The problem of estimation of the mode of a continuous distribution
(unction is considered. The estimation of the mode based on estimators of the density
function is well known, see e.g. Härdle (1991) and Parzen (1962). New parameters
of the continuous distribution function will be defined: the quasi-mode and mean median.
They are parameters of the appropriately truncated random variable. Next, the estimators
of the mode, such as the sample quasi-mode or sample mean-median, are determined.
These statistics are usually biased estimators of the mode. Well known “jackknife”
procedure is adapted to estimate Their mean square error. The accuracy of the mode
estimation is studied on the basis of computer simulation.Zadanie pt. „Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki” nr 885/P-DUN/2014 zostało dofinansowane ze środków MNiSW w ramach działalności upowszechniającej nauk

Wywiał, Janusz

Acta Universitatis Lodziensis Folia oeconomica

Repozytorium Uniwersytetu Łódzkiego (University of Lodz Repository)

A C T A  U N I V E R S I T A T I S  L O D Z I E N S I SFOLIA OECONOMICA 152, 2000II. STATISTICAL INFERENCEJanu sz  W y w ia ł*ESTIM ATION O F M O D E ON T H E  BASIS O F A TRUNCATED SAM PLEAbstract. The problem of estimation of the mode of a continuous distribution (unction is considered. The estimation of the mode based on estimators of the density function is well known, see e.g. H ä r d l c  (1991) and P a r z e n  (1962). New parameters of the continuous distribution function will be defined: the quasi-mode and mean median. They are parameters of the appropriately truncated random variable. Next, the estimators ol the mode, such as the sample quasi-mode or sample mean-median, are determined. These statistics are usually biased estimators of the mode. Well known “jackknife” procedure is adapted to estimate Iheir mean square error. The accuracy of the mode estimation is studied on the basis of computer simulation.1. THE BASIC DEFINITIONS AND NOTATIONThe m ode, the m edian and the mean o f a continuous distribution function are denoted by d, m  and /t, respectively. T he th ird  central m om ent is denoted by >/3. Let A + be the set o f such continuous and one-m odal density functions tha t the inequalities d ^ m ^ ß  and tj3 ^ 0  are fulfilled. The set A + can be treated as the class o f right skewed con­tinuous distribution functions. Similarly, let A _ be such a class o f con­tinuous one-m odal density functions that d ^ m ^ n  and 0. Hence, the set A _ will be called the class o f the left skewed distribution  func­tions.Let us define the following m om ents o f the truncated distribution:K a - h , )= t f (x )dx  (l) where: t =  0, 1, 2, ..., a0 =  a, h0 =  b, F(b) =  1, F(a) =  0 and* Academy of Economies, Chair of Econometrics, Katowice.uF(u) = f f ( x ) d x .“ 00M oreover, let us assume:bl)J xf (x )dx ,  if tj3(a„ h , )>  0 (2)where: a, + i = a„ bt+l = м(а„ b,) and,  . .  i b;Г,̂ а" b,) = F(bt) -  F(a ,) f “  ^ (ар /,,))3dx (3)Í xf (x)dx ,  if ri3(a„ b , )<  0 (4)where: а , +1 =  /i(at, í»,) and bí+ i =  b,.Definition 1. Let / ( x )  be the density function in the interval I  =  <a, b> and let a, and b, be such truncation points that a„ b , e l  andThe param eter g t =  g^a, ,  b,) will be called the quasi-m ode o f the continuous distribution function.Hence, the quasi-m ode is the m ean value o f a distribution function appropriately truncated in such a way that its third central m om ent takes the value zero or the left and right truncated points are equal to  each other.T he m edian m(a„ ht) o f the truncated distribution in the points a, < b, is determ ined in the following way:Let the expected value o f the truncated distribution be defined as follows:gi(a„ bt) = n(at, h t), if rj3(at, bt) =  0, t =  0, 1, 2, ... (5)it.Dt) iJ f ( x ) d x f ( x ) d x  =  Ft(m(a„ b,)) =  -  (6)t -  1J xf (x)dx ,  if m(a„ bt) <  ß(a„ bt) (7)where: al + l =  a„ bl+l =  m(a„ b,),l *'J xf (x )dx ,  if m(a„ bt) < /i(a t, b,) (8)where: a , +1 =  m(a„ bt) and bt+1 =  bt.Definition 2. Let f ( x )  be the density function in the interval I  =  <a, b> and let a, and h, be such truncation points that a„ b ,e l  andg,(a„ b,) =  n(at, bt), if /1(0,, bt) =  m(a„ bt), t =  0 , 1 , 2 , . . .  (9)The param eter g2 =  g2(a„ ht) will be called the m ean-m edian o f the dist­ribution function.The estim ators of the param eters d(u) and m(u) are going to be defined in the next paragraphs. They can be used to  estim ate the m ode o f continuous distribution functions.2. ESTIMATION OF THE MODE BY MEANS OF THE AVERAGE FROM A TRUNCATED SAMPLELet Хц) <  X( 2) < . . .  ^  X (n) be the sequence o f the order statistics. The third central m om ent from the truncated simple sam ple is defined by the expression:1 ‘ _C 3( a , b ) = l YJ( X (l)- X ( a , h ) ) \  \ < k  = b - a + \  (10)K l = awhere:X  (a, b) =  -K l-aLet us define the following statistics:^ з ( а1+ 1> bt+j)  =ÍCj(fl,, b, — Й), if C 3(a,, ht) > 0 and X ^ - ^ ^  X ( a t, bt) < Xb,-i ,+1),= } c 3(a„bt), if C 3(a„ bt) =  0,[ c 3(e t +  h, bt), if C 3(at, bt) < 0 and A!"(&,+*—1 > ^  X ( o t, bt) <  (11)T hen, the sample quasi-m ode Gx is defined as follows:G, = X  (a,, bt), if C 3(flt, bt) =  0 Ф C 3(flf-i ,  b ,_ i) (12)T he statistic Gj will be usually a biased estim ator o f the m ode d. Its bias can be reduced by m eans of the well-known jackknife m ethod. Let G\°, i = l , . . . ,  n, be the quasi-m ode from  the sam ple w ithou t the i-th observation. T he pseudovalues are determ ined by the expression:Z, =  nGi — (л — 1)GP.Hencc, the jackknife type estim ator o f the m ode d is as follows:o . /  =  ž  =  í £ z ,  ( 13)n  1 = 1T he estim ators o f the m ean-square error arc defined by the following expressions:<i4>s 2( G v ) = i i b ) | , ( z ‘ - G v , !  <, s >3. ESTIMATION OF THE M O D E BY MEANS OF TH E MEAN-MEDIAN FROM  THETRUNCATED SAMPLELet us define the m edian from a truncated sample in the following way: M(a„ b,) =  X (e), where e =  Г -*~?» +  Л  +  i (] 6 )1, —M(a„ b, — h), if M(a,, bt) < X ( a t, bt) and X (b, - h) <  M ( a t, bt) < X (bt- h+i), =  < M(a„ bt), if M(a„ bt) = X(a„ b,), (17)M(a,  + h , b t), if M(a„ bt) > X(a„ bt) and X (ai+h)> M ( a t, bt) > X lttl+h- iyT he following estim ators o f the m ean-m edian or the m ode o f a skewed d istribution can be defined. The sample m ean-m edian G2 o f the skewed distribution are determined by the expression:G2(a„ bt) =  X ( a „  bt), if M(a„ bt) =  X (a t, bt), t =  0 , 1 , 2 , ... (18)Then, the sample m edian M (a„ bt) is the m edian o f the truncated sample determ ined by the following sequence o f the order statistics: X M , X (bl) and the sam ple m ean-m edian G2(at, bt) is the average of these statistics.Usually, both statistics Gt and G2 are biased estim ators o f the m ode d o f the continuous distribution function. Just like in the case o f the statistics Gj, we can try to reduce this bias by m eans o f the jackknife method.4. ESTIMATION OF A M ODE OF A M ULTIDIM ENSIONAL DISTRIBUTIONLet f ( x lt ..., xf) be a density function o f an r-dim ensional random  variable. The central m om ents o f the order 3 o f the r-dim ensional variable arc denoted by:00 00Пu Ä  X j ) =  \  I ( x ^ E i X m X j - E i X j W f i x , ,  x r) d x l ........ dxr-  00  —  00(19)and the sample m oments:Cuv( X lt X j ) = l i  (Xu -  X i Y i X j ,  -  X j Y ,  X ,  =  1 £  X lt (20) n t=i n t= iwhere и + v =  3 and u, v =  0 ,1 ,2 ,3 . Let us introduce the following vectors:® — [ 0 1 » • • •» ^и>] =  [ I Чзо> X  i , X  2) I I 7̂о з ( i > г)1 • * • I *71г ( Х r - i ,  X r) | \ti21(Xr, X r^ ) \ ]  and L = [ l„  .... 1J  =  tlC30(A ĵ, X 2)\ IC03(X 2, X,)\.. .  