STATISTICAL ANALYSIS OF TWO-PARAMETER GENERALIZED BIRNBAUM-SAUNDERS CAUCHY DISTRIBUTION∗

The image features of density function and failure rate function are studied in detail for two-parameter generalized Birnbaum-Saunders Cauchy fatigue life distribution. The logarithmic moment estimation and other two point estimations of parameters are proposed under full sample, and the precisions of point estimations are investigated by Monte-Carlo simulations. The approximate interval estimations of parameters are given by using Taylor expansion, and the precisions of approximate interval estimations are investigated by Monte-Carlo simulations. Finally, several examples show the feasibility of the methods.


Introduction
Birnbaum-Saunders model is an important failure distribution model in the probabilistic physical method, which is deduced by Birnbaum and Saunders in 1969 when they studied on the material failure process caused by dominant crack propagation. It is widely applied in the study of mechanical production reliability, mainly used in the study of fatigue failure. Besides, it has the important application in the failure analysis of electronic products performance degradation.
Suppose that T follows two-parameter Birnbaum-Saunders fatigue life distribution BS(α, β), its distribution function and density function are respectively where α > 0 is a shape parameter, β > 0 is a scale parameter and φ(x), Φ(x) are respectively density function and distribution function of standard normal distribution, that is, φ(x) = 1 √ 2π e − x 2 2 , Φ(x) = x −∞ φ(y)dy. Birnbaum-Saunders fatigue life distribution is derived from basic characteristics of fatigue process, so it is more suitable for describing the life regularity of several fatigue failure products than common life distributions such as Weibull distribution and log-normal distribution. Besides, it has become one of common distributions in reliability statistical analysis.
Due to the relations between the two-parameters BS distribution BS(α, β) and standard normal distribution N (0, 1) which is a symmetric distribution, if N (0, 1) is replaced by other symmetric distribution, then the obtained distribution is called generalized BS distribution. For example, if N (0, 1) is replaced by standard Laplace distribution, that is, The researches of this kind of generalized BS distributions see reference [1,[3][4][5][6][7][8]12]. They mostly refer to image features of density function and failure rate function, numerical characteristics and the discussion on MLE of parameter. Xiaojun Zhu and Balakrishnan [12] made a further analysis on GBS − Laplace(α, β) in 2015, and proved that MLE is existent and unique, but the process of proof is not perfect. Ronghua Wang [10] proposed the test statistics of fitting test and two new approximate interval estimation methods of environmental factor for two-parameter BS fatigue life distribution.

Image Features of Density Function and Failure
Rate Function for Two-parameter GBS − Cauchy (α, β) Distribution Suppose non-negative continuous random variable X follows two-parameter Birnbaum-Saunders Cauchy fatigue life distribution GBS − Cauchy(α, β), its distribution func-tion F X (x) and density function f X (x) are respectively
,then the equation g 1 (x) = 0 has no positive root. We have , and then f (x) is strictly monotonic decreasing.

When
is strictly monotonic decreasing.
, and we have the following conclusions:

) is a strictly monotonic increasing function, then λ(t)
is strictly monotonic increasing;

and η(t) is strictly monotonic in-
creasing and then strictly monotonic decreasing, that is, inverse-bathtub shape, then λ(t) may be inverse-bathtub shaped, or strictly monotonic decreasing; (iv) If there is t 0 , t 0 > 0 that satisfies η ′ (t 0 ) = 0 , and η(t) is strictly monotonic decreasing and then strictly monotonic increasing, that is, bathtub shape, then λ(t) may be bathtub shaped, or strictly monotonic increasing.
Proof. The failure rate function is Since β is the scale parameter, we choose β = 1 without loss of generality. Then we have Let the function be
Then the image of λ(x) is strictly monotonic decreasing.
Remark 2.1. By drawing the images of λ(x), it can be concluded that the image of λ(x) is firstly strictly monotonic decreasing and then strictly monotonic increasing and finally strictly monotonic decreasing again for α ≤ 1.1 ; while the image of λ(x) is gradually strictly monotonic decreasing for α > 1.1 .That is, with the increase of shape parameter α , the image of λ(x) gradually changes from firstly strictly monotonic decreasing, then strictly monotonic increasing and finally strictly monotonic decreasing again to strictly monotonic decreasing.

Quantile estimations and maximum likelihood estimations of parameters
Since F (β) = Φ(0) = 0.5, the point estimationβ 1 of scale parameter β can be the sample median, that is, when n is an even number, when n is an odd number. Since Let ∂ ln L(α,β) ∂α = 0, and we get the function n α − After simplifying, we have n If the point estimation of β is the sample medianβ 1 , then the point estimationα 1 of shape parameter α is the root of the following equation

Regression estimations of parameters
Let dQ(β) dβ = 0, and we have the equation , and the point estimationα 2 , of shape parameter α is the root of the following equation

Logarithmic moment estimations of parameters
Let Y = ln X, Y i = ln X i , i = 1, 2, · · · , n, and µ = ln β, then we have Therefore when k is an odd number, we know E(Z k ) = 0.
When k is an even number, we know αt + √ α 2 t 2 + 4 2 The above equation is an integral equation, and it is complex to solve it. Then we prove that it has unique positive root.

