The Shortest Confidence Interval for the Mean of a Normal Distribution

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1 Interntionl Journl of Sttistics nd Proility; Vol. 7, No. 2; Mrch 208 ISSN E-ISSN Pulished y Cndin Center of Science nd Eduction The Shortest Confidence Intervl for the Men of Norml Distriution TroréBoukr, DitéLssin, TouréBelco 2 & FnéAdou 2 Fcultédes Sciences Economiques et de Gestion (F.S.E.G), Bmko-Mli 2 Fcultédes Sciences et Techniques (F.S.T), Bmko-Mli Correspondence: DitéLssin, Fcultédes Sciences Economiques et de Gestion (F.S.E.G), Bmko-Mli. E-mil: fseggroupe@gmil.com Received: Novemer 30, 207 Accepted: Decemer 20, 207 Online Pulished: Jnury 8, 208 doi:0.5539/ijsp.v7n2p33 URL: Astrct An interesting topic in mthemticl sttistics is tht of the construction of the confidence intervls. Two kinds of intervls which re oth sed on the method of pivotl quntity re the shortest confidence intervl nd the equl til confidence intervls. The im of this pper is to clrify nd comment on the finding of such intervls nd to investigtion the reltion etween the two kinds of intervls. In prticulr, we will give construction technique of the shortest confidence intervls for the men of the stndrd norml distriution. Exmples illustrting the use of this technique re given. Keywords: estimtion, point estimtion, estimtion intervl, shortest length, unimodl 200 Mthemtics Suject Clssifictions: 62Exx; 62Fxx; 62Qxx; 93E0.. Introduction Let X e rel vlue rndom vrile from the density f(x; ρ) nd consider the prmeter ρ s fixed unknown quntity. If we seek nd intervl for ρ, then it is well known tht the stndrd method for otining confidence intervls for ρ is the pivotl quntity method. (cf. Huzurzr (955), Guenther (969, 987), Dhiy nd Guttmn (982), Ferentinos (987, 988, 990), Juol (993), Ferentinos nd Kourouklis (990), Kirmni (990), Cesll nd Berger (2002), Rohtgi nd Sleh (200) e.t.c). Let Z(X, X 2,, X n ; ρ) e pivotl quntity where X, X 2,, X n is rndom from the distriution of f(x; ρ). The proility sttement is converted (when possile) to P ρ (z Z z 2 ) = α (.) P ρ (z ρ z 2) = α (.2) If constnts z, z 2 in (.) cn e found so tht (z 2 z ) is minimum, then the intervl [z, z 2] is sid to e the shortest confidence intervl sed on Z. On the other hnd if constnts z, z 2 in (.) cn e determined so tht P ρ (Z < z ) = α nd P 2 ρ (Z > z 2 ) = α 2 then the intervl [z, z 2] is sid to e n equl tils confidence intervl. The im of this work is to clrify nd comment on prolems tht emerge t the process of finding, to investigte the reltion of equlity of length of these. In prticulr, we will give construction technique of the shortest confidence intervls for the men of the stndrd norml distriution. 2. Method for Evluting Intervl Estimtors Generlly, we wnt to hve confidence intervls with high confidence coefficients s well s smll size/length. Prolem is: for given confidence coefficient ( α) find the confidence intervl with the shortest length. Let X e rndom vrile such tht X N(μ = E(X), σ 2 ) (the norml distriution) nd X, X 2,, X n rndom n smple of X with σ 2 known. We then know tht good point estimtor of μ is X. (.3) As derived ove, X = X +X 2 + +X n n stisfying N(μ, σ n ) nd Z(X, μ) = X μ σ n N(0,) is pivotl, therefore ny [; ] 33

2 Interntionl Journl of Sttistics nd Proility Vol. 7, No. 2; 208 P μ ( Z ) = Φ() Φ() = α (.4) Yields corresponding ( α) confidence intervl for the men μ: {μ: X σ n μ X σ n }. Now we wnt to choose [; ] so tht is the shortest length possile for given confidence coefficient ( α). It turns out tht the symmetric solution = is optiml here. The symmetric solution is α = Φ() Φ() = Φ() Φ( ) = 2Φ() = Φ ( α 2 ). This result generlizes to ny smpling distriution tht is unimodl. Theorem : Let f e unimodl proility density function. If intervl [; ] stisfies: (i) f(x)dx = α (ii) f() = f() > 0 (iii) x where x is the mode of f, then [; ] is the shortest of ll intervls tht stisfy (i). Theorem 2: Let θ nd T(X) e rel-vlued. Let U(X) e positive sttistic. Suppose tht T(X) θ U(X) is pivotl hving proility density function f tht is unimodl t x 0 R. Consider the following clss of confidence intervls for θ: Proof: see references. 3. Results C = {[T U, T U]: R, R, f(x) dx = α} If [T U, T U] C; f( ) = f( ) > 0, nd x 0 then the intervl [T U, T U] hs the shortest length within C. Suppose throughout this prt: Let X e rndom vrile such tht X N(μ = E(X), σ 2 ) (the norml distriution) nd X, X 2,, X n rndom n smple of X with σ 2 known. We then know tht good point estimtor of μ is X. Prolem is: choose [; ] so tht L(, ) = is the shortest length possile for given confidence coefficient ( α). 3. Clcultion of nd The following result provides generl method of finding confidence intervls nd covers most cses in prctice. Theorem 3: Let f e unimodl proility density function of the stndrd norml distriution. If the rels nd stisfies: Min ( ) { suject to f(z)dz = α Then [X σ ; X σ ] is the shortest confidence intervl for the men μ. n n Proof: See Theorem : [, ] is the shortest of ll intervls tht stisfy (i) so tht L(, ) = is the shortest length possile for given confidence coefficient ( α). Therefore the confidence intervl for the men is [X σ n ; X (.5) 34

