On Some Generalization of the Normal Distribution

arXiv:2108.00272v2 [math.PR] 5 Aug 2021 eibull W λX is, that decay, e snt that parameter note us Let au ftesaeprmtr twl aetesm re fti dec tail of F order same distribution. the Weibull have the will to it sense, parameter, distribution. some shape in the related, of value be will which tion, bv etoe eeaiain hs onticuetenra dis normal the include not case. do distribut special these normal generalizations inverse and mentioned normal, above folded log-normal, classical as uin;seeg 7.Teaoepoaiiydsrbtosbcm norm become distributions values. probability example above parameter Another distribu The certain log-normal 8]. for [7]. the Appendix is e.g. [5, one see in second butions; authors The the [8]. by compre presented in a form found distribution, power be exponential can d which the be of is can one distribution first normal the the ample, of generalization of the types Several of Notion 1 norms Orlicz (exponential) variables, random exponential 00MteaisSbetCasfiain rmr 00,Se 60E05, Primary Classification: Subject Mathematics 2020 e od:nra itiuin ebl itiuin diff distribution, Weibull distribution, normal words: Key 1 h tnadepnnilydsrbtdrno variable random distributed exponentially standard The nti ae,w ol iet rsn eeaiaino h normal the of generalization a present to like would we paper, this In n a loidct te itiuin eae otenra distr normal the to related distributions other indicate also can One /α niletoy eas aclt h epnnil Orlicz (exponential) the momen calculate α for also expression We show entropy. We ential distribution. normal standard ewl ali the it call will We o some for nra admvariables. random -normal ( nti ae epooesm e eeaiaino h norma the of generalization new some propose we paper this In ,λ α, nsm eeaiaino h omldistribution normal the of generalization some On P α ( W ) n h cl parameter scale the and admvariable random α,λ P ,λ> λ α, ( X ≥ W aut fMteais nvriyo Bialystok of University Mathematics, of Faculty t ≥ α,λ ilosig M 525Baytk Poland Bialystok, 15-245 1M, Ciolkowskiego = ) .Osreta for that Observe 0. t α exp( = ) a h w-aaee ebl itiuinwt h shape the with distribution Weibull two-parameter the has - P oml(asin distribution (Gaussian) normal λX α ryzo Zajkowski Krzysztof [email protected] 1 (i.e., nra distribution -normal /α − ≥ t for ) W t Abstract α,λ = λ opr 6 h8.W ilcl ta the as it call will We Ch.8]. [6, compare ; t ∼ P 1 ≥ t eibull W X ≥ .Cnie admvariable random a Consider 0. 0 ≥ ( t/λ ( ,λ α, ) For . α )). exp = omo h standard the of norm α tbcmsthe becomes it 2 = X rniletoy sub- entropy, erential sadtediffer- the and ts odr 46E30 condary sigihd o ex- For istinguished. − a xoeta tail exponential has esv description hensive osbtulk the unlike but ions distribution. l ( stese distri- skew the is ya h Weibull the as ay t/λ rbto sthe as tribution in u nthe in but tion, ldistributions al ) bto,such ibution, α rtesame the or . distribu- W . α,λ := Similarly as in the case of the Weibull distribution we define a generalization of the normal distribution, which was announced in [9]. Definition 1.1. For the standard normally distributed random variable G and positive number α define a symmetric random variable Gα such that Gα has the same 2/α | | distribution as G . We will call Gα the standard α-normal (α-Gaussian) random variable. | |

The cumulative distribution function FGα (t) of the α-Gaussian random variable Gα, for t> 0, has the form

F (t) = P(G t)=1 P(G > t) Gα α ≤ − α 1 = 1 P( G > t) (by symmetry of G ) − 2 | α| α 1 2 = 1 P( G α > t) (by distribution of G ) − 2 | | α α α α = 1 P(G > t 2 )= P(G t 2 )=Φ(t 2 ), − ≤ where Φ denotes the cumulative distribution function of the standard normal distribution. ′ One can calculate that its density function fα := FGα has the form

α α/2−1 −|x|α/2 fα(x)= x e . 2√2π | | Let us stress that for α = 2 we get the density of the standard normal distribution. Density function description (The following description and the graphic were made by the student Jacek Oszczepali´nski). The density function of the random variable Gα is even. One can calculate that, for x> 0, its derivative has the following form

