Skip to main content

Maximum Likelihood Estimation of Asymmetric Double Type II Pareto Distributions

Abstract

This paper considers a flexible class of asymmetric double Pareto distributions (ADP) that allows for skewness and asymmetric heavy tails. The inference problem is examined for maximum likelihood. Consistency is proven for the general case when all parameters are unknown. After deriving the Fisher information matrix, asymptotic normality and efficiency are established for a restricted model with the location parameter known. The asymptotic properties of the estimators are then examined using Monte Carlo simulations. To assess its goodness of fit, the ADP is applied to companies’ growth rates, for which it is favored over competing models.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Notes

  1. 1.

    The similarity of (1) and (2) can also bee seen in their limiting case of the Laplace distribution. While taking the limit as \(\alpha \rightarrow \infty\) in (1), and as \(\alpha \rightarrow \infty\) and \(\beta \rightarrow 0\) in (2) (provided the limit of \(\alpha \beta\) exists), the DP and CL distribution both reduce to the Laplace distribution.

  2. 2.

    Strong leptokurtosis is well established for, i.a. financial return series [7]; output growth rates, both between and within countries [8]; and the growth rates of business companies [9]. Significant skewness is also known to permeate financial return series [10, 11], and the growth rates of business companies [12].

  3. 3.

    There are two distinct families of stable distributions that are the only non-trivial limits to normalized (i) ordinary sums of random variables and (ii) geometric sums of random variables. The former is perhaps the more commonly known stable distribution that includes the Gaussian (\(\alpha =2\)), but also the Cauchy (\(\alpha =1\)) and Levý (\(\alpha =1.5\)) distribution [13], and the latter is the geometric stable distribution, which include the Laplace (\(\alpha =2\)) distribution as the only known special case [14]. Popularized by Mandelbrot [15] and Fama [16], the \(\alpha\)-stable class has been applied extensively to i.a. financial return series (see [17] for an extensive review).

  4. 4.

    For example by using integration parts and the substitution \(y^{-}=1+(x-\mu )/\sigma \kappa\) for \(x\le \mu\) and \(y^{+}=1+\kappa (\mu -x)/\sigma\) for \(x>\mu\).

  5. 5.

    Different values on \(\mu\) are only used here to better illustrate skewness and have no additional effect on the shape of the density other than shifting its location.

  6. 6.

    How to estimate the power law cutoff in the tail is a question that have entertained a growing applied literature (see [19] for a vivid discussion).

  7. 7.

    If data would be truncated outside of this interval, a reasonably good fit could be achieved by fitting, e.g., an asymmetric Laplace (L) distribution. For practical purposes, it can be convenient to trim data from so-called outliers. Whether appropriate or nor, such an operation could easily mistake an ADP distribution for a double exponential that would appear as two linear segments emanating downwards from the mode in a semi-log plot.

  8. 8.

    Note the particular simple form of the entropy in the fully symmetric DP case with \(\kappa =1\) and \(\alpha _{l}=\alpha _{r}=\alpha\), then

    $$\begin{aligned} H_{\mathrm{ADP}}|_{\kappa =1,\alpha _{l}=\alpha _{r}}=1+\frac{1}{\alpha }+\log \left( 2\frac{\sigma }{\alpha }\right) , \end{aligned}$$
    (13)

    which can be compared to the entropy for the type I Pareto (P) distribution defined over some lower bound \(x_{m}>0\), namely \(H_{P}=1+\frac{1}{\alpha }+\log \left( \frac{x_{m}}{\alpha }\right)\). It shows a direct correspondence between roles of \(\sigma\) and \(x_{m}\) in the DP and P distributions. Moreover, it is not difficult to show that \(f_{\mathrm{ADP}}\left( x;\mathbf p \right) |_{\kappa =1,\alpha _{l}=\alpha _{r}}\) is a maximum entropy distribution subject to the following functional constraints

    $$\begin{aligned} E\left[ \log \left( 1+\left| \frac{X-\mu }{\sigma }\right| \right) \right] =\frac{1}{\alpha },\;\; \text {and}\;\; \int _{-\infty }^{\infty }f\left( x\right) \mathrm{d}x=1, \end{aligned}$$
    (14)

    where the latter is a normalization constraint. As far as I know, this property has yet to be recognized for DP distributions. For a general solution method to the maximum entropy problem, I refer to [20], and for the case of \(f_{\mathrm{ADP}}\left( x;\mathbf p \right) |_{\kappa =1,\alpha _{l}=\alpha _{r}}\), a detailed solution can be provided upon request.

  9. 9.

    The asymptotic properties of ML estimator for the generalized Pareto distribution were first derived by Smith [21]. Since then, much attention has been directed at the non-convergence problem of ML estimation for smaller samples [22, 23]. In a two-part paper [24, 25], the authors provide an extensive review over the estimation problem for the GPD where classical ML approach is compared to other more refined methods.

  10. 10.

    See, e.g., [10] and [11] for a similar use of indicator functions to describe the log-likelihood function.

  11. 11.

    It can be shown that \(\mathcal {H}_{\mu ,\mu }\left( {\mathbf {p}}_{0}\right) ={I}_{\mu \mu }\left( {\mathbf {p}}_{0}\right)\) is the only instance where condition (B) is violated.

  12. 12.

    By requiring that \(g^{T} H g<1\times 10^{-3}\) does not alter the results nor the convergence incidence. However, when considering default tolerance levels (in ml) of \(1\times 10^{-5}\), I find the failure rate to be high even for larger n. For \(n=6400 ,\) e.g., about 1 / 3 of the simulations fail to converge. However, by considering the first 10,000 converging samples under default tolerance, the results appear not to be qualitatively different (not shown).

  13. 13.

    Convergence properties are generally good when the ratio of \(\alpha _{l}/\sigma\) or \(\alpha _{r}/\sigma\) is large. But for small n, numerical derivatives could sometimes fail to compute.

  14. 14.

    Due to their long expressions, I do not show them here, which would take up too much space.

  15. 15.

    All models are estimated with ML using the Stata routine ml as described in the previous section.

  16. 16.

    Estimates for the AL and AEP densities are not presented to save space, but can be made available upon request.

  17. 17.

    Note that the seemingly increased erraticness in the respective tail is mainly an artifact of the logarithmic scaling on the vertical axis.

References

  1. 1.

