Power Normal Distribution Debasis Kundu1 and Rameshwar D. Gupta2 Abstract Recently Gupta and Gupta [10] proposed the power normal distribution for which normal distribution is a special case. The power normal distribution is a skewed distri- bution, whose support is the whole real line. Our main aim of this paper is to consider bivariate power normal distribution, whose marginals are power normal distributions. We obtain the proposed bivariate power normal distribution from Clayton copula, and by making a suitable transformation in both the marginals. Lindley-Singpurwalla dis- tribution also can be used to obtain the same distribution. Different properties of this new distribution have been investigated in details. Two different estimators are proposed. One data analysis has been performed for illustrative purposes. Finally we propose some generalizations to multivariate case also along the same line and discuss some of its properties. Key Words and Phrases: Clayton copula; maximum likelihood estimator; failure rate; approximate maximum likelihood estimator. 1 Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin 208016, India. Corresponding author. e-mail: [email protected]. Part of his work has been supported by a grant from the Department of Science and Technology, Government of India. 2 Department of Computer Science and Applied Statistics. The University of New Brunswick, Saint John, Canada, E2L 4L5. Part of the work was supported by a grant from the Natural Sciences and Engineering Research Council. 1 1 Introduction The skew-normal distribution proposed by Azzalini [3] has received a considerable attention in the recent years. Due to its flexibility, it has been used quite extensively for analyzing skewed data on the entire real line. It has a nice physical interpretation also, as a hid- den truncation model, see Arnold and Beaver [2]. Several generalizations namely to other distributions and also to multivariate cases have been proposed and investigated in details. Although, it is well known that for skew normal distribution the maximum likelihood estimators (MLEs) are consistent and asymptotically normally distributed, the probability of non-existence of the MLEs is quite high, particularly, if the sample size is small. By extensive simulation studies, it has been observed by Gupta and Gupta [10] that even for moderate skewness, one needs a very large sample size to achieve a reasonable estimate of the skewness parameter. Due to this reason, Gupta and Gupta [10] proposed an alternative skewed model for which normal distribution becomes a special case and named it a power normal distribution. The power normal distribution is defined as follows: The random variable X is said to have a power normal distribution if it has the cumulative distribution function (CDF); F (x; α)= P (X x) = (Φ(x))α ; <x< . (1) X ≤ −∞ ∞ Here α> 0, and Φ( ) denoted the CDF of a standard normal distribution. Different proper- · ties of the power normal distribution have been discussed by Gupta and Gupta [10]. The aim of this paper is to introduce a bivariate power normal distribution (BPN) whose marginals are power normal distributions. This new distribution has been obtained from Clayton copula coupled with power normal marginals. Many properties of Clayton copula can be used in establishing different properties of the proposed bivariate power normal dis- tribution. It can also be obtained by making a suitable transformation from the well known 2 Lindley-Singpurwalla distribution, see [23]. Multivariate generalization and the generaliza- tion for the other distribution functions are also possible along the same line. It can be easily seen that the bivariate power normal distribution is an absolute contin- uous distribution. The joint probability density function (PDF) can take different shapes depending on the values of the parameters. The joint PDF can be expressed in terms of Φ( ) function. The conditional PDF can be easily obtained from the joint and marginal · distribution functions. It has been shown that although the power normal distribution has an increasing hazard rate but the bivariate hazard rate in the sense of Johnson and Kotz [19] is a decreasing function. We study different dependency properties using the properties of Clayton copula. It is ob- served that the two variables (marginals) are positive quadrant dependent. It automatically implies that Kendall’s τ, Spearman’s τ, Gini’s γ and Blomqvist’s β are all non-negative. We provide all these concordance measures. It has been shown that one variable is stochastically increasing with respect to the other. It can be verified that Clayton copula satisfies the total positivity of order two (TP2) property in the sense of Karlin [15]. Therefore, if a bivariate random vector has a BPN distribution, it satisfies the left corner set property also. Estimation of the unknown parameters is an important problem for any inferential pro- cedure. We propose to use the maximum likelihood method to estimate the unknown pa- rameters. The maximum likelihood estimators (MLEs) can be obtained by solving a two dimensional optimization process. The expected Fisher information matrix has also been provided which can be used for testing of hypothesis or for constructing confidence intervals. Further we propose to use a two-step procedure as suggested by Joe [18], mainly to avoid the two dimensional optimization process, and we provide the asymptotic distribution of the proposed estimators. One data set has been analyzed for illustrative purposes. Finally we provide the multivariate power normal distribution and study some of its properties. 