Discriminating between gamma and lognormal distributions with applications

1 Introduction

Let X be a random variable has the three-parameter gamma[Gamma(θ, λ, α)] density function (pdf) as

1. Mean:

   (1.5)
   $$E(X)=\alpha \lambda \text{,}\mathrm{and}E(Y)=\exp [\mu +{\sigma }^2/2]\text{.}$$

2. Variance:

   (1.6)
   $$\begin{array}{cc}\hfill & \mathrm{Var}(X)=\alpha {\lambda }^2\mathrm{and}\hfill \\ \hfill & \mathrm{Var}(Y)=\omega (1-\omega )\exp [2\mu ]\text{,}\omega =\exp [{\sigma }^2]\text{.}\hfill \end{array}$$

3. Skewness:

   (1.7)
   $$\mathit{SK}(X)=\frac{2}{\alpha }\text{,}\mathrm{and}\mathit{SK}(Y)=(\omega +2)\sqrt{\omega -1}\text{.}$$

The problem for testing whether some given data come from one of the two probability distributions, is quite old in the statistical literature. Atkinson (1969, 1970), Chen (1980), Chambers and Cox (1967), Cox (1961, 1962), Dyer (1973) have considered this problem in general for discriminating between two models. Due to increasing applications of the lifetime distributions, special attention is given to the problem of discriminating between the lognormal and Weibull distributions by Dumonceaux and Antle (1973) and between the lognormal and gamma by Jackson (1969) and between the gamma and Weibull distribution by Bain and Engelhard (1980) and Fearn and Nebenzahl (1991). Wiens (1999) has discussed a case study when the lognormal and gamma give different results. Recently, Gupta and Kundu (2003a) have discussed the closeness of gamma and the generalized exponential distribution while Gupta and Kundu (2003b) have discriminated between Weibull and the generalized exponential distributions. Gupta and Kundu (2004) have discriminated between gamma and the generalized exponential distribution.

On the other hand, goodness-of-fit tests are very important techniques for data analysis in the sense of check whether the given data fits the distributional assumptions of the statistical model. A variety of goodness-of-fit tests are available in the literature and recently there seems to be significant research on this topic. For more details, see, D’Agostino and Stephens (1986) and Huber-Carol et al. (2002). Correlation coefficient test is considered one of the easiest of such tests, that is because it is only needs special tables introduce from Monte Carlo simulations. The correlation coefficient test was introduced by Filliben (1975) for testing goodness-of-fit to the normal distribution and tables where updated later by Looney and Gulledge (1985). Among others Kinnison (1985, 1989) used the correlation coefficient method to present tables for testing goodness-of-fit to the extreme-value Type-I (Gumbel) and the extreme-value distribution, respectively. Recently, Sultan (2001) has devolved the correlation goodness-of-fit to the logarithmically-decreasing survival distribution. Baklizi (2006) has suggested weighted Kolmogrove–Smirnov type test for grouped Rayleigh data. Chen (2006) has discussed some tests of fit for the three-parameter lognormal distribution.

In this paper, we discuss the motivation of the problem in Section 2 below. In Section 3, we use the single moments of the rth order statistic from the one-parameter gamma distribution to develop goodness-of-fit tests for the two- and three-parameter gamma distributions. In Section 4, we calculate the power of the tests based on some different alternative distributions. In addition, we discuss some simulated examples. Finally, in Section 5, we apply the proposed test for some real data sets were collected from Dalla hospital, Riyadh, Saudi Arabia.

2 Motivation

The problem starts whenever we have a certain data and we need to fit the given data to either gamma or lognormal distributions. In many situations, we have found that gamma distribution fits better than the lognormal distribution. Then a question rises: why we do use the lognormal? Consequently, the answer of such question leads us to discuss some issues they are: (i) different measures of skewness, (ii) nonparametric tests, and (iii) correlation coefficient goodness-of-fit test.