    Wang H et al (2012) Bayesian graphical lasso models and efficient posterior computation. Bayesian Anal 7(4):867–886

    MathSciNet  Article  Google Scholar 

  2. 2.

    Armagan A, Dunson DB, Lee J (2013) Generalized double Pareto shrinkage. Stat Sin 23(1):119

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Nadarajah S, Afuecheta E, Chan S (2013) A double generalized Pareto distribution. Stat Prob Lett 83(12):2656–2663

    MathSciNet  Article  Google Scholar 

  4. 4.

    Papastathopoulos I, Tawn JA (2013) A generalised students t-distribution. Stat Probab Lett 83(1):70–77

    MathSciNet  Article  Google Scholar 

  5. 5.

    Kotz S, Kozubowski T, Podgorski K (2012) The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Springer, Berlin

    MATH  Google Scholar 

  6. 6.

    Punathumparambath B, Kulathinal S, George S (2012) Asymmetric type II compound laplace distribution and its application to microarray gene expression. Comput Stat Data Anal 56(6):1396–1404

    MathSciNet  Article  Google Scholar 

  7. 7.

    Mantegna RN, Stanley HE (2007) Introduction to econophysics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  8. 8.

    Fagiolo G, Napoletano M, Roventini A (2008) Are output growth-rate distributions fat-tailed? some evidence from OECD countries. J Appl Econom 23(5):639–669

    MathSciNet  Article  Google Scholar 

  9. 9.

    Stanley M, Amaral L, Buldyrev S, Havlin S, Leschhorn H, Maass P, Salinger M, Stanley H (1996) Scaling behaviour in the growth of companies. Nature 379(6568):804–806

    Article  Google Scholar 

  10. 10.

    Komunjer I (2007) Asymmetric power distribution: theory and applications to risk measurement. J Appl Econom 22(5):891–921

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bottazzi G, Coad A, Jacoby N, Secchi A (2011) Corporate growth and industrial dynamics: evidence from French manufacturing. Appl Econ 43(1):103–116

    Article  Google Scholar 

  12. 12.

    Holly S, Petrella I, Santoro E (2013) Aggregate fluctuations and the cross-sectional dynamics of firm growth. J R Stat Soc Ser A (Stat Soc) 176(2):459–479

    MathSciNet  Article  Google Scholar 

  13. 13.

    Zolotarev VM (1986) One-dimensional stable distributions, vol 65. American Mathematical Society, Providence

    Book  Google Scholar 

  14. 14.

    Kozubowski TJ, Rachev ST (1999) Univariate geometric stable laws. J Comput Anal Appl 1(2):177–217

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Mandelbrot BB (1997) The variation of certain speculative prices. Fractals and scaling in finance. Springer, Berlin, pp 371–418

    Google Scholar 

  16. 16.

    Fama EF (1965) The behavior of stock-market prices. J Bus 38(1):34–105

    Article  Google Scholar 

  17. 17.

    Mittnik S, Rachev ST, Paolella MS (1998) Stable Paretian modeling in finance: some empirical and theoretical aspects. In: Adler RJ, Feldman RE, Taqqu MS (eds) A practical guide to heavy tails: statistical techniques and applications. Birkhauser Boston Inc., Cambridge, pp 79–110

    MATH  Google Scholar 

  18. 18.

    Fu D, Pammolli F, Buldyrev S, Riccaboni M, Matia K, Yamasaki K, Stanley H (2005) The growth of business firms: theoretical framework and empirical evidence. Proc Natl Acad Sci USA 102(52):18801

    Article  Google Scholar 

  19. 19.

    Clauset A, Shalizi C, Newman M (2009) Power-law distributions in empirical data. SIAM Rev 51(4):661–703

    MathSciNet  Article  Google Scholar 

  20. 20.

    Alfarano S, Milakovic M, Irle A, Kauschke J (2012) A statistical equilibrium model of competitive firms. J Econ Dyn Control 36(1):136–149

    MathSciNet  Article  Google Scholar 

  21. 21.

    Smith RL (1984) Threshold methods for sample extremes. In: de Oliveira J (ed) Statistical extremes and applications. Springer, New York, pp 621–638

    Chapter  Google Scholar 

  22. 22.

    Hosking JR, Wallis JR (1987) Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29(3):339–349

    MathSciNet  Article  Google Scholar 

  23. 23.

    Ashkar F, Tatsambon CN (2007) Revisiting some estimation methods for the generalized Pareto distribution. J Hydrol 346(3–4):136–143

    Article  Google Scholar 

  24. 24.

    de Zea Bermudez P, Kotz S (2010) Parameter estimation of the generalized Pareto distribution—part I. J Stat Plan Inference 140(6):1353–1373

    Article  Google Scholar 

  25. 25.

    de Zea Bermudez P, Kotz S (2010) Parameter estimation of the generalized Pareto distribution—part II. J Stat Plan Inference 140(6):1374–1388

    Article  Google Scholar 

  26. 26.

    Singh V, Guo H (1995) Parameter estimation for 3-parameter generalized Pareto distribution by the principle of maximum entropy (POME). Hydrol Sci J 40(2):165–181

    Article  Google Scholar 

  27. 27.

    Newey WK, McFadden D (1994) Large sample estimation and hypothesis testing. In: Engle R, McFadden D (eds) Handbook of econometrics, vol 4. Elsevier, Amsterdam, pp 2111–2245

    Google Scholar 

  28. 28.

    Lehmann E, Casella G (1998) Theory of point estimation. Springer, Berlin

    MATH  Google Scholar 

  29. 29.

    Bottazzi G, Secchi A (2006) Explaining the distribution of firm growth rates. RAND J Econ 37(2):235–256

    Article  Google Scholar 

  30. 30.

    Axtell R (2001) Zipf distribution of US firm sizes. Science 293(5536):1818

    Article  Google Scholar 

  31. 31.

    Schwarz G et al (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464

    MathSciNet  Article  Google Scholar 

  32. 32.

    Bottazzi G, Secchi A (2011) A new class of asymmetric exponential power densities with applications to economics and finance. Ind Corp Change 20(4):991–1030

    Article  Google Scholar 

  33. 33.

