Sequential Probabilistic Ratio Test for the Scale Parameter of the P-Norm Distribution

We consider a series of independent observations from a P-norm distribution with the position parameter μ and the scale parameter σ. We test the simple hypothesis H0: σ � σ1 versus H1: σ � σ2. Firstly, we give the stop rule and decision rule of sequential probabilistic ratio test (SPRT). Secondly, we prove the existence of h(σ) which needs to satisfy the specific situation in SPRTmethod, and the approximate formula of the mean sample function is derived. Finally, a simulation example is given. 'e simulation shows that the ratio of sample size required by SPRTand the classic Neyman–Pearson (N − P) test is about 50.92% at most,38.30% at least.


Introduction
P-norm distribution is a family of distributions, including normal distribution, Laplace distribution, uniform distribution, degenerate distribution and many unknown distributions. Let the density function of the random variable be given by en, it is said that X follows the P-norm distribution, where λ � (Γ(3/p)/Γ(1/p)) 1/2 , the position parameter is μ, and the scale parameter is σ > 0. Since it contains some important distributions, the P-norm distribution can better describe error distributions to some extent. ere are many articles that studied the properties of the P-norm distribution. For example, Hu and Sun [1] systematically obtained unbiased estimators of parameters for the P-norm distribution. Sun and Hu [2] gave a density function of P-norm distribution and its sampling distribution. However, nobody investigated the sequential probabilistic ratio test for the scale parameter of P-norm distribution.
Likelihood ratio method is a widely used test method, which can get a lot of in-depth results. For example, Self and Liang [3] gave the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics under nonstandard conditions. Fan and Zhang [4] proposed the sieve empirical likelihood ratio test for nonparametric functions. Ferrari and Cysneiros [5] used the Skovgaard's modified likelihood ratio method to study the exponential family nonlinear model and obtained the approximate distribution of modified likelihood ratio statistics. Giampaoli and Singer [6] tested the variance parameter of the linear mixed model by the likelihood ratio method. Huang et al. [7] tested the shape parameter of the generalized extremum distribution with the Lq-likelihood ratio method. Qin and Priebe [8] proposed a robust Lq-likelihood ratio test for the general pollution distribution and obtained the asymptotic distribution of Lq-likelihood ratio test statistics.
In order to meet the requirements of quality inspection of American munitions production during World War II, Wald [9] presented a sequential analysis method. Since then, many authors had studied sequence analysis methods and proposed various sequence tests to test different hypotheses. For example, Whitehead and Jones [10] and Jennison and Turnbull [11] provided extensive applications of sequence and group sequence based techniques in the formation and execution of clinical trials in their books. Darkhovsky [12] studied the sequence examination of two compound hypotheses and proposed a sequence process that minimizes the maximum Bayesian risk on a series of prior parameter distributions. Kachiashvili [13] proposed a sequential method to constrain the multiple test problem in the Bayesian hypothesis test task and proved the high quality of this method. Li et al. [14] extended the sequence probability ratio test and proved that the sequence test was asymptotically optimal when the error probability went to zero, so the sequence test could asymptotically obtain the minimum expected sample size. Wang et al. [15] proposed the weighted expected sample size (WESS) to evaluate the test problem of the composite hypothesis for the overall performance of three different regions. Nakamura et al. [16] proposed a sequential test procedure to determine the minimum dose with threshold effect. Li et al. [17] constructed a general sequence test to detect outliers in all collected observation sequences. Mudholkar et al. [18] deduced the sequential probabilistic ratio test method of M-Gaussian population model under the assumption that the discrete parameters were known. Zou et al. [19] proposed a nonparametric sequential test based on empirical likelihood to test the treatment effect. Tartakovsky et al. [20] discussed in detail recent advances in sequential hypothesis testing in their book.
e P-norm distribution describes the error distribution. In order to use the P-norm distribution, it is necessary to use the sequential likelihood ratio test method to study the P-norm distribution parameter. erefore, this paper will apply the sequential probability ratio test method to study the parameter of P-norm distribution. e specific structure of this paper is as follows. Section 2 introduces the SPRT method. Section 3 gives some properties of SPRT. Section 4 performs a simulation study to confirm results. Proofs of theorems are contained in Section 5.