Let X₁, … , X_n be a random sample has mean M and variance V and assume: $$E(X)=E(Y)=M\text{,}$$ and $$\mathit{Var}(X)=\mathit{Var}(Y)=V\text{,}$$ then it is easy to write

3 Correlation goodness of fit test of gamma pdf

Let x_1:n, … , x_n−r:n represents n order statistics from Gamma(0, 1, α) given in (1.3). Then, the pdf of the rth order statistic is given by:

4 Power calculation

In this section, we calculate the power of the considered tests by replacing the Gamma(θ, λ, α) random variates generator in the simulation program with generators from the alternatives including; normal, lognormal, and the Weibull distributions. Based on different sample size, different censoring ratios and 10,001 simulations, the power is calculated to be $$\mathit{Power}=\frac{\#\mathit{of}\mathit{rejection}\mathit{of}{H}_0}{10\text{,}001}\text{,}$$ where H₀ is rejected if T₁ (T₂) ⩾ the corresponding percentage points given in Table 1 (Table 2), and T₁ (T₂) is evaluated from the alternative distributions.

Tables 3 and 4 represent the power of the test for the two-parameters and three-parameter cases, respectively. The different considered alternative distributions are:

1. Normal distribution N(μ, σ).

2. Lognormal distribution LN(μ, σ).

3. Weibull distribution with shape α, scale parameter σ and location parameter μ, W(μ, σ, α).

4. Chi-square distribution χ²(α).

5. Cauchy distribution with scale parameter σ and location parameter σ, C(μ, σ).

6. Mixtures of two exponential distribution $\mathit{MTE}({\theta }_1\text{,}{\theta }_2\text{,}w)={\mathit{wf}}_1({\theta }_1)+(1-w)f_2({\theta }_2)\text{.}$

Tables 3 and 4 indicate that the correlation test has good power to reject sample from the chosen alternative distributions. Also, the power increases as the sample sizes increase for all given censoring ratios p = 1.0, 0.8, 0.6 as well as the significance level increases.

5 Applications

5.2

5.2 Application 2

In this application, we use some collected data from Dalla hospital, Riyadh, Saudi Arabia. The data represents the cost (in SR) of 50 patients from each different ages they already have visited the outpatients clinic during one year. The summary of the data is given in Table 5. The values of $\hat{\lambda }$ and $\hat{\alpha }$ in Table 5 are estimated by using the mean and variance of the original data.

Table 5 The real data in application 2.

Age (year)	Mean	StDev	Min.	Median	Max.	$\hat{\lambda }$	$\hat{\alpha }$
<1	851.2	407.2	100	824.6	1521.5	195	4
1–5	159.43	52.77	50	162.03	255	17	9
6–15	172.16	57.79	49.5	168.7	276.6	19	9
16–20	967.5	629.4	86.2	860.8	2319.1	409	2
21–30	282.4	153.6	100	262.5	599.5	84	3
31–40	348.9	203.2	100	295.4	821.1	118	3
41–50	348.9	203.2	100	295.4	821.1	118	3
>50	319.1	145	100	292.5	601.2	66	5

Age	α	T1(calculated)	Decision	T1(calculated)	Decision
<1	4	0.9908	A	0.9852	A
1–5	5	0.8136	R	0.7902	R
6–15	5	0.7594	R	0.7259	R
16–20	2	0.7917	R	0.7716	R
21–30	3	0.7924	R	0.7604	R
31–40	3	0.8028	R	0.7667	R
41–50	3	0.8129	R	0.7741	R
>50	5	0.8162	R	0.7753	R

A: Accept and R: Reject.

By using the original data and the moments of order statistics of Gamma(0, 1, 5), we calculate T₁(calculated) and T₂(calculated). Next, we use the corresponding values and at 1% level of significance. We have the decisions in Table 6. From Table 6, we recommend gamma distribution for the age less than one year.

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Distribution	Test statistic Ti	Decision
Gamma(0, 1, 3)	T1 = 0.99633	A
LN(0, 1)	T1 = 0.95277	R
Gamma(1, 5, 3)	T2 = 0.99339	A
LN(0, 1)	T2 = 0.704667	R

A: Accept and R: Reject.