Suppose that a nonnegative random variable
which is nonnegative and has the characterization property that the equality to zero holds iff
To calculate (1.1), we need to determine the bandwidth type parameter in
In survival analysis, the hazard function rather than the probability density function is of interest where the hazard function is defined as
which can be considered in comparing
The comparison of the empirical distribution function
We here propose two test statistics based on the difference and ratio of the cumulative hazard functions, respectively. We first consider the squared difference of the cumulative hazard functions as
and also consider the ratio of the cumulative hazard functions by using the extension of Kullback-Leibler information to the cumulative hazard function as
Then we can establish goodness-of-fit test statistics by using
Suppose that (
where
Since
However, the comparison of
Hence, we can consider the Nelson-Aalen estimator which has been brought up by Nelson (1972) and Aalen (1978), instead of
where
The corresponding distribution function estimator, whose survival function is called the Fleming and Harrington estimator, can be written as
In comparing
which can be arranged as
where
which can be written as
Equation (2.1) can be simplified to
where Li_{2}(
In the previous section, we considered the Nelson-Aalen estimator instead of the empirical distribution function because
for
Subsequently, the cumulative hazard function of the empirical distribution function can be written as
Then the average squared difference is defined as
which can be written as
where
By approximating
Equation (2.2) can be simplified:
The derivation of the last term of the equation (2.3) can be easily done.
The comparison of the cumulative hazard functions,
which can be written as
We note that
where
In Table 1, we provide the empirical critical values at
If the variable
We can compare the cumulative hazard functions in terms of their ratio by extending the Kullback-Leibler function as
which can be written as
We also obtained the critical value estimates of
In this section, we consider the exponential distribution,
Cramer von-Mises test statistics:
where
Anderson-Darling test statistics:
Test statistic based on cumulative residual entropy (Baratpour and Habibi Rad, 2012):
where
To compare the powers of those test statistics, we consider the following alternatives according to the type of the hazard functions.
Monotone decreasing hazard: Gamma (shape parameter: 0.5), Weibull (shape parameter: 0.5), Chi-square (df 1).
Monotone increasing hazard: Uniform, Gamma (shape parameter: 2), Weibull distribution (shape parameter: 2), Chi-square (df 4).
None-monotone hazard: Log normal (shape parameter: 0.5, 1, 1.5).
We employed the Monte Carlo simulation to estimate the powers against the above alternatives for
Similarly, we also consider Type-slowromancapii@ censored case for the same alternatives. Here we suggest the case when
Same as the previous section, we compare the performances of
Cramer von-Mises test statistics:
where
Anderson-Darling test statistics:
Censored version of the test statistic based on cumulative residual entropy (Park and Lim, 2015)
Similarly in censored case, we employed the Monte Carlo simulation to estimate the powers against the above alternatives for
In this section, we consider the real-life data, which consists of the failure times for 36 appliances subjected to an automatic life test (Lawless, 1982)
11 35 49 170 329 381 708 958 1062 1167 1594 1925 1990 2223 2327 2400 2451 2471
2551 2565 2568 2694 2702 2761 2831 3034 3059 3112 3214 3478 3504 4329 6367 6976
7846 13403
and illustrate the use of the proposed tests as the goodness-of-fit test for exponentiality. Table 6 shows the statistics values of four tests (
We proposed some goodness-of-fit test statistics based on the comparison of cumulative hazards, which are omnibus tests applicable to many of distributions. They shows comparable performances with other EDF-based test statistics. It is clearer to give weights on earlier (or later) departures in cumulative hazards rather than in cumulative density functions since the proposed test statistics are expressed in terms of the cumulative hazard functions. It is best to give more weight on later departures when hazards increase and give more weight on earlier departures when they do not.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07042581).