Cn (Xr^ , X r)\ \C2i ( X r, X r- i ) |] ,  where w =  r2.It is well known that if a distribution function is symmetric, then all central m om ents o f the order 3 o f the m arginal one or two dim ensional d istributions are equal to zero. Hence: 0  =  0 .In order to simplify our analysis, let us consider a two-dim ensional random  variable. The da ta  observed in the simple sample will be denoted by {(x„ y,)}, i = l ,  ..., n. F rom  the geometrical point o f view (x„ y t) are the coordinates o f a point A i ( i = l ,  ..., n) in a two dim ensional plane. Let P i,  P 2, Ph, h < n ,  be points selected from  the set A  =  {Л(} in such a way that they are apexes o f a convex polygon and the points o f the set A  are inside this polygon. The set P  =  { P ^  P 2, ..., Ph} determ ines the polygon P. Hence the edge o f the polygon P  can be treated as a convex envelope o f the points {Л,}.Let us construct the polygons: P  =  P (0) э  P (1) э  P (2)... э Р (1). The polygon P (0 is obtained trough rejecting one apex P ^ e P “ " 1), г =  1, 2, ... T he central m om ents o f the d ata  creating the coordinates of points belonging to the polygon P (,) are as follows:Cuv( X i, X J\B(t ,z ) )  = -±r  £  ( x l t - x m x j e - x f y  (21)n  1 ( e X , e « r .  г,)}where:X  Г  =  - Ц  I  *,e>П [ (e:A.eB(l , z )B(t, z) — — Pz and Pze P w, t =  1, 2, ... and B(0, z) =  P. M oreover:L(t, z) =  [1,(1, z) ... IJt ,  z)] =  [|C 30,(X i, X 2\B(t, z))\...\C2i(Xr. u X r\B(t, z))|]The polygon P (') is determ ined through dropping the point Pq from the polygon р (' -1) in such a way thatl(t, q) =  m in max {If t ,  z)} (22)This procedure leads to  truncation o f the two dim ensional sample. Tbis algorithm  is stopped if l(t, q) = 0 or t = n — 2 and G =  (G„ ,Gj) =  (X j'\ X (j0) are estim ators o f dom inants (d!t dj) o f two-dimensional random  variables ( X t, Xj ) .  Hence:G = (X\ ,\ X f )  if l ( t ,q )  = 0 and l ( t - l , q ) > 0  (23)The presented m ethod o f estim ation of the dom inants o f a two dim ensional variable can be easily generalized on a case of distribution o f m ore than two variables. W y w i a ł  (1998) considered a one-dim ensional case o f this m ethod including sim ulation study o f estim ation precision.5. SIMULATION ANALYSISIn order to  study the basic properties o f the introduced estim ators of the m ode, a sim ulation analysis is developed. The estim ation o f the m ode o f the following triangular distribution is considered1:1 The basic properties of the triangular distribution can be found in H e 11 w ig  (1995)./00  =f°J 2(x + b — 2a)b2 — 2a2 2x+2bb2 — 2 a2 b2 — 2 a2for x  < 0 v x  > b > 2a, for 0 <  x  <  a, for a < x < b .(24)The distribution function is as follows:F(x) =10 for x <  0 ,x 2 2  (b - -2a)for 0 <  x <  a,b2 — 2a 2 b2 — 2 a2, ( * - b ) 2 b2 — 2a2for a < x < b ,0 for x > b .T he function inverse to  the distribution function is as follows: F- l (y) = j2 — b + y / ( b - 2 ) 2 + (b2 — 2)y for b - y j ( b 2 - 2 ) ( \ - y )  forwhere:у  о =  F(a) =(2b — 3 a)a b2 - 2 a 2 'T he m om ents are as follows:r m _  fe3~ 2fl3 F (Y > \ b* ~ 2a* F (X )  <j/l2 ~  e tui  л-гч3(b2 -  2a2) 6(b2 -  2a2)E(X>)(25)(26)b 5 -  2a5 I0(b2 — 2a2)In the sim ulation experiment we assumed that a =  1 and h =  5. Hence, the m ode d =  1.0, E(X)  =  1.7826, D \ X )  =  1.3368, t]3(X )  =  0.7646 and the skewness coefficient ß ,  =  0.4947.The pseudo-random  values of the triangular distribution are generated. Values o f the estim ators are determined on the basis o f 2000 samples of a fixed size. The sim ulation experiment was developed on the basis o f the well known SPSS statistical package.Let us assume that the expected value E(.) and the variance D 2(.) are estim ated on the basis o f the com puter simulation. Hence:E(G .) =  ‘ M * , » ;  i m >  -  i  1 ( 9 , ( M )  -  E ( G , ) ) \  k -  1 , 2 ,1У (*|) 1У (X,)MSE( Gk) = D 2(Gk) + [ E ( G k) - d \ \e(Gk) =  ( E i S i G ^ / s / M S E i G J  -  1, b(Gk) =  100%(E(G*) -  á )/M S E (G J,where: к =  1 , 2 , gk(x t) is the value o f the estim ator determ ined on the basis o f the simple sample {x(} o f size n and the num ber o f such samples is denoted by N.  