On testing the log-gamma distribution hypothesis by bootstrap

Abstract

In this paper we propose two bootstrap goodness of fit tests for the log-gamma distribution with three parameters, location, scale and shape. These tests are built using the properties of this distribution family and are based on the sample correlation coefficient which has the property of invariance with respect to location and scale transformations. Two estimators are proposed for the shape parameter and show that both are asymptotically unbiased and consistent in mean-squared error. The test size and power is estimated by simulation. The power of the two proposed tests against several alternative distributions is compared to that of the Kolmogorov-Smirnov, Anderson-Darling, and chi-square tests. Finally, an application to data from a production process of carbon fibers is presented.

Appendix

The following theorem can be found in Sen and Singer (1993).

Theorem 8.1

Let $X_1,\ldots ,X_n$ be a random sample from $f(x;\theta ),\ \theta \in \Theta \subset \mathbb R$, which satisfy the following conditions:

• 1. The derivatives $\frac{\partial }{\partial \theta }f(x;\theta )$ and $\frac{\partial ^2}{\partial \theta ^2}f(x;\theta )$ exist almost everywhere, and are such that

  $$\begin{aligned} \Big |\frac{\partial }{\partial \theta }f(x;\theta )\Big |\le H_1(x)\ \mathrm{and}\ \Big |\frac{\partial ^2}{\partial \theta ^2}f(x;\theta )\Big |\le H_2(x) \end{aligned}$$

  where $\int _\mathbb R H_j(x)dx<\infty$, for $j=1,2$;

• 2. $\frac{\partial }{\partial \theta }\log \big (f(x;\theta )\big )$ and $\frac{\partial ^2}{\partial \theta ^2}\log \big (f(x;\theta )\big )$ exist almost everywhere, and are such that

  • (a) For $X$ with density $f(x;\theta )$

    $$\begin{aligned} 0<I(\theta )=E\left( \Big [\frac{\partial }{\partial \theta }\log \big (f(X;\theta )\big ) \Big ]^2\right) <\infty . \end{aligned}$$

  • (b) For $\delta \rightarrow 0$,

    $$\begin{aligned} E\left( \sup _{\{h:|h|\le \delta \}}\Big |\frac{\partial ^2}{\partial \theta ^2} \log \big (f(X;\theta +h)\big ) - \frac{\partial ^2}{\partial \theta ^2}\log \big (f(X;\theta )\big )\Big |\right) =\psi _\delta \rightarrow 0. \end{aligned}$$

Then the maximum likelihood estimator $\hat{\theta }_n$ of $\theta$ is such that $\sqrt{n}(\hat{\theta }_n-\theta )\rightarrow N\big (0,I^{-1}(\theta )\big )$ in distribution where $N(0,\sigma ^2)$ denotes the normal distribution with zero mean and variance $\sigma$.

Theorem 8.2

Let $\mathbf Z =(Z_1,\ldots ,Z_n)$ be a random sample from $LG(0,1,\kappa )$, then for $\kappa >0, n>20$ and any realization $\mathbf z =(z_1,\ldots ,z_n)$ of the random sample preserves approximately the percentage of negative values in Table 8, the maximum likelihood estimator for $\kappa$ exists.

Table 8 Percentage of negative values for different $\kappa$

Proof

Consider the log-likelihood function given in (8.1)

$$\begin{aligned} \ell (\kappa ;\mathbf z )=n\left( \Big (\kappa -\frac{1}{2}\Big ) \log (\kappa )+\bar{z}\sqrt{\kappa }-\log \big (\varGamma (\kappa )\big ) \right) -\kappa \sum _{i=1}^{n}e^{z_i/\sqrt{\kappa }}.\qquad \end{aligned}$$

(8.1)

This proof is divided in two parts. In part 1 the derivative of (8.1) changes signs for $\kappa >0$. Part 2 shows that the derivative function of part 1 is negative. Part 2 concludes that the likelihood function is continuous in $\kappa$, and follows that it has a unique maximum for $\kappa >0$.$\square$