    Agro G (1995) Maximum likelihood estimation for the exponential power function parameters. Commun Stat Simul Comput 24(2):523–536

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

I thank seminar participants at Örebro University and one anonymous referee for valuable comments. The usual caveats apply.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Daniel Halvarsson.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Proofs

I begin by solving the following useful integrals

$$\begin{aligned} E \left[ \log \left( 1+ \frac{1}{\kappa }\left| \frac{X-\mu }{\sigma }\right| \right) \mathbb {1}_{x\le \mu }\right] \;\; \text {and} \;\; E \left[ \log \left( 1+ \kappa \left| \frac{X-\mu }{\sigma }\right| \right) \mathbb {1}_{x>\mu }\right] , \end{aligned}$$
(34)

given in (20) and (21). The approach is similar for either expression, so I here show the steps only for the case when \(x\le \mu\). Writing the expected value in terms of the PDF given in Definition 3 it results in the following improper integral

$$\begin{aligned} C\int _{-\infty }^{\mu }\log \left( 1+\frac{\mu - x}{\kappa \sigma }\right) \left( 1+\frac{\mu - x}{\kappa \sigma }\right) ^{-1-\alpha _{l}}\mathrm{d}x. \end{aligned}$$
(35)

Using the substitution \(u=1+(\mu -x)/\kappa \sigma\), the integral can be written as follows

$$\begin{aligned} \underset{b\rightarrow \infty }{\lim } C \kappa \sigma \int _{1}^{b}\log \left( u\right) u^{-1-\alpha _{l}}{\mathrm{d}}u. \end{aligned}$$
(36)

Note that the sign of the integral changes twice since the order of integration is reversed. Next, the resulting integral can easily be evaluated by using integration by parts. Define \(v=\log \left( u\right)\) and \(ds=u^{-1-\alpha _{l}}\). It directly follows that

$$\begin{aligned} \underset{b\rightarrow \infty }{\lim } C \kappa \sigma \int _{1}^{b}\log \left( u\right) u^{-1-\alpha _{l}}{\mathrm{d}}&= \underset{b\rightarrow \infty }{\lim } C \kappa \sigma \left[ \frac{\log \left( u\right) u^{-\alpha _{l}}}{\alpha _{l}}\bigg |_{1}^{b} + \int _{1}^{b}u^{-1-\alpha _{l}}{\mathrm{d}} \right] \end{aligned}$$
(37)
$$\begin{aligned} &= 0 + \underset{b\rightarrow \infty }{\lim }C\kappa \sigma \int _{1}^{b}u^{-1-\alpha _{l}}{\mathrm{d}} \end{aligned}$$
(38)
$$\begin{aligned} &= \frac{C\kappa \sigma }{\alpha _{l}}, \end{aligned}$$
(39)

which after plugging in the value for C gives the expression in (20).

Proof of Proposition 1

To show that the density is uniquely identified for a given vector of parameters (i), I consider the converse of the implication in the proposition (c.f. [10]), namely that if \(f_{\mathrm{ADP}}\left( x;{\mathbf {p}}\right) = f_{\mathrm{ADP}}\left( x;\mathbf {p_{0}}\right)\) then \({\mathbf {p}}={\mathbf {p}}_{0}\). Since the ADP density is unimodal it means that \(f_{\mathrm{ADP}}\left( x;{\mathbf {p}}\right)\) and \(f_{\mathrm{ADP}}\left( x;\mathbf {p_{0}}\right)\) have the same mode. Thus \(\mu =\mu _{0}\). To establish identity for \(\alpha _{l}\) and \(\alpha _{r}\), the reasoning is a little bit longer. Begin by noting that \(f_{\mathrm{ADP}}\left( \mu ;{\mathbf {p}}\right) = f_{\mathrm{ADP}}\left( \mu ;\mathbf {p_{0}}\right)\) establishes that equality holds between the normalization constants, i.e., \(C=C_0\). Further, \(f_{\mathrm{ADP}}\left( \mu ;{\mathbf {p}}\right) = f_{\mathrm{ADP}}\left( \mu ;\mathbf {p_{0}}\right)\) also implies that \(F_{\mathrm{ADP}}\left( \mu ;{\mathbf {p}}\right) = F_{\mathrm{ADP}}\left( \mu ;\mathbf {p_{0}}\right)\) and hence that \(\kappa ^2\alpha _{r}/\left( \alpha _{l}+\kappa ^2 \alpha _{r}\right) =\kappa _{0}^2\alpha _{r 0}/\left( \alpha _{l 0}+\kappa _{0}^2 \alpha _{r 0}\right)\). It also follows from \(f_{\mathrm{ADP}}\left( x;{\mathbf {p}}\right) = f_{\mathrm{ADP}}\left( x;\mathbf {p_{0}}\right)\) that \(\log f_{\mathrm{ADP}}\left( x;{\mathbf {p}}\right) = \log f_{\mathrm{ADP}}\left( x;\mathbf {p_{0}}\right)\). If \(\log C=\log C_0\) and \(\mu =\mu _{0}\) it means that \(\left( 1+\alpha _{l}\right) D_{1}^{-}\left( x\right) =\left( 1+\alpha _{l0}\right) D_{10}^{-}\left( x\right)\) and \(\left( 1+\alpha _{r}\right) D_{1}^{+}\left( x\right) =\left( 1+\alpha _{r0}\right) D_{10}^{+}\left( x\right)\). From the uniqueness of the expectation operator in (20) and (21), it then follows that \(\left( \alpha _{l}+1\right) D^{-}=\left( \alpha _{l0}+1\right) D^{+}\) and \(\left( \alpha _{r}+1\right) D^{+}=\left( \alpha _{r0}+1\right) D^{-}\). As a final step, using this information, we get the following equalities

$$\begin{aligned} \frac{\alpha _{l}}{\alpha _{l}+1}&= \frac{ F_{\mathrm{ADP}}\left( \mu ;{\mathbf {p}}\right) }{D^{-}}= \frac{F_{\mathrm{ADP}}\left( \mu ;\mathbf {p_{0}}\right) }{D^{-}} =\frac{\alpha _{l0}}{\alpha _{l0}+1} \end{aligned}$$
(40)
$$\begin{aligned} \frac{\alpha _{r}}{\alpha _{r}+1}&=\frac{1-F_\mathrm{ADP}\left( \mu ;{\mathbf {p}}\right) }{D^{+}} = \frac{1-F_{\mathrm{ADP}}\left( \mu ;\mathbf {p_{0}}\right) }{D^{+}}=\frac{\alpha _{0r}}{\alpha _{r0}+1}. \end{aligned}$$
(41)

Since \(\alpha /\left( \alpha +1\right)\) is strictly monotonically increasing on \(\alpha \in (-1,\infty )\) it follows that \(\alpha _{l}=\alpha _{l0}\) and \(\alpha _{r}=\alpha _{r0}\). To establish the result for \(\kappa\), consider \(D^{-}=D^{-}\), which after factoring out \(\kappa\) from the numerator and denominator together with \(\alpha _{l}=\alpha _{0l}\) and \(\alpha _{r}=\alpha _{0r}\) establishes that \(\kappa =\kappa _{0}\). Finally, given \(\mu =\mu _{0}\), \(\alpha _{l}=\alpha _{0l}\), \(\alpha _{r}=\alpha _{0r}\), and \(\kappa =\kappa _{0}\), it follows from, e.g., \(C= C_{0}\) by comparing cases that \(\sigma =\sigma _{0}\), which verifies identification.

