Published by De Gruyter November 16, 2023

Zero-Inflated Autoregressive Conditional Duration Model for Discrete Trade Durations with Excessive Zeros

Francisco Blasques , Vladimír Holý and Petra Tomanová

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2022-0008

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Abstract

In finance, durations between successive transactions are usually modeled by the autoregressive conditional duration model based on a continuous distribution omitting zero values. Zero or close-to-zero durations can be caused by either split transactions or independent transactions. We propose a discrete model allowing for excessive zero values based on the zero-inflated negative binomial distribution with score dynamics. This model allows to distinguish between the processes generating split and standard transactions. We use the existing theory on score models to establish the invertibility of the score filter and verify that sufficient conditions hold for the consistency and asymptotic normality of the maximum likelihood of the model parameters. In an empirical study, we find that split transactions cause between 92 % and 98 % of zero and close-to-zero values. Furthermore, the loss of decimal places in the proposed approach is less severe than the incorrect treatment of zero values in continuous models.

Keywords: financial high-frequency data; autoregressive conditional duration model; zero-inflated negative binomial distribution; generalized autoregressive score model

Corresponding authors: Vladimír Holý and Petra Tomanová, Department of Econometrics, Prague University of Economics and Business, Prague, Czechia, E-mail: vladimir.holy@vse.cz (V. Holý), petra.tomanova@vse.cz (P. Tomanová) (V. Holý) (P. Tomanová)

Funding source: Czech Science Foundation

Award Identifier / Grant number: 23-06139S

Funding source: Dutch Science Foundation

Award Identifier / Grant number: VI.Vidi.195.099

Funding source: Internal Grant Agency of the Prague University of Economics and Business

Award Identifier / Grant number: F4/21/2018

Acknowledgments

Computational resources were supplied by the project “e-Infrastruktura CZ” (e-INFRA LM2018140) provided within the program Projects of Large Research, Development and Innovations Infrastructures. We would like to thank Michal Černý and Tomáš Cipra for their comments. We would also like to thank participants of the 61st Meeting of EURO Working Group for Commodities and Financial Modelling (Kaunas, May 16–18, 2018) and the second International Conference on Econometrics and Statistics (Hong Kong, June 19–21, 2018) for fruitful discussions.

Research funding: The work of Francisco Blasques was supported by the Dutch Science Foundation (NWO) under project VI.Vidi.195.099. The work of Vladimír Holý was supported by the Internal Grant Agency of the University of Economics, Prague under project F4/21/2018. The work of Petra Tomanová was supported by the Czech Science Foundation under project 23-06,139S.

A. Proofs of Asymptotic Properties

Proof of Proposition 1

Following Straumann and Mikosch (2006) and Blasques et al. (2022), we obtain invertibility by verifying that the conditions of Theorem 3.1 of Bougerol (1993) hold uniformly on a non-empty set Θ, for any initialization f ̂ 1 ( θ ) In particular, we note that a ln⁺ bounded moment holds at i = 1 since

E log + sup θ ∈ Θ c + b f ̂ 1 ( θ ) + a s ( x 1 , f ̂ 1 ( θ ) ) ≤ 4 ⁡ ln ⁡ 2 + E ln + sup θ ∈ Θ | c | + E ln + sup θ ∈ Θ b f ̂ 1 ( θ ) + E ln + sup θ ∈ Θ a s ( x 1 , f ̂ 1 ( θ ) ) ≤ 4 ⁡ ln ⁡ 2 + E ln + sup θ ∈ Θ | c | + E ln + sup θ ∈ Θ | b | + E ln + sup θ ∈ Θ f ̂ 1 ( θ ) + E ln + sup θ ∈ Θ | a | + E sup θ ∈ Θ s ( x 1 , f ̂ 1 ( θ ) ) ≤ 4 ⁡ ln ⁡ 2 + max { | c − | , | c + | } + max { | b − | , | b + | } + sup θ ∈ Θ f ̂ 1 ( θ ) + max { | a − | , | a + | } + E ln + sup θ ∈ Θ s ( x 1 , f ̂ 1 ( θ ) ) < ∞ ,

where the three inequalities follow by norm sub-additivity, as well as the ln⁺ sub-additive and sub-multiplicative inequalities in Lemma 2.2 of Straumann and Mikosch (2006), and the last bound follows since c, b, a are strictly positive and lie on the compact Θ and f ̂ 1 ( θ ) is a given real number. We also have that E ln + sup θ ∈ Θ | s ( x 1 , f ̂ 1 ( θ ) ) | < ∞ as

