If the random sample obeys two-component mixed generalized normal distribution, its logarithmic likelihood function is:

log L ( θ ) = ∑ j = 1 n log { λ ( s 1 2 σ 1 Γ ( 1 / s 1 ) ) exp { − | x j − μ 1 σ 1 | s 1 }       + ( 1 − λ ) ( s 2 2 σ 2 Γ ( 1 / s 2 ) ) exp { − | x j − μ 2 σ 2 | s 2 } } (4)

with reference to Wen et al. (2022) [4], the maximum likelihood estimation of the two-component mixed generalized normal distribution ECM algorithm in the case of complete data is given below. Set the sample X 1 , X 2 , ⋯ , X n with the capacity of n from the two-component mixed generalized normal distribution, x 1 , x 2 , ⋯ , x n are the sample observation values:

f i ( x i | λ , μ 1 , μ 2 , σ 1 , σ 2 , s 1 , s 2 ) = f i ( x i | θ ) = λ f 1 i + ( 1 − λ ) f 2 i (5)

where f 1 i = f 1 i ( x i | μ 1 , σ 1 , s 1 ) = s 1 2 σ 1 Γ ( 1 / s 1 ) exp { − | x i − μ 1 σ 1 | s 1 } , f 2 i = f 2 i ( x i | μ 2 , σ 2 , s 2 ) = s 2 2 σ 2 Γ ( 1 / s 2 ) exp { − | x i − μ 2 σ 2 | s 2 } .

The indicator function I i is introduced. Suppose it follows the two-point distribution:

P ( I i = 1 ) = λ , P ( I i = 0 ) = 1 − λ ,

since the X i of generalized normal distribution population from f 1 i or f 2 i is unknown, the joint distribution of X i and I i is: g ( x i , I i , θ ) = ( λ f 1 i ) I i [ ( 1 − λ ) f 2 i ] 1 − I i , and in a given case X i , the conditional distribution of I i is:

P ( I i = 1 | x i , θ ) = α f 1 i f i , P ( I i = 0 | x i , θ ) = ( 1 − α ) f 2 i f i .

E-Step: seek expectations.

Q ( θ , θ ( m − 1 ) ) = ∑ i = 1 n z 1 i ( m − 1 ) ( log λ s 1 2 σ 1 Γ ( 1 / s 1 ) − | x i − μ 1 σ 1 | s 1 )       + ∑ i = 1 n z 2 i ( m − 1 ) ( log ( 1 − λ ) s 2 2 σ 2 Γ ( 1 / s 2 ) − | x i − μ 2 σ 2 | s 2 ) . (6)

where z 1 i m − 1 = λ ( m − 1 ) f 1 i ( m − 1 ) f i ( m − 1 ) , z 2 i m − 1 = ( 1 − λ ( m − 1 ) ) f 2 i ( m − 1 ) f i ( m − 1 ) .

CM-Step: maximizing conditions.

Ψ ( 1 υ ) represents the digamma function and Ψ ′ ( 1 υ ) represents the trigamma

function. The iterative formula for the seven parameters of the two-component mixed generalized normal distribution is derived as follows:

{ λ ( m ) = ∑ i = 1 n z 1 i ( m − 1 ) ∑ j = 1 2 ∑ i = 1 n z j i ( m − 1 ) = ∑ i = 1 n z 1 i ( m − 1 ) ∑ i = 1 n ( z 1 i ( m − 1 ) + z 2 i ( m − 1 ) ) μ j ( m ) = μ j ( m − 1 ) + ∑ x i ≥ μ j ( m − 1 ) z j i ( m − 1 ) ( x i − μ j ( m − 1 ) ) s j ( m − 1 ) − 1 − ∑ x i < μ j ( m − 1 ) z j i ( m − 1 ) ( μ j ( m − 1 ) − x i ) s j ( m − 1 ) − 1 ( σ j ( m − 1 ) ) υ j ( m − 1 ) − 2 ( s j ( m − 1 ) − 1 ) ∑ i = 1 n z j i ( m − 1 ) | x i − μ j ( m − 1 ) σ j ( m − 1 ) | s j ( m − 1 ) − 2 ,       j = 1 , 2 σ j ( m ) = [ s j ( m − 1 ) ⋅ ∑ i = 1 n z i j ( m − 1 ) | x i − μ j ( m ) | s j ( m − 1 ) ∑ i = 1 n z i j ( m − 1 ) ] 1 s j ( m − 1 ) ,   j = 1 , 2 s j ( m ) = s j ( m − 1 ) − ∑ i = 1 n z i j ( m − 1 ) A 1 − ∑ i = 1 n z i j ( m − 1 ) A 2 ∑ i = 1 n z i j ( m − 1 ) A 3 − ∑ i = 1 n z i j ( m − 1 ) A 4 ,   j = 1 , 2 (7)