Condition (ii) can be trivially verified for \({\mathbf {p}}=\left( \mu ,\kappa ,\sigma ,\alpha _{l},\alpha _{r}\right)\) by defining \(\mathcal {P}\) as the open set \(\left( -\infty ,\infty \right) \times \left( 0,\infty \right) \times \left( 0,\infty \right) \times \left( 0,\infty \right) \times \left( 0,\infty \right)\) such that \({\mathbf {p}}\in \mathcal {P}\). Since we can always consider a compact subset \(\Theta\) of \(\mathcal {P}\) such that \(\mathbf {p_{0}}\in \Theta\)\(\forall \mathbf {p_{0}}\) the condition is verified.

As for the condition (iii), it suffices to note that the density has only one discontinuity at \(x=\mu\) with a measure of 0. It therefore follows that \(f_{\mathrm{ADP}}\left( x;\mathbf {p_{0}}\right)\) is continuous at every \(p_{0}\in \Theta\) with probability 1.

Condition (iv) is verified if we can find a dominating function \(M\left( x;{\mathbf {p}}\right)\) such that \(M\left( x;{\mathbf {p}}\right) \ge \left| \log f_{\mathrm{ADP}}\left( x;{\mathbf {p}}\right) \right|\), \(\forall {\mathbf {p}}\in \Theta\), for which \(E \left[ M\left( X;{\mathbf {p}}\right) \right] <\infty\). To find such \(M\left( x;{\mathbf {p}}\right)\), note that by the triangle inequality

$$\begin{aligned} \left| \log f_{\mathrm{ADP}}\left( x;{\mathbf {p}}\right) \right|&\le M\left( x;{\mathbf {p}}\right) \nonumber \\&:={\left\{ \begin{array}{ll} \left| \log \frac{\kappa \alpha _{l}\alpha _{r}}{\sigma \left( \alpha _{l}+\alpha _{r}\kappa ^{2}\right) }\right| +\left| \left( \alpha _{l}+1\right) \log \left( 1+ \frac{1}{\kappa }\left| \frac{x-\mu }{\sigma }\right| \right) \right| \;\; \text {if}\;x\le \mu \\ \left| \log \frac{\kappa \alpha _{l}\alpha _{r}}{\sigma \left( \alpha _{l}+\alpha _{r}\kappa ^{2}\right) }\right| +\left| \left( \alpha _{r}+1\right) \log \left( 1+ \kappa \left| \frac{x-\mu }{\sigma }\right| \right) \right| \;\; \text {if}\;x>\mu \end{array}\right. } \end{aligned}$$
(42)

Taking the expectation of \(M\left( x;{\mathbf {p}}\right)\), using (20) and (21) results in

$$\begin{aligned} E \left[ M\left( X;{\mathbf {p}}\right) \right] =\left| \log \frac{\kappa \alpha _{l}\alpha _{r}}{\sigma \left( \alpha _{l}+\alpha _{r}\kappa ^{2}\right) }\right| +\frac{\left( \alpha _{l}+1\right) \kappa ^2 \alpha _{r}}{\alpha _{l}\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) }+\frac{\left( \alpha _{r}+1\right) \alpha _{n}}{\alpha _{r}\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) }<\infty , \end{aligned}$$
(43)

which shows (iv) and concludes the proof. \(\square\)

Proof of Proposition 2

Calculating the cross-derivatives, the simplified expressions of the expectations are presented below and can be readily solved using standard techniques.