Lemma 3.2. The equation
has unique positive root with respect to α.

Proof. Let the function be g(α)
1+t 2 dt, α > 0. Firstly we prove that the function g(α) is convergent. Since Then according to Lemma 3.1, we know that the function g(α) is convergent. Next we prove that the equation g(α) = 0 has unique positive root. Since and Y 2 −Ȳ 2 > 0, we know that the equation g(α) = 0 has unique positive root.

of population X is limited. If the forth derivative of the function h(x) is existent and limited, then we have
Theorem 3.1.β 3 is asymptotic unbiased estimation and consistent estimation of β.
Let the function be h(x) = e x , and any order derivation of h(x) is still e x . Then we have .

It is obvious that lim
Thereforeβ 3 is asymptotic unbiased estimation and consistent estimation of β . In order to compare the precisions of various point estimations of parameters α, β , we choose the sample size n = 10(5)30 and the truth values of parameters α = 1, β = 1. Then we obtain the samples from GBS − Cauchy(α, β) by 1000 Monte-Carlo simulations, and calculate mean values and mean square errors of various point estimations for parameters α, β. The results are shown in Table 2. It can be concluded that quantile estimation and maximum likelihood estimation α 1 ,β 1 are best.

Approximate Interval Estimations of Parameters for Two-parameter Distribution GBS−Cauchy(α, β)
Let the parameter be µ = ln β, and it is denoted by Y = ln X, Y i = ln X i , i = 1, 2, · · · , n. Then Y 1 , Y 2 , · · · , Y n is a simple random sample from the distribution } with sample size n, and its order statistics are denoted by Y (1) , Y (2) , · · · , Y (n) .
The first order Taylor expansion of 1 That is, Y = ln X can be approximately regarded as two-parameter Cauchy distribution with location-scale parameters. Let Z = Y −µ α , Z i = Yi−µ α , i = 1, 2, · · · , n, and then Z approximately follows standard Cauchy distribution C(0, 1). Z 1 , Z 2 , · · · , Z n follow the same distribution as a simple random sample from standard Cauchy distribution C(0, 1) with sample size n , and it is sorted from small to large, which is denoted by Z (1) , Z (2) , · · · , Z (n) .

Approximate interval estimations of parameter β
The approximate interval estimation of parameter µ is obtained firstly, and then it is easy to obtain the approximate interval estimation of parameter β.
(1) When n is an even number, it is denoted by j = n 2 , n−j = n 2 , A = 1 j n i=j+1 Y (i) .
Then F(µ) is a pivot that only contains parameter µ. Besides, F(µ) is a strictly monotonic increasing function of µ, and we know Hence, for a given significance level α ′ £¬the upper 1 − α ′ /2, α ′ /2 quantiles of the pivot F(µ) are denoted by F 1−α ′ /2 and F α ′ /2 . Then it is obvious that the approximate interval estimation of parameter µ at the confidence level 1 − α ′ is Furthermore, the approximate interval estimation of parameter β at the confi- (2) When n is an odd number, it is denoted by j = n+1 , −∞ < µ < +∞, and we know .
Then F(µ) is a pivot that only contains parameter µ. Besides, F(µ) is a strictly monotonic increasing function of µ, and we know Hence, for a given significance level α ′ , the upper 1 − α ′ /2, α ′ /2 quantiles of the pivot F(µ) are denoted by F 1−α ′ /2 and F α ′ /2 . Then it is obvious that the approximate interval estimation of parameter µ at the confidence level Furthermore, the approximate interval estimation of parameter β at the confi- Let the sample size be n = 3 (1)

Approximate interval estimation of parameter α
The following pivot that only contains parameter α is constructed and we know Then T (α) is a pivot that only contains parameter α, and T (α) is a strictly monotonic decreasing function of α.
For a given significance level α ′ the upper 1 − α ′ /2, α ′ /2 quantiles of the pivot T (α) are denoted by T 1−α ′ /2 and T α ′ /2 . Then the interval estimation of parameter α at the confidence level 1 − α ′ is Let the sample size be n = 3(1)30 , and through 10000 Monte-Carlo simulations, the upper 0.99, 0.975, 0.95, 0.90, 0.85, 0.15, 0.10, 0.05, 0.025, 0.01 quantiles of T (α) are shown in Table 4. In order to investigate the precisions of approximate α, β interval estimations of parameters , we choose the sample size n = 10(1)15 and the truth values of parameters α = 1, β = 1. Then we obtain the samples from GBS − Cauchy(α, β) by 1000 Monte-Carlo simulations, and calculate mean lower limit, mean upper limit, mean interval length and the number of intervals that contain the truth values of approximate interval estimations for parameters α, β at the confidence level 0.95. The results are shown in Table 5.