# Reliability Analysis of Small-Sample Failure Data for Random Truncation High-Voltage Relay
## 2 Principle of Random Truncation Small-Sample Reliability Analysis
### Optimal Selection of Empirical Distribution Functions  
Assuming \(X_{1},X_{2},\cdots,X_{n}\) is an independent and identically distributed sample from population \(X\), the \(i\)th order statistic of this sample is denoted as \(X_{(i)}\). \(x_{1},x_{2},\cdots,x_{n}\) are observations of \(X_{1},X_{2},\cdots,X_{n}\). We arrange them in ascending order, \(x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}\), where \(x_{(i)}\) is \(i\)th order statistic. The commonly defined empirical distribution function \(Q_{n}(x)\) is as follows:  
\[Q_{n}(x)=\begin{cases}0,x\leq x_{(1)};\\ \frac{k}{n},x_{(k)}\leq x\leq x_{(k+1)},k=1,2,\cdots,n-1;\\ 1,x_{(n)}\leq x.\end{cases} \tag{2}\]  
where \(n\) is the number of the samples; \(k\) is the ordinal number of order statistics.  
From Equation (2), it can be seen that \(Q_{n}(x)\) is a non-decreasing left continuous function with \(Q_{n}(-\infty)=0,Q_{n}(+\infty)=1\). Another empirical distribution function, \(G_{n}(x)\), recommended by reference [17] is as follows:  
\[G_{n}(x)=\left\{\begin{array}{ll}0,\ x\leq x_{(1)};\\ \frac{k-0.375}{n+0.25},\ x_{(k)}\leq x\leq x_{(k+1)},k=1,2,\cdots,n-1;\\ \frac{k-0.375}{n+0.25},\ x_{(n)}\leq x.\end{array}\right. \tag{3}\]  
According to (2) and (3), the general expression for the empirical distribution function is as follows:  
\[M_{n}(x)=\begin{cases}m_{0},\ x\leq x_{(1)};\\ m_{k},\ x_{(k)}\leq x\leq x_{(k+1)},k=1,2,\cdots,n-1;\\ m_{n},\ x_{(n)}\leq x.\end{cases} \tag{4}\]  
In the study of probability graphs, there are many expressions for \(m_{k}\), and the commonly used ones of \(m_{k}\) include expectation estimation \(m_{k}=k/(n+1)\), midpoint estimation \(m_{k}=\left(k-0.5\right)/n\), median rank estimation \(m_{k}=\left(k-0.3\right)/(n+0.4)\), and approximate median rank estimation \(m_{k}=\left(i+1\right)/(n+2)\), where \(m_{0}=0\)[18].  
The selection of \(m_{k}\) depends on the approximation degree between the empirical distribution \(M_{n}(x)\) and the overall distribution \(F(x)\). The maximum linear distance \(d_{n}(M)=\sup\lvert M_{n}(x)-F(x)\rvert\) is intuitive and has practical significance. However, it only considers the points of extreme values and does not examine the approximation degree between \(M_{n}(x)\) and \(F(x)\) from an overall perspective. The square distance \(L(M_{n},F)=\int(M_{n}-F)^{2}dF\) examines the approximation degree from an overall perspective and reflects the difference between \(M_{n}(x)\) and \(F(x)\) more comprehensively. However, square distance depends on the value of the sample, resulting in randomness in the outcome. Therefore, in the average sense, this article introduces the average square distance to measure the overall approximation degree of the sample, which can overcome the impact of small-sample fault data randomness [18]. The average square distance is calculated as follows:  
\[R(M_{n})=E(L(M_{n},F))=\int\ldots\int L(M_{n},F)dP_{n}(x_{1},x_{2},\cdots,x_{ n}) \tag{5}\]  
where \(P_{n}(x_{1},x_{2},\cdots,x_{n})\) is the joint distribution function of \(\left(x_{(1)},x_{(2)},\cdots,x_{(n)}\right)\).  
Therefore, the \(m_{k}\) with the minimum average square distance is the optimal empirical distribution function.