Nakagami Distribution with Heavy Tails and Applications to Mining Engineering Data

Abstract

In this paper we introduce a new extension of the Nakagami distribution. This new distribution is obtained by the quotient of two independent random variables. The quotient consists of a Nakagami distribution divided by a power of the uniform distribution in (0,1). Thus the new distribution has a heavier tail than the Nakagami distribution. In this study we obtain the density function and some important properties for making the inference, such as estimators of moment and maximum likelihood. We examine two sets of real data from the mining industry which show the usefulness of the new model in analyses with high kurtosis.

Appendix

• Gamma distribution

  $$\begin{aligned} f_Y(y, \alpha , \beta )&= \frac{1}{\beta ^{\alpha }\varGamma (\alpha )} \,\, y^{\alpha -1} \, e^{-y/\beta } \qquad y \ge 0, \,\, \alpha>0, \,\, \beta >0. \\&\Rightarrow \quad Y \sim Ga(\alpha , \beta ) \end{aligned}$$

• Chi-square distribution

  $$\begin{aligned} f_Y(y;\nu ) = \frac{1}{2^{\nu /2}\varGamma (\nu /2)} \,\, y^{\nu /2-1} \, e^{-y/2} \qquad y \ge 0, \,\, \nu \ge 0. \quad \Rightarrow \quad Y \sim \chi ^2_{(\nu )} \end{aligned}$$

• Rayleigh distribution

  $$\begin{aligned} f_Y(y;b) = \frac{y}{b^2} \,\, e^{-y^2/2b^2} \qquad y \ge 0, \,\, b \ge 0 \quad \Rightarrow \quad Y \sim R(b) \end{aligned}$$

• Beta distribution

  $$\begin{aligned} f_Y(y, \alpha , \beta )&= \frac{\varGamma (\alpha + \beta )}{\varGamma (\alpha )\varGamma (\beta )} \,\, y^{\alpha -1} \, (1-y)^{\beta - 1} \qquad y \ge 0, \,\, \alpha>0, \quad \beta >0.\\&\Rightarrow \quad Y \sim Beta(\alpha , \beta ) \end{aligned}$$

• Skew-slash distribution

  $$\begin{aligned} f_Y(y, \mu , \sigma , q, \lambda )&= \frac{2}{\sigma \sqrt{2\pi }} \,\, \int _0^1 v^{1/q} \, e^{-\frac{1}{2}(\frac{y-\mu }{\sigma })^2 v^{2/q}} \,\, \varPhi (\lambda \frac{y-\mu }{\sigma }v^{1/q}) \,\, \mathrm{d}v \\&\Rightarrow \quad Y \sim SS(\mu , \sigma , q, \lambda )\\&\quad y \in R,\quad \mu , \lambda \in R, \quad \sigma >0. \end{aligned}$$

• Digamma function

  $$\psi (z) = \frac{{{\text{d}}\ln \Gamma (z)}}{{{\text{d}}z}} = \frac{{\Gamma ^{'} (z)}}{{\Gamma (z)}}$$

• 1. Rubidium concentration dataset

61	145	45	81	179	83	93	115	165	111	110	95	118
5	2	44	86	148	199	116	14	136	85	80	101	77
112	12	56	57	134	79	118	156	144	39	95	32	43
86	93	95	55	61	23	37	64	87	36	43	177	34
102	82	85	155	78	85	140	73	7	95	75	62	76
79	54	95	30	117	54	82	16	112	40	62	406	144
95	101	47	114	49	151	133	59

• 2. Zinc concentration dataset.

31	51	92	1	47	11	29	17	17	47	25	22	45
6	1165	80	12	22	115	455	93	93	81	115	63	75
92	79	113	277	118	113	76	406	69	86	58	79	86
61	69	80	86	74	79	70	65	78	78	66	61	113
12	45	141	111	70	500	67	22	24	63	22	36	29
74	24	43	354	97	30	94	20	311	67	40	47	203
21	17	40	36	78	19	42	77