E ln + sup θ ∈ Θ s ( x i , f ̂ 1 ( θ ) , θ ) = P [ x i = 0 ] ⋅ ln + sup θ ∈ Θ s ( 0 , f ̂ 1 ( θ ) , θ ) + P [ x i > 0 ] ⋅ E x i > 0 ln + sup θ ∈ Θ s ( x i , f ̂ 1 ( θ ) , θ ) ≤ ln + sup θ ∈ Θ s ( 0 , f ̂ 1 ( θ ) , θ ) + E x i > 0 ln + sup θ ∈ Θ s ( x i , f ̂ 1 ( θ ) , θ ) < ∞ ,

where E x i > 0 denotes the conditional expectation E x i > 0 [ ⋅ ] = E [ ⋅ | x i > 0 ] and

which holds as the parameter vector θ lies on the compact set Θ, and f ̂ 1 is a given point in R , and

E x i > 0 ln + sup θ ∈ Θ s ( x i , f ̂ 1 , θ ) = E x 1 > 0 ln + sup θ ∈ Θ x 1 − exp ( f ̂ 1 ) ( α ⁡ exp ( f ̂ 1 ) + 1 ) − 1 ≤ E x 1 > 0 ln + sup θ ∈ Θ x 1 − exp ( f ̂ 1 ) ≤ 2 ⁡ ln ( 2 ) + E x 1 > 0 ln + | x 1 | + ln + | exp ( f ̂ 1 ) | < ∞ ,

since x ₁ has a logarithmic moment, Θ is compact and f ̂ 1 ∈ R .

Finally, the contraction condition of Bougerol (1993) is satisfied uniformly in θ ∈ Θ since

E ln sup f sup θ ∈ Θ a ∂ s ( x i , f , θ ) ∂ f + b < 0 ⇔ P [ x i = 0 ] ⋅ ln sup f sup θ ∈ Θ a ∂ s ( 0 , f , θ ) ∂ f + b + P [ x i > 0 ] ⋅ E x i > 0 ln sup f sup θ ∈ Θ a ∂ s ( x i , f , θ ) ∂ f + b < 0

where

E ln sup f ̂ sup θ ∈ Θ a ∂ s ( x i , f ̂ , θ ) ∂ f ̂ + b < 0 ⇔ P [ x i = 0 ] ⋅ ln sup f ̂ sup θ ∈ Θ a ∂ s ( 0 , f ̂ , θ ) ∂ f ̂ + b + P [ x i > 0 ] ⋅ E x i > 0 ln sup f ̂ sup θ ∈ Θ a ∂ s ( x i , f ̂ , θ ) ∂ f ̂ + b < 0 ⇔ π + ( 1 − π ) α − 1 α − 1 + f ̂ i α − 1 ⋅ ln sup f ̂ sup θ ∈ Θ − a ( π − 1 ) 2 ⁡ exp ( 2 f ̂ ) ( α ⁡ exp ( f ̂ ) + 1 ) 2 π ( α ⁡ exp ( f ̂ ) + 1 ) 1 / α − π + 1 2 − a ( π − 1 ) exp ( f ̂ ) ( exp ( f ̂ ) − 1 ) ( α ⁡ exp ( f ̂ ) + 1 ) 2 π ( α ⁡ exp ( f ̂ ) + 1 ) 1 / α − π + 1 + b + 1 − π − ( 1 − π ) α − 1 α − 1 + f ̂ i α − 1 ⋅ E x i > 0 ln sup f ̂ sup θ ∈ Θ − a ( α x i + 1 ) exp ( f ̂ ) ( α ⁡ exp ( f ̂ ) + 1 ) 2 + b < 0 ⇐ ln sup θ ∈ Θ a ( π − 1 ) 2 2 α + sup θ ∈ Θ a ( π − 1 ) α 2 + sup θ ∈ Θ | b | + E x i > 0 ln sup θ ∈ Θ a α x i + 1 4 α + sup θ ∈ Θ | b | < 0 .