Empirical critical values of
10 | 20 | 30 | 40 | 50 | |
---|---|---|---|---|---|
2.8991 | 3.7004 | 4.2626 | 4.6932 | 5.0342 | |
1.1183 | 1.2149 | 1.2523 | 1.2749 | 1.2850 | |
1.0112 | 1.1165 | 1.1814 | 1.2276 | 1.2575 |
Empirical powers (%) of six tests at
Alternatives | ||||||
---|---|---|---|---|---|---|
Exp(1) | 5.02 | 5.00 | 4.99 | 4.94 | 5.16 | 4.98 |
Gamma(0.5) | 44.33 | 72.82 | 55.92 | 52.59 | 70.74 | 13.87 |
Gamma(2) | 18.36 | 37.25 | 49.16 | 48.52 | 45.25 | 39.16 |
Log normal(0, 0.5) | 64.77 | 98.52 | 99.59 | 99.37 | 99.42 | 85.64 |
Log normal(0,1) | 18.84 | 13.61 | 19.22 | 15.26 | 13.99 | 13.26 |
Log normal(0, 1.5) | 65.52 | 64.51 | 61.82 | 61.89 | 62.67 | 43.67 |
Weibull(0.5) | 85.66 | 96.16 | 91.53 | 90.05 | 95.70 | 53.95 |
Weibull(2) | 71.38 | 89.10 | 94.14 | 93.41 | 92.21 | 91.99 |
44.45 | 72.85 | 56.02 | 52.71 | 70.78 | 13.87 | |
18.46 | 37.11 | 49.21 | 48.56 | 45.32 | 39.23 | |
Unif(0, 1) | 82.38 | 65.83 | 80.05 | 67.52 | 63.18 | 92.97 |
Empirical powers (%) of six tests at
Alternatives | ||||||
---|---|---|---|---|---|---|
Exp(1) | 5.01 | 5.19 | 5.12 | 5.18 | 5.20 | 4.97 |
Gamma(0.5) | 71.58 | 96.63 | 91.23 | 89.94 | 96.38 | 41.34 |
Gamma(2) | 55.03 | 90.01 | 90.32 | 90.22 | 91.77 | 65.80 |
Log normal(0, 0.5) | 98.83 | 100.00 | 100.00 | 100.00 | 100.0 | 99.28 |
Log normal(0, 1) | 30.40 | 32.69 | 40.20 | 30.231 | 34.17 | 28.18 |
Log normal(0, 1.5) | 92.15 | 93.60 | 93.34 | 93.61 | 93.37 | 85.59 |
Weibull(0.5) | 99.29 | 99.98 | 99.95 | 99.92 | 99.99 | 94.93 |
Weibull(2) | 99.83 | 100.00 | 100.00 | 100.00 | 100.00 | 99.98 |
71.69 | 96.64 | 91.14 | 89.83 | 96.38 | 41.36 | |
55.48 | 90.15 | 90.51 | 90.38 | 91.86 | 66.00 | |
Unif(0, 1) | 100.00 | 99.77 | 99.98 | 98.57 | 98.65 | 100.00 |
Empirical powers (%) of six tests at
Alternatives | |||||||
---|---|---|---|---|---|---|---|
Exp(1) | 10 | 4.92 | 4.85 | 4.90 | 4.97 | 4.87 | 4.97 |
15 | 4.95 | 5.00 | 5.04 | 5.03 | 5.02 | 4.99 | |
18 | 4.87 | 4.91 | 4.89 | 4.98 | 4.95 | 4.92 | |
Gamma(0.5) | 10 | 27.40 | 52.30 | 38.62 | 27.23 | 49.88 | 25.66 |
15 | 46.24 | 65.33 | 52.88 | 42.24 | 62.72 | 42.81 | |
18 | 51.77 | 70.88 | 60.28 | 49.12 | 68.52 | 49.17 | |
Gamma(2) | 10 | 23.65 | 19.35 | 17.70 | 28.50 | 23.28 | 24.70 |
15 | 14.80 | 25.83 | 25.67 | 36.32 | 32.90 | 18.92 | |
18 | 5.34 | 28.98 | 26.16 | 42.29 | 37.98 | 5.55 | |
Log normal(0, 0.5) | 10 | 87.70 | 90.30 | 40.66 | 93.88 | 93.11 | 89.12 |
15 | 80.55 | 97.02 | 63.95 | 98.22 | 98.48 | 86.58 | |
18 | 36.99 | 98.06 | 69.07 | 99.08 | 99.18 | 56.20 | |
Log normal(0, 1) | 10 | 13.45 | 11.69 | 9.50 | 17.66 | 14.62 | 14.28 |
15 | 5.12 | 8.11 | 7.57 | 12.42 | 11.24 | 6.33 | |
18 | 8.07 | 7.82 | 7.71 | 11.34 | 9.80 | 8.29 | |
Log normal(0, 1.5) | 10 | 5.24 | 4.63 | 4.35 | 4.97 | 4.38 | 5.15 |
15 | 23.93 | 19.41 | 17.90 | 17.66 | 17.11 | 22.60 | |
18 | 49.74 | 42.03 | 43.09 | 38.83 | 39.42 | 48.95 | |
Weibull(0.5) | 10 | 40.20 | 65.87 | 53.00 | 40.39 | 63.72 | 38.15 |