$$\begin{aligned} {\mathcal {I}}_{\mu \mu }&= \left( \frac{\alpha _{l}+1}{\kappa \sigma }\right) ^{2}\int _{-\infty }^{\mu }\left[ 1+\frac{\mu -x}{\kappa \sigma }\right] ^{-2}f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\&\quad +\left( \frac{\alpha _{r}+1}{\kappa ^{-1}\sigma }\right) ^{2}\int _{\mu }^{\infty }\left[ 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right] ^{-2}f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= C\left( \frac{1}{\kappa \sigma }\frac{\left( 1+\alpha _{l}\right) ^{2}}{\left( 2+\alpha _{l}\right) }+\frac{1}{\kappa ^{-1}\sigma }\frac{\left( 1+\alpha _{r}\right) ^{2}}{\left( 2+\alpha _{r}\right) }\right) \\ {\mathcal {I}}_{\mu \kappa }&= \frac{\alpha _{l}+1}{\kappa ^{2}\sigma }\int _{-\infty }^{\mu }\bigg [-\alpha _{l}\left( \frac{2+\alpha _{l}+\kappa ^{2}\alpha _{r}}{\alpha _{l}+\kappa ^{2}\alpha _{r}}\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\\&\quad +\left( 1+\alpha _{n}\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-2}\bigg ]f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad +\frac{\alpha _{r}+1}{\sigma }\int _{\mu }^{\infty }\bigg [-\alpha _{r}\left( \frac{2\kappa ^{2}+\alpha _{l}+\kappa ^{2}\alpha _{r}}{\alpha _{l}+\kappa ^{2}\alpha _{r}}\right) \left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) ^{-1}\\& \quad +\left( 1+\alpha _{r}\right) \left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) ^{-2}\bigg ]f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= \frac{C}{\kappa }\left( \frac{1}{2+\alpha _{l}}-\frac{2\alpha _{l}}{2\kappa ^{2}\alpha _{r}+\alpha _{l}}\right) +\frac{C}{\kappa }\left( -2+\frac{1}{2+\alpha _{r}}+\frac{2\alpha _{l}}{2\kappa ^{2}\alpha _{r}+\alpha _{l}}\right) \\ &= \frac{C}{\kappa }\left( -2+\frac{1}{2+\alpha _{l}}+\frac{1}{2+\alpha _{r}}\right) \\ {\mathcal {I}}_{\mu \sigma }&= \left( \frac{\alpha _{l}+1}{\kappa \sigma ^{2}}\right) \int _{-\infty }^{\mu }\bigg [-\alpha _{l}\left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\\& \quad +\left( 1+\alpha _{l}\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-2}\bigg ]f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad +\left( \frac{\alpha _{r}+1}{\kappa ^{-1}\sigma ^{2}}\right) \int _{\mu }^{\infty }\bigg [\alpha _{r}\left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) ^{-1}\\ & \quad -\,\left( 1+\alpha _{r}\right) \left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) ^{-2}\bigg ]f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= \frac{C}{\sigma }\left( \frac{1}{2+\alpha _{l}}-\frac{1}{2+\alpha _{r}}\right) \\ \end{aligned}$$
$$\begin{aligned} {\mathcal {I}}_{\mu \alpha _{l}}&= \left( \frac{\alpha _{l}+1}{\kappa \sigma }\right) \int _{-\infty }^{\mu }\left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\bigg [-\frac{\kappa ^{2}\alpha _{r}}{\alpha _{l}\left( \kappa ^2\alpha _{r}+\alpha _{n}\right) }\\& \quad +\log \left( 1+\frac{\mu -x}{\kappa \sigma }\right) \bigg ]f\left( x;{\mathbf {p}}\right) \mathrm{d}x \\& \quad +\left( \frac{\alpha _{r}+1}{\kappa ^{-1}\sigma }\right) \frac{\kappa ^{2}\alpha _{r}}{\alpha _{l}\left( \kappa ^2\alpha _{r}+\alpha _{n}\right) }\int _{\mu }^{\infty }\bigg [\left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) ^{-1}\bigg ]f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= C\left( -\frac{1}{\alpha _{l}+\alpha _{l}^{2}}+\frac{1}{\kappa ^{2}\alpha _{r}+\alpha _{l}}\right) +\frac{C\kappa ^{2}\alpha _{r}}{\alpha _{l}\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) }\\ &= \frac{C}{1+\alpha _{l}}\\ {\mathcal {I}}_{\mu \alpha _{r}}&= -\left( \frac{\alpha _{l}+1}{\kappa \sigma }\right) \frac{\alpha _{l}}{\alpha _{r}\left( \kappa ^2\alpha _{r}+\alpha _{n}\right) }\int _{-\infty }^{\mu }\bigg [\left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\bigg ]f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad +\left( \frac{\alpha _{r}+1}{\kappa ^{-1}\sigma }\right) \int _{\mu }^{\infty }\left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) ^{-1}\bigg [\frac{\alpha _{l}}{\alpha _{r}\left( \kappa ^2\alpha _{r}+\alpha _{n}\right) }\\& \quad -\log \left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) \bigg ]f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= -\frac{C\alpha _{l}}{\alpha _{l}\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) }+C\left( \frac{1}{\alpha _{r}+\alpha _{r}^{2}}-\frac{\kappa ^{2}}{\kappa ^{2}\alpha _{r}+\alpha _{l}}\right) \\ &= -\frac{C}{1+\alpha _{r}}\\ {\mathcal {I}}_{\kappa \kappa }&= \frac{1}{\kappa ^{2}}\int _{-\infty }^{\mu }\left( \frac{\alpha _{l}\left( 2+\kappa ^{2}\alpha _{r}+\alpha _{l}\right) }{\kappa ^{2}\alpha _{r}+\alpha _{l}}-\left( 1+\alpha _{l}\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\right) ^{2}f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad +\frac{1}{\kappa ^{2}}\int _{\mu }^{\infty }\left( -\frac{\alpha _{r}\left( 2\kappa ^{2}+\kappa ^{2}\alpha _{r}+\alpha _{l}\right) }{\kappa ^{2}\alpha _{r}+\alpha _{l}}\right. \\& \quad \left. +\left( 1+\alpha _{r}\right) \left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) ^{-1}\right) ^{2}f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= \frac{C}{\kappa ^{2}}\left( \frac{\kappa \sigma }{\left( 1+\alpha _{l}\right) }+\frac{4\kappa \sigma \alpha _{l}}{\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) ^{2}}\right) + \frac{C}{\kappa ^{2}}\left( \frac{\sigma }{\kappa \left( 1+\alpha _{r}\right) }+\frac{4\kappa ^{3}\sigma \alpha _{r}}{\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) ^{2}}\right) \\ &= \frac{C}{\kappa ^{2}}\left( \frac{\kappa \sigma }{2+\alpha _{l}}+\frac{\sigma }{\kappa \left( 2+\alpha _{r}\right) }+\frac{4\kappa \sigma }{\kappa ^{2}\alpha _{r}+\alpha _{l}}\right) \\ {\mathcal {I}}_{\kappa \sigma }&= \frac{1}{\kappa \sigma }\int _{-\infty }^{\mu }\left( \alpha _l-\left( \alpha _l+1\right) \left( \frac{\mu -x}{\kappa \sigma }+1\right) ^{-1}\right) \left( \frac{\alpha _l \left( 2+\kappa ^2 \alpha _r+\alpha _l\right) }{\kappa ^2 \alpha _r+\alpha _l}\right. \\& \quad \left. -\left( \alpha _l+1\right) \left( \frac{\mu -x}{\kappa \sigma }+1\right) ^{-1}\right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\&\frac{1}{\kappa \sigma }\int _{\mu }^{\infty }\left( \alpha _r-\left( \alpha _r+1\right) \left( \frac{x-\mu }{\kappa ^{-1} \sigma }+1\right) ^{-1}\right) \left( -\frac{\alpha _r \left( 2\kappa ^{2}+\kappa ^2 \alpha _r+\alpha _l\right) }{\kappa ^2 \alpha _r+\alpha _l}\right. \\& \quad \left. +\left( \alpha _r+1\right) \left( \frac{x-\mu }{\kappa ^{-1}\sigma }+1\right) ^{-1}\right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= \frac{C}{\kappa \sigma }\left( \frac{\kappa \sigma }{2+\alpha _{l}}-\frac{\sigma }{\kappa \left( 2+\alpha _{r}\right) }\right) \\ \end{aligned}$$
$$\begin{aligned} {\mathcal {I}}_{\kappa \alpha _{l}}&= \frac{1}{\kappa }\int _{-\infty }^{\mu }\left( \frac{\kappa ^{2}\alpha _{r}}{\alpha _{l}\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) }-\log \left( 1+\frac{\mu -x}{\kappa \sigma }\right) \right) \left( \frac{\alpha _{l}\left( 2+\kappa ^{2}\alpha _{r}+\alpha _{l}\right) }{\kappa ^{2}\alpha _{r}+\alpha _{l}}\right. \\& \quad \left. -\left( 1+\alpha _{l}\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad +\frac{\kappa ^{2}\alpha _{r}}{\alpha _{l}\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) }\int _{\mu }^{\infty }\left( -\frac{\alpha _{r}\left( 2\kappa ^{2}+\kappa ^{2}\alpha _{r}+\alpha _{l}\right) }{\kappa ^{2}\alpha _{r}+\alpha _{l}}\right. \\& \quad \left. +\left( 1+\alpha _{r}\right) \left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) ^{-1}\right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= C\left( -\frac{\sigma }{\alpha _l^2+\alpha _l}-\frac{2\sigma }{\left( \alpha _l+\kappa ^2 \alpha _r\right) ^2}\right) -\frac{2 c \kappa ^2 \sigma \alpha _r}{\alpha _l \left( \alpha _l+\kappa ^2 \alpha _r\right) {}^2}\\ &= \frac{C}{\kappa \alpha _{l}}\left( -\frac{2\kappa \sigma }{\alpha _l+\kappa ^2 \alpha _r}-\frac{\kappa \sigma }{\alpha _l+1}\right) \\ {\mathcal {I}}_{\kappa \alpha _{r}}&= +\frac{\alpha _{l}}{\kappa ^{2}\alpha _{r}\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) }\int _{\mu }^{\infty }\left( \frac{\alpha _{l}\left( 2+\kappa ^{2}\alpha _{r}+\alpha _{l}\right) }{\kappa ^{2}\alpha _{r}+\alpha _{l}}\right. \\& \quad \left. -\left( 1+\alpha _{l}\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad \frac{1}{\kappa }\int _{\mu }^{\infty }\left( \frac{\alpha _{l}}{\alpha _{r}\left( \kappa ^{2}\alpha _{r}+\alpha _{l}\right) }-\log \left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) \right) \left( -\frac{\alpha _{r}\left( 2\kappa ^{2}+\kappa ^{2}\alpha _{r}+\alpha _{l}\right) }{\kappa ^{2}\alpha _{r}+\alpha _{l}}\right. \\ & \quad \left. +\left( 1+\alpha _{r}\right) \left( 1+\frac{\mu -x}{\kappa ^{-1}\sigma }\right) ^{-1}\right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= C\left( -\frac{\sigma }{\alpha _l^2+\alpha _l}-\frac{2\sigma }{\left( \alpha _l+\kappa ^2 \alpha _r\right) ^2}\right) -\frac{2 c \kappa ^2 \sigma \alpha _r}{\alpha _l \left( \alpha _l+\kappa ^2 \alpha _r\right) {}^2}\\ &= \frac{C}{\kappa \alpha _{l}}\left( -\frac{2\kappa \sigma }{\alpha _l+\kappa ^2 \alpha _r}-\frac{\kappa \sigma }{\alpha _l+1}\right) \\ &= \frac{2 C \sigma \alpha _l}{\alpha _r \left( \alpha _l+\kappa ^2 \alpha _r\right) {}^2}+\frac{C \sigma }{\kappa ^2}\left( \frac{2 \kappa ^4}{\left( \alpha _l+\kappa ^2 \alpha _r\right) {}^2}+\frac{1}{\alpha _{r}\left( 1+\alpha _r\right) }\right) \\ &= \frac{c}{\kappa \alpha _r} \left( \frac{2 \kappa \sigma }{\alpha _l+\kappa ^2 \alpha _r}+\frac{\sigma }{\kappa \left( \alpha _r+1\right) }\right) \\ \end{aligned}$$