This can be simplified by noting that

exp ( 2 f ̂ ) ( α ⁡ exp ( f ̂ ) + 1 ) 2 ≤ 1 2 α , π ( α ⁡ exp ( f ̂ ) + 1 ) 1 / α − π + 1 2 ≥ 1 , exp ( f ̂ ) ( exp ( f ̂ ) − 1 ) ( α ⁡ exp ( f ̂ ) + 1 ) 2 ≤ 1 α 2 .

This, in turn, implies that

E ln sup f ̂ sup θ ∈ Θ a ∂ s ( x i , f ̂ , θ ) ∂ f ̂ + b < 0 ⇐ sup θ ∈ Θ a ( π − 1 ) 2 2 α + sup θ ∈ Θ a | π − 1 | α 2 + sup θ ∈ Θ b + < 1 ∧ E x i > 0 ln sup θ ∈ Θ a α x i + 1 4 α + b + < 0 ⇐ a + ( π − − 1 ) 2 2 α − + a + | π − − 1 | ( α − ) 2 + b + < 1 ∧ E x i > 0 ln a + α + x i + 1 4 α − + b + < 0 .

Proof of Lemma 1

This proof follows that of Blasques et al. (2022, Theorem 4.6). The existence and measurability of θ ̂ n is obtained through an application of White (1994, Theorem 2.11) or Gallant and White (1988, Lemma 2.1, Theorem 2.2), as Θ is compact and the log likelihood is continuous in θ and measurable in x _i. The consistency of the ML estimator, θ ̂ n ( f ̂ 1 ) → a s θ 0 , is obtained by White (1994, Theorem 3.4) or Gallant and White (1988, Theorem 3.3). Below, we note that we satisfy the sufficient conditions of uniform convergence of the log likelihood function

sup θ ∈ Θ | L ̂ n ( θ ) − L ∞ ( θ ) | → a s 0 ∀ f ̂ 1 ∈ F as n → ∞ ,

and the identifiable uniqueness of the maximizer θ ₀ ∈ Θ introduced in White (1994),

sup θ : ‖ θ − θ 0 ‖ > ϵ L ∞ ( θ ) < L ∞ ( θ 0 ) ∀ ϵ > 0 .

The uniform convergence of the criterion is obtained since, by norm sub-additivity, we can split the log likelihood as follows

The first term on the right-hand-side of (17) vanishes if | l ̂ i ( θ ) − l i ( θ ) | → a s 0 since

| L ̂ n ( θ ) − L n ( θ ) | ≤ 1 n ∑ n | l ̂ i ( θ ) − l i ( θ ) | → a s 0 ,

and we have that

where sup θ ∈ Θ | f ̂ i ( θ ) − f i ( θ ) | → a s 0 follows from the invertibility of the filter (proved in Proposition 1) and the product vanishes by the bounded logarithmic moment of the score E ln + sup f | ∇ ( x i , f ) | < ∞ (see Lemma 2.1 in Straumann and Mikosch 2006). The logarithmic moment E ln + sup f | ∇ ( x i , f ) | < ∞ follows as

E ln + | s ( 0 , f ̂ i ) | = E ln + exp ( f ̂ i ) ( π − 1 ) ( α ⁡ exp ( f ̂ i ) + 1 ) 1 + π ( α ⁡ exp ( f ̂ i ) + 1 ) α − 1 − π < ∞ , E x i > 0 ln + | s ( x i , f ̂ i ) | = x i − exp ( f ̂ i ) α ⁡ exp ( f ̂ i ) + 1 < ∞ for x i > 0 .

Note that since we use unit scaling in Lemma 1, we have that ∇(x _i, f) = s∇(x _i, f). The uniform convergence of the second term on the right-hand-side of (17)

sup θ ∈ Θ | L n ( θ ) − L ∞ ( θ ) | → a s 0 ∀ f ̂ 1 ∈ F as n → ∞ ,

follows by application of the ergodic theorem for separable Banach spaces in Rao (1962). We note that the { L n ( ⋅ ) } t ∈ N has strictly stationary and ergodic elements as it depends on the limit strictly stationary and ergodic filter taking values in the Banach space of continuous functions C ( Θ , R ) equipped with sup norm. We also note that L _n(⋅) has one bounded moment since E [ L n ( θ ) ] ≤ 1 n ∑ n E [ l i ( θ ) ] < ∞ . In particular, the bounded moment for the log likelihood holds trivially if the data has a bounded moment E[x _i] < ∞ since ln ℓ _i(x _i, θ) is bounded in μ _i and bounded by a linear function in x _i,

