The Gompertz Fréchet distribution: Properties and applications

Figures & data

Figure 1. PDF of the Gomp_ertz Frechet distribution.

Figure 2. CDF of the Gompertz Frechet distribution.

Figure 3. Plots for the hazard function of Gompertz Frechet distribution.

Table 1. The ML estimates, $\mathit{l}$, $\mathit{A}IC$, $\mathit{C}AIC$, $\mathit{B}IC$ and $\mathit{H}QIC$ for the first data set

Model	ML Estim	l	AIC	CAIC	BIC	HQIC
GoW	$\hat{\alpha }$=0.7523	−59.5751	127.1502	127.8399	135.7227	130.5218
	$\hat{\beta }$=−0.1317
	$\hat{\gamma }$=0.3264
	$\hat{\theta }$=6.0883
GoBXII	$\hat{\alpha }$=0.0056	−62.3509	132.7019	133.3915	141.2744	136.0735
	$\hat{\beta }$=2.0355
	$\hat{\gamma }$=0.6809
	$\hat{\theta }$=3.5460
GoLom	$\hat{\alpha }$=0.0044	−64.9569	137.9139	138.6035	146.4864	141.2855
	$\hat{\beta }$=5.3157
	$\hat{\gamma }$=0.4533
	$\hat{\theta }$=1.4547
GoFr	$\hat{\alpha }$=0.6178	−55.8831	119.7662	120.4558	128.3387	123.1378
	$\hat{\beta }$=0.5458
	$\hat{\gamma }$=5.0302
	$\hat{\theta }$=2.6392

Table 2. The $\mathit{W}$, $\mathit{A}$, $\mathit{K}S$ and $\mathit{p}-value$ using the first data set

Model	$\mathit{W}$	$\mathit{A}$	$\mathit{K}S$	$\mathit{p}-value$
GoW	0.0945	0.6353	0.0913	0.6705
GoBXII	0.1275	0.8865	0.1012	0.5385
GoLom	0.1624	1.1128	0.1282	0.2515
GoFr	0.0433	0.2472	0.0687	0.9275

Figure 4. Histogram of the fitted distributions.

Figure 5. Empirical cdf of the fitted distributions.

Table 3. The ML estimates, $\mathit{l}$, $\mathit{A}IC$, $\mathit{C}AIC$, $\mathit{B}IC$ and $\mathit{H}QIC$ for the second data set

Model	ML Estim	$\mathit{l}$	$\mathit{A}IC$	$\mathit{C}AIC$	$\mathit{B}IC$	$\mathit{H}QIC$
GoW	$\hat{\alpha }$=0.4984	−133.559	275.1180	275.6180	284.8886	279.0480
	$\hat{\beta }$=−0.0826
	$\hat{\gamma }$=0.1869
	$\hat{\theta }$=6.1176
GoBXII	$\hat{\alpha }$=0.0020	−142.7397	293.4794	293.9794	303.2500	297.4094
	$\hat{\beta }$=1.8871
	$\hat{\gamma }$=0.9231
	$\hat{\theta }$=2.1938
GoLom	$\hat{\alpha }$=0.0050	−137.9373	283.8747	284.3747	293.6453	287.8047
	$\hat{\beta }$=6.7485
	$\hat{\gamma }$=0.1938
	$\hat{\theta }$=1.3562
GoFr	$\hat{\alpha }$=9.8016	−132.9584	273.9168	274.4168	283.6874	277.8468
	$\hat{\beta }$=3.8554
	$\hat{\gamma }$=1.6332
	$\hat{\theta }$=10.6159

Table 4. The $\mathit{W}$, $\mathit{A}$, $\mathit{K}S$ and $\mathit{p}-value$ using the second data set

Model	$\mathit{W}$	$\mathit{A}$	$\mathit{K}S$	$\mathit{p}-value$
GoW	0.1045	0.5568	0.1261	0.1340
GoBXII	0.1223	0.6611	0.1754	0.0107
GoLom	0.2044	1.1845	0.1515	0.0404
GoFr	0.1284	0.1169	0.1961

Figure 6. Histogram of the fitted distributions.

Figure 7. Empirical cdf of the fitted distributions.