$$\begin{aligned} {\mathcal {I}}_{\sigma \sigma }&= \frac{1}{\sigma ^2}\int _{-\infty }^{\mu }\left( \alpha _l-\left( \alpha _l+1\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\right) ^2 f\left( x;{\mathbf {p}}\right) \mathrm{d}x+\frac{1}{\sigma ^2}\int _{\mu }^{\infty }\left( \alpha _r\right. \\& \quad \left. -\left( \alpha _r+1\right) \left( 1+\frac{x-\mu }{\kappa ^{-1} \sigma }\right) ^{-1}\right) ^2 f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= \frac{c}{\sigma ^{2}}\left( \frac{\kappa \sigma }{\alpha _l+2}\right) +\frac{c}{\sigma ^{2}}\left( \frac{\sigma }{\kappa \left( \alpha _l+2\right) }\right) \\ {\mathcal {I}}_{\sigma \alpha _{l}}&= \frac{1}{\sigma }\int _{-\infty }^{\mu }\left( \alpha _l-\left( \alpha _l+1\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\right) \left( \frac{\kappa ^2 \alpha _r}{\alpha _l \left( \alpha _l+\kappa ^2 \alpha _r\right) }\right. \\& \quad \left. -\log \left( \frac{\mu -x}{\kappa \sigma }+1\right) \right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad +\frac{1}{\sigma }\int _{\mu }^{\infty }\left( \alpha _r-\left( \alpha _r+1\right) \left( 1+\frac{x-\mu }{\kappa ^{-1} \sigma }\right) ^{-1}\right) \left( \frac{\kappa ^2 \alpha _r}{\alpha _l \left( \alpha _l+\kappa ^2 \alpha _r\right) }\right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= -\frac{c}{\sigma \alpha _{l}}\left( \frac{\sigma \kappa }{\alpha _l+1}\right) \\ \end{aligned}$$
$$\begin{aligned} {\mathcal {I}}_{\sigma \alpha _{r}}&= \frac{1}{\sigma }\int _{-\infty }^{\mu }\left( \alpha _l-\left( \alpha _l+1\right) \left( 1+\frac{\mu -x}{\kappa \sigma }\right) ^{-1}\right) \left( \frac{\alpha _l}{\alpha _r \left( \alpha _l+\kappa ^2 \alpha _r\right) }\right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ & \quad +\frac{1}{\sigma }\int _{\mu }^{\infty }\left( \alpha _r-\left( \alpha _r+1\right) \left( 1+\frac{x-\mu }{\kappa ^{-1} \sigma }\right) ^{-1}\right) \left( \frac{\alpha _l}{\alpha _r \left( \alpha _l+\kappa ^2 \alpha _r\right) }\right. \\& \quad \left. -\log \left( \frac{x-\mu }{\kappa ^{-1} \sigma }+1\right) \right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= -\frac{c}{\sigma \alpha _{r}}\left( \frac{\sigma }{\kappa \left( \alpha _l+1\right) }\right) \\ {\mathcal {I}}_{\alpha _{l}\alpha _{l}}&= \int _{-\infty }^{\mu }\left( \frac{\kappa ^2 \alpha _r}{\alpha _l \left( \alpha _l+\kappa ^2 \alpha _r\right) }-\log \left( \frac{\mu -x}{\kappa \sigma }+1\right) \right) ^2f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad +\frac{1}{\sigma }\int _{\mu }^{\infty }\left( \frac{\kappa ^2 \alpha _r}{\alpha _l \left( \alpha _l+\kappa ^2 \alpha _r\right) }\right) ^2f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= \frac{c}{\alpha _l^2} \left( \frac{\kappa \sigma }{\alpha _l}+\frac{\kappa \sigma \alpha _l}{\left( \alpha _l+\kappa ^2 \alpha +_r\right) {}^2}\right) +\frac{c \kappa ^3 \sigma \alpha _r}{\alpha _l^2 \left( \alpha _l+\kappa ^2 \alpha _r\right) ^2}\\ &= \frac{c}{\alpha _l^2} \left( \frac{\kappa \sigma }{\alpha _l}+\frac{1}{\alpha _l+\kappa ^2 \alpha _r}\right) \\ \end{aligned}$$
$$\begin{aligned} {\mathcal {I}}_{\alpha _{l}\alpha _{r}}&= \frac{\alpha _{l}}{\alpha _{r}\left( \alpha _{l}+\kappa ^{2}\alpha _{r}\right) }\int _{-\infty }^{\mu }\left( \frac{\kappa ^2 \alpha _r}{\alpha _l \left( \alpha _l+\kappa ^2 \alpha _r\right) }-\log \left( \frac{\mu -x}{\kappa \sigma }+1\right) \right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\& \quad +\frac{\kappa ^2 \alpha _r}{\alpha _l \left( \alpha _l+\kappa ^2 \alpha _r\right) }\int _{\mu }^{\infty }\left( \frac{\alpha _{l}}{\alpha _{r}\left( \alpha _{l}+\kappa ^{2}\alpha _{r}\right) }-\log \left( \frac{x-\mu }{\kappa ^{-1} \sigma }+1\right) \right) f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= -\frac{C\kappa \sigma \alpha _{l}}{\alpha _{l}\alpha _{r}\left( \alpha _{l}+\kappa ^{2}\alpha _{r}\right) ^{2}}-\frac{C\kappa ^{3}\sigma \alpha _{r}}{\alpha _{l}\alpha _{r}\left( \alpha _{l}+\kappa ^{2}\alpha _{r}\right) ^{2}}\\ &= -\frac{C}{\alpha _{l}\alpha _{r}}\left( \frac{\kappa \sigma }{\alpha _{l}+\kappa ^{2}\alpha _{r}}\right) \\ {\mathcal {I}}_{\alpha _{r}\alpha _{r}}&= \left( \frac{\alpha _{l}}{\alpha _{r}\left( \alpha _{l}+\kappa ^{2}\alpha _{r}\right) }\right) ^{2}\int _{-\infty }^{\mu }f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ & \quad +\int _{\mu }^{\infty }\left( \frac{\alpha _{l}}{\alpha _{r}\left( \alpha _{l}+\kappa ^{2}\alpha _{r}\right) }-\log \left( \frac{x-\mu }{\kappa ^{-1} \sigma }+1\right) \right) ^{2}f\left( x;{\mathbf {p}}\right) \mathrm{d}x\\ &= \frac{C\kappa \sigma \alpha _{l}}{\alpha _{r}^{2}\left( \alpha _{l}+\kappa ^{2}\alpha _{r}\right) ^{2}}+\frac{C \sigma \left( \alpha _l^2+2 \kappa ^2 \alpha _l \alpha _r+2 \kappa ^4 \alpha _r^2\right) }{\kappa \alpha _r^3 \left( \alpha _l+\kappa ^2 \alpha _r\right) ^2}\\ &= \frac{C}{\alpha _{r}^{2}}\left( \frac{\kappa \sigma }{\alpha _{l}+\kappa ^{2}\alpha _{r}}+\frac{\sigma }{\kappa \alpha _{r}}\right) \end{aligned}$$

