of repairing weapons and equipment, and this distribution plays an important role in equipment maintenance theory. The literature [8] analyses the characteristics of Zrlanga distribution. Besides, the maximum likelihood method is used to estimate the parameters of the distribution in the full sample situation, and the examples are used to verify the feasibility and practicability of this distribution. The principle of the simulation to the combat damage parts of armored equipment and the computing method of working time for fixing the combat damage parts of armored equipment has been given in literature [11], according to the need of the prediction of maintenance support in wartime, computer simulation is carried out in a certain type of tanks. Besides, the simulation data of repairing tank's time is produced, the time of repairing injured tank is found to obey Zrlanga distribution by statistical analysis. In the literature [3], they study the interval estimation of small sample and test problems of Zrlanga distribution and point out that the estimated accuracy of the time of equipment maintenance with Zrlanga distribution estimation is higher than exponential distribution. Literature [4] gives the maximum likelihood estimation of the parameters under the type-II censoring and investigates the estimation precision through a lot of Monte-Carlo simulations. Secondly, the inverse moment estimations of the parameters are given in the full sample and compared with the moment estimations and maximum likelihood estimations, the moment estimations and maximum likelihood estimations are slightly better than the inverse moment estimations; Finally, the precise interval estimations and the approximate interval estimations of the parameters are acquired. Comparing the accuracy of these two kinds of interval estimations, the precise interval estimation is better than the approximate interval estimation.
3. Some typical characteristics of the "two-parameter generalized exponential sum distribution" It is easy to obtain Theorem 3.1 and Theorem 3.2 as following.
Theorem 3.1. Suppose the non-negative continuous random variable X ∼ GES(β; m), the image of its density function has the following characteristics: (1) when 0 < m ≤ 1/2, f (x) is monotone decreasing; (2) when m > 1/2, f (x) is increased at first and then decreased.
Theorem 3.2. Suppose the non-negative continuous random variable X ∼ GES(β; m), the image of its failure rate function λ(x) has the following characteristics: (1) when m ≥ 1, λ(x) is monotone increasing and belongs to increasing failure rate; (2) when 0 < m ≤ 1/2, λ(x) is monotone decreasing and belongs to decreasing failure rate; (3) when 1/2 < m < 1, λ(x) is "inverted bathtub" shape with its vertex at The "inverted bathtub" shape of failure rate function is more common in practical problems, such as the "logarithmic normal distribution", which has a wide application in the reliability engineering, and the shape of its failure rate function shows "inverted bathtub". Literature [9] and [6] point out that the "inverted bathtub" shaped of failure rate model can be used to fit the number of survival analysis of organisms (including humans) and so on. Taking into account the bottom of the "inverted bathtub" shape which is the stable stage of the failure rate function, it plays a key role in practical application. Next, the width of the upper bottom of the "inverted bathtub" (referred to as the upper bottom width) is defined as: is the failure rate function of the non-negative continuous random variable X, which is of "inverted bathtub" shape, we obtain its maximum value at x = x 0 . For any given ε > 0, consider the difference l(ε) = x 2 −x 1 between the two roots x 1 , x 2 (x 1 < x 0 < x 2 ) that meets λ(x) − λ(x 0 ) = ε, which reflects the width of the upper bottom of "inverted bathtub" under the given precision ε, namely the upper bottom width. Theorem 3.3. Suppose the non-negative continuous random variable X ∼ GES(β; m), when 1/2 < m < 1, the failure rate function λ(x) is shape of "inverted bathtub", for any given ε > 0, the width l(ε) of the upper bottom of "inverted bathtub" is proportional to the value of the parameter β.
Proof. Since λ(x) has the shape of "inverted bathtub" when 1/2 < m < 1, for any given ε > 0, there are two roots which satisfy λ(x) − λ(x 0 ) = ε, namely x 1 , x 2 (0 < x 1 < x 2 ), and The two roots are t 1 , t 2 , it is easy to see that when β increases, the right side of the equation also increases, then t 2 − t 1 also increases. So the width l(ε) = x 2 − x 1 of the upper bottom of "inverted bathtub" also increases, then l(ε) is proportional to the value of the parameter β.