ℓ i ( 0 , θ ) = ln ⁡ P [ X i = 0 | f ̂ i ( θ ) , θ ] = ln π + ( 1 − π ) α − 1 α − 1 + exp ( f ̂ i ( θ ) ) α − 1 , ℓ i ( x i , θ ) = ln ⁡ P [ X i = x i | f ̂ i ( θ ) , θ ] = ln ( 1 − π ) + ln Γ x i + α − 1 Γ ( x i + 1 ) Γ ( α − 1 ) + 1 α ln α − 1 α − 1 + exp ( f ̂ i ( θ ) ) + x i ⁡ ln exp ( f ̂ i ( θ ) ) α − 1 + exp ( f ̂ i ( θ ) ) for x i > 0 .

The identifiable uniqueness (see e.g. White 1994) follows from the compactness of Θ, the assumed uniqueness of θ ₀, and the continuity of the limit likelihood function E[ℓ _i(θ)] in θ ∈ Θ.

Proof of Lemma 2

This proof follows Blasques et al. (2022, Theorem 4.16). In particular, we obtain the asymptotic normality using the usual expansion argument found e.g. in White (1994, Theorem 6.2) by establishing:

The consistency of θ ̂ n → a s θ 0 ∈ i n t ( Θ ) , , which follows immediately by Lemma 1.
The as twice continuous differentiability of L n ( θ , f ̂ 1 ) in θ ∈ Θ, which holds trivially for our zero-inflated score model.
The asymptotic normality of the score, which can be shown to hold by verifying that,
(18) n ∂ L n ( θ 0 ) ∂ θ → d N 0 , I ( θ 0 ) as n → ∞ ,
and
(19) n ∂ L ̂ ( θ 0 ) ∂ θ − ∂ L ( θ 0 ) ∂ θ → a s 0 as n → ∞ .

The asymptotic normality in (18) is obtained by application of a central limit theorem for martingale difference sequences to the score, after noting that the score
∂ L n ( θ 0 ) ∂ θ = 1 n ∑ n ∂ ℓ i ( x i , θ 0 ) ∂ θ + ∂ ℓ i ( x i , θ 0 ) ∂ f i ∂ f i ( θ 0 ) ∂ θ .
has two bounded moments. In particular,
E ∂ L n ( θ 0 ) ∂ θ 2 ≤ E ∂ ℓ i ( x i , θ 0 ) ∂ θ 2 + E ∂ ℓ i ( x i , θ 0 ) ∂ f i ∂ f i ( θ 0 ) ∂ θ 2 < ∞ ,
where the bounds
E ∂ ℓ i ( x i , θ 0 ) ∂ θ 2 < ∞ and E ∂ ℓ i ( x i , θ 0 ) ∂ f i ∂ f i ( θ 0 ) ∂ θ 2 < ∞ ,
hold, for example, under the assumption that
E ∂ ℓ i ( x i , θ 0 ) ∂ f i 4 < ∞ and E ∂ ℓ i ( x i , θ 0 ) ∂ θ 4 < ∞ ;
by a generalized Holder’s inequality as used e.g. in Blasques et al. (2022). For the negative binomial model it is easy to see for example that the four bounded moments for score term ∂ℓ _i(x _i, θ ₀)/∂f _i can be obtained if the data has four bounded moments, E|x _i|⁴ < ∞, by noting that
E sup θ ∈ Θ ‖ s ( 0 , f ̂ i , θ ) ‖ 4 ≤ sup μ sup θ ∈ Θ ‖ s ( 0 , f ̂ i , θ ) ‖ 4 = sup μ sup θ ∈ Θ ( π − 1 ) exp ( f ̂ i ) ( α ⁡ exp ( f ̂ i ) + 1 ) − 1 1 + π ( α ⁡ exp ( f ̂ i ) + 1 ) α − 1 − π − 1 4 < ∞ ,
since s ( 0 , f ̂ i , θ ) is uniformly bounded in f ̂ i . Furthermore, by application of the so-called c _n-inequality, there exists a finite constant k such that,
E x i > 0 sup θ ∈ Θ | s ( x i , f ̂ i , θ ) | 4 = E x i > 0 sup θ ∈ Θ x i − exp ( f ̂ i ) ( α ⁡ exp ( f ̂ i ) + 1 ) − 1 4 ≤ k sup θ ∈ Θ 1 α E x i > 0 x i 4 + k | α − 1 | 4 < ∞ .

