On a statistical approximation model of probabilitydensity function of non-negative random variables
Tibor KPogany*
Faculty od Maritime Studies, University of Rijeka, Croatia
Institute of Applied Mathematics, Obuda University, Hungary
Submission: July 31, 2018; Published: November 13, 2018
*Corresponding author: Tibor K Pogany, Faculty od Maritime Studies, University of Rijeka, Studentska 2, 51000 Rijeka, Croatia; Institute of Applied Mathematics, Obuda University, Becsiut 96/b, 1034 Budapest, Hungary.
How to cite this article: Tibor KPogany. On a statistical approximation model of probabilitydensity function of non-negative random variables. Biostat Biometrics Open Acc J. 2018; 8(4): 555741. DOI: 10.19080/BBOAJ.2018.08.555741
Abstract
The probability density function of a non–negative random variable is approximated by virtue of a result by Totik [1], when the support domain is truncated to a finite interval. This result is transferred to the case of three–parameter Weibull distribution.
2010 Mathematics Subject Classification: Primary: 60E05, 62E17; Secondary: 41A10, 42C05, 44A60
Keywords: Christoffel function; moments; statistical approximation of probability density; three-parameter Weibull distribution
Introduction and preparation
In continuation we expose a method to estimate any probability density function (pdf) ()fx having finite, compact support set such that consists from n i.i.d. replicae of the initial Moreover, when the ()suppf is infinite, we approximate f with its truncated variant so, that the corresponding approximation error does not skip the prescribed error level ε>0.
Denote a given standard probability space, and let be the class of all complex–valued random variables having finite moment of pth order, p positive integer, that is
Let us consider the sequence of orthogonal polynomials which can be expressed in terms of moments of ()fx [2][7], [5] [3] as follows:
Our second tool is the Christoffel function [1, Eq. (4)]
In his excellent article, solving the so–called Szego˝–problem on the real line, Totik proved [1] that for almost all x ∈ I it holds
being the Christoffel function for orthogonal polynomials associated to a positive measure v>0 giving the Radon–Nikodym derivative v′, when Inv′ is integrable on .I Now, pointing out that for all B Borel
is a probability measure, therefore it can be easily indentified with ,v when f is bounded and positive. Thus, we have re–formulate Totik’s asymptotic result into our setting (also see [1, p. 10, Theorem 4.1]).
Theorem A. Let f(x) be a bounded pdf of , and Then we have
for almost all x∈[ab]
The first approximation of the underlying pdf we deduce from (1.3) for almost all [],:xab
where the approximant ()kfx is not a pdf, so it has to be renormalized.
Main results
Another problem arises with a pdf ()fx having infinite support set which turns out to be only incidentally connected with the so–called Stieltjes moment problem, consult for instance in this respect [3, 4, 6][4,5,6]. To avoid the infinite support interval we truncate it to the finite [],,ab ab< so, that the truncation error should be less then ε>0.
Proposition 1. Let be the unique zero of the equation
Proof: Because is a decreasing function of ,b and pbε monotone increases on the same domain ,ba> , (2.1) has an unique root in .b Thus
which is equivalent to the claim (2.2).
Having in mind the approximation of an initial pdf having infinite support by means of (2.2) and in the same time renormalizing the resulting function approximants (1.4) so, that we get the sequence of approximants
It remains to estimate b by means of the random sample Ξ in approximation formula (2.3).
In this goal we have to estimate all moments involved in (),kpfx by means of certain statistics/Borel functions of .Ξ Let us recall that
is the so–called kth sample moment, the analogue of theoretical moment .km It is a standard result of mathematical statistics, that kA is an unbiased estimator for ,km being EAk=mk
moreover, by the Khintchine’s Law of Large Numbers for all k uch that when n runs to infinity, that is, is a consistent estimator of Ak mk
Now, we transform the approximation procedure (2.3) to an infinite domain pdf, prescribing the admissible approximation error level 0,ε> taking finite support interval where
By some practical reasons, we make use of the first sample mean A1 or the maximum–order statistic evaluating ˆb.
Proposition 2: Replacing jm with jA Aj in (1.1) and (1.2), by means of (2.3) we get the finite pdf approximating sequence
Proof: Let us define the matrices
First, consider the case .kn≥ The columns of kT are linearly dependent so rank ().kTn= The matrix TkTkτ has 1kn+> columns, and every column of TkTkτ is a linear combination of columns of .kT therefore TkTkτ possesses at most n linearly independent columns. Thus, being the columns of TkTkτ linear dependent, we conclude that
Now, it remains the case .kn< The matrix kT has linear independent row–vectors such that we denote
It turns out that where Γ stands for the Gramm determinant of the vector Therefore,
Obvious steps finish the proof.
Moreover, we quote an alternative approach to this problem.
So, we have where F is the cumulative distribution function of the rv ξ with the related pdf .fF′= Then
Assuming that a is the continuity point of ,F it is ()0Fa+= and we have to solve the problem with respect to .b By the way , while
The value ()ˆkfx we approximate with the associated Christoffel function etc. However, if this procedure is not efficient, take ()nX instead of ().nX Namely, in that case the lower bound for b will be moved to the right as
Application to (),,Wαβη distribution
The random variable ξ possesses three–parameter Weibull distribution when the corresponding pdf reads
with the parameter correspondence we write denotes the indicator of the event {x,A}.
The question of estimating the parameter vector by means of random sample i.i.d. replica of ()Wξθ arises frequently. The most common way of given some estimator ˆθ is the maximum likelihood method. However, the ML estimator for the three–parameter Weibull distribution does not necessarily exists, and it is not necessarily unique, see [7] and the references therein.
Corollary 2.1. Let with all three parameters positive [8]. Consider the sample
Proof. It is enough to show that the Szeg˝o–type local integrability condition is satisfied, that is, ()(1ˆIn ,.WfxLbα∈ Indeed, as ()0Wfx> for all 0,xα>> equivalently ,xαδ≥+ 0,δ> we have
Obviously, the integral is convergent.
Remark. Instead of the lower bound 1ˆ,b the estimate
can be used.
References
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