Note that the presence of a squared logarithmic term in the integral in, e.g., \({\mathcal {I}}_{\alpha _{r}\alpha _{r}}\) does not introduce further complication and can be solved using substitution and integration by parts along the lines of the expected logarithms solved above. \(\square\)

Proof of Proposition 3

To verify condition (A) see condition (ii) and (iii) in the proof of Proposition 1. Thus, since \(f_{\mathrm{ADP}}\left( \mathbf {x}|{\mathbf {p}}\right)\) is everywhere differentiable (when \(\mu\) is known), all third-order derivatives are admitted.

Turning to condition (B). Beginning with condition for the expectation of the differentiated log-likelihood function (23), direct calculations result in

$$\begin{aligned} E _{{\mathbf {p}}}\left[ \frac{\partial \log f\left( X;{\mathbf {p}}\right) }{\partial \kappa }\right] &= \frac{1}{\kappa }\int _{-\infty }^{\mu }\left[ \frac{2 \alpha _l}{\kappa }^2 \alpha _r\right. \nonumber \\& \quad \left. +\alpha _l+\left( \alpha _{l}+1\right) \left( 1-\left[ \frac{\mu -x}{\kappa \sigma }+1\right] ^{-1}\right) \right] f\left( x;{\mathbf {p}}\right) \mathrm{d}x\nonumber \\& \quad +\frac{1}{\kappa }\int _{\mu }^{\infty }\left[ \frac{\alpha _l-\kappa ^2 \alpha _r}{\kappa ^2 \alpha _r+ \alpha _l}-\left( \alpha _{r}+1\right) \left( 1\right. \right. \nonumber \\& \quad \left. \left. -\left[ \frac{x-\mu }{\kappa ^{-1}\sigma }+1\right] ^{-1}\right) \right] f\left( x;{\mathbf {p}}\right) \mathrm{d}x\nonumber \\ &= \frac{2 \kappa \alpha _l \alpha _r}{\left( \kappa ^2 \alpha _r+\alpha _l\right) {}^2}-\frac{2\kappa \alpha _l \alpha _r}{\left( \kappa ^2 \alpha _r+\alpha _l\right) {}^2}=0\nonumber \\ E _{{\mathbf {p}}}\left[ \frac{\partial \log f\left( X;{\mathbf {p}}\right) }{\partial \sigma }\right] &= \frac{1}{\sigma }\int _{-\infty }^{\mu }\left[ -1+\left( \alpha _{l}+1\right) \left( 1\right. \right. \nonumber \\& \quad \left. \left. -\left[ \frac{\mu -x}{\kappa \sigma }+1\right] ^{-1}\right) \right] f\left( x;{\mathbf {p}}\right) \mathrm{d}x\nonumber \\&\quad +\frac{1}{\sigma }\int _{\mu }^{\infty }\left[ -1+\left( \alpha _{r}+1\right) \left( 1\right. \right. \nonumber \\&\quad \left. \left. -\left[ \frac{x-\mu }{\kappa ^{-1}\sigma }+1\right] ^{-1}\right) \right] f\left( x;{\mathbf {p}}\right) \mathrm{d}x\nonumber \\ &= 0+0\nonumber \\ E _{{\mathbf {p}}}\left[ \frac{\partial \log f\left( X;{\mathbf {p}}\right) }{\partial \alpha _{l}}\right] &= \int _{-\infty }^{\mu }\left[ \frac{\kappa ^2 \alpha _r}{\alpha _l\left( \kappa ^2 \alpha _r+\alpha _l\right) }-\log \left( 1+\frac{\mu -x}{\kappa \sigma }\right) \right] f\left( x;{\mathbf {p}}\right) \mathrm{d}x\nonumber \\&\quad +\int _{\mu }^{\infty }\left[ \frac{\kappa ^2 \alpha _r}{\alpha _l\left( \kappa ^2 \alpha _r+\alpha _l\right) }\right] f\left( x;{\mathbf {p}}\right) \mathrm{d}x\nonumber \\ &= \frac{\kappa ^2 \alpha _r}{\left( \kappa ^2 \alpha _r+\alpha _l\right) {}^2}-\frac{\kappa ^2 \alpha _r}{\left( \kappa ^2 \alpha _r+\alpha _l\right) {}^2}=0\nonumber \\ E _{{\mathbf {p}}}\left[ \frac{\partial \log f\left( X;{\mathbf {p}}\right) }{\partial \alpha _{r}}\right] &= \int _{-\infty }^{\mu }\left[ \frac{\alpha _l}{\alpha _r \left( \kappa ^2 \alpha _r+\alpha _l\right) } \right] f\left( x;{\mathbf {p}}\right) \mathrm{d}x\nonumber \\&\quad +\int _{\mu }^{\infty }\left[ \frac{\alpha _l}{\alpha _r \left( \kappa ^2 \alpha _r+\alpha _l\right) } -\log \left( 1+\frac{x-\mu }{\kappa ^{-1}\sigma }\right) \right] f\left( x;{\mathbf {p}}\right) \mathrm{d}x\nonumber \\ &= \frac{\kappa ^2 \alpha _l}{\left( \kappa ^2 \alpha _r+\alpha _l\right) {}^2}-\frac{\kappa ^2 \alpha _l}{\left( \kappa ^2 \alpha _r+\alpha _l\right) {}^2}=0 \end{aligned}$$
(44)

as required.

To verify the criteria for Fisher matrix (24), it is noted in [32] that for continuous \(f\left( x;{\mathbf {p}}\right) , \partial ^{2} \log f\left( x;{\mathbf {p}}\right) /\partial p_{i}\partial p_{j}\), as is the case when \(\mu\) is known, is simply a consequence of integration by parts, and thus necessitates \(\mathcal {H}\left( {\mathbf {p}}\right) _{ij}={\mathcal {I}}\left( {\mathbf {p}}\right) _{ij}\).

To verify condition (C), i.e., that the matrix \({\mathcal {I}}\left( {\mathbf {p}}\right)\) is positive definite, we need to show that \(\mathbf {z}{\mathcal {I}}\left( {\mathbf {p}}\right) \mathbf {z}^{T}>0\) for all nonzero real vectors \(\mathbf {z}=\left[ z_{1}\; z_{2}\; z_{3}\; z_{4}\right]\), and that \(\mathbf {z}{\mathcal {I}}\left( {\mathbf {p}}\right) \mathbf {z}^{T}=0\) only in the case when \(z_{1}=z_{2}=z_{3}=z_{4}=0\). Separate calculations in Mathematica confirm that \(\mathbf {z}{\mathcal {I}}\left( {\mathbf {p}}\right) \mathbf {z}^{T}>0\), \(\forall\) nonzero vectors \(\mathbf {z}\). Here, I do not show the expression that is lengthy and would take up too much space.

Finally, to establish condition (D) it suffices to consider any dominating function \(M_{ijk}\), defined for some arbitrary constant \(h>1\) (c.f. [33]), such that

$$\begin{aligned} M_{ijk}\left( x\right) =h\left| \frac{\partial ^{3}}{\partial _{i}\partial _{j}\partial _{k}}\log f_{\mathrm{ADP}}\left( x;{\mathbf {p}}\right) \right| , \end{aligned}$$
(45)

for which it is not difficult to show that

$$\begin{aligned} m_{ijk}= E _{{\mathbf {p}}}\left[ M_{ijk}\left( X\right) \right] <\infty ,\;\;\;\text {for all } i,j,k\ne \mu . \end{aligned}$$
(46)

This concludes the proof. \(\square\)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Halvarsson, D. Maximum Likelihood Estimation of Asymmetric Double Type II Pareto Distributions. J Stat Theory Pract 14, 22 (2020). https://doi.org/10.1007/s42519-019-0080-5

Download citation

Keywords

  • Distribution theory
  • Double Pareto distribution
  • Maximum likelihood
  • Firm growth rates