SPRT Method
Suppose that (X 1 , X 2 , . . .) is an independent and identically distributed (i.i.d.) random sample sequence from the P-norm distribution. Let (x 1 , x 2 , . . .) be the sequence of their observed values. e following is a simple hypothesis test problem (let us assume that parameter μ is known. Without loss of generality, let parameter μ � 0): Here, σ 1 , σ 2 ∈ Θ are known numbers and 0 < σ 1 < σ 2 . Let us give the stopping rule and decision rule of SPRT in turn. First, consider the likelihood ratio statistic: Let S p n � n i�1 |X i | p . Taking the logarithm of equation (3), we can get ln λ n � n ln Given the test level α > 0 and β > 0, α We easily know c > 0, d 1 > 0, and d 2 > 0. It is given by Similarly, it is given by λ n ≥ A that It is not difficult to see that the sufficient and necessary condition for λ n ≥ B is S p n ≥ R n , the sufficient and necessary condition of λ n ≤ A is S p n ≤ A n . en, the stopping rule of the SPRT for the P-norm distribution is τ * � inf n: n ≥ 1 and S p n ∉ A n , R n .
e decision rule of the P-norm distribution SPRT is as follows: (i) When S p n ≥ R n , we should stop the experiment and accept H 0 (ii) When S p n ≤ A n , we should stop the experiment and reject H 0

Sample Size Required for SPRT
From the above analysis, we know that the SPRT method depends on the constants A and B that have been selected beforehand. For the selection of A and B, we need to start from some properties of SPRT. e study of SPRT properties can be turned into the study of random walks. en, this section will study the random walks of the SPRT method for the P-norm distribution and give some properties of SPRT.
Given α and β (which satisfy α + β < 1), we can get constants A and B.
By (4), we can get 2 Discrete Dynamics in Nature and Society Note that E|X i | p � (1/p)(σ/λ) p and E|X i | 2p � ((1 + p)/p 2 )(σ/λ) 2p (Hu and Sun [1]). By (10), we have If EZ 1 ≥ 0, then For simplicity, h � h(σ) is denoted as a function of σ. Let 15) By studying random walks, we can derive some properties of SPRT. Next, we will give some properties of SPRT method for P-norm distributions. At the same time, we will also give the average sample size required for the SPRT method.
In order to find out the average sample size in the SPRT method, we need to verify whether condition in eorem 1 is true. It is given by eorem 2.

Theorem 2. For σ 1 and σ
e operational characteristic (OC) of Wald's SPRT is the probability of accepting the null hypothesis. e average sample number (ASN) is the average number of observations that we would have to collect in order to make a decision regarding the statistical hypotheses put forth. According to Mudholkar et al. [18], we can give the OC function and ANS of SPRT for the P-norm distribution. ese are given by eorems 3 and 4.
us, we can give eorem 4.

Theorem 4. e ANS of SPRT for the P-norm distribution is that
For the given probability α and β which are the type 1 and the type 2 errors, A � β/(1 − α) and B � (1 − β)/α are taken, respectively. According to eorem 4, it can be obtained that the average sample size of the SPRT method is E σ 1 τ * and E σ 2 τ * here are determined in four cases as follows: Discrete Dynamics in Nature and Society 3 where where To compare the methods of the simulation example in Section 4, we briefly introduce the Neyman-Pearson test method to calculate the average sample size. e principle of the N − P test is to control the probability of making the first type of error within a given range and to find the test to make the probability of making the second type of error as small as possible, that is, to maximize the effectiveness of the test. Before proceeding to the results, we introduce these notions: 1, 2, . . . , n) is an independent sequence, and we take S p n � n i�1 |X i | p and X p � (S p n /n). Likewise, we know E|X i | p � (1/p)(σ/λ) p and D|X i | p � (1/p)(σ/λ) 2p (Hu and Sun [1]).
According to the Neyman-Pearson theory, the optimal fixed quantity is as follows. We should look for n and c that satisfy By the central limit theorem, we know that X p approximately follows a normal distribution AN(E|X i | p , We can get   Discrete Dynamics in Nature and Society