Theorem 3.4. Suppose the non-negative continuous random variable X ∼ GES(β; m), the image of its average failure rate function G(x) has the following characteristics: (1) when m ≥ 1, λ(x) is monotone increasing, and it belongs to increasing failure rate average; (2) when 0 < m ≤ 1/2, G(x) is monotone decreasing and it belongs to decreasing average failure rate; (3) when 1/2 < m < 1, G(x) is increased at first and then decreased and its shape presents an "inverted bathtub". Proof.
is monotone increasing and belongs to increasing average failure rate.
Theorem 3.5. Suppose the non-negative continuous random variable X ∼ GES(β; m), the image of its mean residual life function m(x) has the following characteristics: is decreased at first and then increased and it takes on a shape of "bathtub". Proof.
increases at first and then decreases. Thus, there exists a unique decreases at first and then increases.
Theorem 3.6. Suppose the non-negative continuous random variable X obeys the "two-parameter generalized exponential sum distribution" GES(β; m), then 4. Parameter estimation of the "two-parameter generalized exponential sum distribution" 4.1. Moment estimation of parameters in the full samplemethod one Let X 1 , X 2 , · · · , X n be a simple random sample of size n from X ∼ GES(β; m), the sample observations of them are x 1 , x 2 , · · · , x n , the sample mean, the secondary moment and the sample variance are denoted asX = 1 2 respectively the observations of them are denoted byx, x 2 , s 2 n , respectively. The following equations can be established by the method of moment estimation: After simplification, the following transcendental equation with the only parameter m is obtained as follows: Γ(2 + 2/m)Γ −2 (2 + 1/m) = X 2 /X 2 , and the moment estimation of the parameter m is obtained from the solution, which is denoted asm 1 . Then we get the moment estimation of parameter β, denoted asβ 1 = XΓ −1 (2 + 1/m 1 ).
There is only one positive real root on the transcendental equation Furthermore, is uniform convergent and strictly monotone increasing in x, then G(x) is strictly monotone increasing in x. And as well, lim x→+∞ Owing to X 2 /X 2 > 1, the equation Γ(2 + 2/m)Γ −2 (2 + 1/m) = X 2 /X 2 has only one positive real root.

The maximum likelihood estimation (MLE) and interval estimation of parameters under the type-II censoring sample-method two
(1) The maximum likelihood estimation of parameters under the type-II censoring sample Suppose the life of the product X ∼ GES(β; m), the product life test will be stopped until the r products out of n products fail, namely the type-II censoring life test, thereby obtaining the first r order failure data x (1) ≤ x (2) ≤ · · · ≤ x (r) , the likelihood function is Simplify the second equation, it becomes then substitute it into the first equation and simplify the following transcendental equation with the only parameter m:g(m) = 0, where Through a lot of Monte-Carlo simulations, the conclusion can be found that the equation g(m) = 0 has a unique real root. Hence, the root of the transcendental equation g(m) = 0 of parameter m is the maximum likelihood estimationm 2 , then the maximum likelihood estimation of parameter β iŝ .
(2) The maximum likelihood estimation and interval estimation of parameters under the full sample.
In the full sample case, that is to say r = n, the likelihood function is We can work out β m = 1 2n n i=1 x m i from the above formula, and the transcendental equation has only the parameter m The maximum likelihood estimation of the parameter m can be obtained from the above equation, denote asm 2 . Besides, the maximum likelihood estimation of the parameterβ can be obtainedβ has only one positive real root.