15 | 72.14 | 85.50 | 77.89 | 69.67 | 84.01 | 69.31 | |
18 | 84.00 | 92.89 | 88.96 | 82.92 | 92.06 | 82.14 | |
Weibull(2) | 10 | 45.75 | 40.04 | 32.19 | 52.50 | 45.40 | 47.11 |
15 | 46.64 | 63.05 | 57.73 | 74.84 | 70.86 | 53.09 | |
18 | 16.79 | 75.36 | 68.00 | 86.32 | 82.64 | 29.27 | |
10 | 27.17 | 52.13 | 38.32 | 26.96 | 49.70 | 25.48 | |
15 | 46.31 | 65.02 | 52.71 | 42.37 | 62.45 | 42.95 | |
18 | 51.88 | 70.63 | 60.35 | 49.32 | 68.24 | 49.29 | |
10 | 23.76 | 19.56 | 17.62 | 28.74 | 23.40 | 24.76 | |
15 | 15.11 | 26.09 | 25.80 | 36.70 | 33.26 | 19.18 | |
18 | 2.64 | 29.38 | 26.34 | 42.51 | 38.24 | 5.65 | |
Unif(0, 1) | 10 | 10.04 | 6.59 | 8.99 | 10.66 | 7.76 | 10.37 |
15 | 13.60 | 12.92 | 20.54 | 23.44 | 17.37 | 16.23 | |
18 | 4.36 | 24.10 | 33.86 | 42.67 | 32.60 | 9.34 |
Empirical powers (%) of six tests at
Alternatives | |||||||
---|---|---|---|---|---|---|---|
Exp(1) | 25 | 5.03 | 5.02 | 4.96 | 5.01 | 4.98 | 5.04 |
37 | 5.02 | 4.97 | 4.90 | 4.96 | 4.92 | 4.94 | |
45 | 4.94 | 4.95 | 5.02 | 5.01 | 4.96 | 4.91 | |
Gamma(0.5) | 25 | 61.22 | 84.30 | 74.21 | 64.52 | 83.31 | 59.91 |
37 | 77.64 | 92.87 | 86.99 | 80.49 | 92.22 | 75.53 | |
45 | 83.26 | 95.56 | 91.73 | 86.70 | 95.12 | 81.23 | |
Gamma(2) | 25 | 55.74 | 62.94 | 64.31 | 64.04 | 65.54 | 56.83 |
37 | 55.03 | 78.31 | 79.58 | 78.20 | 81.28 | 58.73 | |
45 | 36.92 | 84.95 | 83.76 | 85.75 | 87.60 | 43.33 | |
Log normal(0, 0.5) | 25 | 100.00 | 100.00 | 99.58 | 100.00 | 100.00 | 100.00 |
37 | 100.00 | 100.00 | 99.98 | 100.00 | 100.00 | 100.00 | |
45 | 99.78 | 100.00 | 99.99 | 100.00 | 100.00 | 99.92 | |
Log normal(0, 1) | 25 | 29.58 | 42.39 | 39.18 | 39.22 | 45.45 | 30.83 |
37 | 11.66 | 31.24 | 28.77 | 27.10 | 35.16 | 13.68 | |
45 | 12.86 | 24.28 | 21.82 | 21.31 | 26.89 | 13.11 | |
Log normal(0, 1.5) | 25 | 7.89 | 6.53 | 5.60 | 7.28 | 6.12 | 7.81 |
37 | 41.55 | 35.52 | 33.15 | 36.18 | 32.94 | 40.40 | |
45 | 79.76 | 73.90 | 74.84 | 73.26 | 72.12 | 78.90 | |
Weibull(0.5) | 25 | 80.74 | 94.24 | 89.23 | 83.33 | 93.81 | 79.88 |
37 | 96.60 | 99.37 | 98.65 | 97.46 | 99.31 | 96.06 | |
45 | 99.30 | 99.92 | 99.82 | 99.60 | 99.91 | 99.12 | |
Weibull(2) | 25 | 88.85 | 91.55 | 91.56 | 92.45 | 92.62 | 89.30 |
37 | 96.54 | 99.28 | 99.35 | 99.39 | 99.43 | 97.16 | |
45 | 95.11 | 99.92 | 99.91 | 99.95 | 99.94 | 96.64 | |
25 | 60.92 | 84.07 | 73.72 | 64.21 | 83.09 | 59.57 | |
37 | 77.23 | 92.72 | 86.76 | 80.12 | 92.01 | 75.08 | |
45 | 82.75 | 95.54 | 91.58 | 86.46 | 95.06 | 80.78 | |
25 | 55.54 | 62.72 | 64.20 | 63.74 | 64.40 | 56.56 | |
37 | 54.96 | 78.21 | 79.33 | 78.12 | 81.12 | 58.68 | |
45 | 36.84 | 84.87 | 83.67 | 85.77 | 87.53 | 43.46 | |
Unif(0, 1) | 25 | 16.47 | 11.78 | 16.48 | 16.84 | 13.03 | 16.68 |
37 | 42.01 | 37.11 | 47.42 | 49.79 | 41.48 | 43.76 | |
45 | 57.91 | 73.06 | 81.12 | 85.11 | 77.90 | 62.57 |
Test statistics and
Tests statistics | ||||
---|---|---|---|---|
Values | 1.3730 | 1.2746 | 0.2995 | 1.4970 |
0.9615 | 0.9591 | 0.9838 | 0.9693 |