Data set: Covid-19 data for Saudi Arab (Total deaths in Millions)
0.029,0.057,0.086,0.086,0.115,0.23,0.23,0.287,0.46,0.603,0.718,0.833,0.977,1.092,1.178,1.178,1.264,1.35,1.494,1.695,1.86
7,2.097,2.269,2.384,2.499,2.643,2.786,2.959,3.131,3.275,3.476,3.648,3.906,3.993,4.136,4.366,4.51,4.653,4.854,5.055,5.285
,5.486,5.745,6.003,6.291,6.578,6.865,7.066,7.325,7.583,7.842,8.129,8.387,8.675,8.962,9.192,9.45,9.737,10.082,10.456,10.8
86,11.202,11.461,11.806,12.208,12.667,13.156,13.788,14.448,15.08,15.77,16.631,17.55,18.441,19.418,20.452,21.428,22.49
1,23.525,24.617,25.651,26.771,27.92,29.04,30.218,31.338,32.717,34.009,35.331,36.394,37.543,38.663,39.84,41.018,42.339
,43.402,44.551,45.93,47.366,48.774,50.325,51.761,53.37,55.036,56.529,57.937,59.143,60.321,61.786,62.647,63.854,64.428
,65.577,66.784,68.076,69.139,70.288,71.408,72.471,73.448,,74.712,75.688,76.751,77.641,78.503,79.279,80.112,80.887,81.
634,82.324,82.927,83.788,84.708,85.713,86.747,87.752,88.844,89.907,90.969,91.889,92.865,93.899,94.876,95.881,96.772,
97.892,98.696,99.673,100.707,101.913,102.833,103.953,104.815,106.021,106.911,107.859,108.721,109.525,110.301,111.1
63,111.938,112.857,113.633,114.38115.328,116.304,117.223,117.97,118.832,119.636,120.326,121.015,121.791,122.595,12
3.658,124.606,125.496,126.358,127.248,128.052,128.828,129.604,130.465,131.241,132.103,132.849,133.711,134.515,135.
348,136.124,136.957,137.704,138.537,139.312,140.03,140.691,141.409,142.099,142.817,143.506,144.138,144.856,145.574
,146.12,146.723,147.269,147.757,148.36,148.935,149.394,149.854,150.371,150.802,151.204,151.692,152.123,152.612,153
.071,153.617,154.048,154.622,155.168,155.685,156.173,156.719,157.15,157.667,158.155,158.701,159.132,159.678,160.166
,160.568,160.999,161.574,162.033,162.493,163.038,163.498,164.015,164.561,165.02,165.48,166.026,166.485,166.916,167.
318,167.749,168.238,168.611,169.013,169.358,169.674,170.018,170.334,170.622,171.024,171.34,171.684,172.029,172.403
,172.69,173.006,173.379,173.724,174.04,174.327,174.643,174.959,175.246,175.562,175.849,176.108,176.338,176.596,176.
912,177.171,177.401,177.659,177.975,178.205,178.492,178.751,178.952,179.21,179.411,179.699,179.957
Figure 3 demonstrates the theoretical and empirical plots of the NFW distribution. The data were considered as the total deaths in million for Saudi Arab during covid19. Numerous existing lifetime distributions like exponential, Gull Alpha Power Weibull, Weibull, Weibull- exponential, Algohary Inverse Flexible Weibull and alpha power inverted exponential, and new flexible exponential distribution were matched with the suggested distribution. The theoretical and empirical graphs clearly shows that the suggested distribution fits the data comparatively better than as compared with the existing distributions. The performance of both the graphs can be justified with model selection criteria presented in Table-2 and Table-3.
The Q-Q and P-P plots provided in Fig. 4 below demonstrates the covid-19 deaths data in million for Saudi Arab. The Q-Q and P-P plots shows that the proposed model fits the data more reasonably.
The EE for the covid-19 deaths data can be described with theoretical and empirical densities.
Figure 5 demonstrates the pattern of hazard rate function. The plot clearly suggests that as the curve passes through the diagonal line that mean that the data follows the nonmonotonic failure function. The Boxplot indicates that the data is positively skewed.
Table 2
MLE and standard errors for Covid-19 deaths data of Saudi Arab
Model
|
W
|
A
|
MLE
|
Standard error
|
-log(L)
|
EE
|
1.943391
|
11.56428
|
0.01293112 0.46819304
|
0.0007204026 0.0503989878
|
1553.233
|
W-Exp
|
2.139546
|
12.40669
|
2.124230191 0.004110006 0.851753362
|
NaN NaN 0.03435723
|
1559.575
|
W
|
2.66361
|
15.08307
|
0.0144918 0.9436326
|
0.003117416 0.043776350
|
1582.859
|
E
|
2.604043
|
14.7871
|
0.01114114
|
0.0006511581
|
1583.465
|
APIE
|
NaN
|
NaN
|
39.630234 1.781984
|
5.8143416 0.1147966
|
1962.377
|
AIFW
|
1.284578
|
8.427546
|
0.06524940 0.01262842
|
0.0075814439 0.0006747553
|
1639.086
|
GAPW
|
2.454047
|
14.06546
|
0.50031651 0.02627818 0.87130220
|
0.102253599 0.007150854 0.049569249
|
1576.656
|
Table 3
Goodness of fit measures for Covid-19 data of Saudi Arab
Models
|
AIC
|
CAIC
|
BIC
|
HQIC
|
EE
|
3110.465
|
3110.507
|
3117.791
|
3113.401
|
E
|
3168.931
|
3168.945
|
3172.594
|
3170.399
|
W
|
3169.719
|
3169.761
|
3177.045
|
3172.655
|
W-Exp
|
3125.151
|
3125.235
|
3136.14
|
3129.554
|
APIE
|
3928.914
|
3928.956
|
3936.24
|
3931.85
|
AIFW
|
3282.173
|
3282.215
|
3289.499
|
3285.109
|
GAPW
|
3159.312
|
3159.397
|
3170.301
|
3163.716
|
Table-2 and Table-3 represents the various model selection criteria including maximum likelihood estimates, standard errors, log-likelihood, Anderson darling (A), Cramer-von mises (W), AIC, CAIC, BIC, and HQIC. The results of Table-2 and Table-3 clearly shows that based on these model selection criteria, the EE provides a better fit as compared exponential, Gull Alpha Power Weibull, Weibull, Weibull- exponential, Algohary Inverse Flexible Weibull and alpha power inverted exponential distribution.