3 Rest of the paper is organized as follows. In Section 2, we briefly describe the power normal distribution. The bivariate power normal distribution is introduced in Section 3. Different properties of the proposed bivariate power normal are discussed in Section 4. In Section 5, we provide the two estimation procedures. Analysis of one data set is presented in Section 6. In Section 7 we provide the multivariate power normal distribution. Finally we conclude the paper in Section 8. The elements of the Fisher information matrix and the elements of the asymptotic variance covariance matrix of the proposed estimators are provided in the Appendix. 2 Power Normal Distribution: A Brief Review Power Normal distribution was proposed by Gupta and Gupta [10], as an alternative to the Azzalini’s skew normal distribution. A random variable X is said to have a power normal distribution with parameter α> 0, if X has the CDF (1), and therefore, it has the probability density function (PDF); f (x; α)= α (Φ(x))α−1 φ(x); <x< . (2) X −∞ ∞ It is clear that the PDF (2) is a weighted normal PDF with weight function (Φ(x))α−1. For α = 1, it coincides with the standard normal density function. The power normal density function is a unimodal density function, which is skewed to the right if α> 1 and to the left if α< 1. It has some nice physical interpretations when α is an integer. It can be observed as the lifetime of a parallel system, see Gupta and Gupta [10] for a detailed discussions on this issue. Note that a class of distribution functions F ( ; α); α > 0 is said to be a proportional { · } reversed hazard rate family, if F (x; α)=(F (x))α ; <x< . (3) 0 −∞ ∞ 4 Here F ( ) is known as the baseline distribution function. Proportional reversed hazard rate 0 · model was introduced as an alternative to the celebrated proportional hazard rate model of Cox. Proportional reversed hazard rate model was originally introduced by Lehmann [20], in a testing of hypothesis problem, and it is known as Lehmann alternatives also. Power normal distribution can be seen as a member of the proportional reversed hazard rate family. Many general properties of the proportional reversed hazard rate models can be easily translated for the power normal distribution. It can be easily seen that the power normal distribution has an increasing hazard rate function and hence a decreasing mean residual life for all α > 0. It has likelihood ratio ordering, and hence it has hazard rate ordering and mean residual life ordering. Furthermore, unlike the skew normal distribution, the maximum likelihood estimator of the power normal distribution always exist. Therefore, for data analysis purposes power normal distribution can be used more effectively than the skew normal distribution. 3 Bivariate Power Normal Distribution In this section first we introduce the bivariate power normal distribution and discuss its different properties. It may be mentioned that every bivariate distribution function, FX1,X2 with continuous marginals F and F , corresponds a unique function C : [0, 1]2 [0, 1], X1 X2 → called a copula such that F (x , x )= C F (x ),F (x ) , for (x , x ) ( , ) ( , ). (4) X1,X2 1 2 { X1 1 X2 2 } 1 2 ∈ −∞ ∞ × −∞ ∞ Conversely, it is possible to construct a bivariate distribution function having the desired marginal distributions and a chosen description structure, i.e. copula. Let us consider the following copula −α C (u, v)= u−1/α + v−1/α 1 ; 0 <u,v < 1, (5) α − 5 known as Clayton copula, see for example Nelsen [28]. The following properties of the random vector (U, V ) with the joint CDF (5) can be easily established, and therefore the proofs are omitted. Theorem 3.1: If (U, V ) has the joint CDF (5) for α> 0, then (i) U, V uniform[0, 1]. ∼ (ii) The joint PDF of (U, V ) for 0 u, v 1 is ≤ ≤ (α + 1) 1 1 fU,V (u, v)= 1+α α+2 α × (uv) α × (u−1/α + v−1/α 1) · − (iii) The joint survival function of (U, V ) for 0 <u,v < 1 is −α S (u, v)= P (U u, V v) = 1 u v + u−1/α + v−1/α 1 . U,V ≥ ≥ − − − (iv) U V = v has the CDF | 1 1 P (U u V = v)= ≤ | v(α+1)/α × (u−1/α + v−1/α 1)α+1 · − Now we would like to define a bivariate power normal distribution using this Clayton cop- ula so that the marginals are univariate power normal distributions.