2 α α−4 α 2 1 α ′ 2 α − 2 x fα(x)= − x x − e . 4√2π − α Let us observe that for 0 <α< 2 the density function has inﬁnite negative slope at 0 ′ ′ (limx→0+ fα(x)= ) and it is negative for any x> 0. If α = 2 then limx→0+ f2(x)= ′ −∞ ′ f2(0)=0. If2 <α< 4 then the slope at 0 is inﬁnite positive, for α = 4 limx→0+ f4(x)= ′ 2/√π, and fα(0) = 0 if α > 4. In general, for any 2 < α and x > 0 the slope of the α α−2 ′ α α−2 density function is positive until the value α , fα α = 0, and it is negative q q above α α−2 . q α By the above we get that the form of the density function of the α-normal distribution changes drastically with the value of α. And so, for 0 <α< 2, the density

2 function has the vertical asymptote at zero. For α = 2, we have the density of the standard normal distribution. For α > 2, the density function have a local minimum at zero with a value zero and two maxima at α α−2 . ± q α The following graphic shows examples of α-normal density functions depending on the shape parameter, such as α = 1 (red), α = 2 (blue), α = 3 (purple) and α = 5 (green).

Figure 1: Density function fα depending on the value of parameter α.

Remark 1.2. Because the cumulative distribution function of Gα has the form FGα (x)= α α 2 2 Φ(x ) if x 0 and FGα (x)=1 Φ( x ) if x< 0 then we see that Gα tends in distribution to Rademacher’s≥ distribution− | as| α . →∞ Moments and the moment generating function. Since, for G (0, 1) and p> 0, ∼N 2p/2 p +1 E( G p)= Γ , | | √π 2 we immediately get p/α p 2p/α 2 p 1 E( Gα )= E( G )= Γ + . | | | | √π α 2

Thus the moment generating function of Gα equals 1 ∞ (41/αs)k 2k 1 E exp(sG )= Γ + . α √π k! α 2 Xk=0

Now we compare the distribution of Gα with the distributions of the Weibull(α,λ) random variables for some λ’s. It is deﬁne that a random variable X majorizes a random variable Y in distribution, if there exists t0 0 such that ≥ P( X t) P( Y t), | | ≥ ≥ | | ≥ 3 for any t > t0; see for instance [1, Def. 1.1.2]. Proposition 1.3. The standard α-normal random variable majorizes the Weibull(α,1) random variable and it is majorized by the Weibull(α, 21/α) random variable. Proof. It is known that the tails of the Gaussian random variable can be estimated from above in the following way

P( G t) exp( t2/2) | | ≥ ≤ − for any t 0; see for instance [2, Prop.2.2.1]) . Hence for the α-normal random variable we get ≥ P( G t)= P( G tα/2) exp (t/21/α)α . (1) | α| ≥ | | ≥ ≤ − 1 Let us observe that the right hand side is the tails of the Weibull(α, 2 /α) random variable. It means that the Weibull(α, 21/α) random variable majorizes the α-normal random variable. In the same source [2, Prop.2.2.1]) one can ﬁnd the following lower estimate of the tails of the Gaussian random variable 1 1 P( G t) exp( t2/2) | | ≥ ≥ t √2π − 2 for t 1. Because √2πt exp( t /2) tends to 0 as t then there exists t0 such that ≥ 2 − 2 →∞ √2πt exp( t /2) 1 for t t0. It gives exp( t /2) 1/√2πt and, in consequence, − ≤ ≥ − ≤ 1 1 P( G t) exp( t2/2) exp( t2) | | ≥ ≥ t √2π − ≥ − for t t0. By the above ≥ 2/α α/2 α P( G t)= P( G t)= P( G t ) exp( t )= P(W 1 t), | α| ≥ | | ≥ | | ≥ ≥ − α, ≥ 2/α for t t0 , which means that the α-normal random variable majorizes the Weibull(α, 1) random≥ variable.

Although the the standard α-normal distribution is comparable to the Weibull distribution in the above sense, it is significantly different. We will show it on the example of the entropy function. Recall that the differential entropy of the two-parameter Weibull distribution is given by the formula 1 λ H(Wα,λ)= γ 1 + ln +1, − α α where γ is the Euler-Mascheroni constant. In the following proposition we derive a formula on the differential entropy for Gα.

4 Proposition 1.4. The diﬀerential entropy of Gα has the following form

1 1 2√2π 1 H(Gα)= (γ + ln 2) + ln + , α − 2 α 2 where γ denotes the Euler-Mascheroni constant.