3 Interntionl Journl of Sttistics nd Proility Vol. 7, No. 2; 208 σ n ]. The length of this confidence intervl t level α is K = ( ) σ n. To find the shortest confidence intervl for the men t level α is to minimize the length L(, ) = suject to: f(z)dz = α. Theorem 4: Let f e unimodl proility density function of the stndrd norml distriution. If [X σ n ; X σ n ] is the shortest confidence intervl for the men μ, then = 2ln (λ 2π) nd = 2ln (λ 2π) with λ ]0, Proof: [. 2π f() = f() From (i) nd (ii), we hve: { f(z)dz = α f() = f() = or = with (to reject = ). Therefore =. Finding nd mounts to solving the eqution f(z) = λ tht is to sy (f(x)dx λ) After studying the function f(z), we find the tle of vrition of f(z): = α. z f f 2 35

4 Interntionl Journl of Sttistics nd Proility Vol. 7, No. 2; Grph of the stndrd norml distriution nd the line f(z) (f(z) λ)dz = α z Solve f(z) = λ for λ ]0, [: e z2 2π 2π 2 = λ z = ± 2ln (λ 2π). Therefore Then = 2ln (λ 2π) nd = 2ln (λ 2π) with λ ]0, Corollry: Let f e unimodl proility density function of the stndrd norml distriution. If [X σ ; X σ ] is the shortest confidence intervl for the men μ, then n n [. 2π (i) = 2ln (λ 2π) nd = 2ln (λ 2π) with λ ]0, [ nd 2π (ii) λ = αx x 2. x k for 0,0 α 0, nd x i {0,,2,,9}, i k. 36

5 Interntionl Journl of Sttistics nd Proility Vol. 7, No. 2; Exmples α λ ]0, 2π [ Equl -Til Length L Shortest Confidence intervl Length L 2 L L 2 Reltive error 0,0 λ =0,0446 = 2,5760 = 2,5760 5,520 = 2, =2, ,5644 0, , ,02 λ = 0, = 2,326 = 2,326 4,652 = 2,32635 =2, ,6527 0,0007 0, ,03 λ = 0,03787 = 2,70 = 2,70 4,340 = 2,70096 =2, , , , ,04 λ = 0,04848 = 2,054 = 2,054 4,08 = 2,05375 =2, ,075 0,0005 0, ,05 λ =0, =,9600 =,9600 3,9200 =, =, ,9993 0, , ,06 λ = 0, =,88 =,88 3,762 =, =, , , , ,07 λ = 0,07727 =,82 =,82 3,624 =,89 =,89 3, , , ,08 λ = 0,08674 =,75 =,75 3,502 =, =, ,5037 0, , ,09 λ = 0, =,695 =,695 3,390 =, =, , , , , λ =0,0336 =,6450 =,6450 3,2900 =,6448 =,6448 3,2896 0,0004 0,

6 Interntionl Journl of Sttistics nd Proility Vol. 7, No. 2; Conclusions The technique given in th for is pper for constructing shortest-length confidence intervls is esy to pply. This technique cn lso e used solving the similr prolems. The existence of confidence intervls with the shortest length do not lwys exist, even when the distriution of the pivotl quntity is symmetric. References Dhiy, R., & Guttmn, I. (982). Shortest confidence intervls nd prediction intervls for log-norml. The Cndin Journl of Sttistics, 0(4), Evns, M., & Shkhtreh, M. (2008). Optiml properties of some Byesin inferences. Electronic Journl of Sttistics, 2, Ferentinos, K. K., & Krkosts, K. X. More on shortest nd equl tils confidence intervls. Deprtment of mthemtics University of Ionnin - Greece. Ferentions, K. (987). Shortest confidence intervls nd UMVU estimtors for fmilies of distriutions involving trunction prmeters. Metrik, 34, Ferentions, K. (990). Shortest confidence intervls for fmilies of distriutions involving trunction prmeters. The Americn Sttisticin, 44, Ferentions, K., & Kouroukli, S. (990). Shortest confidence intervls for fmilies of distriutions involving two trunction prmeters. Metrik, 37, George Csell. Sttisticl inference: Intervl estimtion. Sec:9.3, 44. Guenther, W. C. (969). Shortest confidence intervls. The Americn Sttisticin, 23, Guenther, W. C. (97). Unised confidence intervls. The Americn Sttisticin, 25, John, H. M, & Kurtis, D. F. Numericl methods using mtl. Sec: 7.2, Konstntin, N. N., Nichols, A. N., & Edgrs, K. V. (2002). Constructing shortest-length confidence intervls. Deprtment of Mthemticl Sttistics, University of Ltvi., 3(). Roert, V. H, & Elliot, A. T. Proility nd sttisticl inference. Chp6, Troendle, J. F., & Frnk, J. (200). Unised confidence intervls for the odds rtio of two independent inomil smples with ppliction to cse-control dt. Biometrics, 57, Wll, M. M., Boen, J., & Tweedie, R. (200). An effective confidence intervl for the men with smples of size one nd two. The Americn Sttisticin, 55(2), Copyrights Copyright for this rticle is retined y the uthor(s), with first puliction rights grnted to the journl. This is n open-ccess rticle distriuted under the terms nd conditions of the Cretive Commons Attriution license ( 38