Proof. Sorting the sample observations x 1 , x 2 , · · · , x n from smallest to largest, denote as x (1) ≤ x (2) ≤ · · · ≤ x (n) , in this case the equation becomes Therefore, g (m) > 0, furthermore the equation g(m) = 1 n n i=1 ln x (i) has only one positive root. Lemma 4.3. Suppose the non-negative continuous random variable X ∼ GES(β; m), X 1 , X 2 , · · · , X n is a simple random sample of size n from population X, the sample observations of them are denoted as x 1 , x 2 , · · · , x n ,m 2 is the maximum likelihood estimation of the parameter m, thenm 2 /m,m 2 (lnβ 2 − ln β) are pivotal quantities.
Proof.m 2 is the root of the following equation: Y n are independently and obey GES(1; 1), and ln y (i) .
Thus,m 2 /m is a pivotal quantity, as well aŝ Denote a α , a α as the upper α quartiles of the pivotal quantitiesm 2 /m,m 2 (lnβ 2 − ln β), respectively. We can get the upper α quartiles with different sample size n through 10000 times Monte-Carlo simulations.

The inverse moment estimation and interval estimation of parameters under the full sample-method three
can also be denoted by It's easy to see lim m→0 g(m) = 1/n, and if X (n−k+1) = X (n−k+2) = · · · = X (n) , then lim m→+∞ g(m) = 1/k, According to Cauchy mean value theorem, there exists ξ, min(X i , X j ) < ξ < max(X i , X j ), such that ln Xi−ln Xj X m i −X m j = 1 mξ m > 0, then g (m) > 0 and usually 1/n < 3/(2n) < 1/k, so the equation has only one positive root. Lemma 4.5. Suppose the non-negative continuous random variable X ∼ GES (β; m), X 1 , X 2 , · · · , X n is a simple random sample of size n from population X, then is a pivotal quantity, which is strictly monotone increasing in m.
The upper α quartile of the pivotal quantity Then the interval estimation of parameter m of the confidence level 1 − α is obtained: [m 31 ,m 32 ], wherem 31 ,m 32 are the roots of the

The inverse moment estimation and interval estimation of parameters under the type-II censoring sample-method four
Suppose X (1) ≤ X (2) ≤ · · · ≤ X (r) are the first r order statistic of a sample of size n from the population X ∼ GES(β, m), using the method of literature [10] to construct the pivotal quantity of parameter m.
Then, we can easily know ln Sr Si is a pivotal quantity. Let . Let is monotone increasing in m and ln Sr Si is also monotone increasing in m.
At the same time it is easy to see, for any c > 0, the equation ln Sr Si = c has a unique positive real root.
Denote the mean value of the pivotal quantity r−1 i=1 ln Sr Si by c, the inverse moment estimate of parameter m ism 4 , which is the root of the equation And is a pivotal quantity, we can get the mean value c of it by numerical simulation. If now it is in the full sample size case, i.e. r = n, at this time c = 2n. So we can establish the following equation according to the literature [6]: Proof. Sincem 4 is the root of the equation We can see thatm 4 (lnβ 4 − ln β) is a pivotal quantity from the above equation.
Denote c α , c α as the upper α quartiles of the pivotal quantitiesm 4 /m,m 4 (lnβ 4 − ln β), respectively, that is P (m 4 /m > c α ) = α, we can get the upper α quartiles for different sample size n and truncating failure number r through 10000 times Monte-Carlo simulation.

The best linear unbiased estimation(BLUE) and interval estimation of parameters under the type-II censoring sample-method five
Suppose X (1) ≤ X (2) ≤ · · · ≤ X (r) are the first r order statistics of a sample size n from the population X ∼ GES(β, m), let Z = ln X, µ = ln β, σ = 1/m, then Therefore, Z is the distribution of location-scale parameter. Thus, according to Gauss-Markov theorem, the best linear unbiased estimations (BLUE) of parameters σ, µ areσ Particularly, in the full sample size r = n, C(n, r, j) is denoted by C(n, j) and D(n, r, j) is denoted by D(n, j), then the best linear unbiased estimator (BLUE) of the parameters β, m arê Lemma 4.8. Suppose X (1) ≤ X (2) ≤ · · · ≤ X (r) are the first r order statistic of a sample size n from the population X ∼ GES(β, m),σ/σ, (μ − µ)/σ are the pivotal quantities.