Proof. By the deﬁnition of the diﬀerential entropy and the form of density fα of Gα we get

∞ ∞ α α/2−1 −|x|α/2 ln x fα(x)dx = ln x x e dx. (4) Z−∞ | | 2√2π Z−∞ | || |

Substituting u = xα/2 (x> 0) we obtain

∞ ∞ 2 1 α 2 8 1 2 α/ − −|x| / − 2 u ln x x e dx = 2 e ln udu. (5) Z−∞ | || | α Z0 By [4, 4.333] we have that

∞ 1 2 1 e− 2 u ln udu = (γ + ln 2)√2π (6) Z0 −4 Summing up (6), (5), (4), (3) and substituting into (2) we obtain

1 1 2√2π 1 H(Gα)= (γ + ln 2) + ln + . α − 2 α 2

Let us note that for α = 2 we obtain the diﬀerential entropy of the standard Gaussian density.

5 2 Orlicz norm of the α-normal distribution

Let us emphasize that the Weibull random variables form the model examples of random variables with α-sub-exponential tail decay. We say that a random variable X has α-sub-exponential tail decay if there exist two constant c,C such that for t 0 it holds ≥ P( X t) c exp (t/C)α . | | ≥ ≤ − Since α P(Wα,λ t) = exp (t/λ) , ≥ − the Weibull random Wα,λ has α-sub-exponential tail decay with c = 1 and C = λ. 1/α Whereas the estimate (1) means that Gα has such tail decay with c = 1 and C =2 . The property of α-sub-exponential tail decay can be equivalently expressed in terms of so-called (exponential) Orlicz norms. Recall that for any random variable X deﬁne the ψα-norm X := inf K > 0 : E exp( X/K α) 2 ; k kψα | | ≤ according to the standard convention inf = . We will call the above functional ψ -norm but let us emphasize that only for∅ α ∞1 it is a proper norm. For 0 <α< 1 α ≥ it is so-called quasi-norm. It do not satisfy the triangle inequality (see Appendix A in [3] for more details).

One can easy observe that X ψα = X ψα and, moreover, one can easy check β k| β|k k k that, for α,β > 0, X ψ = X ; see Lemma 2.3 in [9]. k| | k α k kψαβ Since the closed form of the moment generating function of random variable G2 is 2 known, we can calculate the ψα-norm of α-normal random variable Gα. Because G 2 has χ1-distribution with one degree of freedom whose moment generating function is E exp(sG)=(1 2s)−1/2 for s< 1/2 then − E exp(G2/K2)=(1 2/K2)−1/2, − which is less or equal 2 if K 8/3. It gives that G = 8/3. The ψ2-norm ≥ k kψ2 of G is equal to ψ2-norm of G.p By Lemma 2.3 in [9] and the deﬁnitionp of α-normal distribution| | we get 2 G = G 2/α = G /α = (8/3)1/α. k αkψα k| | kψα k kψ2 Remark 2.1. Using the closed form of the moment generating function of the standard exponential random variable and the above mentioned deﬁnition of the two-parameter Weibull distribution, similarly as for the standard α-Gaussian random variable, one can obtain its ψ -norm W = λ21/α. α k α,λkψα Remark 2.2. Although the Weibull(α,λ) random variables provide model examples of random variables with α-sub-exponential tail decay (they are model elements of spaces generated by the ψα-norms), it can nevertheless be argued that the standard α-Gaussian variables play a central role among these variables (in these spaces).

6 References

[1] V. Buldygin, Yu. Kozachenko, Metric Characterization of Random Variables and Random Processes, Amer.Math.Soc., Providence, 2000.

[2] R.M. Dudley, Uniform Central Limit Theorems, Cambridge University Press, 1999.

[3] F. G¨otze, H. Sambale, A. Sinulis (2021) Concentration inequalities for polynomials in α-sub-exponential random variables, Electron. J. Probab. 26, article no. 48, 1-22.

[4] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Academic Press, 2007.

[5] J.R.M. Hosking, Wallis, J.R. Regional Frequency Analysis: an Approach Based on L-moments, Cambridge University Press, 1997.

[6] N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, Vol. 1, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics (2nd ed.), New York: John Wiley & Sons, 1994.

[7] A. O’Hagan, T. Leonard (1976) Bayes estimation subject to uncertainty about parameter constraints, Biometrika 63 (1): 201-203.

[8] N. Saralees (2005) A generalized normal distribution, Journal of Applied Statistics, 32:7, 685-694.

[9] K. Zajkowski (2020) Concentration of norms of random vectors with independent p-sub-exponential coordinates, arXiv:1909.06776.