Additionally, following the argument of Blasques et al. (2022, Theorem 4.14) and Straumann and Mikosch (2006, Lemma 2.1), the as convergence in (19) follows by the invertibility of the filter and its derivatives. The invertibility of the first derivative process can be verified by applying Theorem 2.10 in Straumann and Mikosch (2006). This theorem is analogue to Theorem 3.1 of Bougerol (1993), also used in the proof of Proposition 1 above, but it applies to perturbed stochastic sequences. For example, the updating equation for derivative process ∂ f i / ∂ c = ∂ f ̂ i / ∂ c takes the form
∂ f ̂ i + 1 ∂ c = 1 + b ∂ f ̂ i ∂ c + ∂ s ( x i , f ̂ i ) ∂ f ̂ i ∂ f ̂ i ∂ c = 1 + b + ∂ s ( x i , f ̂ i ) ∂ f ̂ i ∂ f ̂ i ∂ c .

Hence, by application of Theorem 2.10 in Straumann and Mikosch (2006), the invertibility of this filter is ensured by (a) the invertibility of the filter { f ̂ i } i ∈ N (shown in Proposition 1); (b) the contraction condition E [ ln | b + ∂ s ( x i , f ̂ i ) / ∂ f ̂ i | ] < 0 ; and a logarithmic moment for ∂ 2 ⁡ s ( x i , f ̂ i ) / ∂ f ̂ i 2 .
The uniform convergence of the Hessian, is obtained through the invertibility of the filter and its derivative processes. In particular, a sufficient condition is for the first and second derivatives of the filtering process to converge almost surely, exponentially fast, to a limit stationary and ergodic sequence,
∂ f ̂ i ( θ 0 ) ∂ θ − ∂ f i ( θ 0 ) ∂ θ → e a s 0 and sup θ ∈ Θ ∂ 2 f ̂ i ( θ ) ∂ θ ∂ θ ′ − ∂ 2 f i ( θ ) ∂ θ ∂ θ ′ → e a s 0 as i → ∞ ,
with four bounded moments
E ∂ f i ( θ 0 ) ∂ θ 4 < ∞ and E sup θ ∈ Θ ∂ 2 f i ( θ ) ∂ θ ∂ θ ′ 4 < ∞ .
and to have logarithmic moments for cross derivatives,
E sup θ ∈ Θ ∂ 2 ℓ i ( x i , θ ) ∂ f i ∂ θ ′ < ∞ , E sup θ ∈ Θ ∂ 2 ℓ i ( x i , θ ) ∂ f i 2 < ∞ and E sup θ ∈ Θ ∂ 2 ℓ i ( x i , θ ) ∂ θ ∂ θ ′ < ∞ ;
and also for the third-order derivatives of the log likelihood to have a uniform logarithmic bounded moment,
E ln + sup θ ∈ Θ ∂ 3 ℓ i ( x i , θ 0 ) ∂ f i 2 ∂ θ ′ < ∞ , E ln + sup θ ∈ Θ ∂ 3 ℓ i ( x i , θ 0 ) ∂ f i 3 < ∞ .

and E l n + sup θ ∈ Θ ∂ 3 ℓ i ( x i , θ 0 ) ∂ θ ∂ θ ′ ⁡ ∂ f < ∞ ;

Then by application of the ergodic theorem for separable Banach spaces in Rao (1962) to the limit Hessian (see also Blasques et al. 2022; Straumann and Mikosch 2006, Theorem 2.7 for additional details), we have,
(20) sup θ ∈ Θ ∂ 2 L n ( θ ) ∂ θ ∂ θ ′ − E ∂ 2 ℓ i ( θ ) ∂ θ ∂ θ ′ → a s 0 as n → ∞ .
The non-singularity of the limit L ∞ ′ ′ ( θ ) = E [ ℓ i ′ ′ ( θ ) ] = I ( θ ) follows by the uniqueness of θ ₀ and the independence of derivative processes (Straumann and Mikosch 2006, Theorem 2.7).