The best linear invariant estimation (BLIE) and the interval estimation of parameters under the type-II censoring sample-method six
It is easy to see that the best linear invariant estimation (BLIE) of parameters σ, µ areσ The coefficients of the best linear invariant estimation of the parameters σ, µ are C I (n, r, j) = C(n, r, j)/(1 + l rn ), D I (n, r, j) = D(n, r, j) − C(n, r, j)B rn /(1 + l rn ).
Then, the estimations of the parameters β, m arê In particular, in the full sample size r = n,C I (n, r, j) is denoted by C I (n, j) and D I (n, r, j) is denoted by D I (n, j), then the invariant estimations of the parameters β, m arê where C I (n, j) = C(n, j)/(1 + l nn ), D I (n, j) = D(n, j) − C(n, j)B nn /(1 + l nn ).
The upper α quartiles of the pivotal quantityσ/σ, (μ − µ)/σ are denoted by e α , e α , respectively. We can obtain the upper α quartiles for different sample size n and truncating failure number r though 10000 times Monte-Carlo simulation.
Moreover, the approximate maximum likelihood estimation of the parameters β, m areβ  Table 1 and Table 2. We can see: (1) when the sample size increases, the mean square error of the various point estimation method gradually decreases; (2) for all point estimations of parameter m, the mean square error of BLUE is relatively smaller, and mean square error of each point estimation method is gradually approaching with the increase of sample size; (3) for all point estimations of parameter β, the mean square error of BLUE is relatively smaller, and mean square error of each point estimation method is gradually approaching with the increase of sample size.

Simulation comparison of interval estimation of parameter
In the full sample, the accuracy of interval estimation of parameter is investigated by Monte-Carlo simulations. Taking the true value of parameters β = 1, m = 1, 1000 times Monte-Carlo simulations are carried out under different sample sizes. At the confidence level of 0.95, we calculate the average lower limit, the average upper limit, the average interval length and the number of the intervals containing the true value of parameter. The simulation results are shown in Table 3 and Table 4. It can be found that it is better to use the best linear unbiased estimation and the best linear invariant estimation to construct interval estimation when the sample size is smaller from the view of the average interval length. That is to say, the method five and method six of interval estimations of parameters m and β are better. Besides, the average interval length of each interval estimation method is gradually close as the sample size increases.
4.11. Case analysis-the estimation of parameter for the distribution regularity of maintenance time of a certain type tank The literature [8] investigates a type of tank in the process of maintenance, after the 47 observations of the basic level I preventive maintenance of class two maintenance time, and the field observations are (unit: hours) For "two-parameter generalized exponential sum distribution", let m = 1, β = β 0 /2, that is Zrlanga distribution, and the distribution function and density function are F (x) = 1 − (1 + 2x/β 0 ) e −2x/β0 , f (x) = (4x/β 2 0 )e −2x/β0 , x > 0, β 0 > 0.
The literature [8] considers that the associated fault repair maintenance time of a certain type of tank obeys Zrlanga distribution by χ 2 test, and the maximum likelihood estimation of parameter β 0 isβ 0 = 12.1. In the literature [4], the inverse moment estimation of parameter β 0 is given by 5.469, and the precise interval estimation is [4.5104, 6.762] and the approximate interval estimation is [4.5416, 6.8432] at the confidence level 95%.
Parameter estimation and interval estimation are as follows.
(1) We can get each point estimation of parameters m, β by using the method presented in this paper, as shown in Table 5. From the above point estimates of the parameters, we can find that the point estimation of parameter m is close to 1 and the point estimation of parameter β is close toβ 0 /2 = 2.7297. From another point of view, it is illustrates that it is reliable to fit the above data by using the GES(β; m) distribution.
(2) At a confidence level of 95%, we can get each interval estimation of parameters m, β by using the method presented in this paper, as shown in Table 6.