The m ean square error o f the estim ator Gk is denoted by M SE (G J. The results of the sim ulation study o f the quasi-m ode distribution are shown in the T ab. 1 and 2.T a b l e  1The simulation results for the distribution of the estimator G, in the case of the left side truncated triangular distribution for the parameters a = 1 and b -  5n £(G.) 0 2(G.) V m sec g ,) й(С,)% e(Glf)%10 1.591 1.146 1.289 21.0 113.320 1.373 0.983 1.051 12.6 246.430 1.296 0.857 0.907 10.7 218.650 1.235 0.673 0.713 10.9 295.1S o u r c e :  the author’s own elaboration.T a b l e  2The simulation results for the distribution of the estimator G2 in the case of the left side truncated triangular distribution for the parameters a =  1 and b =  5n E(G2) D2(Gj) v/ m SE(G3) b(G2)% e(Ga/)%10 1.950 0.943 1.339 50.4 - 1 1 920 1.669 0.928 1.144 34.2 20.530 1.516 0.826 0.974 28.1 54.650 1.134 0.655 0.669 4.0 126.3100 1.147 0.516 0.537 7.5 1778S o u r c e :  the author’s own elaboration.The analysis o f the T ab. 1 and 2 lead to  the following conclusions: The bias o f both estim ators are rather large. T he bias o f the estim ator G L is larger then the bias o f the G2 only for the size of the sam ple n =  50. Similarly, the relative efficiency b(Gx) is larger than h(G2) for the standard  deviations, and the mean square errors o f both estim ators decrease when the sample size increases. T he relative biases e(Gf ) and e(G2) increase when the sample size increases. Especially, the bias o f the estim ator S^Gy) is too large. In conclusion, neither statistic S(GX) nor S(G2) are useful as estim ators o f the mean square errors v/M SE(G j) and > /M SE(C j), respectively. Generally however, the estim ator G2 is slightly better than GtREFERENCESH ä r d l e  W. (1991), Smoothing Techniques, Springer Verlag, New York-Berlin-Heidel- berg-London-Paris-Tokyo-Hong Kong-Barcelona.H e l l  w i g  Z. (1995), Elementy rachunku prawdopodobieństwa i statystyki matematycznej, PWN, Warszawa.P a r z e n  E. (1962), On Estimation o f a Probability Density and Mode, “Annals of Mathematical Statistics”, 35, 1005-1076.W у wi l l  J. (1998), Estimation o f Distribution Function Mode on the Basis o f Sample Moment or Sample Median, Submitted to Badania Operacyjne i Decyzje.

Estimation of Mode on the Basis of a Truncated Sample

The problem of estimation of the mode of a continuous distribution
(unction is considered. The estimation of the mode based on estimators of the density
function is well known, see e.g. Härdle (1991) and Parzen (1962). New parameters
of the continuous distribution function will be defined: the quasi-mode and mean median.
They are parameters of the appropriately truncated random variable. Next, the estimators
of the mode, such as the sample quasi-mode or sample mean-median, are determined.
These statistics are usually biased estimators of the mode. Well known “jackknife”
procedure is adapted to estimate Their mean square error. The accuracy of the mode
estimation is studied on the basis of computer simulation

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Estimation of Mode on the Basis of a Truncated Sample

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Estimation of Mode on the Basis of a Truncated Sample

Abstract

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Full text

Available Versions

Repozytorium Uniwersytetu Łódzkiego (University of Lodz Repository)

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Biblioteka Nauki - repozytorium artykuÅÃ³w