where

A 1 = 1 s j ( m − 1 ) ( 1 s j ( m − 1 ) Ψ ( 1 s j ( m − 1 ) ) + 1 ) , A 2 = | x i − μ j ( m ) σ j ( m ) | s j ( m - 1 ) log | x i − μ j ( m ) σ j ( m ) | , A 3 = − 1 s j 2 ( m − 1 ) [ 1 + 2 s j ( m − 1 ) Ψ ( 1 s j ( m − 1 ) ) + 1 s j 2 ( m − 1 ) Ψ ′ ( 1 s j ( m − 1 ) ) ] , A 4 = | x i − μ j ( m ) σ j ( m ) | s j ( m − 1 ) ( log | x i − μ j ( m ) σ j ( m ) | ) 2 .

On the basis of known observation data, numerical iterative method can be used to solve the above equations, because the transcendental equation is involved, the solution process is difficult. Two propositions of consistency and asymptotic normality for maximum likelihood estimation of two-component mixed generalized normal distribution are given below.

Proposition 1: (Consistency) For the two-component mixed generalized normal distribution, given arbitrarily θ 0 , its maximum likelihood estimates θ ^ are continuous in the interval, and then θ ^ converge to in probability θ 0 , that is θ ^ → p θ 0 .

It is proved that the parameters θ = ( α , μ 1 , σ 1 , s 1 , μ 2 , σ 2 , s 2 ) of the two-component mixed generalized normal distribution are assumed to be open sets:

Θ = ( 0 , 1 ) × ( − ∞ , + ∞ ) × ( 0 , + ∞ ) × ( 0 , + ∞ ) × ( − ∞ , + ∞ ) × ( 0 , + ∞ ) × ( 0 , + ∞ )

Let θ 0 = ( α 0 , μ 01 , σ 01 , s 01 , μ 02 , σ 02 , s 02 ) denote the true parameter value. Assume that for any θ 0 = ( α 0 , μ 01 , σ 01 , s 01 , μ 02 , σ 02 , s 02 ) , there is a compact set θ ⊂ Θ that satisfies:

1) θ 0 ∈ θ ,

2) ∀ ξ ≠ ξ 0 , ξ ∈ θ , f ( x i | θ ) ≠ f ( x i | θ 0 ) ,

3) ∀ ξ ∈ θ , log f ( x i | ξ ) is a continuous function,

4) E [ sup ξ | f ( x i | ξ ) | ] < ∞ .

According to the relevant theorem content of Newey and McFadden (1994) [9], it can be proved that the two-component mixed generalized normal distribution satisfies the above four conditions.

With reference to some bounded conditions given by Redner and Walker (1984) [10], such as the set value range with s 1 , s 2 , it can be proved that the maximum likelihood estimator of the two-component mixed generalized normal distribution satisfies the asymptotic normality.

Proposition 2: (Asymptotic normality) If υ 1 > 1 , υ 2 > 1 , I ( θ 0 ) is an information matrix, then the maximum likelihood estimate θ ^ of θ 0 satisfies the asymptotic normality, that is

T ( θ ^ − θ 0 ) → d N ( 0 , I − 1 ( θ 0 ) ) .

To prove proposition 2, refer to Redner and Walker (1984) [10] and McLachlan and Peel (2004) [11] for the derivation process. It should be noted that the information matrix expression of the two component mixed generalized normal distribution is more complex. In the process of seeking expectations, it is a difficult problem how to effectively simplify an analytic expression.

I ( θ 0 ) ≡ E [ ∂ ln f ∂ θ i ⋅ ∂ ln f ∂ θ j ] = − E [ ∂ 2 ln f ∂ θ i ∂ θ j ] .

Before the analysis of real data, numerical simulation experiments are conducted. In this section, we will evaluate the performance of ECM algorithm for parameter estimation of two-component mixed generalized normal distribution. First, the formula of skewness and mean square error is given:

Bias ( θ ^ ) = | 1 N ∑ j = 1 N θ ^ j − θ | ,     MSE ( θ ^ ) = 1 N ∑ j = 1 N ( θ ^ j − θ ) 2 (8)

where θ represents the real parameter value and θ ^ j represents the θ estimated value for the j time. Skewness and mean square error are measures that reflect the difference between the estimator and the estimated. The smaller the skewness and mean square error, the better the estimation effect.