B Model Evaluation

It is well know that ranking models based on their expected log-likelihood E[ℓ _i(θ ₀)] evaluated at the best (pseudo-true) parameter θ ₀ is equivalent to model selection based on minimizing the expected Kullback-Leibler divergence between the true distribution of the data and the model-implied distribution. The sample log-likelihood is however an asymptotically biased estimator of the expected log likelihood. Under restrictive conditions, Akaike (1973, 1974 showed that the bias is approximately given by the number of parameters of the model dim(θ). Since then, the AIC has been shown to consistently rank models according to the Kullback-Leibler divergence under considerably weaker conditions (Sin and White 1996; Konishi and Kitagawa 2008). Unfortunately, model specification and identification issues still exert a strong influence over the performance of in-sample information criteria.

For this reason, it could be interesting to consider criteria based on a validation sample. Lemma 3 highlights that log-likelihood based on an independent validation sample of m observations, n L ̂ m ( θ ̂ n ) , is asymptotically unbiased for nE[ℓ _i(θ ₀)]. A proof can be found in Andrée, Blasques, and Koomen (2017).^[5]

Lemma 3

Let the conditions of Lemma 1 hold. Then lim n , m → ∞ E n L ̂ m ( θ ̂ n ) − n E [ ℓ i ( θ 0 ) ] = 0 .

Lemma 4 uses a Diebold-Mariano test statistic (Diebold and Mariano 1995) to test for differences in log-likelihoods across different models obtained from the validation sample (see Andrée, Blasques, and Koomen 2017, for a proof). This test is also known as a logarithmic scoring rule, see e.g. Diks, Panchenko, and van Dijk (2011); Amisano and Giacomini (2007); Bao, Lee, and Saltoglu (2007). Given two models, A and B, let ℓ ̃ i A θ 0 A and ℓ ̃ i B θ 0 B denote their respective log-likelihood contributions at a certain time i (in the validation sample) evaluated at each model’s pseudo-true parameter. Define the log-likelihood difference

D i A , B ≔ ℓ ̃ i A θ 0 A − ℓ ̃ i B θ 0 B .

Finally, define the Diebold-Mariano test statistic

D M m , n = m μ D A , B σ D A , B , μ D A , B = 1 m ∑ i = n + 1 n + m D i A , B , σ D A , B = 1 m − 1 ∑ i = n + 1 n + m D i A , B − μ D A , B 2 .

Lemma 4

(Validation-Sample Test). Let Lemma 1 hold for both models A and B, such that θ ̂ n A → a s θ 0 A and θ ̂ n B → a s θ 0 B as n → ∞. Then we have that

DM m , n → d N ( 0,1 ) as n , m → ∞ ,

under the null hypothesis H 0 : E D m A , B = 0 , where σ D A , B is a consistent estimator of the standard deviation of D m A , B . If E D m A , B > 0 then DM_m,n → ∞ as n, m → ∞. Finally, if E D m A , B < 0 , then DM_m,n → −∞.

C Generalized Gamma Distribution

The generalized gamma distribution is a continuous probability distribution and a three-parameter generalization of the two-parameter gamma distribution (Stacy 1962). It also contains the exponential distribution and the Weibull distribution as special cases. It uses the scale parameter β and two shape parameters θ and φ. The probability density function is

p ( x | β , θ , φ ) = 1 Γ θ φ β x β θ φ − 1 e − x β φ for x ∈ ( 0 , ∞ ) .

The expected value and variance is

E [ X ] = β Γ θ + φ − 1 Γ θ , v a r [ X ] = β 2 Γ θ + 2 φ − 1 Γ θ − β Γ θ + φ − 1 Γ θ 2 .

The score vector is

∇ ( x ; β , θ , φ ) = φ β − 1 x φ β − φ − θ φ ⁡ ln x β − 1 − ψ 0 ( θ ) θ ⁡ ln x β − 1 − x φ β − φ ⁡ ln x β − 1 + φ − 1 for x ∈ ( 0 , ∞ ) .

Special cases of the generalized gamma distribution include the gamma distribution for φ = 1, the Weibull distribution for θ = 1 and the exponential distribution for θ = 1 and φ = 1.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2022-0008).

Received: 2022-01-31

Accepted: 2023-10-17

Published Online: 2023-11-16