Based on the iterative formula given in Section 3.1, and referring to the rounding sampling method proposed by Tadikamalla (1980) [12] and Chiodi (1995) [13], random numbers of two-component mixed generalized normal distribution are generated. Since the sample size of higher mathematics and linear algebra examination scores selected in this paper is 2035 and 2417 respectively, in order to better simulate real data, we choose to generate 2000 and 2500 random numbers for each simulation.

For λ , we select the initial value λ ∈ ( 0 , 1 ) ; for s 1 , s 2 , we select the initial values by s 1 ∈ [ 1 , 4 ] , s 2 ∈ [ 1 , 4 ] ; for μ 1 , μ 2 and σ 1 , σ 2 , we use the initial values by μ 1 ∈ [ 1 , 4 ] , μ 2 ∈ [ 1 , 4 ] , σ 1 ∈ [ 1 , 3 ] , σ 2 ∈ [ 1 , 3 ] . The convergence criterion is set as | θ ^ ( m + 1 ) − θ ^ ( m ) | < 10 − 4 , the numerical simulation experiment is carried out for 30 times, and the average value is calculated for analysis. Programming calculation with Python software.

Table 1 shows that the skewness of the seven parameters is within 0.25 and the mean square error is within 7% when estimating the parameters of the two-component mixed generalized normal distribution; Table 2 shows that when estimating the five parameters of the two-component mixed normal distribution, the skewness of the parameters is within 0.09 and the mean square error is within 4%. Under the same sample size, it is found that the fewer parameters to be estimated, the better the estimation effect. In general, when the sample size is 2000 and 2500, the parameter estimates of the two distributions have reached convergence. Through the analysis of skewness and mean square error, the parameter estimates are also relatively ideal.

We choose the linear algebra (2417 samples) and advanced mathematics (2035 samples) test scores of students of relevant majors in F University in the second semester of 2019-2020 academic year, and the column chart of the distribution is shown in Figure 1. In order to facilitate data analysis, the scores of linear algebra test (abbreviated as XXDS) and advanced mathematics test (abbreviated as GDSX) are normalized. To generate new data for descriptive statistics, the scores

of each examinee are set as x i , y i = x i 100 ∈ [ 0 , 1 ] .

It can be seen from Table 3 that linear algebra (XXDS) and advanced mathematics (GDSX) have common features, such as a small difference between their mean values; The skewness coefficients are all less than 0, showing the characteristics of left bias, and the kurtosis are all less than 3; At the 5% significance level, the results of J-B statistics are far greater than 5.99, and the assumption of normal distribution is rejected.

AIC and BIC criteria are generally used to evaluate the model fitting effect:

{ AIC = 2 φ − 2 log L ( θ ^ | x ) , BIC = φ log ( n ) − 2 log L ( θ ^ | x ) , (9)

where φ represents the number of parameters and the sample size. The smaller the AIC and BIC values, the better the fitting effect. To achieve more accurate results, the parameter estimation is repeated for 30 times to get the final result.

Table 4 shows the parameter estimation and AIC and BIC calculation results of the two-component mixed normal distribution and two-component generalized normal distribution using ECM algorithm. By analyzing the calculation results of AIC and BIC, it can be seen that the fitting effect of the two-component mixed generalized normal distribution is obviously better than the two-component mixed normal distribution for the test results of linear algebra (XXDS), and for the test results of advanced mathematics (GDSX), the fitting effect of the two-component mixed generalized normal distribution is slightly better than the two-component mixed normal distribution, and there is little difference between the two distributions.

Through the analysis of the parameter estimation s 1 , s 2 results, we can see that for the linear algebra test scores, the estimation s 1 results are significantly less than 2, and the s 2 results are significantly greater than 2. For the estimation of the higher mathematics scores, we find that the s 1 , s 2 results are all near 2, which also explains the reason why the AIC and BIC results of the two-component mixed generalized normal distribution and the two-component mixed normal distribution are very close.

Figure 3 is the histogram of the two-component mixed normal distribution and the two-component generalized normal distribution. It can be found that the two-component generalized mixed normal distribution better fits the peak on the left, and the two-component mixed normal distribution better fits the peak on the right. In general, the fitting effect of the two-component generalized mixed normal distribution is better than the two-component mixed normal distribution, which is consistent with the conclusion drawn from the analysis of AIC and BIC calculation results.